i Tyn Myint-U Lokenath Debnath Linear Partial Differential Equations for Scientists and Engineers Fourth Edition Birkhauser ¨ Boston • Basel • Berlin Tyn Myint-U 5 Sue Terrace Westport, CT 06880 USA Lokenath Debnath Department of Mathematics University of Texas-Pan American 1201 W. University Drive Edinburgh, TX 78539 USA Cover design by Alex Gerasev. Mathematics Subject Classification (2000): 00A06, 00A69, 34B05, 34B24, 34B27, 34G20, 35-01, 35-02, 35A15, 35A22, 35A25, 35C05, 35C15, 35Dxx, 35E05, 35E15, 35Fxx, 35F05, 35F10, 35F15, 35F20, 35F25, 35G10, 35G20, 35G25, 35J05, 35J10, 35J20, 35K05, 35K10, 35K15, 35K55, 35K60, 35L05, 35L10, 35L15, 35L20, 35L25, 35L30, 35L60, 35L65, 35L67, 35L70, 35Q30, 35Q35, 35Q40, 35Q51, 35Q53, 35Q55, 35Q58, 35Q60, 35Q80, 42A38, 44A10, 44A35 49J40, 58E30, 58E50, 65L15, 65M25, 65M30, 65R10, 70H05, 70H20, 70H25, 70H30, 76Bxx, 76B15, 76B25, 76D05, 76D33, 76E30, 76M30, 76R50, 78M30, 81Q05 Library of Congress Control Number: 2006935807 ISBN-10: 0-8176-4393-1 e-ISBN-10: 0-8176-4560-8 ISBN-13: 978-0-8176-4393-5 e-ISBN-13: 978-0-8176-4560-1 Printed on acid-free paper. c 2007 Birkhauser Boston ¨ All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer Science ¨ +Business Media LLC, 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 www.birkhauser.com (SB) To the Memory of U and Mrs. Hla Din U and Mrs. Thant Tyn Myint-U In Loving Memory of My Mother and Father Lokenath Debnath “True Laws of Nature cannot be linear.” “The search for truth is more precious than its possession.” “Everything should be made as simple as possible, but not a bit simpler.” Albert Einstein “No human investigation can be called real science if it cannot be demonstrated mathematically.” Leonardo Da Vinci “First causes are not known to us, but they are subjected to simple and constant laws that can be studied by observation and whose study is the goal of Natural Philosophy ... Heat penetrates, as does gravity, all the substances of the universe; its rays occupy all regions of space. The aim of our work is to expose the mathematical laws that this element follows ... The differential equations for the propagation of heat express the most general conditions and reduce physical questions to problems in pure Analysis that is properly the object of the theory.” James Clerk Maxwell “One of the properties inherent in mathematics is that any real progress is accompanied by the discovery and development of new methods and simplifications of previous procedures ... The unified character of mathematics lies in its very nature; indeed, mathematics is the foundation of all exact natural sciences.” David Hilbert “ ... partial differential equations are the basis of all physical theorems. In the theory of sound in gases, liquid and solids, in the investigations of elasticity, in optics, everywhere partial differential equations formulate basic laws of nature which can be checked against experiments.” Bernhard Riemann “The effective numerical treatment of partial differential equations is not a handicraft, but an art.” Folklore “The advantage of the principle of least action is that in one and the same equation it relates the quantities that are immediately relevant not only to mechanics but also to electrodynamics and thermodynamics; these quantities are space, time and potential.” Max Planck “The thorough study of nature is the most ground for mathematical discoveries.” “The equations for the flow of heat as well as those for the oscillations of acoustic bodies and of fluids belong to an area of analysis which has recently been opened, and which is worth examining in the greatest detail.” Joseph Fourier “Of all the mathematical disciplines, the theory of differential equation is the most important. All branches of physics pose problems which can be reduced to the integration of differential equations. More generally, the way of explaining all natural phenomena which depend on time is given by the theory of differential equations.” Sophus Lie “Differential equations form the basis for the scientific view of the world.” V.I. Arnold “What we know is not much. What we do not know is immense.” “The algebraic analysis soon makes us forget the main object [of our research] by focusing our attention on abstract combinations and it is only at the end that we return to the original objective. But in abandoning oneself to the operations of analysis, one is led to the generality of this method and the inestimable advantage of transforming the reasoning by mechanical procedures to results often inaccessible by geometry ... No other language has the capacity for the elegance that arises from a long sequence of expressions linked one to the other and all stemming from one fundamental idea.” “It is India that gave us the ingenious method of expressing all numbers by ten symbols, each symbol receiving a value of position, as well as an absolute value. We shall appreciate the grandeur of the achievement when we remember that it escaped the genius of Archimedes and Appolonius.” P.S. Laplace “The mathematician’s best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another.” G˙osta Mittag-Leffler Contents Preface to the Fourth Edition . . . . . . . . . . . . . . . . . xv Preface to the Third Edition . . . . . . . . . . . . . . . . . xix 1 Introduction 1 1.1 Brief Historical Comments . . . . . . . . . . . . . . . . . 1 1.2 Basic Concepts and Definitions . . . . . . . . . . . . . . . 12 1.3 Mathematical Problems . . . . . . . . . . . . . . . . . . . 15 1.4 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Superposition Principle . . . . . . . . . . . . . . . . . . . 20 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 First-Order, Quasi-Linear Equations and Method of Characteristics 27 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Classification of First-Order Equations . . . . . . . . . . . 27 2.3 Construction of a First-Order Equation . . . . . . . . . . 29 2.4 Geometrical Interpretation of a First-Order Equation . . 33 2.5 Method of Characteristics and General Solutions . . . . . 35 2.6 Canonical Forms of First-Order Linear Equations . . . . 49 2.7 Method of Separation of Variables . . . . . . . . . . . . . 51 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3 Mathematical Models 63 3.1 Classical Equations . . . . . . . . . . . . . . . . . . . . . 63 3.2 The Vibrating String . . . . . . . . . . . . . . . . . . . . 65 3.3 The Vibrating Membrane . . . . . . . . . . . . . . . . . . 67 3.4 Waves in an Elastic Medium . . . . . . . . . . . . . . . . 69 3.5 Conduction of Heat in Solids . . . . . . . . . . . . . . . . 75 3.6 The Gravitational Potential . . . . . . . . . . . . . . . . . 76 3.7 Conservation Laws and The Burgers Equation . . . . . . 79 3.8 The Schr¨odinger and the Korteweg–de Vries Equations . 81 3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4 Classification of Second-Order Linear Equations 91 4.1 Second-Order Equations in Two Independent Variables . 91 x Contents 4.2 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . 93 4.3 Equations with Constant Coefficients . . . . . . . . . . . 99 4.4 General Solutions . . . . . . . . . . . . . . . . . . . . . . 107 4.5 Summary and Further Simplification . . . . . . . . . . . . 111 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5 The Cauchy Problem and Wave Equations 117 5.1 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . 117 5.2 The Cauchy–Kowalewskaya Theorem . . . . . . . . . . . 120 5.3 Homogeneous Wave Equations . . . . . . . . . . . . . . . 121 5.4 Initial Boundary-Value Problems . . . . . . . . . . . . . . 130 5.5 Equations with Nonhomogeneous Boundary Conditions . 134 5.6 Vibration of Finite String with Fixed Ends . . . . . . . . 136 5.7 Nonhomogeneous Wave Equations . . . . . . . . . . . . . 139 5.8 The Riemann Method . . . . . . . . . . . . . . . . . . . . 142 5.9 Solution of the Goursat Problem . . . . . . . . . . . . . . 149 5.10 Spherical Wave Equation . . . . . . . . . . . . . . . . . . 153 5.11 Cylindrical Wave Equation . . . . . . . . . . . . . . . . . 155 5.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6 Fourier Series and Integrals with Applications 167 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.2 Piecewise Continuous Functions and Periodic Functions . 168 6.3 Systems of Orthogonal Functions . . . . . . . . . . . . . . 170 6.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.5 Convergence of Fourier Series . . . . . . . . . . . . . . . . 173 6.6 Examples and Applications of Fourier Series . . . . . . . 177 6.7 Examples and Applications of Cosine and Sine Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.8 Complex Fourier Series . . . . . . . . . . . . . . . . . . . 194 6.9 Fourier Series on an Arbitrary Interval . . . . . . . . . . 196 6.10 The Riemann–Lebesgue Lemma and Pointwise Convergence Theorem . . . . . . . . . . . . . . . . . . . . 201 6.11 Uniform Convergence, Differentiation, and Integration . . 208 6.12 Double Fourier Series . . . . . . . . . . . . . . . . . . . . 212 6.13 Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . 214 6.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7 Method of Separation of Variables 231 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.2 Separation of Variables . . . . . . . . . . . . . . . . . . . 232 7.3 The Vibrating String Problem . . . . . . . . . . . . . . . 235 7.4 Existence and Uniqueness of Solution of the Vibrating String Problem . . . . . . . . . . . . . . . . . . . . . . . . 243 7.5 The Heat Conduction Problem . . . . . . . . . . . . . . . 248 Contents xi 7.6 Existence and Uniqueness of Solution of the Heat Conduction Problem . . . . . . . . . . . . . . . . . . . . . 251 7.7 The Laplace and Beam Equations . . . . . . . . . . . . . 254 7.8 Nonhomogeneous Problems . . . . . . . . . . . . . . . . . 258 7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 8 Eigenvalue Problems and Special Functions 273 8.1 Sturm–Liouville Systems . . . . . . . . . . . . . . . . . . 273 8.2 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . 277 8.3 Eigenfunction Expansions . . . . . . . . . . . . . . . . . . 283 8.4 Convergence in the Mean . . . . . . . . . . . . . . . . . . 284 8.5 Completeness and Parseval’s Equality . . . . . . . . . . . 286 8.6 Bessel’s Equation and Bessel’s Function . . . . . . . . . . 289 8.7 Adjoint Forms and Lagrange Identity . . . . . . . . . . . 295 8.8 Singular Sturm–Liouville Systems . . . . . . . . . . . . . 297 8.9 Legendre’s Equation and Legendre’s Function . . . . . . . 302 8.10 Boundary-Value Problems Involving Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 8.11 Green’s Functions for Ordinary Differential Equations . . 310 8.12 Construction of Green’s Functions . . . . . . . . . . . . . 315 8.13 The Schr¨odinger Equation and Linear Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 8.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 9 Boundary-Value Problems and Applications 329 9.1 Boundary-Value Problems . . . . . . . . . . . . . . . . . . 329 9.2 Maximum and Minimum Principles . . . . . . . . . . . . 332 9.3 Uniqueness and Continuity Theorems . . . . . . . . . . . 333 9.4 Dirichlet Problem for a Circle . . . . . . . . . . . . . . . . 334 9.5 Dirichlet Problem for a Circular Annulus . . . . . . . . . 340 9.6 Neumann Problem for a Circle . . . . . . . . . . . . . . . 341 9.7 Dirichlet Problem for a Rectangle . . . . . . . . . . . . . 343 9.8 Dirichlet Problem Involving the Poisson Equation . . . . 346 9.9 The Neumann Problem for a Rectangle . . . . . . . . . . 348 9.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 10 Higher-Dimensional Boundary-Value Problems 361 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 361 10.2 Dirichlet Problem for a Cube . . . . . . . . . . . . . . . . 361 10.3 Dirichlet Problem for a Cylinder . . . . . . . . . . . . . . 363 10.4 Dirichlet Problem for a Sphere . . . . . . . . . . . . . . . 367 10.5 Three-Dimensional Wave and Heat Equations . . . . . . . 372 10.6 Vibrating Membrane . . . . . . . . . . . . . . . . . . . . . 372 10.7 Heat Flow in a Rectangular Plate . . . . . . . . . . . . . 375 10.8 Waves in Three Dimensions . . . . . . . . . . . . . . . . . 379 xii Contents 10.9 Heat Conduction in a Rectangular Volume . . . . . . . . 381 10.10 The Schr¨odinger Equation and the Hydrogen Atom . . . 382 10.11 Method of Eigenfunctions and Vibration of Membrane . . 392 10.12 Time-Dependent Boundary-Value Problems . . . . . . . . 395 10.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 11 Green’s Functions and Boundary-Value Problems 407 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 407 11.2 The Dirac Delta Function . . . . . . . . . . . . . . . . . . 409 11.3 Properties of Green’s Functions . . . . . . . . . . . . . . . 412 11.4 Method of Green’s Functions . . . . . . . . . . . . . . . . 414 11.5 Dirichlet’s Problem for the Laplace Operator . . . . . . . 416 11.6 Dirichlet’s Problem for the Helmholtz Operator . . . . . . 418 11.7 Method of Images . . . . . . . . . . . . . . . . . . . . . . 420 11.8 Method of Eigenfunctions . . . . . . . . . . . . . . . . . . 423 11.9 Higher-Dimensional Problems . . . . . . . . . . . . . . . . 425 11.10 Neumann Problem . . . . . . . . . . . . . . . . . . . . . . 430 11.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 12 Integral Transform Methods with Applications 439 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 439 12.2 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . 440 12.3 Properties of Fourier Transforms . . . . . . . . . . . . . . 444 12.4 Convolution Theorem of the Fourier Transform . . . . . . 448 12.5 The Fourier Transforms of Step and Impulse Functions . 453 12.6 Fourier Sine and Cosine Transforms . . . . . . . . . . . . 456 12.7 Asymptotic Approximation of Integrals by Stationary Phase Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 12.8 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . 460 12.9 Properties of Laplace Transforms . . . . . . . . . . . . . . 463 12.10 Convolution Theorem of the Laplace Transform . . . . . 467 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 12.12 Hankel Transforms . . . . . . . . . . . . . . . . . . . . . . 488 12.13 Properties of Hankel Transforms and Applications . . . . 491 12.14 Mellin Transforms and their Operational Properties . . . 495 12.15 Finite Fourier Transforms and Applications . . . . . . . . 499 12.16 Finite Hankel Transforms and Applications . . . . . . . . 504 12.17 Solution of Fractional Partial Differential Equations . . . 510 12.18 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 13 Nonlinear Partial Differential Equations with Applications 535 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 535 Contents xiii 13.2 One-Dimensional Wave Equation and Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 536 13.3 Linear Dispersive Waves . . . . . . . . . . . . . . . . . . . 540 13.4 Nonlinear Dispersive Waves and Whitham’s Equations . . 545 13.5 Nonlinear Instability . . . . . . . . . . . . . . . . . . . . . 548 13.6 The Traffic Flow Model . . . . . . . . . . . . . . . . . . . 549 13.7 Flood Waves in Rivers . . . . . . . . . . . . . . . . . . . . 552 13.8 Riemann’s Simple Waves of Finite Amplitude . . . . . . . 553 13.9 Discontinuous Solutions and Shock Waves . . . . . . . . . 561 13.10 Structure of Shock Waves and Burgers’ Equation . . . . . 563 13.11 The Korteweg–de Vries Equation and Solitons . . . . . . 573 13.12 The Nonlinear Schr¨odinger Equation and Solitary Waves. 581 13.13 The Lax Pair and the Zakharov and Shabat Scheme . . . 590 13.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 14 Numerical and Approximation Methods 601 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 601 14.2 Finite Difference Approximations, Convergence, and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 14.3 Lax–Wendroff Explicit Method . . . . . . . . . . . . . . . 605 14.4 Explicit Finite Difference Methods . . . . . . . . . . . . . 608 14.5 Implicit Finite Difference Methods . . . . . . . . . . . . . 624 14.6 Variational Methods and the Euler–Lagrange Equations . 629 14.7 The Rayleigh–Ritz Approximation Method . . . . . . . . 647 14.8 The Galerkin Approximation Method . . . . . . . . . . . 655 14.9 The Kantorovich Method . . . . . . . . . . . . . . . . . . 659 14.10 The Finite Element Method . . . . . . . . . . . . . . . . . 663 14.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 668 15 Tables of Integral Transforms 681 15.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . 681 15.2 Fourier Sine Transforms . . . . . . . . . . . . . . . . . . . 683 15.3 Fourier Cosine Transforms . . . . . . . . . . . . . . . . . 685 15.4 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . 687 15.5 Hankel Transforms . . . . . . . . . . . . . . . . . . . . . . 691 15.6 Finite Hankel Transforms . . . . . . . . . . . . . . . . . . 695 Answers and Hints to Selected Exercises 697 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 704 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 5.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 6.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 xiv Contents 8.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 9.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 10.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 731 11.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 12.18 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 740 14.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Appendix: Some Special Functions and Their Properties 749 A-1 Gamma, Beta, Error, and Airy Functions . . . . . . . . . 749 A-2 Hermite Polynomials and Weber–Hermite Functions . . . 757 Bibliography 761 Index 771 Preface to the Fourth Edition “A teacher can never truly teach unless he is still learning himself. A lamp can never light another lamp unless it continues to burn its own flame. The teacher who has come to the end of his subject, who has no living traffic with his knowledge but merely repeats his lessons to his students, can only load their minds; he cannot quicken them.” Rabindranath Tagore An Indian Poet 1913 Nobel Prize Winner for Literature The previous three editions of our book were very well received and used as a senior undergraduate or graduate-level text and research reference in the United States and abroad for many years. We received many comments and suggestions from many students, faculty and researchers around the world. These comments and criticisms have been very helpful, beneficial, and encouraging. This fourth edition is the result of the input. Another reason for adding this fourth edition to the literature is the fact that there have been major discoveries of new ideas, results and methods for the solution of linear and nonlinear partial differential equations in the second half of the twentieth century. It is becoming even more desirable for mathematicians, scientists and engineers to pursue study and research on these topics. So what has changed, and will continue to change is the nature of the topics that are of interest in mathematics, applied mathematics, physics and engineering, the evolution of books such is this one is a history of these shifting concerns. This new and revised edition preserves the basic content and style of the third edition published in 1989. As with the previous editions, this book has been revised primarily as a comprehensive text for senior undergraduates or beginning graduate students and a research reference for professionals in mathematics, science and engineering, and other applied sciences. The main goal of the book is to develop required analytical skills on the part of the xvi Preface to the Fourth Edition reader, rather than to focus on the importance of more abstract formulation, with full mathematical rigor. Indeed, our major emphasis is to provide an accessible working knowledge of the analytical and numerical methods with proofs required in mathematics, applied mathematics, physics, and engineering. The revised edition was greatly influenced by the statements that Lord Rayleigh and Richard Feynman made as follows: “In the mathematical investigation I have usually employed such methods as present themselves naturally to a physicist. The pure mathematician will complain, and (it must be confessed) sometimes with justice, of defi- cient rigor. But to this question there are two sides. For, however important it may be to maintain a uniformly high standard in pure mathematics, the physicist may occasionally do well to rest content with arguments, which are fairly satisfactory and conclusive from his point of view. To his mind, exercised in a different order of ideas, the more severe procedure of the pure mathematician may appear not more but less demonstrative. And further, in many cases of difficulty to insist upon highest standard would mean the exclusion of the subject altogether in view of the space that would be required.” Lord Rayleigh “... However, the emphasis should be somewhat more on how to do the mathematics quickly and easily, and what formulas are true, rather than the mathematicians’ interest in methods of rigorous proof.” Richard P. Feynman We have made many additions and changes in order to modernize the contents and to improve the clarity of the previous edition. We have also taken advantage of this new edition to correct typographical errors, and to update the bibliography, to include additional topics, examples of applications, exercises, comments and observations, and in some cases, to entirely rewrite and reorganize many sections. This is plenty of material in the book for a year-long course. Some of the material need not be covered in a course work and can be left for the readers to study on their own in order to prepare them for further study and research. This edition contains a collection of over 900 worked examples and exercises with answers and hints to selected exercises. Some of the major changes and additions include the following: 1. Chapter 1 on Introduction has been completely revised and a new section on historical comments was added to provide information about the historical developments of the subject. These changes have been made to provide the reader to see the direction in which the subject has developed and find those contributed to its developments. 2. A new Chapter 2 on first-order, quasi-linear, and linear partial differential equations, and method of characteristics has been added with many new examples and exercises. Preface to the Fourth Edition xvii 3. Two sections on conservation laws, Burgers’ equation, the Schr¨odinger and the Korteweg-de Vries equations have been included in Chapter 3. 4. Chapter 6 on Fourier series and integrals with applications has been completely revised and new material added, including a proof of the pointwise convergence theorem. 5. A new section on fractional partial differential equations has been added to Chapter 12 with many new examples of applications. 6. A new section on the Lax pair and the Zakharov and Shabat Scheme has been added to Chapter 13 to modernize its contents. 7. Some sections of Chapter 14 have been revised and a new short section on the finite element method has been added to this chapter. 8. A new Chapter 15 on tables of integral transforms has been added in order to make the book self-contained. 9. The whole section on Answers and Hints to Selected Exercises has been expanded to provide additional help to students. All figures have been redrawn and many new figures have been added for a clear understanding of physical explanations. 10. An Appendix on special functions and their properties has been expanded. Some of the highlights in this edition include the following: • The book offers a detailed and clear explanation of every concept and method that is introduced, accompanied by carefully selected worked examples, with special emphasis given to those topics in which students experience difficulty. • A wide variety of modern examples of applications has been selected from areas of integral and ordinary differential equations, generalized functions and partial differential equations, quantum mechanics, fluid dynamics and solid mechanics, calculus of variations, linear and nonlinear stability analysis. • The book is organized with sufficient flexibility to enable instructors to select chapters appropriate for courses of differing lengths, emphases, and levels of difficulty. • A wide spectrum of exercises has been carefully chosen and included at the end of each chapter so the reader may further develop both rigorous skills in the theory and applications of partial differential equations and a deeper insight into the subject. • Many new research papers and standard books have been added to the bibliography to stimulate new interest in future study and research. Index of the book has also been completely revised in order to include a wide variety of topics. • The book provides information that puts the reader at the forefront of current research. With the improvements and many challenging worked-out problems and exercises, we hope this edition will continue to be a useful textbook for xviii Preface to the Fourth Edition students as well as a research reference for professionals in mathematics, applied mathematics, physics and engineering. It is our pleasure to express our grateful thanks to many friends, colleagues, and students around the world who offered their suggestions and help at various stages of the preparation of the book. We offer special thanks to Dr. Andras Balogh, Mr. Kanadpriya Basu, and Dr. Dambaru Bhatta for drawing all figures, and to Mrs. Veronica Martinez for typing the manuscript with constant changes and revisions. In spite of the best efforts of everyone involved, some typographical errors doubtless remain. Finally, we wish to express our special thanks to Tom Grasso and the staff of Birkh¨auser Boston for their help and cooperation. Tyn Myint-U Lokenath Debnath Preface to the Third Edition The theory of partial differential equations has long been one of the most important fields in mathematics. This is essentially due to the frequent occurrence and the wide range of applications of partial differential equations in many branches of physics, engineering, and other sciences. With much interest and great demand for theory and applications in diverse areas of science and engineering, several excellent books on PDEs have been published. This book is written to present an approach based mainly on the mathematics, physics, and engineering problems and their solutions, and also to construct a course appropriate for all students of mathematical, physical, and engineering sciences. Our primary objective, therefore, is not concerned with an elegant exposition of general theory, but rather to provide students with the fundamental concepts, the underlying principles, a wide range of applications, and various methods of solution of partial differential equations. This book, a revised and expanded version of the second edition published in 1980, was written for a one-semester course in the theory and applications of partial differential equations. It has been used by advanced undergraduate or beginning graduate students in applied mathematics, physics, engineering, and other applied sciences. The prerequisite for its study is a standard calculus sequence with elementary ordinary differential equations. This revised edition is in part based on lectures given by Tyn Myint-U at Manhattan College and by Lokenath Debnath at the University of Central Florida. This revision preserves the basic content and style of the earlier editions, which were written by Tyn Myint-U alone. However, the authors have made some major additions and changes in this third edition in order to modernize the contents and to improve clarity. Two new chapters added are on nonlinear PDEs, and on numerical and approximation methods. New material emphasizing applications has been inserted. New examples and exercises have been provided. Many physical interpretations of mathematical solutions have been added. Also, the authors have improved the exposition by reorganizing some material and by making examples, exercises, and ap- xx Preface to the Third Edition plications more prominent in the text. These additions and changes have been made with the student uppermost in mind. The first chapter gives an introduction to partial differential equations. The second chapter deals with the mathematical models representing physical and engineering problems that yield the three basic types of PDEs. Included are only important equations of most common interest in physics and engineering. The third chapter constitutes an account of the classifi- cation of linear PDEs of second order in two independent variables into hyperbolic, parabolic, and elliptic types and, in addition, illustrates the determination of the general solution for a class of relatively simple equations. Cauchy’s problem, the Goursat problem, and the initial boundary-value problems involving hyperbolic equations of the second order are presented in Chapter 4. Special attention is given to the physical significance of solutions and the methods of solution of the wave equation in Cartesian, spherical polar, and cylindrical polar coordinates. The fifth chapter contains a fuller treatment of Fourier series and integrals essential for the study of PDEs. Also included are proofs of several important theorems concerning Fourier series and integrals. Separation of variables is one of the simplest methods, and the most widely used method, for solving PDEs. The basic concept and separability conditions necessary for its application are discussed in the sixth chapter. This is followed by some well-known problems of applied mathematics, mathematical physics, and engineering sciences along with a detailed analysis of each problem. Special emphasis is also given to the existence and uniqueness of the solutions and to the fundamental similarities and differences in the properties of the solutions to the various PDEs. In Chapter 7, self-adjoint eigenvalue problems are treated in depth, building on their introduction in the preceding chapter. In addition, Green’s function and its applications to eigenvalue problems and boundary-value problems for ordinary differential equations are presented. Following the general theory of eigenvalues and eigenfunctions, the most common special functions, including the Bessel, Legendre, and Hermite functions, are discussed as examples of the major role of special functions in the physical and engineering sciences. Applications to heat conduction problems and the Schr¨odinger equation for the linear harmonic oscillator are also included. Boundary-value problems and the maximum principle are described in Chapter 8, and emphasis is placed on the existence, uniqueness, and wellposedness of solutions. Higher-dimensional boundary-value problems and the method of eigenfunction expansion are treated in the ninth chapter, which also includes several applications to the vibrating membrane, waves in three dimensions, heat conduction in a rectangular volume, the threedimensional Schr¨odinger equation in a central field of force, and the hydrogen atom. Chapter 10 deals with the basic concepts and construction of Green’s function and its application to boundary-value problems. Preface to the Third Edition xxi Chapter 11 provides an introduction to the use of integral transform methods and their applications to numerous problems in applied mathematics, mathematical physics, and engineering sciences. The fundamental properties and the techniques of Fourier, Laplace, Hankel, and Mellin transforms are discussed in some detail. Applications to problems concerning heat flows, fluid flows, elastic waves, current and potential electric transmission lines are included in this chapter. Chapters 12 and 13 are entirely new. First-order and second-order nonlinear PDEs are covered in Chapter 12. Most of the contents of this chapter have been developed during the last twenty-five years. Several new nonlinear PDEs including the one-dimensional nonlinear wave equation, Whitham’s equation, Burgers’ equation, the Korteweg–de Vries equation, and the nonlinear Schr¨odinger equation are solved. The solutions of these equations are then discussed with physical significance. Special emphasis is given to the fundamental similarities and differences in the properties of the solutions to the corresponding linear and nonlinear equations under consideration. The final chapter is devoted to the major numerical and approximation methods for finding solutions of PDEs. A fairly detailed treatment of explicit and implicit finite difference methods is given with applications The variational method and the Euler–Lagrange equations are described with many applications. Also included are the Rayleigh–Ritz, the Galerkin, and the Kantorovich methods of approximation with many illustrations and applications. This new edition contains almost four hundred examples and exercises, which are either directly associated with applications or phrased in terms of the physical and engineering contexts in which they arise. The exercises truly complement the text, and answers to most exercises are provided at the end of the book. The Appendix has been expanded to include some basic properties of the Gamma function and the tables of Fourier, Laplace, and Hankel transforms. For students wishing to know more about the subject or to have further insight into the subject matter, important references are listed in the Bibliography. The chapters on mathematical models, Fourier series and integrals, and eigenvalue problems are self-contained, so these chapters can be omitted for those students who have prior knowledge of the subject. An attempt has been made to present a clear and concise exposition of the mathematics used in analyzing a variety of problems. With this in mind, the chapters are carefully organized to enable students to view the material in an orderly perspective. For example, the results and theorems in the chapters on Fourier series and integrals and on eigenvalue problems are explicitly mentioned, whenever necessary, to avoid confusion with their use in the development of PDEs. A wide range of problems subject to various boundary conditions has been included to improve the student’s understanding. In this third edition, specific changes and additions include the following: xxii Preface to the Third Edition 1. Chapter 2 on mathematical models has been revised by adding a list of the most common linear PDEs in applied mathematics, mathematical physics, and engineering science. 2. The chapter on the Cauchy problem has been expanded by including the wave equations in spherical and cylindrical polar coordinates. Examples and exercises on these wave equations and the energy equation have been added. 3. Eigenvalue problems have been revised with an emphasis on Green’s functions and applications. A section on the Schr¨odinger equation for the linear harmonic oscillator has been added. Higher-dimensional boundary-value problems with an emphasis on applications, and a section on the hydrogen atom and on the three-dimensional Schr¨odinger equation in a central field of force have been added to Chapter 9. 4. Chapter 11 has been extensively reorganized and revised in order to include Hankel and Mellin transforms and their applications, and has new sections on the asymptotic approximation method and the finite Hankel transform with applications. Many new examples and exercises, some new material with applications, and physical interpretations of mathematical solutions have also been included. 5. A new chapter on nonlinear PDEs of current interest and their applications has been added with considerable emphasis on the fundamental similarities and the distinguishing differences in the properties of the solutions to the nonlinear and corresponding linear equations. 6. Chapter 13 is also new. It contains a fairly detailed treatment of explicit and implicit finite difference methods with their stability analysis. A large section on the variational methods and the Euler–Lagrange equations has been included with many applications. Also included are the Rayleigh–Ritz, the Galerkin, and the Kantorovich methods of approximation with illustrations and applications. 7. Many new applications, examples, and exercises have been added to deepen the reader’s understanding. Expanded versions of the tables of Fourier, Laplace, and Hankel transforms are included. The bibliography has been updated with more recent and important references. As a text on partial differential equations for students in applied mathematics, physics, engineering, and applied sciences, this edition provides the student with the art of combining mathematics with intuitive and physical thinking to develop the most effective approach to solving problems. In preparing this edition, the authors wish to express their sincere thanks to those who have read the manuscript and offered many valuable suggestions and comments. The authors also wish to express their thanks to the editor and the staff of Elsevier–North Holland, Inc. for their kind help and cooperation. Tyn Myint-U Lokenath Debnath 1 Introduction “If you wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.” Henri Poincar´e “However varied may be the imagination of man, nature is a thousand times richer, ... Each of the theories of physics ... presents (partial differential) equations under a new aspect ... without the theories, we should not know partial differential equations.” Henri Poincar´e 1.1 Brief Historical Comments Historically, partial differential equations originated from the study of surfaces in geometry and a wide variety of problems in mechanics. During the second half of the nineteenth century, a large number of famous mathematicians became actively involved in the investigation of numerous problems presented by partial differential equations. The primary reason for this research was that partial differential equations both express many fundamental laws of nature and frequently arise in the mathematical analysis of diverse problems in science and engineering. The next phase of the development of linear partial differential equations was characterized by efforts to develop the general theory and various methods of solution of linear equations. In fact, partial differential equations have been found to be essential to the theory of surfaces on the one hand and to the solution of physical problems on the other. These two areas of mathematics can be seen as linked by the bridge of the calculus of variations. With the discovery of the basic concepts and properties of distributions, the modern theory of linear partial differential equations is now 2 1 Introduction well established. The subject plays a central role in modern mathematics, especially in physics, geometry, and analysis. Almost all physical phenomena obey mathematical laws that can be formulated by differential equations. This striking fact was first discovered by Isaac Newton (1642–1727) when he formulated the laws of mechanics and applied them to describe the motion of the planets. During the three centuries since Newton’s fundamental discoveries, many partial differential equations that govern physical, chemical, and biological phenomena have been found and successfully solved by numerous methods. These equations include Euler’s equations for the dynamics of rigid bodies and for the motion of an ideal fluid, Lagrange’s equations of motion, Hamilton’s equations of motion in analytical mechanics, Fourier’s equation for the diffusion of heat, Cauchy’s equation of motion and Navier’s equation of motion in elasticity, the Navier–Stokes equations for the motion of viscous fluids, the Cauchy–Riemann equations in complex function theory, the Cauchy–Green equations for the static and dynamic behavior of elastic solids, Kirchhoff’s equations for electrical circuits, Maxwell’s equations for electromagnetic fields, and the Schr¨odinger equation and the Dirac equation in quantum mechanics. This is only a sampling, and the recent mathematical and scientific literature reveals an almost unlimited number of differential equations that have been discovered to model physical, chemical and biological systems and processes. From the very beginning of the study, considerable attention has been given to the geometric approach to the solution of differential equations. The fact that families of curves and surfaces can be defined by a differential equation means that the equation can be studied geometrically in terms of these curves and surfaces. The curves involved, known as characteristic curves, are very useful in determining whether it is or is not possible to find a surface containing a given curve and satisfying a given differential equation. This geometric approach to differential equations was begun by Joseph-Louis Lagrange (1736–1813) and Gaspard Monge (1746–1818). Indeed, Monge first introduced the ideas of characteristic surfaces and characteristic cones (or Monge cones). He also did some work on second-order linear, homogeneous partial differential equations. The study of first-order partial differential equations began to receive some serious attention as early as 1739, when Alex-Claude Clairaut (1713– 1765) encountered these equations in his work on the shape of the earth. On the other hand, in the 1770s Lagrange first initiated a systematic study of the first-order nonlinear partial differential equations in the form f (x, y, u, ux, uy)=0, (1.1.1) where u = u (x, y) is a function of two independent variables. Motivated by research on gravitational effects on bodies of different shapes and mass distributions, another major impetus for work in partial differential equations originated from potential theory. Perhaps the most 1.1 Brief Historical Comments 3 important partial differential equation in applied mathematics is the potential equation, also known as the Laplace equation uxx +uyy = 0, where subscripts denote partial derivatives. This equation arose in steady state heat conduction problems involving homogeneous solids. James Clerk Maxwell (1831–1879) also gave a new initiative to potential theory through his famous equations, known as Maxwell’s equations for electromagnetic fields. Lagrange developed analytical mechanics as the application of partial differential equations to the motion of rigid bodies. He also described the geometrical content of a first-order partial differential equation and developed the method of characteristics for finding the general solution of quasi-linear equations. At the same time, the specific solution of physical interest was obtained by formulating an initial-value problem (or a Cauchy Problem) that satisfies certain supplementary conditions. The solution of an initial-value problem still plays an important role in applied mathematics, science and engineering. The fundamental role of characteristics was soon recognized in the study of quasi-linear and nonlinear partial differential equations. Physically, the first-order, quasi-linear equations often represent conservation laws which describe the conservation of some physical quantities of a system. In its early stages of development, the theory of second-order linear partial differential equations was concentrated on applications to mechanics and physics. All such equations can be classified into three basic categories: the wave equation, the heat equation, and the Laplace equation (or potential equation). Thus, a study of these three different kinds of equations yields much information about more general second-order linear partial differential equations. Jean d’Alembert (1717–1783) first derived the onedimensional wave equation for vibration of an elastic string and solved this equation in 1746. His solution is now known as the d’Alembert solution. The wave equation is one of the oldest equations in mathematical physics. Some form of this equation, or its various generalizations, almost inevitably arises in any mathematical analysis of phenomena involving the propagation of waves in a continuous medium. In fact, the studies of water waves, acoustic waves, elastic waves in solids, and electromagnetic waves are all based on this equation. A technique known as the method of separation of variables is perhaps one of the oldest systematic methods for solving partial differential equations including the wave equation. The wave equation and its methods of solution attracted the attention of many famous mathematicians including Leonhard Euler (1707–1783), James Bernoulli (1667–1748), Daniel Bernoulli (1700–1782), J.L. Lagrange (1736–1813), and Jacques Hadamard (1865–1963). They discovered solutions in several different forms, and the merit of their solutions and relations among these solutions were argued in a series of papers extending over more than twenty-five years; most concerned the nature of the kinds of functions that can be represented by trigonometric (or Fourier) series. These controversial problems were finally resolved during the nineteenth century. 4 1 Introduction It was Joseph Fourier (1768–1830) who made the first major step toward developing a general method of solutions of the equation describing the conduction of heat in a solid body in the early 1800s. Although Fourier is most celebrated for his work on the conduction of heat, the mathematical methods involved, particularly trigonometric series, are important and very useful in many other situations. He created a coherent mathematical method by which the different components of an equation and its solution in series were neatly identified with the different aspects of the physical solution being analyzed. In spite of the striking success of Fourier analysis as one of the most useful mathematical methods, J.L. Lagrange and S.D. Poisson (1781–1840) hardly recognized Fourier’s work because of its lack of rigor. Nonetheless, Fourier was eventually recognized for his pioneering work after publication of his monumental treatise entitled La Th´eorie Auatytique de la Chaleur in 1822. It is generally believed that the concept of an integral transform originated from the Integral Theorem as stated by Fourier in his 1822 treatise. It was the work of Augustin Cauchy (1789–1857) that contained the exponential form of the Fourier Integral Theorem as f (x) = 1 2π  ∞ −∞ e ikx  ∞ −∞ e −ikξ f (ξ) dξ dk. (1.1.2) This theorem has been expressed in several slightly different forms to better adapt it for particular applications. It has been recognized, almost from the start, however, that the form which best combines mathematical simplicity and complete generality makes use of the exponential oscillating function exp (ikx). Indeed, the Fourier integral formula (1.1.2) is regarded as one of the most fundamental results of modern mathematical analysis, and it has widespread physical and engineering applications. The generality and importance of the theorem is well expressed by Kelvin and Tait who said: “ ... Fourier’s Theorem, which is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along a telegraph wire, and the conduction of heat by the earth’s crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance.” This integral formula (1.1.2) is usually used to define the classical Fourier transform of a function and the inverse Fourier transform. No doubt, the scientific achievements of Joseph Fourier have not only provided the fundamental basis for the study of heat equation, Fourier series, and Fourier integrals, but for the modern developments of the theory and applications of the partial differential equations. One of the most important of all the partial differential equations involved in applied mathematics and mathematical physics is that associated with the name of Pierre-Simon Laplace (1749–1827). This equation was first discovered by Laplace while he was involved in an extensive study of 1.1 Brief Historical Comments 5 gravitational attraction of arbitrary bodies in space. Although the main field of Laplace’s research was celestial mechanics, he also made important contributions to the theory of probability and its applications. This work introduced the method known later as the Laplace transform, a simple and elegant method of solving differential and integral equations. Laplace first introduced the concept of potential, which is invaluable in a wide range of subjects, such as gravitation, electromagnetism, hydrodynamics, and acoustics. Consequently, the Laplace equation is often referred to as the potential equation. This equation is also an important special case of both the wave equation and the heat equation in two or three dimensions. It arises in the study of many physical phenomena including electrostatic or gravitational potential, the velocity potential for an imcompossible fluid flows, the steady state heat equation, and the equilibrium (time independent) displacement field of a two- or three-dimensional elastic membrane. The Laplace equation also occurs in other branches of applied mathematics and mathematical physics. Since there is no time dependence in any of the mathematical problems stated above, there are no initial data to be satisfied by the solutions of the Laplace equation. They must, however, satisfy certain boundary conditions on the boundary curve or surface of a region in which the Laplace equation is to be solved. The problem of finding a solution of Laplace’s equation that takes on the given boundary values is known as the Dirichlet boundary-value problem, after Peter Gustav Lejeune Dirichlet (1805–1859). On the other hand, if the values of the normal derivative are prescribed on the boundary, the problem is known as Neumann boundary-value problem, in honor of Karl Gottfried Neumann (1832–1925). Despite great efforts by many mathematicians including Gaspard Monge (1746–1818), AdrienMarie Legendre (1752–1833), Carl Friedrich Gauss (1777–1855), SimeonDenis Poisson (1781–1840), and Jean Victor Poncelet (1788–1867), very little was known about the general properties of the solutions of Laplace’s equation until 1828, when George Green (1793–1841) and Mikhail Ostrogradsky (1801–1861) independently investigated properties of a class of solutions known as harmonic functions. On the other hand, Augustin Cauchy (1789–1857) and Bernhard Riemann (1826–1866) derived a set of first-order partial differential equations, known as the Cauchy–Riemann equations, in their independent work on functions of complex variables. These equations led to the Laplace equation, and functions satisfying this equation in a domain are called harmonic functions in that domain. Both Cauchy and Riemann occupy a special place in the history of mathematics. Riemann made enormous contributions to almost all areas of pure and applied mathematics. His extraordinary achievements stimulated further developments, not only in mathematics, but also in mechanics, physics, and the natural sciences as a whole. Augustin Cauchy is universally recognized for his fundamental contributions to complex analysis. He also provided the first systematic and rigorous 6 1 Introduction investigation of differential equations and gave a rigorous proof for the existence of power series solutions of a differential equation in the 1820s. In 1841 Cauchy developed what is known as the method of majorants for proving that a solution of a partial differential equation exists in the form of a power series in the independent variables. The method of majorants was also introduced independently by Karl Weierstrass (1815–1896) in that same year in application to a system of differential equations. Subsequently, Weierstrass’s student Sophie Kowalewskaya (1850–1891) used the method of majorants and a normalization theorem of Carl Gustav Jacobi (1804–1851) to prove an exceedingly elegant theorem, known as the Cauchy–Kowalewskaya theorem. This theorem quite generally asserts the local existence of solutions of a system of partial differential equations with initial conditions on a noncharacteristic surface. This theorem seems to have little practical importance because it does not distinguish between well-posed and ill-posed problems; it covers situations where a small change in the initial data leads to a large change in the solution. Historically, however, it is the first existence theorem for a general class of partial differential equations. The general theory of partial differential equations was initiated by A.R. Forsyth (1858–1942) in the fifth and sixth volumes of his Theory of Differential Equations and by E.J.B. Goursat (1858–1936) in his book entitled Cours d’ analyse mathematiques (1918) and his Lecons sur l’ integration des equations aux d´eriv´ees, volume 1 (1891) and volume 2 (1896). Another notable contribution to this subject was made by E. Cartan’s book, Lecons sur les invariants int´egraux, published in 1922. Joseph Liouville (1809– 1882) formulated a more tractable partial differential equation in the form uxx + uyy = k exp (au), (1.1.3) and obtained a general solution of it. This equation has a large number of applications. It is a special case of the equation derived by J.L. Lagrange for the stream function ψ in the case of two-dimensional steady vortex motion in an incompossible fluid, that is, ψxx + ψyy = F (ψ), (1.1.4) where F (ψ) is an arbitrary function of ψ. When ψ = u and F (u) = keau , equation (1.1.4) reduces to the Liouville equation (1.1.3). In view of the special mathematical interest in the nonhomogeneous nonlinear equation of the type (1.1.4), a number of famous mathematicians including Henri Poincar´e, E. Picard (1856–1941), Cauchy (1789–1857), Sophus Lie (1842– 1899), L.M.H. Navier (1785–1836), and G.G. Stokes (1819–1903) made many major contributions to partial differential equations. Historically, Euler first solved the eigenvalue problem when he developed a simple mathematical model for describing the the ‘buckling’ modes of a vertical elastic beam. The general theory of eigenvalue problems for second-order differential equations, now known as the Sturm–Liouville Theory, originated from the study of a class of boundary-value problems due to 1.1 Brief Historical Comments 7 Charles Sturm (1803–1855) and Joseph Liouville (1809–1882). They showed that, in general, there is an infinite set of eigenvalues satisfying the given equation and the associated boundary conditions, and that these eigenvalues increase to infinity. Corresponding to these eigenvalues, there is an infinite set of orthogonal eigenfunctions so that the linear superposition principle can be applied to find the convergent infinite series solution of the given problem. Indeed, the Sturm–Liouville theory is a natural generalization of the theory of Fourier series that greatly extends the scope of the method of separation of variables. In 1926, the WKB approximation method was developed by Gregor Wentzel, Hendrik Kramers, and MarcelLouis Brillouin for finding the approximate eigenvalues and eigenfunctions of the one-dimensional Schr¨odinger equation in quantum mechanics. This method is now known as the short-wave approximation or the geometrical optics approximation in wave propagation theory. At the end of the seventeenth century, many important questions and problems in geometry and mechanics involved minimizing or maximizing of certain integrals for two reasons. The first of these were several existence problems, such as, Newton’s problem of missile of least resistance, Bernoulli’s isoperimetric problem, Bernoulli’s problem of the brachistochrone (brachistos means shortest, chronos means time), the problem of minimal surfaces due to Joseph Plateau (1801–1883), and Fermat’s principle of least time. Indeed, the variational principle as applied to the propagation and reflection of light in a medium was first enunciated in 1662 by one of the greatest mathematicians of the seventeenth century, Pierre Fermat (1601–1665). According to his principle, a ray of light travels in a homogeneous medium from one point to another along a path in a minimum time. The second reason is somewhat philosophical, that is, how to discover a minimizing principle in nature. The following 1744 statement of Euler is characteristic of the philosophical origin of what is known as the principle of least action: “As the construction of the universe is the most perfect possible, being the handiwork of all-wise Maker, nothing can be met with in the world in which some maximal or minimal property is not displayed. There is, consequently, no doubt but all the effects of the world can be derived by the method of maxima and minima from their final causes as well as from their efficient ones.” In the middle of the eighteenth century, Pierre de Maupertius (1698–1759) stated a fundamental principle, known as the principle of least action, as a guide to the nature of the universe. A still more precise and general formulation of Maupertius’ principle of least action was given by Lagrange in his Analytical Mechanics published in 1788. He formulated it as δS = δ  t2 t1 (2T) dt = 0, (1.1.5) where T is the kinematic energy of a dynamical system with the constraint that the total energy, (T + V ), is constant along the trajectories, and V is 8 1 Introduction the potential energy of the system. He also derived the celebrated equation of motion for a holonomic dynamical system d dt  ∂T ∂q˙i  − ∂T ∂qi = Qi , (1.1.6) where qi are the generalized coordinates, ˙qi is the velocity, and Qi is the force. For a conservative dynamical system, Qi = − ∂V ∂qi , V = V (qi), ∂V ∂q˙i = 0, then (1.1.6) can be expressed in terms of the Lagrangian, L = T − V , as d dt  ∂L ∂q˙i  − ∂L ∂qi = 0. (1.1.7) This principle was then reformulated by Euler in a way that made it useful in mathematics and physics. The work of Lagrange remained unchanged for about half a century until William R. Hamilton (1805–1865) published his research on the general method in analytical dynamics which gave a new and very appealing form to the Lagrange equations. Hamilton’s work also included his own variational principle. In his work on optics during 1834–1835, Hamilton elaborated a new principle of mechanics, known as Hamilton’s principle, describing the stationary action for a conservative dynamical system in the form δA = δ  t1 t0 (T − V ) dt = δ  t1 t0 L dt = 0. (1.1.8) Hamilton’s principle (1.1.8) readily led to the Lagrange equation (1.1.6). In terms of time t, the generalized coordinates qi , and the generalized momenta pi = (∂L/q˙i) which characterize the state of a dynamical system, Hamilton introduced the function H (qi , pi , t) = piq˙i − L(qi , pi , t), (1.1.9) and then used it to represent the equation of motion (1.1.6) as a system of first order partial differential equations q˙i = ∂H ∂pi , p˙i = − ∂H ∂q˙i . (1.1.10) These equations are known as the celebrated Hamilton canonical equations of motion, and the function H (qi , pi , t) is referred to as the Hamiltonian which is equal to the total energy of the system. Following the work of Hamilton, Karl Jacobi, Mikhail Ostrogradsky (1801–1862), and Henri Poincar´e (1854–1912) put forth new modifications of the variational principle. Indeed, the action integral S can be regarded as a function of generalized coordinates and time provided the terminal point is not fixed. In 1842, Jacobi showed that S satisfies the first-order partial differential equation 1.1 Brief Historical Comments 9 ∂S ∂t + H  qi , ∂S ∂qi , t = 0, (1.1.11) which is known as the Hamilton–Jacobi equation. In 1892, Poincar´e defined the action integral on the trajectories in phase space of the variable qi and pi as S =  t1 t0 [piq˙i − H (pi , qi)] dt, (1.1.12) and then formulated another modification of the Hamilton variational principle which also yields the Hamilton canonical equations (1.1.10). From (1.1.12) also follows the celebrated Poincar´e–Cartan invariant I =  C (piδqi − Hδt), (1.1.13) where C is an arbitrary closed contour in the phase space. Indeed, the discovery of the calculus of variations in a modern sense began with the independent work of Euler and Lagrange. The first necessary condition for the existence of an extremum of a functional in a domain leads to the celebrated Euler–Lagrange equation. This equation in its various forms now assumes primary importance, and more emphasis is given to the first variation, mainly due to its power to produce significant equations, than to the second variation, which is of fundamental importance in answering the question of whether or not an extremal actually provides a minimum (or a maximum). Thus, the fundamental concepts of the calculus of variations were developed in the eighteenth century in order to obtain the differential equations of applied mathematics and mathematical physics. During its early development, the problems of the calculus of variations were reduced to questions of the existence of differential equations problems until David Hilbert developed a new method in which the existence of a minimizing function was established directly as the limit of a sequence of approximations. Considerable attention has been given to the problem of finding a necessary and sufficient condition for the existence of a function which extremized the given functional. Although the problem of finding a sufficient condition is a difficult one, Legendre and C.G.J. Jacobi (1804–1851) discovered a second necessary condition and a third necessary condition respectively. Finally, it was Weierstrass who first provided a satisfactory foundation to the theory of calculus of variations in his lectures at Berlin between 1856 and 1870. His lectures were essentially concerned with a complete review of the work of Legendre and Jacobi. At the same time, he reexamined the concepts of the first and second variations and looked for a sufficient condition associated with the problem. In contrast to the work of his predecessors, Weierstrass introduced the ideas of ‘strong variations’ and ‘the excess function’ which led him to discover a fourth necessary condition 10 1 Introduction and a satisfactory sufficient condition. Some of his outstanding discoveries announced in his lectures were published in his collected work. At the conclusion of his famous lecture on ‘Mathematical Problems’ at the Paris International Congress of Mathematicians in 1900, David Hilbert (1862–1943), perhaps the most brilliant mathematician of the late nineteenth century, gave a new method for the discussion of the minimum value of a functional. He obtained another derivation of Weierstrass’s excess function and a new approach to Jacobi’s problem of determining necessary and sufficient conditions for the existence of a minimum of a functional; all this without the use of the second variation. Finally, the calculus of variations entered the new and wider field of ‘global’ problems with the original work of George D. Birkhoff (1884–1944) and his associates. They succeeded in liberating the theory of calculus of variations from the limitations imposed by the restriction to ‘small variations’, and gave a general treatment of the global theory of the subject with large variations. In 1880, George Fitzgerald (1851–1901) probably first employed the variational principle in electromagnetic theory to derive Maxwell’s equations for an electromagnetic field in a vacuum. Moreover, the variational principle received considerable attention in electromagnetic theory after the work of Karl Schwarzchild in 1903 as well as the work of Max Born (1882–1970) who formulated the principle of stationary action in electrodynamics in a symmetric four-dimensional form. On the other hand, Poincar´e showed in 1905 that the action integral is invariant under the Lorentz transformations. With the development of the special theory of relativity and the relativistic theory of gravitation in the beginning of the twentieth century, the variational principles received tremendous attention from many great mathematicians and physicists including Albert Einstein (1879–1955), Hendrix Lorentz (1853–1928), Hermann Weyl (1885–1955), Felix Klein (1849– 1925), Amalie Noether (1882–1935), and David Hilbert. Even before the use of variational principles in electrodynamics, Lord Rayleigh (1842–1919) employed variational methods in his famous book, The Theory of Sound, for the derivation of equations for oscillations in plates and rods in order to calculate frequencies of natural oscillations of elastic systems. In his pioneering work in the 1960’s, Gerald Whitham first developed a general approach to linear and nonlinear dispersive waves using a Lagrangian. He successfully formulated the averaged variational principle, which is now known as the Whitham averaged variational principle, which was employed to derive the basic equations for linear and nonlinear dispersive wave propagation problems. In 1967, Luke first explicitly formulated a variational principle for nonlinear water waves. In 1968, Bretherton and Garret generalized the Whitham averaged variational principle to describe the conservation law for the wave action in a moving medium. Subsequently, Ostrovsky and Pelinovsky (1972) also generalized the Whitham averaged variational principle to nonconservative systems. 1.1 Brief Historical Comments 11 With the rapid development of the theory and applications of differential equations, the closed form analytical solutions of many different types of equations were hardly possible. However, it is extremely important and absolutely necessary to provide some insight into the qualitative and quantitative nature of solutions subject to initial and boundary conditions. This insight usually takes the form of numerical and graphical representatives of the solutions. It was E. Picard (1856–1941) who first developed the method of successive approximations for the solutions of differential equations in most general form and later made it an essential part of his treatment of differential equations in the second volume of his Trait´e d’Analyse published in 1896. During the last two centuries, the calculus of finite differences in various forms played a significant role in finding the numerical solutions of differential equations. Historically, many well known integration formulas and numerical methods including the Euler–Maclaurin formula, Gregory integration formula, the Gregory–Newton formula, Simpson’s rule, Adam– Bashforth’s method, the Jacobi iteration, the Gauss–Seidel method, and the Runge–Kutta method have been developed and then generalized in various forms. With the development of modern calculators and high-speed electronic computers, there has been an increasing trend in research toward the numerical solution of ordinary and partial differential equations during the twentieth century. Special attention has also given to in depth studies of convergence, stability, error analysis, and accuracy of numerical solutions. Many well-known numerical methods including the Crank–Nicolson methods, the Lax–Wendroff method, Richtmyer’s method, and Stone’s implicit iterative technique have been developed in the second half of the twentieth century. All finite difference methods reduce differential equations to discrete forms. In recent years, more modern and powerful computational methods such as the finite element method and the boundary element method have been developed in order to handle curved or irregularly shaped domains. These methods are distinguished by their more general character, which makes them more capable of dealing with complex geometries, allows them to use non-structured grid systems, and allows more natural imposition of the boundary conditions. During the second half of the nineteenth century, considerable attention was given to problems concerning the existence, uniqueness, and stability of solutions of partial differential equations. These studies involved not only the Laplace equation, but the wave and diffusion equations as well, and were eventually extended to partial differential equations with variable coefficients. Through the years, tremendous progress has been made on the general theory of ordinary and partial differential equations. With the advent of new ideas and methods, new results and applications, both analytical and numerical studies are continually being added to this subject. Partial differential equations have been the subject of vigorous mathematical research for over three centuries and remain so today. This is an active 12 1 Introduction area of research for mathematicians and scientists. In part, this is motivated by the large number of problems in partial differential equations that mathematicians, scientists, and engineers are faced with that are seemingly intractable. Many of these equations are nonlinear and come from such areas of applications as fluid mechanics, plasma physics, nonlinear optics, solid mechanics, biomathematics, and quantum field theory. Owing to the ever increasing need in mathematics, science, and engineering to solve more and more complicated real world problems, it seems quite likely that partial differential equations will remain a major area of research for many years to come. 1.2 Basic Concepts and Definitions A differential equation that contains, in addition to the dependent variable and the independent variables, one or more partial derivatives of the dependent variable is called a partial differential equation. In general, it may be written in the form f (x, y, . . . , u, ux, uy,...,uxx, uxy,...)=0, (1.2.1) involving several independent variables x, y, ..., an unknown function u of these variables, and the partial derivatives ux, uy, ..., uxx, uxy, ..., of the function. Subscripts on dependent variables denote differentiations, e.g., ux = ∂u/∂x, uxy = ∂ 2 /∂y ∂x. Here equation (1.2.1) is considered in a suitable domain D of the ndimensional space Rn in the independent variables x, y, .... We seek functions u = u (x, y, . . .) which satisfy equation (1.2.1) identically in D. Such functions, if they exist, are called solutions of equation (1.2.1). From these many possible solutions we attempt to select a particular one by introducing suitable additional conditions. For instance, uuxy + ux = y, uxx + 2yuxy + 3xuyy = 4 sin x, (1.2.2) (ux) 2 + (uy) 2 = 1, uxx − uyy = 0, are partial differential equations. The functions u (x, y)=(x + y) 3 , u (x, y) = sin (x − y), are solutions of the last equation of (1.2.2), as can easily be verified. 1.2 Basic Concepts and Definitions 13 The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation. For example uxx + 2xuxy + uyy = e y is a second-order partial differential equation, and uxxy + xuyy + 8u = 7y is a third-order partial differential equation. A partial differential equation is said to be linear if it is linear in the unknown function and all its derivatives with coefficients depending only on the independent variables; it is said to be quasi-linear if it is linear in the highest-ordered derivative of the unknown function. For example, the equation yuxx + 2xyuyy + u = 1 is a second-order linear partial differential equation, whereas uxuxx + xuuy = sin y is a second-order quasi-linear partial differential equation. The equation which is not linear is called a nonlinear equation. We shall be primarily concerned with linear second-order partial differential equations, which frequently arise in problems of mathematical physics. The most general second-order linear partial differential equation in n independent variables has the form n i,j=1 Aijuxixj + n i=1 Biuxi + F u = G, (1.2.3) where we assume without loss of generality that Aij = Aji. We also assume that Bi , F, and G are functions of the n independent variables xi . If G is identically zero, the equation is said to be homogeneous; otherwise it is nonhomogeneous. The general solution of a linear ordinary differential equation of nth order is a family of functions depending on n independent arbitrary constants. In the case of partial differential equations, the general solution depends on arbitrary functions rather than on arbitrary constants. To illustrate this, consider the equation uxy = 0. If we integrate this equation with respect to y, we obtain ux (x, y) = f (x). 14 1 Introduction A second integration with respect to x yields u (x, y) = g (x) + h (y), where g (x) and h (y) are arbitrary functions. Suppose u is a function of three variables, x, y, and z. Then, for the equation uyy = 2, one finds the general solution u (x, y, z) = y 2 + yf (x, z) + g (x, z), where f and g are arbitrary functions of two variables x and z. We recall that in the case of ordinary differential equations, the first task is to find the general solution, and then a particular solution is determined by finding the values of arbitrary constants from the prescribed conditions. But, for partial differential equations, selecting a particular solution satisfying the supplementary conditions from the general solution of a partial differential equation may be as difficult as, or even more difficult than, the problem of finding the general solution itself. This is so because the general solution of a partial differential equation involves arbitrary functions; the specialization of such a solution to the particular form which satisfies supplementary conditions requires the determination of these arbitrary functions, rather than merely the determination of constants. For linear homogeneous ordinary differential equations of order n, a linear combination of n linearly independent solutions is a solution. Unfortunately, this is not true, in general, in the case of partial differential equations. This is due to the fact that the solution space of every homogeneous linear partial differential equation is infinite dimensional. For example, the partial differential equation ux − uy = 0 (1.2.4) can be transformed into the equation 2uη = 0 by the transformation of variables ξ = x + y, η = x − y. The general solution is u (x, y) = f (x + y), where f (x + y) is an arbitrary function. Thus, we see that each of the functions 1.3 Mathematical Problems 15 (x + y) n , sin n (x + y), cos n (x + y), exp n (x + y), n = 1, 2, 3,... is a solution of equation (1.2.4). The fact that a simple equation such as (1.2.4) yields infinitely many solutions is an indication of an added difficulty which must be overcome in the study of partial differential equations. Thus, we generally prefer to directly determine the particular solution of a partial differential equation satisfying prescribed supplementary conditions. 1.3 Mathematical Problems A problem consists of finding an unknown function of a partial differential equation satisfying appropriate supplementary conditions. These conditions may be initial conditions (I.C.) and/or boundary conditions (B.C.). For example, the partial differential equation (PDE) ut − uxx = 0, 0 <x<l, t=""> 0, with I.C. u (x, 0) = sin x, 0 ≤ x ≤ l, t > 0, B.C. u (0, t) = 0, t ≥ 0, B.C. u (l, t) = 0, t ≥ 0, constitutes a problem which consists of a partial differential equation and three supplementary conditions. The equation describes the heat conduction in a rod of length l. The last two conditions are called the boundary conditions which describe the function at two prescribed boundary points. The first condition is known as the initial condition which prescribes the unknown function u (x, t) throughout the given region at some initial time t, in this case t = 0. This problem is known as the initial boundary-value problem. Mathematically speaking, the time and the space coordinates are regarded as independent variables. In this respect, the initial condition is merely a point prescribed on the t-axis and the boundary conditions are prescribed, in this case, as two points on the x-axis. Initial conditions are usually prescribed at a certain time t = t0 or t = 0, but it is not customary to consider the other end point of a given time interval. In many cases, in addition to prescribing the unknown function, other conditions such as their derivatives are specified on the boundary and/or at time t0. In considering the problem of unbounded domain, the solution can be determined uniquely by prescribing initial conditions only. The corresponding problem is called the initial-value problem or the Cauchy problem. The mathematical definition is given in Chapter 5. The solution of such a prob- 16 1 Introduction lem may be interpreted physically as the solution unaffected by the boundary conditions at infinity. For problems affected by the boundary at infinity, boundedness conditions on the behavior of solutions at infinity must be prescribed. A mathematical problem is said to be well-posed if it satisfies the following requirements: 1. Existence: There is at least one solution. 2. Uniqueness: There is at most one solution. 3. Continuity: The solution depends continuously on the data. The first requirement is an obvious logical condition, but we must keep in mind that we cannot simply state that the mathematical problem has a solution just because the physical problem has a solution. We may well be erroneously developing a mathematical model, say, consisting of a partial differential equation whose solution may not exist at all. The same can be said about the uniqueness requirement. In order to really reflect the physical problem that has a unique solution, the mathematical problem must have a unique solution. For physical problems, it is not sufficient to know that the problem has a unique solution. Hence the last requirement is not only useful but also essential. If the solution is to have physical significance, a small change in the initial data must produce a small change in the solution. The data in a physical problem are normally obtained from experiment, and are approximated in order to solve the problem by numerical or approximate methods. It is essential to know that the process of making an approximation to the data produces only a small change in the solution. 1.4 Linear Operators An operator is a mathematical rule which, when applied to a function, produces another function. For example, in the expressions L[u] = ∂ 2u ∂x2 + ∂ 2u ∂y2 , M [u] = ∂ 2u ∂x2 − ∂u ∂x + x ∂u ∂y , L =  ∂ 2/∂x2 + ∂ 2/∂y2 and M =  ∂ 2/∂x2 − ∂/∂x + x (∂/∂y) are called the differential operators. An operator is said to be linear if it satisfies the following: 1. A constant c may be taken outside the operator: L[cu] = cL[u] . (1.4.1) 1.4 Linear Operators 17 2. The operator operating on the sum of two functions gives the sum of the operator operating on the individual functions: L[u1 + u2] = L[u1] + L[u2] . (1.4.2) We may combine (1.4.1) and (1.4.2) as L[c1u1 + c2u2] = c1L[u1] + c2L[u2] , (1.4.3) where c1 and c2 are any constants. This can be extended to a finite number of functions. If u1, u2, ..., uk are k functions and c1, c2, ..., ck are k constants, then by repeated application of equation (1.4.3) L ⎡ ⎣  k j=1 cjuj ⎤ ⎦ =  k j=1 cjL[uj ] . (1.4.4) We may now define the sum of two linear differential operators formally. If L and M are two linear operators, then the sum of L and M is defined as (L + M) [u] = L[u] + M [u] , (1.4.5) where u is a sufficiently differentiable function. It can be readily shown that L + M is also a linear operator. The product of two linear differential operators L and M is the operator which produces the same result as is obtained by the successive operations of the operators L and M on u, that is, LM [u] = L(M [u]), (1.4.6) in which we assume that M [u] and L(M [u]) are defined. It can be readily shown that LM is also a linear operator. In general, linear differential operators satisfy the following: 1. L + M = M + L (commutative) (1.4.7) 2. (L + M) + N = L + (M + N) (associative) (1.4.8) 3. (LM) N = L(MN) (associative) (1.4.9) 4. L(c1M + c2N) = c1LM + c2LN (distributive). (1.4.10) For linear differential operators with constant coefficients, 5. LM = ML (commutative). (1.4.11) Example 1.4.1. Let L = ∂ 2 ∂x2 + x ∂ ∂y and M = ∂ 2 ∂y2 − y ∂ ∂y . 18 1 Introduction LM [u] =  ∂ 2 ∂x2 + x ∂ ∂y ∂ 2u ∂y2 − y ∂u ∂y  = ∂ 4u ∂x2∂y2 − y ∂ 3u ∂x2∂y + x ∂ 3u ∂y3 − xy ∂ 2u ∂y2 , ML[u] =  ∂ 2 ∂y2 − y ∂ ∂y ∂ 2u ∂x2 + x ∂u ∂y  = ∂ 4u ∂y2∂x2 + x ∂ 3u ∂y3 − y ∂ 3u ∂y∂x2 − xy ∂ 2u ∂y2 , which shows that LM = ML. Now let us consider a linear second-order partial differential equation. In the case of two independent variables, such an equation takes the form A (x, y) uxx + B (x, y) uxy + C (x, y) uyy +D (x, y) ux + E (x, y) uy + F (x, y) u = G (x, y), (1.4.12) where A, B, C, D, E, and F are the coefficients, and G is the nonhomogeneous term. If we denote L = A ∂ 2 ∂x2 + B ∂ 2 ∂x∂y + C ∂ 2 ∂y2 + D ∂ ∂x + E ∂ ∂y + F, then equation (1.4.12) may be written in the form L[u] = G. (1.4.13) Very often the square bracket is omitted and we simply write Lu = G. Let v1, v2, ..., vn be n functions which satisfy L[vj ] = Gj , j = 1, 2,...,n and let w1, w2, ..., wn be n functions which satisfy L[wj ]=0, j = 1, 2, . . . , n. If we let uj = vj + wj then, the function u = n j=1 uj 1.4 Linear Operators 19 satisfies the equation L[u] = n j=1 Gj . This is called the principle of linear superposition. In particular, if v is a particular solution of equation (1.4.13), that is, L[v] = G, and w is a solution of the associated homogeneous equation, that is, L[w] = 0, then u = v + w is a solution of L[u] = G. The principle of linear superposition is of fundamental importance in the study of partial differential equations. This principle is used extensively in solving linear partial differential equations by the method of separation of variables. Suppose that there are infinitely many solutions u1 (x, y), u2 (x, y), ... un (x, y), ... of a linear homogeneous partial differential equation Lu = 0. Can we say that every infinite linear combination c1u1 +c2u2 +···+cnun + ··· of these solutions, where c1, c2, ..., cn, ... are any constants, is again a solution of the equation? Of course, by an infinite linear combination, we mean an infinite series and we must require that the infinite series ∞ k=0 ck uk = limn→∞ n k=0 ck uk (1.4.14) must be convergent to u. In general, we state that the infinite series is a solution of the homogeneous equation. There is another kind of infinite linear combination which is also used to find the solution of a given linear equation. This is concerned with a family of solutions u (x, y; k) of the linear equation, where k is any real number, not just the values 1, 2, 3,.... If ck = c (k) is any function of the real parameter k such that  b a c (k) u (x, y; k) dk or  ∞ −∞ c (k) u (x, y; k) dk (1.4.15) is convergent, then, under suitable conditions, the integral (1.4.15), again, is a solution. This may be called the linear integral superposition principle. To illustrate these ideas, we consider the equation Lu = ux + 2uy = 0. (1.4.16) It is easy to verify that, for every real k, the function u (x, y; k) = e k(2x−y) (1.4.17) is a solution of (1.4.16). Multiplying (1.4.17) by e −k and integrating with respect to k over −1 ≤ k ≤ 1 gives 20 1 Introduction u (x, y) =  1 −1 e −k e k(2x−y) dk = e 2x−y−1 2x − y − 1 (1.4.18) It is easy to verify that u (x, y) given by (1.4.18) is also a solution of (1.4.16). It is also easy to verify that u (x, y; k) = e −ky cos (k x), k ∈ R is a one-parameter family of solutions of the Laplace equation ∇2u ≡ uxx + uyy = 0 (1.4.19) It is also easy to check that v (x, y; k) = ∂ ∂k u (x, y; k) (1.4.20) is also a one-parameter family of solutions of (1.4.19), k ∈ R. Further, for any (x, y) in the upper half-plane y > 0, the integral v (x, y) ≡  ∞ 0 u (x, y, k) dk =  ∞ 0 e −ky cos (k x) dk, (1.4.21) is convergent, and v (x, y) is a solution of (1.4.19) for x ∈ R and y > 0. This follows from direct computation of vxx and vyy. The solution (1.4.21) is another example of the linear integral superposition principle. 1.5 Superposition Principle We may express supplementary conditions using the operator notation. For instance, the initial boundary-value problem utt − c 2uxx = G (x, t) 0 < x < l, t > 0, u (x, 0) = g1 (x) 0 ≤ x ≤ l, ut (x, 0) = g2 (x) 0 ≤ x ≤ l, (1.5.1) u (0, t) = g3 (t) t ≥ 0, u (l, t) = g4 (t) t ≥ 0, may be written in the form L[u] = G, M1 [u] = g1, M2 [u] = g2, (1.5.2) M3 [u] = g3, M4 [u] = g4, where gi are the prescribed functions and the subscripts on operators are assigned arbitrarily. 1.5 Superposition Principle 21 Now let us consider the problem L[u] = G, M1 [u] = g1, M2 [u] = g2, (1.5.3) . . . Mn [u] = gn. By virtue of the linearity of the equation and the supplementary conditions, we may divide problem (1.5.3) into a series of problems as follows: L[u1] = G, M1 [u1]=0, M2 [u1]=0, (1.5.4) . . . Mn [u1]=0, L[u2]=0, M1 [u2] = g1, M2 [u2]=0, (1.5.5) . . . Mn [u2]=0, L[un]=0, M1 [un]=0, M2 [un]=0, (1.5.6) . . . Mn [un] = gn. Then the solution of problem (1.5.3) is given by u = n i=1 ui . (1.5.7) Let us consider one of the subproblems, say, (1.5.5). Suppose we find a sequence of functions φ1,φ2, ..., which may be finite or infinite, satisfying the homogeneous system 22 1 Introduction L[φi ]=0, M2 [φi ]=0, (1.5.8) . . . Mn [φi ]=0, i = 1, 2, 3,... and suppose we can express g1 in terms of the series g1 = c1M1 [φ1] + c2M1 [φ2] + .... (1.5.9) Then the linear combination u2 = c1φ1 + c2φ2 + ..., (1.5.10) is the solution of problem (1.5.5). In the case of an infinite number of terms in the linear combination (1.5.10), we require that the infinite series be uniformly convergent and sufficiently differentiable, and that all the series Nk (ui) where N0 = L, Nj = Mj for j = 1, 2, ..., n convergence uniformly. 1.6 Exercises 1. For each of the following, state whether the partial differential equation is linear, quasi-linear or nonlinear. If it is linear, state whether it is homogeneous or nonhomogeneous, and gives its order. (a) uxx + xuy = y, (b) uux − 2xyuy = 0, (c) u 2 x + uuy = 1, (d) uxxxx + 2uxxyy + uyyyy = 0, (e) uxx + 2uxy + uyy = sin x, (f) uxxx + uxyy + log u = 0, (g) u 2 xx + u 2 x + sin u = e y , (h) ut + uux + uxxx = 0. 2. Verify that the functions u (x, y) = x 2 − y 2 u (x, y) = e x sin y u (x, y)=2xy are the solutions of the equation uxx + uyy = 0. 3. Show that u = f (xy), where f is an arbitrary differentiable function satisfies 1.6 Exercises 23 xux − yuy = 0 and verify that the functions sin (xy), cos (xy), log (xy), e xy, and (xy) 3 are solutions. 4. Show that u = f (x) g (y) where f and g are arbitrary twice differentiable functions satisfies uuxy − uxuy = 0. 5. Determine the general solution of the differential equation uyy + u = 0. 6. Find the general solution of uxx + ux = 0, by setting ux = v. 7. Find the general solution of uxx − 4uxy + 3uyy = 0, by assuming the solution to be in the form u (x, y) = f (λx + y), where λ is an unknown parameter. 8. Find the general solution of uxx − uyy = 0. 9. Show that the general solution of ∂ 2u ∂t2 − c 2 ∂ 2u ∂x2 = 0, is u (x, t) = f (x − ct) + g (x + ct), where f and g are arbitrary twice differentiable functions. 10. Verify that the function u = φ (xy) + x ψ 4y x 5 , is the general solution of the equation x 2uxx − y 2uyy = 0. 11. If ux = vy and vx = −uy, show that both u and v satisfy the Laplace equations ∇2u = 0 and ∇2 v = 0. 24 1 Introduction 12. If u (x, y) is a homogeneous function of degree n, show that u satisfies the first-order equation xux + yuy = nu. 13. Verify that u (x, y, t) = A cos (kx) cos (ly) cos (nct) + B sin (kx) sin (ly) sin (nct), where k 2 + l 2 = n 2 , is a solution of the equation utt = c 2 (uxx + uyy). 14. Show that u (x, y; k) = e −ky sin (kx), x ∈ R, y> 0, is a solution of the equation ∇2u ≡ uxx + uyy = 0 for any real parameter k. Verify that u (x, y) =  ∞ 0 c (k) e −ky sin (kx) dk is also a solution of the above equation. 15. Show, by differentiation that, u (x, t) = 1 √ 4πkt exp  − x 2 4kt , x ∈ R, t> 0, is a solution of the diffusion equation ut = k uxx, where k is a constant. 16. (a) Verify that u (x, y) = log 4 x 2 + y 2 5 , satisfies the equation uxx + uyy = 0 for all (x, y) = (0, 0). (b) Show that 1.6 Exercises 25 u (x, y, z) =  x 2 + y 2 + z 2 − 1 2 is a solution of the Laplace equation uxx + uyy + uzz = 0 except at the origin. (c) Show that u (r) = a rn satisfies the equation r 2u ′′ + 2ru′ − n (n + 1) u = 0. 17. Show that un (r, θ) = r n cos (nθ) and un (r, θ) = r n sin (nθ), n = 0, 1, 2, 3, ··· are solutions of the Laplace equation ∇2u ≡ urr + 1 r ur + 1 r 2 uθθ = 0. 18. Verify by differentiation that u (x, y) = cos x cosh y satisfies the Laplace equation uxx + uyy = 0. 19. Show that u (x, y) = f  2y + x 2 + g  2y − x 2 is a general solution of the equation uxx − 1 x ux − x 2uyy = 0. 20. If u satisfies the Laplace equation ∇2u ≡ uxx+uyy = 0, show that both xu and yu satisfy the biharmonic equation ∇4 ⎛ ⎝ xu yu ⎞ ⎠ = 0, but xu and yu will not satisfy the Laplace equation. 21. Show that u (x, y, t) = f (x + iky − iωt) + g (x − iky − iωt) is a general solution of the wave equation utt = c 2 (uxx + uyy), where f and g are arbitrary twice differentiable functions, and ω 2 = c 2  k 2 − 1 , k, ω, c are constants. 26 1 Introduction 22. Verify that u (x, y) = x 3 + y 2 + e x (cos x sin y cosh y − sin x cos y sinh y) is a classical solution of the Poisson equation uxx + uyy = (6x + 2). 23. Show that u (x, y) = exp 4 − x b 5 f (ax − by) satisfies the equation b ux + a uy + u = 0. 24. Show that utt − c 2uxx + 2b ut = 0 has solutions of the form u (x, t)=(A cos kx + B sin kx) V (t), where c, b, A and B are constants. 25. Show that c 2  urr + 1 r ur  − utt = 0 has solutions of the form u (r, t) = V (r) r cos (nct), n = 0, 1, 2,.... Find a differential equation for V (r). 2 First-Order, Quasi-Linear Equations and Method of Characteristics “As long as a branch of knowledge offers an abundance of problems, it is full of vitality.” David Hilbert “Since a general solution must be judged impossible from want of analysis, we must be content with the knowledge of some special cases, and that all the more, since the development of various cases seems to be the only way to bringing us at last to a more perfect knowledge.” Leonhard Euler 2.1 Introduction Many problems in mathematical, physical, and engineering sciences deal with the formulation and the solution of first-order partial differential equations. From a mathematical point of view, first-order equations have the advantage of providing a conceptual basis that can be utilized for second-, third-, and higher-order equations. This chapter is concerned with first-order, quasi-linear and linear partial differential equations and their solution by using the Lagrange method of characteristics and its generalizations. 2.2 Classification of First-Order Equations The most general, first-order, partial differential equation in two independent variables x and y is of the form 28 2 First-Order, Quasi-Linear Equations and Method of Characteristics F (x, y, u, ux, uy)=0, (x, y) ∈ D ⊂ R 2 , (2.2.1) where F is a given function of its arguments, and u = u (x, y) is an unknown function of the independent variables x and y which lie in some given domain D in R2 , ux = ∂u ∂x and uy = ∂u ∂y . Equation (2.2.1) is often written in terms of standard notation p = ux and q = uy so that (2.2.1) takes the form F (x, y, u, p, q)=0. (2.2.2) Similarly, the most general, first-order, partial differential equation in three independent variables x, y, z can be written as F (x, y, z, u, ux, uy, uz)=0. (2.2.3) Equation (2.2.1) or (2.2.2) is called a quasi-linear partial differential equation if it is linear in first-partial derivatives of the unknown function u (x, y). So, the most general quasi-linear equation must be of the form a (x, y, u) ux + b (x, y, u) uy = c (x, y, u), (2.2.4) where its coefficients a, b, and c are functions of x, y, and u. The following are examples of quasi-linear equations: x  y 2 + u ux − y  x 2 + u uy =  x 2 − y 2 u, (2.2.5) uux + ut + nu2 = 0, (2.2.6)  y 2 − u 2 ux − xy uy = xu. (2.2.7) Equation (2.2.4) is called a semilinear partial differential equation if its coefficients a and b are independent of u, and hence, the semilinear equation can be expressed in the form a (x, y) ux + b (x, y) uy = c (x, y, u). (2.2.8) Examples of semilinear equations are xux + yuy = u 2 + x 2 , (2.2.9) (x + 1)2 ux + (y − 1)2 uy = (x + y) u 2 , (2.2.10) ut + aux + u 2 = 0, (2.2.11) where a is a constant. Equation (2.2.1) is said to be linear if F is linear in each of the variables u, ux, and uy, and the coefficients of these variables are functions only of the independent variables x and y. The most general, first-order, linear partial differential equation has the form a (x, y) ux + b (x, y) uy + c (x, y) u = d (x, y), (2.2.12) 2.3 Construction of a First-Order Equation 29 where the coefficients a, b, and c, in general, are functions of x and y and d (x, y) is a given function. Unless stated otherwise, these functions are assumed to be continuously differentiable. Equations of the form (2.2.12) are called homogeneous if d (x, y) ≡ 0 or nonhomogeneous if d (x, y) = 0. Obviously, linear equations are a special kind of the quasi-linear equation (2.2.4) if a, b are independent of u and c is a linear function in u. Similarly, semilinear equation (2.2.8) reduces to a linear equation if c is linear in u. Examples of linear equations are xux + yuy − nu = 0, (2.2.13) nux + (x + y) uy − u = e x , (2.2.14) yux + xuy = xy, (2.2.15) (y − z) ux + (z − x) uy + (x − y) uz = 0. (2.2.16) An equation which is not linear is often called a nonlinear equation. So, first-order equations are often classified as linear and nonlinear. 2.3 Construction of a First-Order Equation We consider a system of geometrical surfaces described by the equation f (x, y, z, a, b)=0, (2.3.1) where a and b are arbitrary parameters. We differentiate (2.3.1) with respect to x and y to obtain fx + p fz = 0, fy + q fz = 0, (2.3.2) where p = ∂z ∂x and q = ∂z ∂y . The set of three equations (2.3.1) and (2.3.2) involves two arbitrary parameters a and b. In general, these two parameters can be eliminated from this set to obtain a first-order equation of the form F (x, y, z, p, q)=0. (2.3.3) Thus the system of surfaces (2.3.1) gives rise to a first-order partial differential equation (2.3.3). In other words, an equation of the form (2.3.1) containing two arbitrary parameters is called a complete solution or a complete integral of equation (2.3.3). Its role is somewhat similar to that of a general solution for the case of an ordinary differential equation. On the other hand, any relationship of the form f (φ, ψ)=0, (2.3.4) 30 2 First-Order, Quasi-Linear Equations and Method of Characteristics which involves an arbitrary function f of two known functions φ = φ (x, y, z) and ψ = ψ (x, y, z) and provides a solution of a first-order partial differential equation is called a general solution or general integral of this equation. Clearly, the general solution of a first-order partial differential equation depends on an arbitrary function. This is in striking contrast to the situation for ordinary differential equations where the general solution of a firstorder ordinary differential equation depends on one arbitrary constant. The general solution of a partial differential equation can be obtained from its complete integral. We obtain the general solution of (2.3.3) from its complete integral (2.3.1) as follows. First, we prescribe the second parameter b as an arbitrary function of the first parameter a in the complete solution (2.3.1) of (2.3.3), that is, b = b (a). We then consider the envelope of the one-parameter family of solutions so defined. This envelope is represented by the two simultaneous equations f (x, y, z, a, b (a)) = 0, (2.3.5) fa (x, y, z, a, b (a)) + fb (x, y, z, b (a)) b ′ (a)=0, (2.3.6) where the second equation (2.3.6) is obtained from the first equation (2.3.5) by partial differentiation with respect to a. In principle, equation (2.3.5) can be solved for a = a (x, y, z) as a function of x, y, and z. We substitute this result back in (2.3.5) to obtain f {x, y, z, a (x, y, z), b (a (x, y, z))} = 0, (2.3.7) where b is an arbitrary function. Indeed, the two equations (2.3.5) and (2.3.6) together define the general solution of (2.3.3). When a definite b (a) is prescribed, we obtain a particular solution from the general solution. Since the general solution depends on an arbitrary function, there are infinitely many solutions. In practice, only one solution satisfying prescribed conditions is required for a physical problem. Such a solution may be called a particular solution. In addition to the general and particular solutions of (2.3.3), if the envelope of the two-parameter system (2.3.1) of surfaces exists, it also represents a solution of the given equation (2.3.3); the envelope is called the singular solution of equation (2.3.3). The singular solution can easily be constructed from the complete solution (2.3.1) representing a two-parameter family of surfaces. The envelope of this family is given by the system of three equations f (x, y, z, a, b)=0, fa (x, y, z, a, b)=0, fb (x, y, z, a, b)=0. (2.3.8) In general, it is possible to eliminate a and b from (2.3.8) to obtain the equation of the envelope which gives the singular solution. It may be pointed out that the singular solution cannot be obtained from the general 2.3 Construction of a First-Order Equation 31 solution. Its nature is similar to that of the singular solution of a first-order ordinary differential equation. Finally, it is important to note that solutions of a partial differential equation are expected to be represented by smooth functions. A function is called smooth if all of its derivatives exist and are continuous. However, in general, solutions are not always smooth. A solution which is not everywhere differentiable is called a weak solution. The most common weak solution is the one that has discontinuities in its first partial derivatives across a curve, so that the solution can be represented by shock waves as surfaces of discontinuity. In the case of a first-order partial differential equation, there are discontinuous solutions where z itself and not merely p = ∂z ∂x and q = ∂z ∂y are discontinuous. In fact, this kind of discontinuity is usually known as a shock wave. An important feature of quasi-linear and nonlinear partial differential equations is that their solutions may develop discontinuities as they move away from the initial state. We close this section by considering some examples. Example 2.3.1. Show that a family of spheres x 2 + y 2 + (z − c) 2 = r 2 , (2.3.9) satisfies the first-order linear partial differential equation yp − xq = 0. (2.3.10) Differentiating the equation (2.3.9) with respect to x and y gives x + p (z − c) = 0 and y + q (z − c)=0. Eliminating the arbitrary constant c from these equations, we obtain the first-order, partial differential equation yp − xq = 0. Example 2.3.2. Show that the family of spheres (x − a) 2 + (y − b) 2 + z 2 = r 2 (2.3.11) satisfies the first-order, nonlinear, partial differential equation z 2  p 2 + q 2 + 1 = r 2 . (2.3.12) We differentiate the equation of the family of spheres with respect to x and y to obtain (x − a) + z p = 0, (y − b) + z q = 0. Eliminating the two arbitrary constants a and b, we find the nonlinear partial differential equation 32 2 First-Order, Quasi-Linear Equations and Method of Characteristics z 2  p 2 + q 2 + 1 = r 2 . All surfaces of revolution with the z-axis as the axis of symmetry satisfy the equation z = f  x 2 + y 2 , (2.3.13) where f is an arbitrary function. Writing u = x 2 + y 2 and differentiating (2.3.13) with respect to x and y, respectively, we obtain p = 2x f′ (u), q = 2y f′ (u). Eliminating the arbitrary function f (u) from these results, we find the equation yp − xq = 0. Theorem 2.3.1. If φ = φ (x, y, z) and ψ = ψ (x, y, z) are two given functions of x, y, and z and if f (φ, ψ) = 0, where f is an arbitrary function of φ and ψ, then z = z (x, y) satisfies a first-order, partial differential equation p ∂ (φ, ψ) ∂ (y, z) + q ∂ (φ, ψ) ∂ (z, x) = ∂ (φ, ψ) ∂ (x, y) , (2.3.14) where ∂ (φ, ψ) ∂ (x, y) =     φx φy ψx ψy     . (2.3.15) Proof. We differentiate f (φ, ψ) = 0 with respect to x and y respectively to obtain the following equations: ∂f ∂φ  ∂φ ∂x + p ∂φ ∂z  + ∂f ∂ψ  ∂ψ ∂x + p ∂ψ ∂z  = 0, (2.3.16) ∂f ∂φ  ∂φ ∂y + q ∂φ ∂z  + ∂f ∂ψ  ∂ψ ∂y + q ∂ψ ∂z  = 0. (2.3.17) Nontrivial solutions for ∂f ∂φ and ∂f ∂ψ can be found if the determinant of the coefficients of these equations vanishes, that is,       φx + pφz ψx + pψz φy + qφz ψy + qψz       = 0. (2.3.18) Expanding this determinant gives the first-order, quasi-linear equation (2.3.14). 2.4 Geometrical Interpretation of a First-Order Equation 33 2.4 Geometrical Interpretation of a First-Order Equation To investigate the geometrical content of a first-order, partial differential equation, we begin with a general, quasi-linear equation a (x, y, u) ux + b (x, y, u) uy − c (x, y, u)=0. (2.4.1) We assume that the possible solution of (2.4.1) in the form u = u (x, y) or in an implicit form f (x, y, u) ≡ u (x, y) − u = 0 (2.4.2) represents a possible solution surface in (x, y, u) space. This is often called an integral surface of the equation (2.4.1). At any point (x, y, u) on the solution surface, the gradient vector ∇f = (fx, fy, fu)=(ux, uy, −1) is normal to the solution surface. Clearly, equation (2.4.1) can be written as the dot product of two vectors a ux + b uy − c = (a, b, c) · (ux, uy − 1) = 0. (2.4.3) This clearly shows that the vector (a, b, c) must be a tangent vector of the integral surface (2.4.2) at the point (x, y, u), and hence, it determines a direction field called the the characteristic direction or Monge axis. This direction is of fundamental importance in determining a solution of equation (2.4.1). To summarize, we have shown that f (x, y, u) = u (x, y) − u = 0, as a surface in the (x, y, u)-space, is a solution of (2.4.1) if and only if the direction vector field (a, b, c) lies in the tangent plane of the integral surface f (x, y, u) = 0 at each point (x, y, u), where ∇f = 0, as shown in Figure 2.4.1. A curve in (x, y, u)-space, whose tangent at every point coincides with the characteristic direction field (a, b, c), is called a characteristic curve. If the parametric equations of this characteristic curve are x = x (t), y = y (t), u = u (t), (2.4.4) then the tangent vector to this curve is 4 dx dt , dy dt , du dt 5 which must be equal to (a, b, c). Therefore, the system of ordinary differential equations of the characteristic curve is given by dx dt = a (x, y, u), dy dt = b (x, y, u), du dt = c (x, y, u). (2.4.5) These are called the characteristic equations of the quasi-linear equation (2.4.1). 34 2 First-Order, Quasi-Linear Equations and Method of Characteristics Figure 2.4.1 Tangent and normal vector fields of solution surface at a point (x, y, u). In fact, there are only two independent ordinary differential equations in the system (2.4.5); therefore, its solutions consist of a two-parameter family of curves in (x, y, u)-space. The projection on u = 0 of a characteristic curve on the (x, t)-plane is called a characteristic base curve or simply characteristic. Equivalently, the characteristic equations (2.4.5) in the nonparametric form are dx a = dy b = du c . (2.4.6) The typical problem of solving equation (2.4.1) with a prescribed u on a given plane curve C is equivalent to finding an integral surface in (x, y, u) space, satisfying the equation (2.4.1) and containing the three-dimensional space curve Γ defined by the values of u on C, which is the projection on u = 0 of Γ. Remark 1. The above geometrical interpretation can be generalized for higher-order partial differential equations. However, it is not easy to visualize geometrical arguments that have been described for the case of three space dimensions. Remark 2. The geometrical interpretation is more complicated for the case of nonlinear partial differential equations, because the normals to possible 2.5 Method of Characteristics and General Solutions 35 solution surfaces through a point do not lie in a plane. The tangent planes no longer intersect along one straight line, but instead, they envelope along a curved surface known as the Monge cone. Any further discussion is beyond the scope of this book. We conclude this section by adding an important observation regarding the nature of the characteristics in the (x, t)-plane. For a quasi-linear equation, characteristics are determined by the first two equations in (2.4.5) with their slopes dy dx = b (x, y, u) a (x, y, u) . (2.4.7) If (2.4.1) is a linear equation, then a and b are independent of u, and the characteristics of (2.4.1) are plane curves with slopes dy dx = b (x, y) a (x, y) . (2.4.8) By integrating this equation, we can determine the characteristics which represent a one-parameter family of curves in the (x, t)-plane. However, if a and b are constant, the characteristics of equation (2.4.1) are straight lines. 2.5 Method of Characteristics and General Solutions We can use the geometrical interpretation of first-order, partial differential equations and the properties of characteristic curves to develop a method for finding the general solution of quasi-linear equations. This is usually referred to as the method of characteristics due to Lagrange. This method of solution of quasi-linear equations can be described by the following result. Theorem 2.5.1. The general solution of a first-order, quasi-linear partial differential equation a (x, y, u) ux + b (x, y, u) uy = c (x, y, u) (2.5.1) is f (φ, ψ)=0, (2.5.2) where f is an arbitrary function of φ (x, y, u) and ψ (x, y, u), and φ = constant = c1 and ψ = constant = c2 are solution curves of the characteristic equations dx a = dy b = du c . (2.5.3) The solution curves defined by φ (x, y, u) = c1 and ψ (x, y, u) = c2 are called the families of characteristic curves of equation (2.5.1). 36 2 First-Order, Quasi-Linear Equations and Method of Characteristics Proof. Since φ (x, y, u) = c1 and ψ (x, y, u) = c2 satisfy equations (2.5.3), these equations must be compatible with the equation dφ = φxdx + φydy + φudu = 0. (2.5.4) This is equivalent to the equation a φx + b φy + c φu = 0. (2.5.5) Similarly, equation (2.5.3) is also compatible with a ψx + b ψy + c ψu = 0. (2.5.6) We now solve (2.5.5), (2.5.6) for a, b, and c to obtain a ∂(φ,ψ) ∂(y,u) = b ∂(φ,ψ) ∂(u,x) = c ∂(φ,ψ) ∂(x,y) . (2.5.7) It has been shown earlier that f (φ, ψ) = 0 satisfies an equation similar to (2.3.14), that is, p ∂ (φ, ψ) ∂ (y, u) + q ∂ (φ, ψ) ∂ (u, x) = ∂ (φ, ψ) ∂ (x, y) . (2.5.8) Substituting, (2.5.7) in (2.5.8), we find that f (φ, ψ) = 0 is a solution of (2.5.1). This completes the proof. Note that an analytical method has been used to prove Theorem 2.5.1. Alternatively, a geometrical argument can be used to prove this theorem. The geometrical method of proof is left to the reader as an exercise. Many problems in applied mathematics, science, and engineering involve partial differential equations. We seldom try to find or discuss the properties of a solution to these equations in its most general form. In most cases of interest, we deal with those solutions of partial differential equations which satisfy certain supplementary conditions. In the case of a first-order partial differential equation, we determine the specific solution by formulating an initial-value problem or a Cauchy problem. Theorem 2.5.2. (The Cauchy Problem for a First-Order Partial Differential Equation). Suppose that C is a given curve in the (x, y)-plane with its parametric equations x = x0 (t), y = y0 (t), (2.5.9) where t belongs to an interval I ⊂ R, and the derivatives x ′ 0 (t) and y ′ 0 (t) are piecewise continuous functions, such that (x ′ 0 ) 2 + (y ′ 0 ) 2 = 0. Also, suppose that u = u0 (t) is a given function on the curve C. Then, there exists a solution u = u (x, y) of the equation 2.5 Method of Characteristics and General Solutions 37 F (x, y, u, ux, uy) = 0 (2.5.10) in a domain D of R2 containing the curve C for all t ∈ I, and the solution u (x, y) satisfies the given initial data, that is, u (x0 (t), y0 (t)) = u0 (t) (2.5.11) for all values of t ∈ I. In short, the Cauchy problem is to determine a solution of equation (2.5.10) in a neighborhood of C, such that the solution u = u (x, y) takes a prescribed value u0 (t) on C. The curve C is called the initial curve of the problem, and u0 (t) is called the initial data. Equation (2.5.11) is called the initial condition of the problem. The solution of the Cauchy problem also deals with such questions as the conditions on the functions F, x0 (t), y0 (t), and u0 (t) under which a solution exists and is unique. We next discuss a method for solving a Cauchy problem for the firstorder, quasi-linear equation (2.5.1). We first observe that geometrically x = x0 (t), y = y0 (t), and u = u0 (t) represent an initial curve Γ in (x, y, u)-space. The curve C, on which the Cauchy data is prescribed, is the projection of Γ on the (x, y)-plane. We now present a precise formulation of the Cauchy problem for the first-order, quasi-linear equation (2.5.1). Theorem 2.5.3. (The Cauchy Problem for a Quasi-linear Equation). Suppose that x0 (t), y0 (t), and u0 (t) are continuously differentiable functions of t in a closed interval, 0 ≤ t ≤ 1, and that a, b, and c are functions of x, y, and u with continuous first-order partial derivatives with respect to their arguments in some domain D of (x, y, u)-space containing the initial curve Γ : x = x0 (t), y = y0 (t), u = u0 (t), (2.5.12) where 0 ≤ t ≤ 1, and satisfying the condition y ′ 0 (t) a (x0 (t), y0 (t), u0 (t)) − x ′ 0 (t) b (x0 (t), y0 (t), u0 (t)) = 0. (2.5.13) Then there exists a unique solution u = u (x, y) of the quasi-linear equation (2.5.1) in the neighborhood of C : x = x0 (t), y = y0 (t), and the solution satisfies the initial condition u0 (t) = u (x0 (t), y0 (t)), for 0 ≤ t ≤ 1. (2.5.14) Note: The condition (2.5.13) excludes the possibility that C could be a characteristic. Example 2.5.1. Find the general solution of the first-order linear partial differential equation. 38 2 First-Order, Quasi-Linear Equations and Method of Characteristics x ux + y uy = u. (2.5.15) The characteristic curves of this equation are the solutions of the characteristic equations dx x = dy y = du u . (2.5.16) This system of equations gives the integral surfaces φ = y x = C1 and ψ = u x = C2, where C1 and C2 are arbitrary constants. Thus, the general solution of (2.5.15) is f 4y x , u x 5 = 0, (2.5.17) where f is an arbitrary function. This general solution can also be written as u (x, y) = x g 4y x 5 , (2.5.18) where g is an arbitrary function. Example 2.5.2. Obtain the general solution of the linear Euler equation x ux + y uy = nu. (2.5.19) The integral surfaces are the solutions of the characteristic equations dx x = dy y = du nu . (2.5.20) From these equations, we get y x = C1, u x n = C2, where C1 and C2 are arbitrary constants. Hence, the general solution of (2.5.19) is f 4y x , u x n 5 = 0. (2.5.21) This can also be written as u x n = g 4y x 5 or u (x, y) = x n g 4y x 5 . (2.5.22) This shows that the solution u (x, y) is a homogeneous function of x and y of degree n. 2.5 Method of Characteristics and General Solutions 39 Example 2.5.3. Find the general solution of the linear equation x 2 ux + y 2 uy = (x + y) u. (2.5.23) The characteristic equations associated with (2.5.23) are dx x 2 = dy y 2 = du (x + y) u . (2.5.24) From the first two of these equations, we find x −1 − y −1 = C1, (2.5.25) where C1 is an arbitrary constant. It follows from (2.5.24) that dx − dy x 2 − y 2 = du (x + y) u or d (x − y) x − y = du u . This gives x − y u = C2, (2.5.26) where C2 is a constant. Furthermore, (2.5.25) and (2.5.26) also give xy u = C3, (2.5.27) where C3 is a constant. Thus, the general solution (2.5.23) is given by f  xy u , x − y u  = 0, (2.5.28) where f is an arbitrary function. This general solution representing the integral surface can also be written as u (x, y) = xy g  x − y u  , (2.5.29) where g is an arbitrary function, or, equivalently, u (x, y) = xy h  x − y xy  , (2.5.30) where h is an arbitrary function. 40 2 First-Order, Quasi-Linear Equations and Method of Characteristics Example 2.5.4. Show that the general solution of the linear equation (y − z) ux + (z − x) uy + (x − y) uz = 0 (2.5.31) is u (x, y, z) = f  x + y + z, x2 + y 2 + z 2 , (2.5.32) where f is an arbitrary function. The characteristic curves satisfy the characteristic equations dx y − z = dy z − x = dz x − y = du 0 (2.5.33) or du = 0, dx + dy + dz = 0, xdx + ydy + zdz = 0. Integration of these equations gives u = C1, x + y + z = C2, and x 2 + y 2 + z 2 = C3, where C1, C2 and C3 are arbitrary constants. Thus, the general solution can be written in terms of an arbitrary function f in the form u (x, y, z) = f  x + y + z, x2 + y 2 + z 2 . We next verify that this is a general solution by introducing three independent variables ξ, η, ζ defined in terms of x, y, and z as ξ = x + y + z, η = x 2 + y 2 + z 2 , and ζ = y + z, (2.5.34) where ζ is an arbitrary combination of y and z. Clearly the general solution becomes u = f (ξ,η), and hence, uζ = ux ∂x ∂ζ + uy ∂y ∂ζ + uz ∂z ∂ζ . (2.5.35) It follows from (2.5.34) that 0 = ∂x ∂ζ + ∂y ∂ζ + ∂z ∂ζ , 0=2  x ∂x ∂ζ + y ∂y ∂ζ + z ∂z ∂ζ  , ∂y ∂ζ + ∂z ∂ζ = 1. It follows from the first and the third results that ∂x ∂ζ = −1 and, therefore, x = y ∂y ∂ζ + z ∂z ∂ζ , y = y ∂y ∂ζ + y ∂z ∂ζ , z = z ∂y ∂ζ + z ∂z ∂ζ . 2.5 Method of Characteristics and General Solutions 41 Clearly, it follows by subtracting that x − y = (z − y) ∂z ∂ζ , x − z = (y − z) ∂y ∂ζ . Using the values for ∂x ∂ζ , ∂z ∂ζ , and ∂y ∂ζ in (2.5.35), we obtain (z − y) ∂u ∂ζ = (y − z) ∂u ∂x + (z − x) ∂u ∂y + (x − y) ∂u ∂z . (2.5.36) If u = u (ξ,η) satisfies (2.5.31), then ∂u ∂ζ = 0 and, hence, (2.5.36) reduces to (2.5.31). This shows that the general solution (2.5.32) satisfies equation (2.5.31). Example 2.5.5. Find the solution of the equation u (x + y) ux + u (x − y) uy = x 2 + y 2 , (2.5.37) with the Cauchy data u = 0 on y = 2x. The characteristic equations are dx u (x + y) = dy u (x − y) = du x 2 + y 2 = ydx + xdy − udu 0 = xdx − ydy − udu 0 . Consequently, d xy − 1 2 u 2  = 0 and d 1 2  x 2 − y 2 − u 2 = 0. (2.5.38) These give two integrals u 2 − x 2 + y 2 = C1 and 2xy − u 2 = C2, (2.5.39) where C1 and C2 are constants. Hence, the general solution is f  x 2 − y 2 − u 2 , 2xy − u 2 = 0, where f is an arbitrary function. Using the Cauchy data in (2.5.39), we obtain 4C1 = 3C2. Therefore 4  u 2 − x 2 + y 2 = 3  2xy − u 2 . Thus, the solution of equation (2.5.37) is given by 7u 2 = 6xy + 4  x 2 − y 2 . (2.5.40) Example 2.5.6. Obtain the solution of the linear equation ux − uy = 1, (2.5.41) 42 2 First-Order, Quasi-Linear Equations and Method of Characteristics with the Cauchy data u (x, 0) = x 2 . The characteristic equations are dx 1 = dy −1 = du 1 . (2.5.42) Obviously, dy dx = −1 and du dx = 1. Clearly, x + y = constant = C1 and u − x = constant = C2. Thus, the general solution is given by u − x = f (x + y), (2.5.43) where f is an arbitrary function. We now use the Cauchy data to find f (x) = x 2 − x, and hence, the solution is u (x, y)=(x + y) 2 − y. (2.5.44) The characteristics x + y = C1 are drawn in Figure 2.5.1. The value of u must be given at one point on each characteristic which intersects the line y = 0 only at one point, as shown in Figure 2.5.1. Figure 2.5.1 Characteristics of equation (2.5.41). 2.5 Method of Characteristics and General Solutions 43 Example 2.5.7. Obtain the solution of the equation (y − u) ux + (u − x) uy = x − y, (2.5.45) with the condition u = 0 on xy = 1. The characteristic equations for equation (2.5.45) are dx y − u = dy u − x = du x − y . (2.5.46) The parametric forms of these equations are dx dt = y − u, dy dt = u − x, du dt = x − y. These lead to the following equations: x˙ + ˙y + ˙u = 0 and xx˙ + yy˙ + uu˙ = 0, (2.5.47) where the dot denotes the derivative with respect to t. Integrating (2.5.47), we obtain x + y + u = const. = C1 and x 2 + y 2 + u 2 = const. = C2. (2.5.48) These equations represent circles. Using the Cauchy data, we find that C 2 1 = (x + y) 2 = x 2 + y 2 + 2xy = C2 + 2. Thus, the integral surface is described by (x + y + u) 2 = x 2 + y 2 + u 2 + 2. Hence, the solution is given by u (x, y) = 1 − xy x + y . (2.5.49) Example 2.5.8. Solve the linear equation y ux + x uy = u, (2.5.50) with the Cauchy data u (x, 0) = x 3 and u (0, y) = y 3 . (2.5.51) The characteristic equations are dx y = dy x = du u 44 2 First-Order, Quasi-Linear Equations and Method of Characteristics or du u = dx − dy y − x = dx + dy y + x . Solving these equations, we obtain u = C1 x − y = C2 (x + y) or u = C2 (x + y), x2 − y 2 = C1 C2 = constant = C. So the characteristics are rectangular hyperbolas for C > 0 or C < 0. Thus, the general solution is given by f  u x + y , x2 − y 2  = 0 or, equivalently, u (x, y)=(x + y) g  x 2 − y 2 . (2.5.52) Using the Cauchy data, we find that g  x 2 = x 2 , that is, g (x) = x. Consequently, the solution becomes u (x, y)=(x + y)  x 2 − y 2 on x 2 − y 2 = C > 0. Similarly, u (x, y)=(x + y)  y 2 − x 2 on y 2 − x 2 = C > 0. It follows from these results that u → 0 in all regions, as x → ± y (or y → ± x), and hence, u is continuous across y = ± x which represent asymptotes of the rectangular hyperbolas x 2 − y 2 = C. However, ux and uy are not continuous, as y → ± x. For x 2 − y 2 = C > 0, ux = 3x 2 + 2xy − y 2 = (x + y) (3x − y) → 0, as y → −x. uy = −3y 2 − 2xy + x 2 = (x + y) (x − 3y) → 0, as y → −x. Hence, both ux and uy are continuous as y → −x. On the other hand, ux → 4x 2 , uy → −4x 2 as y → x. This implies that ux and uy are discontinuous across y = x. Combining all these results, we conclude that u (x, y) is continuous everywhere in the (x, t)-plane, and ux, uy are continuous everywhere in the (x, t)-plane except on the line y = x. Hence, the partial derivatives ux, uy are discontinuous on y = x. Thus, the development of discontinuities across characteristics is a significant feature of the solutions of partial differential equations. 2.5 Method of Characteristics and General Solutions 45 Example 2.5.9. Determine the integral surfaces of the equation x  y 2 + u ux − y  x 2 + u uy =  x 2 − y 2 u, (2.5.53) with the data x + y = 0, u = 1. The characteristic equations are dx x (y 2 + u) = dy −y (x 2 + u) = du (x 2 − y 2) u (2.5.54) or dx x (y 2 + u) = dy y − (x 2 + u) = du u (x 2 − y 2) = dx x + dy y + du u 0 . Consequently, log (xyu) = log C1 or xyu = C1. From (2.5.54), we obtain xdx x 2 (y 2 + u) = ydy −y 2 (x 2 + u) = du (x 2 − y 2) u = xdx + ydy − du 0 , whence we find that x 2 + y 2 − 2u = C2. Using the given data, we obtain C1 = −x 2 and C2 = 2x 2 − 2, so that C2 = −2 (C1 + 1). Thus the integral surface is given by x 2 + y 2 − 2u = −2 − 2xyu or 2xyu + x 2 + y 2 − 2u +2=0. (2.5.55) 46 2 First-Order, Quasi-Linear Equations and Method of Characteristics Example 2.5.10. Obtain the solution of the equation x ux + y uy = x exp (−u) (2.5.56) with the data u = 0 on y = x 2 . The characteristic equations are dx x = dy y = du x exp (−u) (2.5.57) or y x = C1. We also obtain from (2.5.57) that dx = e udu which can be integrated to find e u = x + C2. Thus, the general solution is given by f 4 e u − x, y x 5 = 0 or, equivalently, e u = x + g 4y x 5 . (2.5.58) Applying the Cauchy data, we obtain g (x)=1 − x. Thus, the solution of (2.5.56) is given by e u = x + 1 − y x or u = log 4 x + 1 − y x 5 . (2.5.59) Example 2.5.11. Solve the initial-value problem ut + u ux = x, u (x, 0) = f (x), (2.5.60) where (a) f (x) = 1 and (b) f (x) = x. The characteristic equations are dt 1 = dx u = du x = d (x + u) x + u . (2.5.61) 2.5 Method of Characteristics and General Solutions 47 Integration gives t = log (x + u) − log C1 or (u + x) e −t = C1. Similarly, we get u 2 − x 2 = C2. For case (a), we obtain 1 + x = C1 and 1 − x 2 = C2, and hence C2 = 2C1 − C 2 1 . Thus,  u 2 − x 2 = 2(u + x) e −t − (u + x) 2 e −2t or u − x = 2e −t − (u + x) e −2t . A simple manipulation gives the solution u (x, t) = x tanh t + sech t. (2.5.62) Case (b) is left to the reader as an exercise. Example 2.5.12. Find the integral surface of the equation u ux + uy = 1, (2.5.63) so that the surface passes through an initial curve represented parametrically by x = x0 (s), y = y0 (s), u = u0 (s), (2.5.64) where s is a parameter. The characteristic equations for the given equations are dx u = dy 1 = du 1 , which are, in the parametric form, dx dτ = u, dy dτ = 1, du dτ = 1, (2.5.65) 48 2 First-Order, Quasi-Linear Equations and Method of Characteristics where τ is a parameter. Thus the solutions of this parametric system in general depend on two parameters s and τ . We solve this system (2.5.65) with the initial data x (s, 0) = x0 (s), y (s, 0) = y0 (s), u (s, 0) = u0 (s). The solutions of (2.5.65) with the given initial data are x (s, τ ) = τ 2 2 + τ u0 (s) + x0 (s) y (s, τ ) = τ + y0 (s) u (s, τ ) = τ + u0 (s) ⎫ ⎬ ⎭ . (2.5.66) We choose a particular set of values for the initial data as x (s, 0) = 2s 2 , y (s, 0) = 2s, u (s, 0) = 0, s> 0. Therefore, the solutions are given by x = 1 2 τ 2 + 2s 2 , y = τ + 2s, u = τ. (2.5.67) Eliminating τ and s from (2.5.67) gives the integral surface (u − y) 2 + u 2 = 2x or 2u = y ±  4x − y 2 1 2 . (2.5.68) The solution surface satisfying the data u = 0 on y 2 = 2x is given by 2u = y −  4x − y 2 1 2 . (2.5.69) This represents the solution surface only when y 2 < 4x. Thus, the solution does not exist for y 2 > 4x and is not differentiable when y 2 = 4x. We verify that y 2 = 4x represents the envelope of the family of characteristics in the (x, t)-plane given by the τ -eliminant of the first two equations in (2.5.67), that is, F (x, y, s)=2x − (y − 2s) 2 − 4s 2 = 0. (2.5.70) This represents a family of parabolas for different values of the parameter s. Thus, the envelope is obtained by eliminating s from equations ∂F ∂s = 0 and F = 0. This gives y 2 = 4x, which is the envelope of the characteristics for different s, as shown in Figure 2.5.2. 2.6 Canonical Forms of First-Order Linear Equations 49 Figure 2.5.2 Dotted curve is the envelope of the characteristics. 2.6 Canonical Forms of First-Order Linear Equations It is often convenient to transform the more general first-order linear partial differential equation (2.2.12) a (x, y) ux + b (x, y) uy + c (x, y) u = d (x, y), (2.6.1) into a canonical (or standard) form which can be easily integrated to find the general solution of (2.6.1). We use the characteristics of this equation (2.6.1) to introduce the new transformation by equations ξ = ξ (x, y), η = η (x, y), (2.6.2) where ξ and η are once continuously differentiable and their Jacobian J (x, y) ≡ ξxηy − ξyηx is nonzero in a domain of interest so that x and y can be determined uniquely from the system of equations (2.6.2). Thus, by chain rule, ux = uξξx + uηηx, uy = uξξy + uηηy, (2.6.3) we substitute these partial derivatives (2.6.3) into (2.6.1) to obtain the equation A uξ + B uη + cu = d, (2.6.4) where A = uξx + bξy, B = aηx + bηy. (2.6.5) From (2.6.5) we see that B = 0 if η is a solution of the first-order equation aηx + bηy = 0. (2.6.6) This equation has infinitely many solutions. We can obtain one of them by assigning initial condition on a non-characteristic initial curve and solving the resulting initial-value problem according to the method described 50 2 First-Order, Quasi-Linear Equations and Method of Characteristics earlier. Since η (x, y) satisfies equation (2.6.6), the level curves η (x, y) = constant are always characteristic curves of equation (2.6.1). Thus, one set of the new transformations are the characteristic curves of (2.6.1). The second set, ξ (x, y) = constant, can be chosen to be any one parameter family of smooth curves which are nowhere tangent to the family of the characteristic curves. We next assert that A = 0 in a neighborhood of some point in the domain D in which η (x, y) is defined and J = 0. For, if A = 0 at some point of D, then B = 0 at the same point. Consequently, equations (2.6.5) would form a system of linear homogeneous equations in a and b, where the Jacobian J is the determinant of its coefficient matrix. Since J = 0, both a and b must be zero at that point which contradicts the original assumption that a and b do not vanish simultaneously. Finally, since B = 0 and A = 0 in D, we can divide (2.6.4) by A to obtain the canonical form uξ + α (ξ,η) u = β (ξ,η), (2.6.7) where α (ξ,η) = c A and β (ξ,η) = d A . Equation (2.6.7) represents an ordinary differential equation with ξ as the independent variable and η as a parameter which may be treated as constant. This equation (2.6.7) is called the canonical form of equation (2.6.1) in terms of the coordinates (ξ,η). Generally, the canonical equation (2.6.7) can easily be integrated and the general solution of (2.6.1) can be obtained after replacing ξ and η by the original variables x and y. We close this section by considering some examples that illustrate this procedure. In practice, it is convenient to choose ξ = ξ (x, y) and η (x, y) = y or ξ = x and η = η (x, y) so that J = 0. Example 2.6.1. Reduce each of the following equations ux − uy = u, (2.6.8) yux + uy = x, (2.6.9) to canonical form, and obtain the general solution. In (2.6.8), a = 1, b = −1, c = −1 and d = 0. The characteristic equations are dx 1 = dy −1 = du u . The characteristic curves are ξ = x + y = c1, and we choose η = y = c2 where c1 and c2 are constants. Consequently, ux = uξ and uy = uξ + uη, and hence, equation (2.6.8) becomes uη = u. Integrating this equation gives 2.7 Method of Separation of Variables 51 ln u (ξ,η) = −η + ln f (ξ), where f (ξ) is an arbitrary function of ξ only. Equivalently, u (ξ,η) = f (ξ) e −η . In terms of the original variables x and y, the general solution of equation (2.6.8) is u (x, y) = f (x + y) e −y , (2.6.10) where f is an arbitrary function. The characteristic equations of (2.6.9) are dx y = dy 1 = du x . It follows from the first two equations that ξ (x, y) = x− y 2 2 = c1; we choose η (x, y) = y = c2. Consequently, ux = uξ and uy = −y uξ + uη and hence, equation (2.6.9) reduces to uη = ξ + 1 2 η 2 . Integrating this equation gives the general solution u (ξ,η) = ξη + 1 6 η 3 + f (ξ), where f is an arbitrary function. Thus, the general solution of (2.6.9) in terms of x and y is u (x, y) = xy − 1 3 y 3 + f  x − y 2 2  . 2.7 Method of Separation of Variables During the last two centuries several methods have been developed for solving partial differential equations. Among these, a technique known as the method of separation of variables is perhaps the oldest systematic method for solving partial differential equations. Its essential feature is to transform the partial differential equations by a set of ordinary differential equations. The required solution of the partial differential equations is then exposed as a product u (x, y) = X (x) Y (y) = 0, or as a sum u (x, y) = X (x) + Y (y), where X (x) and Y (y) are functions of x and y, respectively. Many signifi- cant problems in partial differential equations can be solved by the method 52 2 First-Order, Quasi-Linear Equations and Method of Characteristics of separation of variables. This method has been considerably refined and generalized over the last two centuries and is one of the classical techniques of applied mathematics, mathematical physics and engineering science. Usually, the first-order partial differential equation can be solved by separation of variables without the need for Fourier series. The main purpose of this section is to illustrate the method by examples. Example 2.7.1. Solve the initial-value problem ux + 2uy = 0, u (0, y)=4 e −2y . (2.7.1ab) We seek a separable solution u (x, y) = X (x) Y (y) = 0 and substitute into the equation to obtain X′ (x) Y (y)+2X (x) Y ′ (y)=0. This can also be expressed in the form X′ (x) 2X (x) = − Y ′ (y) Y (y) . (2.7.2) Since the left-hand side of this equation is a function of x only and the right-hand is a function of y only, it follows that (2.7.2) can be true if both sides are equal to the same constant value λ which is called an arbitrary separation constant. Consequently, (2.7.2) gives two ordinary differential equations X′ (x) − 2λX (x)=0, Y ′ (y) + λY (y)=0. (2.7.3) These equations have solutions given, respectively, by X (x) = A e2λx and Y (y) = B e−λy , (2.7.4) where A and B are arbitrary integrating constants. Consequently, the general solution is given by u (x, y) = AB exp (2λx − λy) = C exp (2λx − λy), (2.7.5) where C = AB is an arbitrary constant. Using the condition (2.7.1b), we find 4 e −2y = u (0, y) = Ce−λy , and hence, we deduce that C = 4 and λ = 2. Therefore, the final solution is u (x, y) = 4 exp (4x − 2y). (2.7.6) 2.7 Method of Separation of Variables 53 Example 2.7.2. Solve the equation y 2u 2 x + x 2u 2 y = (xyu) 2 . (2.7.7) We assume u (x, y) = f (x) g (y) = 0 is a separable solution of (2.7.7), and substitute into the equation. Consequently, we obtain y 2 {f ′ (x) g (y)} 2 + x 2 {f (x) g ′ (y)} 2 = x 2 y 2 {f (x) g (y)} 2 , or, equivalently, 1 x 2  f ′ (x) f (x) 02 + 1 y 2  g ′ (y) g (y) 02 = 1, or 1 x 2  f ′ (x) f (x) 02 = 1 − 1 y 2  g ′ (y) g (y) 02 = λ 2 , where λ 2 is a separation constant. Thus, 1 x f ′ (x) f (x) = λ and g ′ (y) y g (y) =  1 − λ2 . (2.7.8) Solving these ordinary differential equations, we find f (x) = A exp  λ 2 x 2  and g (y) = B exp  1 2 y  1 − λ2  , where A and B are arbitrary constant. Thus, the general solution is u (x, y) = C exp  λ 2 x 2 + 1 2 y 2  1 − λ2  , (2.7.9) where C = AB is an arbitrary constant. Using the condition u (x, 0) = 3 exp  x 2/4 , we can determine both C and λ in (2.7.9). It turns out that C = 3 and λ = (1/2), and the solution becomes u (x, y) = 3 exp 1 4 4 x 2 + y 2 √ 3 5 . (2.7.10) Example 2.7.3. Use the separation of variables u (x, y) = f (x) + g (y) to solve the equation u 2 x + u 2 y = 1. (2.7.11) Obviously, 54 2 First-Order, Quasi-Linear Equations and Method of Characteristics {f ′ (x)} 2 = 1 − {g ′ (y)} 2 = λ 2 , where λ 2 is a separation constant. Thus, we obtain f ′ (x) = λ and g ′ (y) =  1 − λ2 . Solving these ordinary differential equations, we find f (x) = λx + A and g (y) = y  1 − λ2 + B, where A and B are constants of integration. Finally, the solution of (2.7.11) is given by u (x, y) = λx + y  1 − λ2 + C, (2.7.12) where C = A + B is an arbitrary constant. Example 2.7.4. Use u (x, y) = f (x) + g (y) to solve the equation u 2 x + uy + x 2 = 0. (2.7.13) Obviously, equation (2.7.13) has the separable form {f ′ (x)} 2 + x 2 = −g ′ (y) = λ 2 , where λ 2 is a separation constant. Consequently, f ′ (x) =  λ2 − x 2 and g ′ (y) = −λ 2 . These can be integrated to obtain f (x) =   λ2 − x 2 dx + A = λ 2  cos2 θ dθ + A, (x = λ sin θ) = 1 2 λ 2 1 sin−1 4x λ 5 + x λ 2 1 − x 2 λ2 3 + A and g (y) = −λ 2 y + B. Finally, the general solution is given by u (x, y) = 1 2 λ 2 sin−1 4x λ 5 + x 2  λ2 − x 2 − λ 2 y + C, (2.7.14) where C = A + B is an arbitrary constant. 2.8 Exercises 55 Example 2.7.5. Use v = ln u and v = f (x) + g (y) to solve the equation x 2u 2 x + y 2u 2 y = u 2 . (2.7.15) In view of v = ln u, vx = 1 u ux and vy = 1 u uy, and hence, equation (2.7.15) becomes x 2 v 2 x + y 2 v 2 y = 1. (2.7.16) Substitute v (x, y) = f (x) + g (y) into (2.7.16) to obtain x 2 {f ′ (x)} 2 + y 2 {g ′ (y)} 2 = 1 or x 2 {f ′ (x)} 2 = 1 − y 2 {g ′ (y)} 2 = λ 2 , where λ 2 is a separation constant. Thus, we obtain f ′ (x) = λ x and g ′ (y) = 1 y  1 − λ2 . Integrating these equations gives f (x) = λ ln x + A and g (y) =  1 − λ2 ln y + B, where A and B are integrating constants. Therefore, the general solution of (2.7.16) is given by v (x, y) = λ ln x +  1 − λ2 ln y + ln C = ln 4 x λ · y √ 1−λ2 · C 5 , (2.7.17) where ln C = A + B. The final solution is u (x, y) = e v = C xλ · y √ 1−λ2 , (2.7.18) where C is an integrating constant. 2.8 Exercises 1. (a) Show that the family of right circular cones whose axes coincide with the z-axis x 2 + y 2 = (z − c) 2 tan2 α satisfies the first-order, partial differential equation 56 2 First-Order, Quasi-Linear Equations and Method of Characteristics yp − xq = 0. (b) Show that all the surfaces of revolution, z = f  x 2 + y 2 with the z-axis as the axis of symmetry, where f is an arbitrary function, satisfy the partial differential equation yp − xq = 0. (c) Show that the two-parameter family of curves u − ax − by − ab = 0 satisfies the nonlinear equation xp + yq + pq = u. 2. Find the partial differential equation arising from each of the following surfaces: (a) z = x + y + f (xy), (b) z = f (x − y), (c) z = xy + f  x 2 + y 2 , (d) 2z = (αx + y) 2 + β. 3. Find the general solution of each of the following equations: (a) ux = 0, (b) a ux + b uy = 0; a, b, are constant, (c) ux + y uy = 0, (d)  1 + x 2 ux + uy = 0, (e) 2xy ux +  x 2 + y 2 uy = 0, (f) (y + u) ux + y uy = x − y, (g) y 2ux − xy uy = x (u − 2y), (h) yuy − xux = 1, (i) y 2up + u 2xq = −xy2 , (j) (y − xu) p + (x + yu) q = x 2 + y 2 . 4. Find the general solution of the equation ux + 2xy2uy = 0. 5. Find the solution of the following Cauchy problems: (a) 3ux + 2uy = 0, with u (x, 0) = sin x, (b) y ux + x uy = 0, with u (0, y) = exp  −y 2 , (c) x ux + y uy = 2xy, with u = 2 on y = x 2 , (d) ux + x uy = 0, with u (0, y) = sin y, (e) y ux + x uy = xy, x ≥ 0, y ≥ 0, with u (0, y) = exp  −y 2 for y > 0, and u (x, 0) = exp  −x 2 for x > 0, 2.8 Exercises 57 (f) ux + x uy =  y − 1 2 x 2 2 , with u (0, y) = exp (y), (g) x ux + y uy = u + 1, with u (x, y) = x 2 on y = x 2 , (h) u ux − u uy = u 2 + (x + y) 2 , with u = 1 on y = 0, (i) x ux + (x + y) uy = u + 1, with u (x, y) = x 2 on y = 0. (j) √ x ux + u uy + u 2 = 0, u (x, 0) = 1, 0 <x< ∞.="" (k)="" u="" x2ux="" +="" e="" −yuy="" 2="0," (x,="" 0)="1," 0="" <x<="" 6.="" solve="" the="" initial-value="" problem="" ut="" ux="0" with="" initial="" curve="" x="1" τ="" ,="" t="τ,u" =="" τ.="" 7.="" find="" solution="" of="" cauchy="" 2xy="" ="" y="" uy="0," ="" −="" ="" on="" 8.="" following="" equations:="" (a)="" z="" uz="0," (b)="" (x="" y)="" (c)="" (y="" z)="" (z="" x)="" (d)="" yz="" xz="" xy="" (e)="" 9.="" equation="" data="" (0,="" (1,="" 58="" first-order,="" quasi-linear="" equations="" and="" method="" characteristics="" 10.="" show="" that="" u1="e" u2="e" −y="" are="" solutions="" nonlinear="" (ux="" uy)="" but="" their="" sum="" (e="" )="" is="" not="" a="" equation.="" 11.="" u)="" y),="" 12.="" integral="" surfaces="" for="" each="" data:="" (s,="" draw="" in="" case.="" 13.="" containing="" does="" exist.="" 14.="" problems:="" 2ux="" 2uy="0," →="" as="" ∞,="" −x="" <x<y,="" 2x="" 1)=""> 0, u = 2y on x = 1, (e) x ux − 2y uy = x 2 + y 2 for x > 0, y > 0, u = x 2 on y = 1, (f) ux + 2 uy =1+ u, u (x, y) = sin x on y = 3x + 1, (g) ux + 3uy = u, u (x, y) = cos x on y = x, (h) ux + 2x uy = 2xu, u (x, 0) = x 2 , (i) u ux + uy = u, u (x, 0) = 2x, 1 ≤ x ≤ 2, (j) ux + uy = u 2 , u (x, 0) = tanh x. 2.8 Exercises 59 Show that the solution of (j) is unbounded on the critical curve y tanh (x − y) = 1. 15. Find the solution surface of the equation  u 2 − y 2 ux + xy uy + xu = 0, with u = y = x, x > 0. 16. (a) Solve the Cauchy problem ux + uuy = 1, u (0, y) = ay, where a is a constant. (b) Find the solution of the equation in (a) with the data x (s, 0) = 2s, y (s, 0) = s 2 , u  0, s2 = s. 17. Solve the following equations: (a) (y + u) ux + (x + u) uy = x + y, (b) x u  u 2 + xy ux − y u  u 2 + xy uy = x 4 , (c) (x + y) ux + (x − y) uy = 0, (d) y ux + x uy = xy  x 2 − y 2 , (e) (cy − bz) zx + (az − cx) zy = bx − ay. 18. Solve the equation x zx + y zy = z, and find the curves which satisfy the associated characteristic equations and intersect the helix x + y 2 = a 2 , z = b tan−1  y x . 19. Obtain the family of curves which represent the general solution of the partial differential equation (2x − 4y + 3u) ux + (x − 2y − 3u) uy = −3 (x − 2y). Determine the particular member of the family which contains the line u = x and y = 0. 20. Find the solution of the equation y ux − 2xy uy = 2xu with the condition u (0, y) = y 3 . 60 2 First-Order, Quasi-Linear Equations and Method of Characteristics 21. Obtain the general solution of the equation (x + y + 5z) p + 4zq + (x + y + z)=0 (p = zx, q = zy), and find the particular solution which passes through the circle z = 0, x2 + y 2 = a 2 . 22. Obtain the general solution of the equation  z 2 − 2yz − y 2 p + x (y + z) q = x (y − z) (p = zx, q = zy). Find the integral surfaces of this equation passing through (a) the x-axis, (b) the y-axis, and (c) the z-axis. 23. Solve the Cauchy problem (x + y) ux + (x − y) uy = 1, u (1, y) = 1 √ 2 . 24. Solve the following Cauchy problems: (a) 3 ux + 2 uy = 0, u (x, 0) = f (x), (b) a ux + b uy = c u, u (x, 0) = f (x), where a, b, c are constants, (c) x ux + y uy = c u, u (x, 0) = f (x), (d) u ux + uy = 1, u (s, 0) = αs, x (s, 0) = s, y (s, 0) = 0. 25. Apply the method of separation of variables u (x, y) = f (x) g (y) to solve the following equations: (a) ux + u = uy, u (x, 0) = 4e −3x , (b) uxuy = u 2 , (c) ux + 2uy = 0, u (0, y)=3e −2y , (d) y 2u 2 x + x 2u 2 y = (xyu) 2 , (e) x 2uxy + 9y 2u = 0, u (x, 0) = exp  1 x , (f) y ux − x uy = 0, (g) ut = c 2 (uxx + uyy), (h) uxx + uyy = 0. 26. Use a separable solution u (x, y) = f (x) + g (y) to solve the following equations: (a) u 2 x + u 2 y = 1, (b) u 2 x + u 2 y = u, (c) u 2 x + uy + x 2 = 0, (d) x 2u 2 x + y 2u 2 y = 1, (e) y ux + x uy = 0, u (0, y) = y 2 . 2.8 Exercises 61 27. Apply v = ln u and then v (x, y) = f (x) + g (y) to solve the following equations: (a) x 2u 2 x + y 2u 2 y = u 2 , (b) x 2u 2 x + y 2u 2 y = (xyu) 2 . 28. Apply √ u = v and v (x, y) = f (x) + g (y) to solve the equation x 4u 2 x + y 2u 2 y = 4u. 29. Using v = ln u and v = f (x) + g (y), show that the solution of the Cauchy problem y 2u 2 x + x 2u 2 y = (xyu) 2 , u (x, 0) = e x 2 is u (x, y) = exp 4 x 2 + i √ 3 2 y 2 5 . 30. Reduce each of the following equations into canonical form and find the general solution: (a) ux + uy = u, (b) ux + x uy = y, (c) ux + 2xy uy = x, (d) ux − y uy − u = 1. 31. Find the solution of each of the following equations by the method of separation of variables: (a) ux − uy = 0, u (0, y)=2e 3y , (b) ux − uy = u, u (x, 0) = 4e −3x , (c) a ux + b uy = 0, u (x, 0) = αeβx , where a, b, α and β are constants. 32. Find the solution of the following initial-value systems (a) ut + 3uux = v − x, vt − cvx = 0 with u (x, 0) = x and v (x, 0) = x. (b) ut + 2uux = v − x, vt − cvx = 0 with u (x, 0) = x and v (x, 0) = x. 62 2 First-Order, Quasi-Linear Equations and Method of Characteristics 33. Solve the following initial-value systems (a) ut + uux = e −xv, vt − avx = 0 with u (x, 0) = x and v (x, 0) = e x . (b) ut − 2uux = v − x, vt + cvx = 0 with u (x, 0) = x and v (x, 0) = x. 34. Consider the Fokker–Planck equation (See Reif (1965)) in statistical mechanics to describe the evolution of the probability distribution function in the form ut = uxx + (x u)x , u (x, 0) = f (x). Neglecting the term uxx, solve the first-order linear equation ut − x ux = u with u (x, 0) = f (x). 3 Mathematical Models “Physics can’t exist without mathematics which provides it with the only language in which it can speak. Thus, services are continuously exchanged between pure mathematical analysis and physics. It is really remarkable that among works of analysis most useful for physics were those cultivated for their own beauty. In exchange, physics, exposing new problems, is as useful for mathematics as it is a model for an artist.” Henri Poincar´e “It is no paradox to say in our most theoretical models we may be nearest to our most practical applications.” A. N. Whitehead “... builds models based on data from all levels: gene expression, protein location in the cell, models of cell function, and computer representations of organs and organisms.” E. Pennisi 3.1 Classical Equations Partial differential equations arise frequently in formulating fundamental laws of nature and in the study of a wide variety of physical, chemical, and biological models. We start with a special type of second-order linear partial differential equation for the following reasons. First, second-order linear equations arise more frequently in a wide variety of applications. Second, their mathematical treatment is simpler and easier to understand than that of first-order equations in general. Usually, in almost all physical 64 3 Mathematical Models phenomena (or physical processes), the dependent variable u = u (x, y, z, t) is a function of three space variables, x, y, z and time variable t. The three basic types of second-order partial differential equations are: (a) The wave equation utt − c 2 (uxx + uyy + uzz)=0. (3.1.1) (b) The heat equation ut − k (uxx + uyy + uzz)=0. (3.1.2) (c) The Laplace equation uxx + uyy + uzz = 0. (3.1.3) In this section, we list a few more common linear partial differential equations of importance in applied mathematics, mathematical physics, and engineering science. Such a list naturally cannot ever be complete. Included are only equations of most common interest: (d) The Poisson equation ∇2u = f (x, y, z). (3.1.4) (e) The Helmholtz equation ∇2u + λu = 0. (3.1.5) (f) The biharmonic equation ∇4u = ∇2  ∇2u = 0. (3.1.6) (g) The biharmonic wave equation utt + c 2∇4u = 0. (3.1.7) (h) The telegraph equation utt + aut + bu = c 2uxx. (3.1.8) (i) The Schr¨odinger equations in quantum physics iψt = −  2 2m  ∇2 + V (x, y, z) ψ, (3.1.9) ∇2Ψ + 2m  2 [E − V (x, y, z)] Ψ = 0. (3.1.10) (j) The Klein–Gordon equation u + λ 2u = 0, (3.1.11) 3.2 The Vibrating String 65 where ∇2 ≡ ∂ 2 ∂x2 + ∂ 2 ∂y2 + ∂ 2 ∂z2 , (3.1.12) is the Laplace operator in rectangular Cartesian coordinates (x, y, z),  ≡ ∇2 − 1 c 2 ∂ 2 ∂t2 , (3.1.13) is the d’Alembertian, and in all equations λ, a, b, c, m, E are constants and h = 2π is the Planck constant. (k) For a compressible fluid flow, Euler’s equations ut + (u · ∇) u = − 1 ρ ∇p, ρt + div (ρu)=0, (3.1.14) where u = (u, v, w) is the fluid velocity vector, ρ is the fluid density, and p = p (ρ) is the pressure that relates p and ρ (the constitutive equation or equation of state). Many problems in mathematical physics reduce to the solving of partial differential equations, in particular, the partial differential equations listed above. We will begin our study of these equations by first examining in detail the mathematical models representing physical problems. 3.2 The Vibrating String One of the most important problems in mathematical physics is the vibration of a stretched string. Simplicity and frequent occurrence in many branches of mathematical physics make it a classic example in the theory of partial differential equations. Let us consider a stretched string of length l fixed at the end points. The problem here is to determine the equation of motion which characterizes the position u (x, t) of the string at time t after an initial disturbance is given. In order to obtain a simple equation, we make the following assumptions: 1. The string is flexible and elastic, that is the string cannot resist bending moment and thus the tension in the string is always in the direction of the tangent to the existing profile of the string. 2. There is no elongation of a single segment of the string and hence, by Hooke’s law, the tension is constant. 3. The weight of the string is small compared with the tension in the string. 4. The deflection is small compared with the length of the string. 5. The slope of the displaced string at any point is small compared with unity. 66 3 Mathematical Models 6. There is only pure transverse vibration. We consider a differential element of the string. Let T be the tension at the end points as shown in Figure 3.2.1. The forces acting on the element of the string in the vertical direction are T sin β − T sin α. By Newton’s second law of motion, the resultant force is equal to the mass times the acceleration. Hence, T sin β − T sin α = ρ δs utt (3.2.1) where ρ is the line density and δs is the smaller arc length of the string. Since the slope of the displaced string is small, we have δs ≃ δx. Since the angles α and β are small sin α ≃ tan α, sin β ≃ tan β. Figure 3.2.1 An Element of a vertically displaced string. 3.3 The Vibrating Membrane 67 Thus, equation (3.2.1) becomes tan β − tan α = ρ δx T utt. (3.2.2) But, from calculus we know that tan α and tan β are the slopes of the string at x and x + δx: tan α = ux (x, t) and tan β = ux (x + δx, t) at time t. Equation (3.2.2) may thus be written as 1 δx (ux)x+δx − (ux)x ! = ρ T utt, 1 δx [ux (x + δx, t) − ux (x, t)] = ρ T utt. In the limit as δx approaches zero, we find utt = c 2uxx (3.2.3) where c 2 = T /ρ. This is called the one-dimensional wave equation. If there is an external force f per unit length acting on the string. Equation (3.2.3) assumes the form utt = c 2uxx + F, F = f /ρ, (3.2.4) where f may be pressure, gravitation, resistance, and so on. 3.3 The Vibrating Membrane The equation of the vibrating membrane occurs in a large number of problems in applied mathematics and mathematical physics. Before we derive the equation for the vibrating membrane we make certain simplifying assumptions as in the case of the vibrating string: 1. The membrane is flexible and elastic, that is, the membrane cannot resist bending moment and the tension in the membrane is always in the direction of the tangent to the existing profile of the membrane. 2. There is no elongation of a single segment of the membrane and hence, by Hooke’s law, the tension is constant. 3. The weight of the membrane is small compared with the tension in the membrane. 4. The deflection is small compared with the minimal diameter of the membrane. 68 3 Mathematical Models 5. The slope of the displayed membrane at any point is small compared with unity. 6. There is only pure transverse vibration. We consider a small element of the membrane. Since the deflection and slope are small, the area of the element is approximately equal to δxδy. If T is the tensile force per unit length, then the forces acting on the sides of the element are T δx and T δy, as shown in Figure 3.3.1. The forces acting on the element of the membrane in the vertical direction are T δx sin β − T δx sin α + T δy sin δ − T δy sin γ. Since the slopes are small, sines of the angles are approximately equal to their tangents. Thus, the resultant force becomes T δx (tan β − tan α) + T δy (tan δ − tan γ). By Newton’s second law of motion, the resultant force is equal to the mass times the acceleration. Hence, T δx (tan β − tan α) + T δy (tan δ − tan γ) = ρ δA utt (3.3.1) where ρ is the mass per unit area, δA ≃ δxδy is the area of this element, and utt is computed at some point in the region under consideration. But from calculus, we have Figure 3.3.1 An element of vertically displaced membrane. 3.4 Waves in an Elastic Medium 69 tan α = uy (x1, y) tan β = uy (x2, y + δy) tan γ = ux (x, y1) tan δ = ux (x + δx, y2) where x1 and x2 are the values of x between x and x+δx, and y1 and y2 are the values of y between y and y + δy. Substituting these values in (3.3.1), we obtain T δx [uy (x2, y + δy) − uy (x1, y)] + T δy [ux (x + δx, y2) − ux (x, y1)] = ρ δxδy utt. Division by ρ δxδy yields T ρ uy (x2, y + δy) − uy (x1, y) δy + ux (x + δx, y2) − ux (x, y1) δx = utt. (3.3.2) In the limit as δx approaches zero and δy approaches zero, we obtain utt = c 2 (uxx + uyy), (3.3.3) where c 2 = T /ρ. This equation is called the two-dimensional wave equation. If there is an external force f per unit area acting on the membrane. Equation (3.3.3) takes the form utt = c 2 (uxx + uyy) + F, (3.3.4) where F = f /ρ. 3.4 Waves in an Elastic Medium If a small disturbance is originated at a point in an elastic medium, neighboring particles are set into motion, and the medium is put under a state of strain. We consider such states of motion to extend in all directions. We assume that the displacements of the medium are small and that we are not concerned with translation or rotation of the medium as a whole. Let the body under investigation be homogeneous and isotropic. Let δV be a differential volume of the body, and let the stresses acting on the faces of the volume be τxx, τyy, τzz, τxy, τxz, τyx, τyz, τzx, τzy. The first three stresses are called the normal stresses and the rest are called the shear stresses. (See Figure 3.4.1). We shall assume that the stress tensor τij is symmetric describing the condition of the rotational equilibrium of the volume element, that is, 70 3 Mathematical Models Figure 3.4.1 Volume element of an elastic body. τij = τji, i = j, i, j = x, y, z. (3.4.1) Neglecting the body forces, the sum of all the forces acting on the volume element in the x-direction is (τxx)x+δx − (τxx)x ! δyδz + " (τxy) y+δy − (τxy) y # δzδx + (τxz) z+δz − (τxz) z ! δxδy. By Newton’s law of motion this resultant force is equal to the mass times the acceleration. Thus, we obtain (τxx)x+δx − (τxx)x ! δyδz + " (τxy) y+δy − (τxy) y # δzδx + (τxz) z+δz − (τxz) z ! δxδy = ρ δxδyδz utt (3.4.2) where ρ is the density of the body and u is the displacement component in the x-direction. Hence, in the limit as δV approaches zero, we obtain ∂τxx ∂x + ∂τxy ∂y + ∂τxz ∂z = ρ ∂ 2u ∂t2 . (3.4.3) Similarly, the following two equations corresponding to y and z directions are obtained: 3.4 Waves in an Elastic Medium 71 ∂τyx ∂x + ∂τyy ∂y + ∂τyz ∂z = ρ ∂ 2v ∂t2 , (3.4.4) ∂τzx ∂x + ∂τzy ∂y + ∂τzz ∂z = ρ ∂ 2w ∂t2 , (3.4.5) where v and w are the displacement components in the y and z directions respectively. We may now define linear strains [see Sokolnikoff (1956)] as εxx = ∂u ∂x, εyz = 1 2  ∂w ∂y + ∂v ∂z  , εyy = ∂v ∂y , εzx = 1 2  ∂u ∂z + ∂w ∂x  , (3.4.6) εzz = ∂w ∂z , εxy = 1 2  ∂v ∂x + ∂u ∂y  , in which εxx, εyy, εzz represent unit elongations and εyz, εzx, εxy represent unit shearing strains. In the case of an isotropic body, generalized Hooke’s law takes the form τxx = λθ + 2µεxx, τyz = 2µεyz, τyy = λθ + 2µεyy, τzx = 2µεzx, (3.4.7) τzz = λθ + 2µεzz, τxy = 2µεxy, where θ = εxx + εyy + εzz is called the dilatation, and λ and µ are Lame’s constants. Expressing stresses in terms of displacements, we obtain τxx = λθ + 2µ ∂u ∂x, τxy = µ  ∂v ∂x + ∂u ∂y  , (3.4.8) τxz = µ  ∂w ∂x + ∂u ∂z  . By differentiating equations (3.4.8), we obtain ∂τxx ∂x = λ ∂θ ∂x + 2µ ∂ 2u ∂x2 , ∂τxy ∂y = µ ∂ 2v ∂x∂y + µ ∂ 2u ∂y2 , (3.4.9) ∂τxz ∂z = µ ∂ 2w ∂x∂z + µ ∂ 2u ∂z2 . Substituting equation (3.4.9) into equation (3.4.3) yields 72 3 Mathematical Models λ ∂θ ∂x + µ  ∂ 2u ∂x2 + ∂ 2v ∂x∂y + ∂ 2w ∂x∂z  + µ  ∂ 2u ∂x2 + ∂ 2u ∂y2 + ∂ 2u ∂z2  = ρ ∂ 2u ∂t2 . (3.4.10) We note that ∂ 2u ∂x2 + ∂ 2v ∂x∂y + ∂ 2w ∂x∂z = ∂ ∂x  ∂u ∂x + ∂v ∂y + ∂w ∂z  = ∂θ ∂x, and introduce the notation △ = ∇2 = ∂ 2 ∂x2 + ∂ 2 ∂y2 + ∂ 2 ∂z2 . The symbol △ or ∇2 is called the Laplace operator. Hence, equation (3.4.10) becomes (λ + µ) ∂θ ∂x + µ∇2u = ρ ∂ 2u ∂t2 . (3.4.11) In a similar manner, we obtain the other two equations which are (λ + µ) ∂θ ∂y + µ∇2 v = ρ ∂ 2v ∂t2 . (3.4.12) (λ + µ) ∂θ ∂z + µ∇2w = ρ ∂ 2w ∂t2 . (3.4.13) The set of equations (3.4.11)–(3.4.13) is called the Navier equations of motion. In vector form, the Navier equations of motion assume the form (λ + µ) grad div u + µ∇2u = ρ utt, (3.4.14) where u = ui + vj + wk and θ = div u. (i) If div u = 0, the general equation becomes µ∇2u = ρ utt, or utt = c 2 T ∇2u, (3.4.15) where cT is called the transverse wave velocity given by cT =  µ/ρ. This is the case of an equivoluminal wave propagation, since the volume expansion θ is zero for waves moving with this velocity. Sometimes these waves are called waves of distortion because the velocity of propagation depends on µ and ρ; the shear modulus µ characterizes the distortion and rotation of the volume element. 3.4 Waves in an Elastic Medium 73 (ii) When curl u = 0, the vector identity curl curl u = grad div u − ∇2u, gives grad div u = ∇2u, Then the general equation becomes (λ + 2µ) ∇2u = ρ utt, or utt = c 2 L∇2u, (3.4.16) where cL is called the longitudinal wave velocity given by cL = $ λ + 2µ ρ . This is the case of irrotational or dilatational wave propagation, since curl u = 0 describes irrotational motion. Equations (3.4.15) and (3.4.16) are called the three-dimensional wave equations. In general, the wave equation may be written as utt = c 2∇2u, (3.4.17) where the Laplace operator may be one, two, or three dimensional. The importance of the wave equation stems from the facts that this type of equation arises in many physical problems; for example, sound waves in space, electrical vibration in a conductor, torsional oscillation of a rod, shallow water waves, linearized supersonic flow in a gas, waves in an electric transmission line, waves in magnetohydrodynamics, and longitudinal vibrations of a bar. To give a more general method of decomposing elastic waves into transverse and longitudinal wave forms, we write the Navier equations of motion in the form c 2 T ∇2u +  c 2 L − c 2 T grad (div u) = utt. (3.4.18) We now decompose this equation into two vector equations by defining u = uT + uL, where uT and uL satisfy the equations div uT = 0 and curl uL = 0. (3.4.19ab) Since uT is defined by (3.4.19a) that is divergenceless, it follows from vector analysis that there exists a rotation vector ψ such that 74 3 Mathematical Models uT = curl ψ, (3.4.20) where ψ is called the vector potential. On the other hand, uL is irrotational as given by (3.4.19b), so there exists a scalar function φ (x, t), called the scalar potential such that uL = grad φ. (3.4.21) Using (3.4.20) and (3.4.21), we can write u = curl ψ + grad φ. (3.4.22) This means that the displacement vector field is decomposed into a divergenceless vector and irrotational vector. Inserting u = uT + uL into (3.4.18), taking the divergence of each term of the resulting equation, and then using (3.4.19a) gives div c 2 L∇2uL − (uL) tt! = 0. (3.4.23) It is noted that the curl of the square bracket in (3.4.23) is also zero. Clearly, any vector whose divergence and curl both vanish is identically a zero vector. Consequently, c 2 L∇2uL = (uL) tt . (3.4.24) This shows that uL satisfies the vector wave equation with the wave velocity cL. Since uL = grad φ, it is clear that the scalar potential φ also satisfies the wave equation with the same wave speed. All solutions of (3.4.24) represent longitudinal waves that are irrotational (since ψ = 0). Similarly, we substitute u = uL + uT into (3.4.18), take the curl of the resulting equation, and use the fact that curl uL = 0 to obtain curl c 2 T ∇2uT − (uT ) tt! = 0. (3.4.25) Since the divergence of the expression inside the square bracket is also zero, it follows that c 2 T ∇2uT = (uT ) tt . (3.4.26) This is a vector wave equation for uT whose solutions represent transverse waves that are irrotational but are accompanied by no change in volume (equivoluminal, transverse, rotational waves). These waves propagate with a wave velocity cT . We close this section by seeking time-harmonic solutions of (3.4.18) in the form u = Re U (x, y, z) e iωt! . (3.4.27) 3.5 Conduction of Heat in Solids 75 Invoking (3.4.27) into equation (3.4.18) gives the following equation for the function U cT ∇2U + (cL − cT ) grad (div U) + ω 2U = 0. (3.4.28) Inserting, u = uT + uL, and using the above method of taking the divergence and curl of (3.4.28) respectively leads to equation for UL and UT as follows ∇2UL + k 2 L ∇2UL = 0, ∇2UT + k 2 T UT = 0, (3.4.29) where k 2 L = ω 2 c 2 L and k 2 T = ω 2 c 2 T . (3.4.30) Equations (3.4.29) are called the reduced wave equations (or the Helmholtz equations) for UL and UT . Obviously, equations (3.4.29) can also be derived by assuming time-harmonic solutions for uL and uT in the form ⎛ ⎝ uL uT ⎞ ⎠ = e iωt ⎛ ⎝ UL UT ⎞ ⎠ , (3.4.31) and substituting these results into (3.4.24) and (3.4.26) respectively. 3.5 Conduction of Heat in Solids We consider a domain D∗ bounded by a closed surface B∗ . Let u (x, y, z, t) be the temperature at a point (x, y, z) at time t. If the temperature is not constant, heat flows from places of higher temperature to places of lower temperature. Fourier’s law states that the rate of flow is proportional to the gradient of the temperature. Thus the velocity of the heat flow in an isotropic body is v = −Kgradu, (3.5.1) where K is a constant, called the thermal conductivity of the body. Let D be an arbitrary domain bounded by a closed surface B in D∗ . Then the amount of heat leaving D per unit time is  B vnds, where vn = v · n is the component of v in the direction of the outer unit normal n of B. Thus, by Gauss’ theorem (Divergence theorem) 76 3 Mathematical Models  B vnds =  D div (−Kgradu) dx dy dz = −K  D ∇2u dx dy dz. (3.5.2) But the amount of heat in D is given by  D σρu dx dy dz, (3.5.3) where ρ is the density of the material of the body and σ is its specific heat. Assuming that integration and differentiation are interchangeable, the rate of decrease of heat in D is −  D σρ ∂u ∂t dx dy dz. (3.5.4) Since the rate of decrease of heat in D must be equal to the amount of heat leaving D per unit time, we have −  D σρut dx dy dz = −K  D ∇2u dx dy dz, or −  D σρut − K∇2u ! dx dy dz = 0, (3.5.5) for an arbitrary D in D∗ . We assume that the integrand is continuous. If we suppose that the integrand is not zero at a point (x0, y0, z0) in D, then, by continuity, the integrand is not zero in a small region surrounding the point (x0, y0, z0). Continuing in this fashion we extend the region encompassing D. Hence the integral must be nonzero. This contradicts (3.5.5). Thus, the integrand is zero everywhere, that is, ut = κ∇2u, (3.5.6) where κ = K/σρ. This is known as the heat equation. This type of equation appears in a great variety of problems in mathematical physics, for example the concentration of diffusing material, the motion of a tidal wave in a long channel, transmission in electrical cables, and unsteady boundary layers in viscous fluid flows. 3.6 The Gravitational Potential In this section, we shall derive one of the most well-known equations in the theory of partial differential equations, the Laplace equation. 3.6 The Gravitational Potential 77 Figure 3.6.1 Two particles at P and Q. We consider two particles of masses m and M, at P and Q as shown in Figure 3.6.1. Let r be the distance between them. Then, according to Newton’s law of gravitation, a force proportional to the product of their masses, and inversely proportional to the square of the distance between them, is given in the form F = G mM r 2 , (3.6.1) where G is the gravitational constant. It is customary in potential theory to choose the unit of force so that G = 1. Thus, F becomes F = mM r 2 . (3.6.2) If r represents the vector P Q, the force per unit mass at Q due to the mass at P may be written as F = −mr r 3 = ∇ 4m r 5 , (3.6.3) which is called the intensity of the gravitational field of force. We suppose that a particle of unit mass moves under the attraction of the particle of mass m at P from infinity up to Q. The work done by the force F is 78 3 Mathematical Models  r ∞ Fdr =  r ∞ ∇ 4m r 5 dr = m r . (3.6.4) This is called the potential at Q due to the particle at P. We denote this by V = − m r , (3.6.5) so that the intensity of force at P is F = ∇ 4m r 5 = −∇V. (3.6.6) We shall now consider a number of masses m1, m2, ..., mn, whose distances from Q are r1, r2, ..., rn, respectively. Then the force of attraction per unit mass at Q due to the system is F = n k=1 ∇ mk rk = ∇ n k=1 mk rk . (3.6.7) The work done by the forces acting on a particle of unit mass is  r ∞ F · dr = n k=1 mk rk = −V. (3.6.8) Then the potential satisfies the equation ∇2V = −∇2n k=1 mk rk = − n k=1 ∇2  mk rk  = 0, rk = 0. (3.6.9) In the case of a continuous distribution of mass in some volume R, we have, as in Figure 3.6.2. V (x, y, z) =  R ρ (ξ, η, ζ) r dR, (3.6.10) where r = % (x − ξ) 2 + (y − η) 2 + (z − ζ) 2 and Q is outside the body. It immediately follows that ∇2V = 0. (3.6.11) This equation is called the Laplace equation, also known as the potential equation. It appears in many physical problems, such as those of electrostatic potentials, potentials in hydrodynamics, and harmonic potentials in the theory of elasticity. We observe that the Laplace equation can be viewed as the special case of the heat and the wave equations when the dependent variables involved are independent of time. 3.7 Conservation Laws and The Burgers Equation 79 Figure 3.6.2 Continuous Mass Distribution. 3.7 Conservation Laws and The Burgers Equation A conservation law states that the rate of change of the total amount of material contained in a fixed domain of volume V is equal to the flux of that material across the closed bounding surface S of the domain. If we denote the density of the material by ρ (x, t) and the flux vector by q (x, t), then the conservation law is given by d dt  V ρ dV = −  S (q · n) dS, (3.7.1) where dV is the volume element and dS is the surface element of the boundary surface S, n denotes the outward unit normal vector to S as shown in Figure 3.7.1, and the right-hand side measures the outward flux — hence, the minus sign is used. Applying the Gauss divergence theorem and taking d dt inside the integral sign, we obtain  V  ∂ρ ∂t + div q  dV = 0. (3.7.2) This result is true for any arbitrary volume V , and, if the integrand is continuous, it must vanish everywhere in the domain. Thus, we obtain the differential form of the conservation law ρt + div q = 0. (3.7.3) 80 3 Mathematical Models Figure 3.7.1 Volume V of a closed domain bounded by a surface S with surface element dS and outward normal vector n. The one-dimensional version of the conservation law (3.7.3) is ∂ρ ∂t + ∂q ∂x = 0. (3.7.4) To investigate the nature of the discontinuous solution or shock waves, we assume a functional relation q = Q (ρ) and allow a jump discontinuity for ρ and q. In many physical problems of interest, it would be a better approximation to assume that q is a function of the density gradient ρx as well as ρ. A simple model is to take q = Q (ρ) − νρx, (3.7.5) where ν is a positive constant. Substituting (3.7.5) into (3.7.4), we obtain the nonlinear diffusion equation ρt + c (ρ) ρx = νρxx, (3.7.6) where c (ρ) = Q′ (ρ). We multiply (3.7.6) by c ′ (ρ) to obtain ct + c cx = ν c′ (ρ) ρxx, = ν & cxx − c ′′ (ρ) ρ 2 x ' . (3.7.7) If Q (ρ) is a quadratic function in ρ, then c (ρ) is linear in ρ, and c ′′ (ρ) = 0. Consequently, (3.7.7) becomes ct + c cx = ν cxx. (3.7.8) 3.8 The Schr¨odinger and the Korteweg–de Vries Equations 81 As a simple model of turbulence, c is replaced by the fluid velocity field u (x, t) to obtain the well-known Burgers equation ut + u ux = ν uxx, (3.7.9) where ν is the kinematic viscosity. Thus the Burgers equation is a balance between time evolution, nonlinearity, and diffusion. This is the simplest nonlinear model equation for diffusive waves in fluid dynamics. Burgers (1948) first developed this equation primarily to shed light on the study of turbulence described by the interaction of the two opposite effects of convection and diffusion. However, turbulence is more complex in the sense that it is both three dimensional and statistically random in nature. Equation (3.7.9) arises in many physical problems including one-dimensional turbulence (where this equation had its origin), sound waves in a viscous medium, shock waves in a viscous medium, waves in fluid-filled viscous elastic tubes, and magnetohydrodynamic waves in a medium with finite electrical conductivity. We note that (3.7.9) is parabolic provided the coefficient of ux is constant, whereas the resulting (3.7.9) with ν = 0 is hyperbolic. More importantly, the properties of the solution of the parabolic equation are significantly different from those of the hyperbolic equation. 3.8 The Schr¨odinger and the Korteweg–de Vries Equations We consider the following Fourier integral representation of a quasi-monochromatic plane wave solution u (x, t) =  ∞ −∞ F (k) exp [i {kx − ω (k)t}] dk, (3.8.1) where the spectrum function F (k) is determined from the given initial or boundary conditions and has the property F (−k) = F ∗ (k), and ω = ω (k) is the dispersion relation. We assume that the initial wave is slowly modulated as it propagates in a dispersive medium. For such a quasimonochromatic wave, most of the energy is confined in a neighborhood of a specified wave number k = k0, so that spectrum function F (k) has a sharp peak around the point k = k0 with a narrow wave number width k − k0 = δk = O (ε), and the dispersion relation ω (k) can be expanded about k0 in the form ω = ω0 + ω ′ 0 (δk) + 1 2! ω ′′ 0 (δk) 2 + 1 3! ω ′′′ 0 (δk) 3 + ··· , (3.8.2) where ω0 = ω (k0), ω ′ 0 = ω ′ (k0), ω ′′ 0 = ω ′′ (k0), and ω ′′′ 0 = ω ′′′ (k0). 82 3 Mathematical Models Substituting (3.8.2) into (3.8.1) gives a new form u (x, t) = A (x, t) exp [i(k0x − ω0t)] + c.c., (3.8.3) where c.c. stands for the complex conjugate and A (x, t) is the complex wave amplitude given by A (x, t) =  ∞ 0 F (k0 + δk) exp i  (x − ω ′ 0 t) (δk) − 1 2 ω ′′ 0 (δk) 2 t − 1 3 ω ′′′ 0 (δk) 3 t 0 d (δk), (3.8.4) where it has been assumed that ω (−k) = − ω (k). Since (3.8.4) depends on (x − ω ′ 0 t) δk, (δk) 2 t, (δk) 3 t where δk = O (ε) is small, the wave amplitude A (x, t) is a slowly varying function of x ∗ = (x − ω ′ 0 t) and time t. We keep only the term with (δk) in (3.8.4) and neglect all terms with (δk) n , n = 2, 3, ··· , so that (3.8.4) becomes A (x, t) =  ∞ 0 F (k0 + δk) exp [i {(x − ω ′ 0 t)} (δk)] d (δk). (3.8.5) A simple calculation reveals that A (x, t) satisfies the evolution equation ∂A ∂t + cg ∂A ∂x = 0, (3.8.6) where cg = ω ′ 0 is the group velocity. In the next step, we retain only terms with (δk) and (δk) 2 in (3.8.4) to obtain A (x, t) =  ∞ 0 F (k0 + δk) exp i  (x − ω ′ 0 t) (δk) − 1 2 ω ′′ 0 (δk) 2 0 d (δk). (3.8.7) A simple calculation shows that A (x, t) satisfies the linear Schr¨odinger equation i  ∂A ∂t + ω ′ 0 ∂A ∂x  + 1 2 ω ′′ 0 ∂ 2A ∂x2 = 0. (3.8.8) Using the slow variables ξ = ε (x − ω ′ 0 t), τ = ε 2 t, (3.8.9) the modulated wave amplitude A (ξ, τ ) satisfies the linear Schr¨odinger equation i Aτ + 1 2 ω ′′ 0Aξξ = 0. (3.8.10) 3.9 Exercises 83 On the other hand, for the frequencies at which the group velocity ω ′ 0 reaches an extremum, ω ′′ 0 = 0. In this case, the cubic term in the dispersion relation (3.7.2) plays an important role. Consequently, equation (3.8.4) reduces to a form similar to (3.8.7) with ω ′′ 0 = 0 in the exponential factor. Once again, a simple calculation from the resulting integral (3.8.4) reveals that A (x, t) satisfies the linearized Korteweg–de Vries (KdV) equation ∂A ∂t + ω ′ 0 ∂A ∂x + 1 6 ω ′′′ 0 ∂ 3A ∂x3 = 0. (3.8.11) By transferring to the new variables ξ = x−ω ′ 0 t and τ = t which correspond to a reference system moving with the group velocity ω ′ 0 , we obtain the linearized KdV equation ∂A ∂τ + 1 6 ω ′′′ 0 ∂ 3A ∂ξ3 = 0. (3.8.12) This describes waves in a dispersive medium with a weak high frequency dispersion. One of the remarkable nonlinear model equations is the Korteweg–de Vries (KdV) equation in the form ut + αu ux + βuxxx = 0, −∞ <x< ∞,="" t=""> 0. (3.8.13) This equation arises in many physical problems including water waves, ion acoustic waves in a plasma, and longitudinal dispersive waves in elastic rods. The exact solution of this equation is called the soliton which is remarkably stable. We shall discuss the soliton solution in Chapter 13. Another remarkable nonlinear model equation describing solitary waves is known as the nonlinear Schr¨odinger (NLS) equation written in the standard form i ut + 1 2 ω ′′ 0 uxx + γ |u| 2 u = 0, −∞ <x< ∞,="" t=""> 0. (3.8.14) This equation admits a solution called the solitary waves and describes the evolution of the water waves; it arises in many other physical systems that include nonlinear optics, hydromagnetic and plasma waves, propagation of heat pulse in a solid, and nonlinear instability problems. The solution of this equation will be discussed in Chapter 13. 3.9 Exercises 1. Show that the equation of motion of a long string is utt = c 2uxx − g, where g is the gravitational acceleration. 84 3 Mathematical Models 2. Derive the damped wave equation of a string utt + a ut = c 2uxx, where the damping force is proportional to the velocity and a is a constant. Considering a restoring force proportional to the displacement of a string, show that the resulting equation is utt + aut + bu = c 2uxx, where b is a constant. This equation is called the telegraph equation. 3. Consider the transverse vibration of a uniform beam. Adopting Euler’s beam theory, the moment M at a point can be written as M = −EI uxx, where EI is called the flexural rigidity, E is the elastic modulus, and I is the moment of inertia of the cross section of the beam. Show that the transverse motion of the beam may be described by utt + c 2uxxxx = 0, where c 2 = EI/ρA, ρ is the density, and A is the cross-sectional area of the beam. 4. Derive the deflection equation of a thin elastic plate ∇4u = q/D, where q is the uniform load per unit area, D is the flexural rigidity of the plate, and ∇4u = uxxxx + 2uxxyy + uyyyy. 5. Derive the one-dimensional heat equation ut = κuxx, where κ is a constant. Assuming that heat is also lost by radioactive exponential decay of the material in the bar, show that the above equation becomes ut = κuxx + he−αx , where h and α are constants. 6. Starting from Maxwell’s equations in electrodynamics, show that in a conducting medium electric intensity E, magnetic intensity H, and current density J satisfy ∇2X = µεXtt + µσXt, where X represents E, H, and J, µ is the magnetic inductive capacity, ε is the electric inductive capacity, and σ is the electrical conductivity. 3.9 Exercises 85 7. Derive the continuity equation ρt + div (ρu)=0, and Euler’s equation of motion ρ [ut + (u · grad) u] + grad p = 0, in fluid dynamics. 8. In the derivation of the Laplace equation (3.6.11), the potential at Q which is outside the body is ascertained. Now determine the potential at Q when it is inside the body, and show that it satisfies the Poisson equation ∇2u = −4πρ, where ρ is the density of the body. 9. Setting U = e iktu in the wave equation Utt = ∇2U and setting U = e −k 2 tu in the heat equation Ut = ∇2U, show that u (x, y, z) satisfies the Helmholtz equation ∇2u + k 2u = 0. 10. The Maxwell equations in vacuum are ∇ × E = − ∂B ∂t , ∇ × B = µε ∂E ∂t , ∇ · E = 0, ∇ · B = 0, where µ and ε are universal constants. Show that the magnetic field B = (0, By (x, t), 0) and the electric field E = (0, 0, Ez (x, t)) satisfy the wave equation ∂ 2u ∂t2 = c 2 ∂ 2u ∂x2 , where u = By or Ez and c = (µε) − 1 2 is the speed of light. 11. The equations of gas dynamics are linearized for small perturbations about a constant state u = 0, ρ = ρ0, and p0 = p (ρ0) with c 2 0 = p ′ (ρ0). In terms of velocity potential φ defined by u = ∇φ, the perturbation equations are ρt + ρ0 div u = 0, p − p0 = −ρ0φt = c 2 0 (ρ − ρ0), ρ − ρ0 = − ρ0 c 2 0 φt. 86 3 Mathematical Models Show that f and u satisfy the three dimensional wave equations ftt = c 2 0 ∇2 f, and utt = c 2 0 ∇2u, where f = p, ρ, or φ and ∇2 ≡ ∂ 2 ∂x2 + ∂ 2 ∂y2 + ∂ 2 ∂z2 . 12. Consider a slender body moving in a gas with arbitrary constant velocity U, and suppose (x1, x2, x3) represents the frame of reference in which the motion of the gas is small and described by the equations of problem 11. The body moves in the negative x1 direction, and (x, y, z) denotes the coordinates fixed with respect to the body so that the coordinate transformation is (x, y, z)=(x1 + U t, x2, x3). Show that the wave equation φtt = c 2 0∇2φ reduces to the form  M2 − 1 Φxx = Φyy + Φzz, where M ≡ U/c0 is the Mach number and Φ is the potential in the new frame of reference (x, y, z). 13. Consider the motion of a gas in a taper tube of cross section A (x). Show that the equation of continuity and the equation of motion are ρ = ρ0  1 − ∂ξ ∂x − ξ A ∂A ∂x  = ρ0 1 − 1 A ∂ ∂x (Aξ) , and ρ0 ∂ 2 ξ ∂t2 = − ∂p ∂x, where x is the distance along the length of the tube, ξ (x) is the displacement function, p = p (ρ) is the pressure-density relation, ρ0 is the average density, and ρ is the local density of the gas. Hence derive the equation of motion ξtt = c 2 ∂ ∂x 1 A ∂ ∂x (Aξ) , c2 = ∂p ∂ρ. Find the equation of motion when A is constant. If A (x) = a0 exp (2αx) where a0 and α are constants, show that the above equation takes the form ξtt = c 2 (ξxx + 2αξx). 14. Consider the current I (x, t) and the potential V (x, t) at a point x and time t of a uniform electric transmission line with resistance R, 3.9 Exercises 87 inductance L, capacity C, and leakage conductance G per unit length. (a) Show that both I and V satisfy the system of equations LIt + RI = −Vx, CVt + GV = −Ix. Derive the telegraph equation uxx − c −2 utt − a ut − bu = 0, for u = I or V, where c 2 = (LC) −1 , a = RC + LG and b = RG. (b) Show that the telegraph equation can be written in the form utt − c 2uxx + (p + q) ut + pq u = 0, where p = G C and q = R L . (c) Apply the transformation u = v exp − 1 2 (p + q)t to transform the above equation into the form vtt − c 2 vxx = 1 4 (p − q) 2 v. (d) When p = q, show that there exists an undisturbed wave solution in the form u (x, t) = e −ptf (x + ct), which propagates in either direction, where f is an arbitrary twice differentiable function of its argument. If u (x, t) = A exp [i(kx − ωt)] is a solution of the telegraph equation utt − c 2uxx − αut − βu = 0, α = p + q, β = pq, show that the dispersion relation holds ω 2 + iαω −  c 2 k 2 + β 2 = 0. Solve the dispersion relation to show that u (x, t) = exp  − 1 2 p t exp i  kx − t 2  4c 2k 2 + (4q − p 2)  . When p 2 = 4q, show that the solution represents attenuated nondispersive waves. 88 3 Mathematical Models (e) Find the equations for I and V in the following cases: (i) Lossless transmission line (R = G = 0), (ii) Ideal submarine cable (L = G = 0), (iii) Heaviside’s distortionless line (R/L = G/C = constant = k). 15. The Fermi–Pasta–Ulam model is used to describe waves in an anharmonic lattice of length l consisting of a row of n identical masses m, each connected to the next by nonlinear springs of constant κ. The masses are at a distance h = l/n apart, and the springs when extended or compressed by an amount d exert a force F = κ  d + α d2 where α measures the strength of nonlinearity. The equation of motion of the ith mass is my¨i = κ " (yi+1 − yi) − (yi − yi−1) + α ( (yi+1 − yi) 2 − (yi − yi−1) 2 )# , where i = 1, 2, 3...n, yi is the displacement of the ith mass from its equilibrium position, and κ, α are constants with y0 = yn = 0. Assume a continuum approximation of this discrete system so that the Taylor expansions yi+1 − yi = hyx + h 2 2! yxx + h 3 3! yxxx + h 4 4! yxxxx + o  h 5 , yi − yi−1 = hyx − h 2 2! yxx + h 3 3! yxxx − h 4 4! yxxxx + o  h 5 , can be used to derive the nonlinear differential equation ytt = c 2 [1 + 2αhyx] yxx + o  h 4 , ytt = c 2 [1 + 2αhyx] yxx + c 2h 2 12 yxxxx + o  h 5 , where c 2 = κh2 m . Using a change of variables ξ = x − ct, τ = cαht, show that u = yξ satisfies the Korteweg–de Vries (KdV) equation uτ + uuξ + βuξξξ = o  ε 2 , ε = αh, β = h 24α . 16. The one-dimensional isentropic fluid flow is obtained from Euler’s equations (3.1.14) in the form ut + u ux = − 1 ρ px, ρt + (ρu)x = 0, p = p (ρ). 3.9 Exercises 89 (a) Show that u and ρ satisfy the one-dimensional wave equation  u ρ  tt − c 2  u ρ  xx = 0, where c 2 = dp dρ is the velocity of sound. (b) For a compressible adiabatic gas, the equation of state is p = Aργ , where A and γ are constants; show that c 2 = γp ρ . 17. (a) Obtain the two-dimensional unsteady fluid flow equations from (3.1.14). (b) Find the two-dimensional steady fluid flow equations from (3.1.14). Hence or otherwise, show that  c 2 − u 2 ux − u v (uy + vx) +  c 2 − v 2 vy = 0, where c 2 = p ′ (ρ). (c) Show that, for an irrotational fluid flow (u = ∇φ), the above equation reduces to the quasi-linear partial differential equations  c 2 − φ 2 x φxx − 2φxφyφxy +  c 2 − φ 2 y φyy = 0. (d) Show that the slope of the characteristic C satisfies the quadratic equation  c 2 − u 2  dy dx2 + 2uv  dy dx +  c 2 − v 2 = 0. Hence or otherwise derive  c 2 − v 2  dv du2 − 2uv  dv du +  c 2 − u 2 = 0. 18. For an inviscid incompressible fluid flow under the body force, F = −∇Φ, the Euler equations are ∂u ∂t + u · ∇u = −∇Φ − 1 ρ ∇p, divu = 0. (a) Show that the vorticity ω = ∇ × u satisfies the vorticity equation Dω Dt = ∂ω ∂t + u · ∇ω = ω · ∇u. (b) Give the interpretation of this vorticity equation. (c) In two dimensions, show that Dω Dt = 0 (conservation of vorticity). 90 3 Mathematical Models 19. The evolution of the probability distribution function u (x, t) in nonequilibrium statistical mechanics is described by the Fokker–Planck equation (See Reif (1965)) ∂u ∂t = ∂ ∂x  ∂u ∂t + x  u. (a) Use the change of variables ξ = x et and v = u e−t to show that the Fokker–Planck equation assumes the form with u (x, t) = e t v (ξ, τ ) vt = e 2t vξξ. (b) Make a suitable change of variable t to τ (t), and transform the above equation into the standard diffusion equation vt = vξξ. 20. The electric field E (x) and the electromagnetic field H (x) in free space (a vacuum) satisfy the Maxwell equations Et = c curl H, Ht = −c curl H, divE = 0 = divH, where c is the constant speed of light in a vacuum. Show that both E and H the three-dimensional wave equations Ett = c 2∇2E and Htt = c 2∇2H, where x = (x, y, z) and ∇2 is the three-dimensional Laplacian. 21. Consider longitudinal vibrations of a free elastic rod with a variable cross section A (x) with x measured along the axis of the rod from the origin. Assuming that the material of the rod satisfies Hooke’s law, show that the displacement function u (x, t) satisfies the generalized wave equation utt = c 2uxx + c 2 A (x)  dA dx  ux, where c 2 = (λ/ρ), λ is a constant that describes the elastic nature of the material, and ρ is the line density of the rod. When A (x) is constant, the above equation reduces to one-dimensional wave equation. 4 Classification of Second-Order Linear Equations “When we have a good understanding of the problem, we are able to clear it of all auxiliary notions and to reduce it to simplest element.” Ren´e Descartes “The first process ... in the effectual study of sciences must be one of simplification and reduction of the results of previous investigations to a form in which the mind can grasp them.” James Clerk Maxwell 4.1 Second-Order Equations in Two Independent Variables The general linear second-order partial differential equation in one dependent variable u may be written as n i,j=1 Aijuxixj + n i=1 Biuxi + F u = G, (4.1.1) in which we assume Aij = Aji and Aij , Bi , F, and G are real-valued functions defined in some region of the space (x1, x2,...,xn). Here we shall be concerned with second-order equations in the dependent variable u and the independent variables x, y. Hence equation (4.1.1) can be put in the form Auxx + Buxy + Cuyy + Dux + Euy + F u = G, (4.1.2) where the coefficients are functions of x and y and do not vanish simultaneously. We shall assume that the function u and the coefficients are twice continuously differentiable in some domain in R 2 . 92 4 Classification of Second-Order Linear Equations The classification of partial differential equations is suggested by the classification of the quadratic equation of conic sections in analytic geometry. The equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, represents hyperbola, parabola, or ellipse accordingly as B2 − 4AC is positive, zero, or negative. The classification of second-order equations is based upon the possibility of reducing equation (4.1.2) by coordinate transformation to canonical or standard form at a point. An equation is said to be hyperbolic, parabolic, or elliptic at a point (x0, y0) accordingly as B 2 (x0, y0) − 4A (x0, y0) C (x0, y0) (4.1.3) is positive, zero, or negative. If this is true at all points, then the equation is said to be hyperbolic, parabolic, or elliptic in a domain. In the case of two independent variables, a transformation can always be found to reduce the given equation to canonical form in a given domain. However, in the case of several independent variables, it is not, in general, possible to find such a transformation. To transform equation (4.1.2) to a canonical form we make a change of independent variables. Let the new variables be ξ = ξ (x, y), η = η (x, y). (4.1.4) Assuming that ξ and η are twice continuously differentiable and that the Jacobian J =       ξx ξy ηx ηy       , (4.1.5) is nonzero in the region under consideration, then x and y can be determined uniquely from the system (4.1.4). Let x and y be twice continuously differentiable functions of ξ and η. Then we have ux = uξξx + uηηx, uy = uξξy + uηηy, uxx = uξξξ 2 x + 2uξηξxηx + uηηη 2 x + uξξxx + uηηxx, (4.1.6) uxy = uξξξxξy + uξη (ξxηy + ξyηx) + uηηηxηy + uξξxy + uηηxy, uyy = uξξξ 2 y + 2uξηξyηy + uηηη 2 y + uξξyy + uηηyy. Substituting these values in equation (4.1.2) we obtain A ∗uξξ + B ∗uξη + C ∗uηη + D∗uξ + E ∗uη + F ∗u = G ∗ , (4.1.7) where 4.2 Canonical Forms 93 A ∗ = Aξ2 x + Bξxξy + Cξ2 y , B ∗ = 2Aξxηx + B (ξxηy + ξyηx)+2Cξyηy, C ∗ = Aη2 x + Bηxηy + Cη2 y , D∗ = Aξxx + Bξxy + Cξyy + Dξx + Eξy, (4.1.8) E ∗ = Aηxx + Bηxy + Cηyy + Dηx + Eηy, F ∗ = F, G∗ = G. The resulting equation (4.1.7) is in the same form as the original equation (4.1.2) under the general transformation (4.1.4). The nature of the equation remains invariant under such a transformation if the Jacobian does not vanish. This can be seen from the fact that the sign of the discriminant does not alter under the transformation, that is, B ∗2 − 4A ∗C ∗ = J 2  B 2 − 4AC , (4.1.9) which can be easily verified. It should be noted here that the equation can be of a different type at different points of the domain, but for our purpose we shall assume that the equation under consideration is of the single type in a given domain. The classification of equation (4.1.2) depends on the coefficients A (x, y), B (x, y), and C (x, y) at a given point (x, y). We shall, therefore, rewrite equation (4.1.2) as Auxx + Buxy + Cuyy = H (x, y, u, ux, uy), (4.1.10) and equation (4.1.7) as A ∗uξξ + B ∗uξη + C ∗uηη = H∗ (ξ, η, u, uξ, uη). (4.1.11) 4.2 Canonical Forms In this section we shall consider the problem of reducing equation (4.1.10) to canonical form. We suppose first that none of A, B, C, is zero. Let ξ and η be new variables such that the coefficients A∗ and C ∗ in equation (4.1.11) vanish. Thus, from (4.1.8), we have A ∗ = Aξ2 x + Bξxξy + Cξ2 y = 0, C ∗ = Aη2 x + Bηxηy + Cη2 y = 0. These two equations are of the same type and hence we may write them in the form Aζ2 x + Bζxζy + Cζ2 y = 0, (4.2.1) 94 4 Classification of Second-Order Linear Equations in which ζ stand for either of the functions ξ or η. Dividing through by ζ 2 y , equation (4.2.1) becomes A  ζx ζy 2 + B  ζx ζy  + C = 0. (4.2.2) Along the curve ζ = constant, we have dζ = ζxdx + ζydy = 0. Thus, dy dx = − ζx ζy , (4.2.3) and therefore, equation (4.2.2) may be written in the form A  dy dx2 − B  dy dx + C = 0, (4.2.4) the roots of which are dy dx = 4 B +  B2 − 4AC5 /2A, (4.2.5) dy dx = 4 B −  B2 − 4AC5 /2A. (4.2.6) These equations, which are known as the characteristic equations, are ordinary differential equations for families of curves in the xy-plane along which ξ = constant and η = constant. The integrals of equations (4.2.5) and (4.2.6) are called the characteristic curves. Since the equations are first-order ordinary differential equations, the solutions may be written as φ1 (x, y) = c1, c1 = constant, φ2 (x, y) = c2, c2 = constant. Hence the transformations ξ = φ1 (x, y), η = φ2 (x, y), will transform equation (4.1.10) to a canonical form. (A) Hyperbolic Type If B2 − 4AC > 0, then integration of equations (4.2.5) and (4.2.6) yield two real and distinct families of characteristics. Equation (4.1.11) reduces to uξη = H1, (4.2.7) 4.2 Canonical Forms 95 where H1 = H∗/B∗ . It can be easily shown that B∗ = 0. This form is called the first canonical form of the hyperbolic equation. Now if new independent variables α = ξ + η, β = ξ − η, (4.2.8) are introduced, then equation (4.2.7) is transformed into uαα − uββ = H2 (α, β, u, uα, uβ). (4.2.9) This form is called the second canonical form of the hyperbolic equation. (B) Parabolic Type In this case, we have B2 − 4AC = 0, and equations (4.2.5) and (4.2.6) coincide. Thus, there exists one real family of characteristics, and we obtain only a single integral ξ = constant (or η = constant). Since B2 = 4AC and A∗ = 0, we find that A ∗ = Aξ2 x + Bξxξy + Cξ2 y = 4√ A ξx + √ C ξy 52 = 0. From this it follows that A ∗ = 2Aξxηx + B (ξxηy + ξyηx)+2Cξyηy = 2 4√ A ξx + √ C ξy 54√ A ηx + √ C ηy 5 = 0, for arbitrary values of η (x, y) which is functionally independent of ξ (x, y); for instance, if η = y, the Jacobian does not vanish in the domain of parabolicity. Division of equation (4.1.11) by C ∗ yields uηη = H3 (ξ, η, u, uξ, uη), C∗ = 0. (4.2.10) This is called the canonical form of the parabolic equation. Equation (4.1.11) may also assume the form uξξ = H∗ 3 (ξ, η, u, uξ, uη), (4.2.11) if we choose η = constant as the integral of equation (4.2.5). (C) Elliptic Type For an equation of elliptic type, we have B2 − 4AC < 0. Consequently, the quadratic equation (4.2.4) has no real solutions, but it has two complex conjugate solutions which are continuous complex-valued functions of the real variables x and y. Thus, in this case, there are no real characteristic curves. However, if the coefficients A, B, and C are analytic functions of x and y, then one can consider equation (4.2.4) for complex x and y. A function of two real variables x and y is said to be analytic in a certain domain if in some neighborhood of every point (x0, y0) of this domain, the 96 4 Classification of Second-Order Linear Equations function can be represented as a Taylor series in the variables (x − x0) and (y − y0). Since ξ and η are complex, we introduce new real variables α = 1 2 (ξ + η), β = 1 2i (ξ − η), (4.2.12) so that ξ = α + iβ, η = α − iβ. (4.2.13) First, we transform equations (4.1.10). We then have A ∗∗ (α, β) uαα + B ∗∗ (α, β) uαβ + C ∗∗ (α, β) uββ = H4 (α, β, u, uα, uβ), (4.2.14) in which the coefficients assume the same form as the coefficients in equation (4.1.11). With the use of (4.2.13), the equations A∗ = C ∗ = 0 become  Aα2 x + Bαxαy + Cα2 y −  Aβ2 x + Bβxβy + Cβ2 y +i[2Aαxβx + B (αxβy + αyβx)+2Cαyβy]=0,  Aα2 x + Bαxαy + Cα2 y −  Aβ2 x + Bβxβy + Cβ2 y −i[2Aαxβx + B (αxβy + αyβx)+2Cαyβy]=0, or, (A ∗∗ − C ∗∗) + iB∗∗ = 0, (A ∗∗ − C ∗∗) − iB∗∗ = 0. These equations are satisfied if and only if A ∗∗ = C ∗∗ and B ∗∗ = 0. Hence, equation (4.2.14) transforms into the form A ∗∗uαα + A ∗∗uββ = H4 (α, β, u, uα, uβ). Dividing through by A∗∗, we obtain uαα + uββ = H5 (α, β, u, uα, uβ), (4.2.15) where H5 = (H4/A∗∗). This is called the canonical form of the elliptic equation. We close this discussion of canonical forms by adding an important comment. From mathematical and physical points of view, characteristics or characteristic coordinates play a very important physical role in hyperbolic equations. However, they do not play a particularly physical role in parabolic and elliptic equations, but their role is somewhat mathematical 4.2 Canonical Forms 97 in solving these equations. In general, first-order partial differential equations such as advection-reaction equations are regarded as hyperbolic because they describe propagation of waves like the wave equation. On the other hand, second-order linear partial differential equations with constant coefficients are sometimes classified by the associated dispersion relation ω = ω (κ) as defined in Section 13.3. In one-dimensional case, ω = ω (k). If ω (k) is real and ω ′′ (k) = 0, the equation is called dispersive. The word dispersive simply means that the phase velocity cp = (ω/k) of a plane wave solution, u (x, t) = A exp [i(kx − ωt)] depends on the wavenumber k. This means that waves of different wavelength propagate with different phase velocities and hence, disperse in the medium. If ω = ω (k) = σ (k) + iµ (k) is complex, the associated partial differential equation is called diffusive. From a physical point of view, such a classification of equations is particularly useful. Both dispersive and diffusive equations are physically important, and such equations will be discussed in Chapter 13. Example 4.2.1. Consider the equation y 2uxx − x 2uyy = 0. Here A = y 2 , B = 0, C = −x 2 . Thus, B 2 − 4AC = 4x 2 y 2 > 0. The equation is hyperbolic everywhere except on the coordinate axes x = 0 and y = 0. From the characteristic equations (4.2.5) and (4.2.6), we have dy dx = x y , dy dx = − x y . After integration of these equations, we obtain 1 2 y 2 − 1 2 x 2 = c1, 1 2 y 2 + 1 2 x 2 = c2. The first of these curves is a family of hyperbolas 1 2 y 2 − 1 2 x 2 = c1, and the second is a family of circles 1 2 y 2 + 1 2 x 2 = c2. To transform the given equation to canonical form, we consider 98 4 Classification of Second-Order Linear Equations ξ = 1 2 y 2 − 1 2 x 2 , η = 1 2 y 2 + 1 2 x 2 . From the relations (4.1.6), we have ux = uξξx + uηηx = −xuξ + xuη, uy = uξξy + uηηy = yuξ + yuη, uxx = uξξξ 2 x + 2uξηξxηx + uηηη 2 x + uξξxx + uηηxx = x 2uξξ − 2x 2uξη + x 2uηη − uξ + uη. uyy = uξξξ 2 y + 2uξηξyηy + uηηη 2 y + uξξyy + uηηyy = y 2uξξ + 2y 2uξη + y 2uηη + uξ + uη. Thus, the given equation assumes the canonical form uξη = η 2 (ξ 2 − η 2) uξ − ξ 2 (ξ 2 − η 2) uη. Example 4.2.2. Consider the partial differential equation x 2uxx + 2xy uxy + y 2uyy = 0. In this case, the discriminant is B 2 − 4AC = 4x 2 y 2 − 4x 2 y 2 = 0. The equation is therefore parabolic everywhere. The characteristic equation is dy dx = y x , and hence, the characteristics are y x = c, which is the equation of a family of straight lines. Consider the transformation ξ = y x , η = y, where η is chosen arbitrarily. The given equation is then reduced to the canonical form y 2uηη = 0. Thus, uηη = 0 for y = 0. 4.3 Equations with Constant Coefficients 99 Example 4.2.3. The equation uxx + x 2uyy = 0, is elliptic everywhere except on the coordinate axis x = 0 because B 2 − 4AC = −4x 2 < 0, x = 0. The characteristic equations are dy dx = ix, dy dx = −ix. Integration yields 2y − ix2 = c1, 2y + ix2 = c2. Thus, if we write ξ = 2y − ix2 , η = 2y + ix2 , and hence, α = 1 2 (ξ + η)=2y, β = 1 2i (ξ − η) = −x 2 , we obtain the canonical form uαα + uββ = − 1 2β uβ. It should be remarked here that a given partial differential equation may be of a different type in a different domain. Thus, for example, Tricomi’s equation uxx + xuyy = 0, (4.2.16) is elliptic for x > 0 and hyperbolic for x < 0, since B2 − 4AC = −4x. For a detailed treatment, see Hellwig (1964). 4.3 Equations with Constant Coefficients In this case of an equation with real constant coefficients, the equation is of a single type at all points in the domain. This is because the discriminant B2 − 4AC is a constant. From the characteristic equations dy dx = 4 B +  B2 − 4AC5 /2A, (4.3.1) 100 4 Classification of Second-Order Linear Equations we can see that the characteristics y = 4 B + √ B2 − 4AC 2A 5 x + c1, y = 4 B − √ B2 − 4AC 2A 5 x + c2, (4.3.2) are two families of straight lines. Consequently, the characteristic coordinates take the form ξ = y − λ1x, η = y − λ2x, (4.3.3) where λ1,2 = B+ √ B2 − 4AC 2A . (4.3.4) The linear second-order partial differential equation with constant coeffi- cients may be written in the general form as Auxx + Buxy + Cuyy + Dux + Euy + F u = G (x, y). (4.3.5) In particular, the equation Auxx + Buyy + Cuyy = 0, (4.3.6) is called the Euler equation. (A) Hyperbolic Type If B2 − 4AC > 0, the equation is of hyperbolic type, in which case the characteristics form two distinct families. Using (4.3.3), equation (4.3.5) becomes uξη = D1uξ + E1uη + F1u + G1 (ξ,η), (4.3.7) where D1, E1, and F1 are constants. Here, since the coefficients are constants, the lower order terms are expressed explicitly. When A = 0, equation (4.3.1) does not hold. In this case, the characteristic equation may be put in the form −B (dx/dy) + C (dx/dy) 2 = 0, which may again be rewritten as dx/dy = 0, and − B + C (dx/dy)=0. Integration gives x = c1, x = (B/C) y + c2, where c1 and c2 are integration constants. Thus, the characteristic coordinates are 4.3 Equations with Constant Coefficients 101 ξ = x, η = x − (B/C) y. (4.3.8) Under this transformation, equation (4.3.5) reduces to the canonical form uξη = D∗ 1uξ + E ∗ 1uη + F ∗ 1 u + G ∗ 1 (ξ,η), (4.3.9) where D∗ 1 , E∗ 1 , and F ∗ 1 are constants. The canonical form of the Euler equation (4.3.6) is uξη = 0. (4.3.10) Integrating this equation gives the general solution u = φ (ξ) + ψ (η) = φ (y − λ1, x) + ψ (y − λ2, x), (4.3.11) where φ and ψ are arbitrary functions, and λ1 and λ2 are given by (4.3.3). (B) Parabolic Type When B2 − 4AC = 0, the equation is of parabolic type, in which case only one real family of characteristics exists. From equation (4.3.4), we find that λ1 = λ2 = (B/2A), so that the single family of characteristics is given by y = (B/2A) x + c1, where c1 is an integration constant. Thus, we have ξ = y − (B/2A) x, η = hy + kx, (4.3.12) where η is chosen arbitrarily such that the Jacobian of the transformation is not zero, and h and k are constants. With the proper choice of the constants h and k in the transformation (4.3.12), equation (4.3.5) reduces to uηη = D2uξ + E2uη + F2u + G2 (ξ,η), (4.3.13) where D2, E2, and F2 are constants. If B = 0, we can see at once from the relation B 2 − 4AC = 0, that C or A vanishes. The given equation is then already in the canonical form. Similarly, in the other cases when A or C vanishes, B vanishes. The given equation is is then also in canonical form. The canonical form of the Euler equation (4.3.6) is uηη = 0. (4.3.14) 102 4 Classification of Second-Order Linear Equations Integrating twice gives the general solution u = φ (ξ) + η ψ (ξ), (4.3.15) where ξ and η are given by (4.3.12). Choosing h = 1, k = 0 and λ =  B 2A for simplicity, the general solution of the Euler equation in the parabolic case is u = φ (y − λx) + y ψ (y − λx). (4.3.16) (C) Elliptic Type When B2 − 4AC < 0, the equation is of elliptic type. In this case, the characteristics are complex conjugates. The characteristic equations yield y = λ1x + c1, y = λ2x + c2, (4.3.17) where λ1 and λ2 are complex numbers. Accordingly, c1 and c2 are allowed to take on complex values. Thus, ξ = y − (a + ib) x, η = y − (a − ib) x, (4.3.18) where λ1,2 = a + ib in which a and b are real constants, and a = B 2A , and b = 1 2A  4AC − B2 . Introduce the new variables α = 1 2 (ξ + η) = y − ax, β = 1 2i (ξ − η) = −bx. (4.3.19) Application of this transformation readily reduces equation (4.3.5) to the canonical form uαα + uββ = D3uα + E3uβ + F3u + G3 (α, β), (4.3.20) where D3, E3, F3 are constants. We note that B2 − AC < 0, so neither A nor C is zero. In this elliptic case, the Euler equation (4.3.6) gives the complex characteristics (4.3.18) which are ξ = (y − ax) − ibx, η = (y − ax) + ibx = ξ. (4.3.21) Consequently, the Euler equation becomes uξξ = 0, (4.3.22) with the general solution u = φ (ξ) + ψ  ξ . (4.3.23) The appearance of complex arguments in the general solution (4.3.23) is a general feature of elliptic equations. 4.3 Equations with Constant Coefficients 103 Example 4.3.1. Consider the equation 4 uxx + 5 uxy + uyy + ux + uy = 2. Since A = 4, B = 5, C = 1, and B2 − 4AC = 9 > 0, the equation is hyperbolic. Thus, the characteristic equations take the form dy dx = 1, dy dx = 1 4 , and hence, the characteristics are y = x + c1, y = (x/4) + c2. The linear transformation ξ = y − x, η = y − (x/4), therefore reduces the given equation to the canonical form uξη = 1 3 uη − 8 9 . This is the first canonical form. The second canonical form may be obtained by the transformation α = ξ + η, β = ξ − η, in the form uαα − uββ = 1 3 uα − 1 3 uβ − 8 9 . Example 4.3.2. The equation uxx − 4 uxy + 4 uyy = e y , is parabolic since A = 1, B = −4, C = 4, and B2 − 4AC = 0. Thus, we have from equation (4.3.12) ξ = y + 2x, η = y, in which η is chosen arbitrarily. By means of this mapping, the equation transforms into uηη = 1 4 e η . Example 4.3.3. Consider the equation uxx + uxy + uyy + ux = 0. 104 4 Classification of Second-Order Linear Equations Since A = 1, B = 1, C = 1, and B2 − 4AC = −3 < 0, the equation is elliptic. We have λ1,2 = B + √ B2 − 4AC 2A = 1 2 + i √ 3 2 , and hence, ξ = y − 4 1 2 + i √ 3 2 5 x, η = y − 4 1 2 − i √ 3 2 5 x. Introducing the new variables α = 1 2 (ξ + η) = y − 1 2 x, β = 1 2i (ξ − η) = − √ 3 2 x, the given equation is then transformed into canonical form uαα + uββ = 2 3 uα + 2 √ 3 uβ. Example 4.3.4. Consider the wave equation utt − c 2uxx = 0, c is constant. Since A = −c 2 , B = 0, C = 1, and B2 − 4AC = 4c 2 > 0, the wave equation is hyperbolic everywhere. According to (4.2.4), the equation of characteristics is −c 2  dt dx2 +1=0, or dx2 − c 2 dt2 = 0. Therefore, x + ct = ξ = constant, x − ct = η = constant. Thus, the characteristics are straight lines, which are shown in Figure 4.3.1. The characteristics form a natural set of coordinates for the hyperbolic equation. In terms of new coordinates ξ and η defined above, we obtain uxx = uξξ + 2uξη + uηη, utt = c 2 (uξξ − 2uξη + uηη), so that the wave equation becomes 4.3 Equations with Constant Coefficients 105 Figure 4.3.1 Characteristics for the wave equation. −4c 2uξη = 0. Since c = 0, we have uξη = 0. Integrating with respect to ξ, we obtain uη = ψ1 (η). where ψ1 is the arbitrary function of η. Integrating with respect to η, we obtain u (ξ,η) =  ψ1 (η) dη + φ (ξ). If we set ψ (η) = * ψ1 (η) dη, the general solution becomes u (ξ,η) = φ (ξ) + ψ (η), which is, in terms of the original variables x and t, u (x, t) = φ (x + ct) + ψ (x − ct), provided φ and ψ are arbitrary but twice differentiable functions. 106 4 Classification of Second-Order Linear Equations Note that φ is constant on “wavefronts” x = −ct + ξ that travel toward decreasing x as t increases, whereas ψ is constant on wavefronts x = ct + η that travel toward increasing x as t increases. Thus, any general solution can be expressed as the sum of two waves, one traveling to the right with constant velocity c and the other traveling to the left with the same velocity c. Example 4.3.5. Find the characteristic equations and characteristics, and then reduce the equations uxx +  sech4 x uyy = 0, (4.3.24ab) to the canonical forms. In equation (4.3.24a), A = 1, B = 0 and C = −sech4 x. Hence, B 2 − 4AC = 4 sech4 x > 0. Hence, the equation is hyperbolic. The characteristic equations are dy dx = B + √ B2 − 4AC 2A = + sech2 x. Integration gives y + tanh x = constant. Hence, ξ = y + tanh x, η = y − tanh x. Using these characteristic coordinates, the given equation can be transformed into the canonical form uξη = (η − ξ) " 4 − (ξ − η) 2 # (uξ − uη). (4.3.25) In equation (4.3.24b), A = 1, B = 0 and C = sech4 x. Hence, B 2 − 4AC = + isech2 x. Integrating gives y + i tanh x = constant. Thus, ξ = y + i tanh x, η = y − i tanh x. The new real variables α and β are 4.4 General Solutions 107 α = 1 2 (ξ + η) = y, β = 1 2i (ξ − η) = tanh x. In terms of these new variables, equation (4.3.24b) can be transformed into the canonical form uαα + uββ = 2β 1 − β 2 uβ, |β| < 1. (4.3.26) Example 4.3.6. Consider the equation uxx + (2 cosecy) uxy +  cosec2 y uyy = 0. (4.3.27) In this case, A = 1, B = 2 cosecy and C = cosec2y. Hence, B2 − 4AC = 0, and dy dx = B 2A = cosec y. The characteristic curves are therefore given by ξ = x + cos y and η = y. Using these variables, the canonical form of (4.3.27) is uηη =  sin2 η cos η uξ. (4.3.28) 4.4 General Solutions In general, it is not so simple to determine the general solution of a given equation. Sometimes further simplification of the canonical form of an equation may yield the general solution. If the canonical form of the equation is simple, then the general solution can be immediately ascertained. Example 4.4.1. Find the general solution of x 2uxx + 2xy uxy + y 2uyy = 0. In Example 4.2.2, using the transformation ξ = y/x, η = y, this equation was reduced to the canonical form uηη = 0, for y = 0. Integrating twice with respect to η, we obtain u (ξ,η) = ηf (ξ) + g (ξ), where f (ξ) and g (ξ) are arbitrary functions. In terms of the independent variables x and y, we have u (x, y) = y f 4y x 5 + g 4y x 5 . 108 4 Classification of Second-Order Linear Equations Example 4.4.2. Determine the general solution of 4 uxx + 5 uxy + uyy + ux + uy = 2. Using the transformation ξ = y − x, η = y − (x/4), the canonical form of this equation is (see Example 4.3.1) uξη = 1 3 uη − 8 9 . By means of the substitution v = uη, the preceding equation reduces to vξ = 1 3 v − 8 9 . This can be easily integrated by separating the variables. Integrating with respect to ξ, we have v = 8 3 + 1 3 e (ξ/3)F (η). Integrating with respect to η, we obtain u (ξ,η) = 8 3 η + 1 3 g (η) e ξ/3 + f (ξ), where f (ξ) and g (η) are arbitrary functions. The general solution of the given equation becomes u (x, y) = 8 3  y − 1 4  + 1 3 g 4 y − x 4 5 e 1 3 (y−x) + f (y − x). Example 4.4.3. Obtain the general solution of 3 uxx + 10 uxy + 3 uyy = 0. Since B2 − 4AC = 64 > 0, the equation is hyperbolic. Thus, from equation (4.3.2), the characteristics are y = 3x + c1, y = 1 3 x + c2. Using the transformations ξ = y − 3x, η = y − 1 3 x, the given equation can be reduced to the form  64 3  uξη = 0. 4.4 General Solutions 109 Hence, we obtain uξη = 0. Integration yields u (ξ,η) = f (ξ) + g (η). In terms of the original variables, the general solution is u (x, y) = f (y − 3x) + g 4 y − x 3 5 . Example 4.4.4. Find the general solution of the following equations y uxx + 3 y uxy + 3 ux = 0, y = 0, (4.4.1) uxx + 2 uxy + uyy = 0, (4.4.2) uxx + 2 uxy + 5 uyy + ux = 0. (4.4.3) In equation (4.4.1), A = y, B = 3y, C = 0, D = 3, E = F = G = 0. Hence B2 − 4AC = 9y 2 > 0 and the equation is hyperbolic for all points (x, y) with y = 0. Consequently, the characteristic equations are dy dx = B + √ B2 − 4AC 2A = 3y + 3y 2y = 3, 0. Integrating gives y = c1 and y = 3x + c2. The characteristic curves are ξ = y and η = y − 3x. In terms of these variables, the canonical form of (4.4.1) is ξ uξη + uη = 0. Writing v = uη and using the integrating factor gives v = uη = 1 ξ C (η), where C (η) is an arbitrary function. Integrating again with respect to η gives u (ξ,η) = 1 ξ  C (η) dη + g (ξ) = 1 ξ f (η) + g (ξ), where f and g are arbitrary functions. Finally, in terms of the original variables, the general solution is 110 4 Classification of Second-Order Linear Equations u (x, y) = 1 y f (y − 3x) + g (y). (4.4.4) Equation (4.4.2) has coefficients A = 1, B = 2, C = 1, D = E = F = G = 0. Hence, B2 − 4AC = 0, the equation is parabolic. The characteristic equation is dy dx = 1, and the characteristics are ξ = y − x = c1 and η = y. Using these variables, equation (4.4.2) takes the canonical form uηη = 0. Integrating twice gives the general solution u (ξ,η) = η f (ξ) + g (ξ), where f and g are arbitrary functions. In terms of x and y, this solution becomes u (x, y) = y f (y − x) + g (y − x). (4.4.5) The coefficients of equation (4.4.3) are A = 1, B = 2, C = 5, E = 1, F = G = 0 and hence B2 − 4AC = −16 < 0, equation (4.4.3) is elliptic. The characteristic equations are dy dx = (1 + 2i). The characteristics are y = (1 − 2i) x + c1, y = (1 + 2i) x + c2, and hence, ξ = y − (1 − 2i) x, η = y − (1 + 2i) x, and new real variables α and β are α = 1 2 (ξ + η) = y − x, η = 1 2i (ξ − η)=2x. The canonical form is given by (uαα + uββ) = 1 4 (uα − 2 uβ). (4.4.6) It is not easy to find a general solution of (4.4.6). 4.5 Summary and Further Simplification 111 Example 4.4.5. Use u = f (ξ), ξ = √x 4κt to solve the parabolic system ut = κ uxx, −∞ <x< ∞,="" t=""> 0, (4.4.7) u (x, 0) = 0, x< 0; u (x, 0) = u0, x> 0, (4.4.8) where κ and u0 are constant. We use the given transformations to obtain ut = f ′ (ξ) ξt = − 1 2 x √ 4κt3 f ′ (ξ), uxx = ∂ ∂x (ux) = ∂ ∂x (f ′ (ξ) · ξx) = 1 4κt f ′′ (ξ). Consequently, equation (4.4.7) becomes f ′′ (ξ)+2 ξf′ (ξ)=0. The solution of this equation is f ′ (ξ) = A exp  −ξ 2 , where A is a constant of integration. Integrating again gives f (ξ) = A  ξ 0 e −α 2 dα + B, where B is an integrating constant. Using the given conditions yields 0 = A  −∞ 0 e −α 2 dα + B, u0 = A  ∞ 0 e −α 2 dα + B, which give A = u0 √ π and B = 1 2 u0. Thus, the final solution is u (x, t) = u0 1 1 √ π  √x 4κt 0 e −α 2 dα + 1 2 3 . 4.5 Summary and Further Simplification We summarize the classification of linear second-order partial differential equations with constant coefficients in two independent variables. 112 4 Classification of Second-Order Linear Equations hyperbolic: urs = a1ur + a2us + a3u + f1, (4.5.1) urr − uss = a ∗ 1ur + a ∗ 2us + a ∗ 3u + f ∗ 1 , (4.5.2) parabolic: urs = b1ur + b2us + b3u + f2, (4.5.3) elliptic: urr + uss = c1ur + c2us + c3u + f3, (4.5.4) where r and s represent the new independent variables in the linear transformations r = r (x, y), s = s (x, y), (4.5.5) and the Jacobian J = 0. To simplify equation (4.5.1) further, we introduce the new dependent variable v = u e−(ar+bs) , (4.5.6) where a and b are undetermined coefficients. Finding the derivatives, we obtain ur = (vr + av) e ar+bs , us = (vs + bv) e ar+bs , urr =  vrr + 2avr + a 2 v e ar+bs , urs = (vrs + avs + bvr + abv) e ar+bs , uss =  vss + 2bvs + b 2 v e ar+bs . Substitution of these equation (4.5.1) yields vrs + (b − a1) vr + (a − a2) vs + (ab − a1a − a2b − a3) v = f1 e −(ar+bs) . In order that the first derivatives vanish, we set b = a1 and a = a2. Thus, the above equation becomes vrs = (a1a2 + a3) v + g1, where g1 = f1 e −(a2r+a1s) . In a similar manner, we can transform equations (4.5.2)–(4.5.4). Thus, we have the following transformed equations corresponding to equations (4.5.1)–(4.5.4). hyperbolic: vrs = h1v + g1, vrr − vss = h ∗ 1 v + g ∗ 1 , (4.5.7) parabolic: vss = h2v + g2, elliptic: vrr + vss = h3v + g3. In the case of partial differential equations in several independent variables or in higher order, the classification is considerably more complex. For further reading, see Courant and Hilbert (1953, 1962). 4.6 Exercises 113 4.6 Exercises 1. Determine the region in which the given equation is hyperbolic, parabolic, or elliptic, and transform the equation in the respective region to canonical form. (a) xuxx + uyy = x 2 , (b) uxx + y 2uyy = y, (c) uxx + xyuyy = 0, (d) x 2uxx − 2xyuxy + y 2uyy = e x , (e) uxx + uxy − xuyy = 0, (f) e xuxx + e yuyy = u, (g) uxx − √y uxy +  x 4 uyy + 2x ux − 3y uy + 2u = exp  x 2 − 2y , y ≥ 0, (h) uxx − √y uxy + xuyy = cos  x 2 − 2y , y ≥ 0, (i) uxx − yuxy + xux + yuy + u = 0, (j) sin2 x uxx + sin 2x uxy + cos2 x uyy = x, 2. Obtain the general solution of the following equations: (i) x 2uxx + 2xyuxy + y 2uyy + xyux + y 2uy = 0, (ii) rutt − c 2 rurr − 2c 2ur = 0, c = constant, (iii) 4ux + 12uxy + 9uyy − 9u = 9, (iv) uxx + uxy − 2uyy − 3ux − 6uy = 9 (2x − y), (v) yux + 3y uxy + 3ux = 0, y = 0. (vi) uxx + uyy = 0, (vii) 4 uxx + uyy = 0, (viii) uxx − 2 uxy + uyy = 0, (ix) 2 uxx + uyy = 0, (x) uxx + 4 uxy + 4 uyy = 0, (xi) 3 uxx + 4 uxy − 3 4 uyy = 0. 114 4 Classification of Second-Order Linear Equations 3. Find the characteristics and characteristic coordinates, and reduce the following equations to canonical form: (a) uxx + 2uxy + 3uyy + 4ux + 5uy + u = e x , (b) 2uxx − 4uxy + 2uyy + 3u = 0, (c) uxx + 5uxy + 4uyy + 7uy = sin x, (d) uxx + uyy + 2ux + 8uy + u = 0, (e) uxy + 2uyy + 9ux + uy = 2, (f) 6uxx − uxy + u = y 2 , (g) uxy + ux + uy = 3x, (h) uyy − 9ux + 7uy = cos y, (i) x 2uxx − y 2uyy − ux =1+2y 2 , (j) uxx + yuyy + 1 2 uy + 4yux = 0, (k) x 2y 2uxx + 2xyuxy + uyy = 0, (l) uxx + yuyy = 0. 4. Determine the general solutions of the following equations: (i) uxx − 1 c 2 uyy = 0, c = constant, (ii) uxx + uyy = 0, (iii) uxxxx + 2uxxyy + uyyyy = 0, (iv) uxx − 3uxy + 2uyy = 0, (v) uxx + uxy = 0, (vi) uxx + 10uxy + 9uyy = y. 5. Transform the following equations to the form vξη = cv, c = constant, (i) uxx − uyy + 3ux − 2uy + u = 0, (ii) 3uxx + 7uxy + 2uyy + uy + u = 0, by introducing the new variables v = u e−(aξ+bη) , where a and b are undetermined coefficients. 6. Given the parabolic equation uxx = aut + bux + cu + f, where the coefficients are constants, by the substitution u = v e 1 2 bx, for the case c = −  b 2/4 , show that the given equation is reduced to the heat equation vxx = avt + g, g = fe−bx/2 . 7. Reduce the Tricomi equation uxx + xuyy = 0, 4.6 Exercises 115 to the canonical form (i) uξη − [6 (ξ − η)]−1 (uξ − uη)=0, for x < 0, (ii) uαα + uββ + 1 3β = 0, x > 0. Show that the characteristic curves for x < 0 are cubic parabolas. 8. Use the polar coordinates r and θ (x = r cos θ, y = r sin θ) to transform the Laplace equation uxx + uyy = 0 into the polar form ∇2u = urr + 1 r ur + 1 r 2 uθθ = 0. 9. (a) Using the cylindrical polar coordinates x = r cos θ, y = r sin θ, z = z, transform the three-dimensional Laplace equation uxx + uyy + uzz = 0 into the form urr + 1 r ur + 1 r 2 uθθ + uzz = 0. (b) Use the spherical polar coordinates (r, θ, φ) so that x = r sin φ cos θ, y = r sin φ sin θ, z = r cos φ to transform the three-dimensional Laplace equation uxx + uyy + uzz = 0 into the form urr + 2 r ur + 1 r 2 sin φ (sin φ uφ)φ + 1 r 2 sin2 φ uθθ = 0. (c) Transform the diffusion equation ut = κ (uxx + uyy), into the axisymmetric form ut = κ  urr + 1 r ur  . 10. (a) Apply a linear transformation ξ = ax + by and η = cx + dy, to transform the Euler equation A uxx + 2B uxy + C uyy = 0 into canonical form, where a, b, c, d, A, B and C are constants . (b) Show that the same transformation as in (a) can be used to transform the nonhomogeneous Euler equation A uxx + 2B uxy + C uyy = F (x, y, u, ux, uy) into canonical form. 116 4 Classification of Second-Order Linear Equations 11. Obtain the solution of the Cauchy problem uxx + uyy = 0, u (x, 0) = f (x) and uy (x, 0) = g (x). 12. Classify each of the following equations and reduce it to canonical form: (a) y uxx − x uyy = 0, x> 0, y> 0; (b) uxx +  sech4 x uyy = 0, (c) y 2uxx + x 2uyy = 0, (d) uxx −  sech4 x uyy = 0, (e) uxx + 6uxy + 9uyy + 3y uy = 0, (f) y 2uxx + 2xy uxy + 2x 2uyy + xux = 0, (g) uxx − (2 cos x) uxy +  1 + cos2 x uyy + u = 0, (h) uxx + (2 cosec y) uxy +  cosec2y uyy = 0. (i) uxx − 2 uxy + uyy + 3 ux − u + 1 = 0, (j) uxx − y 2uyy + ux − u + x 2 = 0, (k) uxx + y uyy − x uy + y = 0. 13. Transform the equation uxy + y uyy + sin (x + y)=0 into the canonical form. Use the canonical form to find the general solution. 14. Classify each of the following equations for u (x, t): (a) ut = (p ux)x , (b) utt − c 2uxx + αu = 0, (c) (a ux)x + (a ut) t = 0, (d) uxt − a ut = 0, where p (x), c (x, t), a (x, t), and α (x) are given functions that take only positive values in the (x, t) plane. Find the general solution of the equation in (d). 5 The Cauchy Problem and Wave Equations “Since a general solution must be judged impossible from want of analysis, we must be content with the knowledge of some special cases, and that all the more, since the development of various cases seems to be the only way to bringing us at last to a more perfect knowledge.” Leonhard Euler “What would geometry be without Gauss, mathematical logic without Boole, algebra without Hamilton, analysis without Cauchy?” George Temple 5.1 The Cauchy Problem In the theory of ordinary differential equations, by the initial-value problem we mean the problem of finding the solutions of a given differential equation with the appropriate number of initial conditions prescribed at an initial point. For example, the second-order ordinary differential equation d 2u dt2 = f  t, u, du dt  and the initial conditions u (t0) = α,  du dt  (t0) = β, constitute an initial-value problem. An analogous problem can be defined in the case of partial differential equations. Here we shall state the problem involving second-order partial differential equations in two independent variables. 118 5 The Cauchy Problem and Wave Equations We consider a second-order partial differential equation for the function u in the independent variables x and y, and suppose that this equation can be solved explicitly for uyy, and hence, can be represented in the from uyy = F (x, y, u, ux, uy, uxx, uxy). (5.1.1) For some value y = y0, we prescribe the initial values of the unknown function and of the derivative with respect to y u (x, y0) = f (x), uy (x, y0) = g (x). (5.1.2) The problem of determining the solution of equation (5.1.1) satisfying the initial conditions (5.1.2) is known as the initial-value problem. For instance, the initial-value problem of a vibrating string is the problem of finding the solution of the wave equation utt = c 2uxx, satisfying the initial conditions u (x, t0) = u0 (x), ut (x, t0) = v0 (x), where u0 (x) is the initial displacement and v0 (x) is the initial velocity. In initial-value problems, the initial values usually refer to the data assigned at y = y0. It is not essential that these values be given along the line y = y0; they may very well be prescribed along some curve L0 in the xy plane. In such a context, the problem is called the Cauchy problem instead of the initial-value problem, although the two names are actually synonymous. We consider the Euler equation Auxx + Buxy + Cuyy = F (x, y, u, ux, uy), (5.1.3) where A, B, C are functions of x and y. Let (x0, y0) denote points on a smooth curve L0 in the xy plane. Also let the parametric equations of this curve L0 be x0 = x0 (λ), y0 = y0 (λ), (5.1.4) where λ is a parameter. We suppose that two functions f (λ) and g (λ) are prescribed along the curve L0. The Cauchy problem is now one of determining the solution u (x, y) of equation (5.1.3) in the neighborhood of the curve L0 satisfying the Cauchy conditions u = f (λ), (5.1.5a) ∂u ∂n = g (λ), (5.1.5b) 5.1 The Cauchy Problem 119 on the curve L0 where n is the direction of the normal to L0 which lies to the left of L0 in the counterclockwise direction of increasing arc length. The function f (λ) and g (λ) are called the Cauchy data. For every point on L0, the value of u is specified by equation (5.1.5a). Thus, the curve L0 represented by equation (5.1.4) with the condition (5.1.5a) yields a twisted curve L in (x, y, u) space whose projection on the xy plane is the curve L0. Thus, the solution of the Cauchy problem is a surface, called an integral surface, in the (x, y, u) space passing through L and satisfying the condition (5.1.5b), which represents a tangent plane to the integral surface along L. If the function f (λ) is differentiable, then along the curve L0, we have du dλ = ∂u ∂x dx dλ + ∂u ∂y dy dλ = df dλ, (5.1.6) and ∂u ∂n = ∂u ∂x dx dn + ∂u ∂y dy dn = g, (5.1.7) but dx dn = − dy ds and dy dn = dx ds . (5.1.8) Equation (5.1.7) may be written as ∂u ∂n = − ∂u ∂x dy ds + ∂u ∂y dx ds = g. (5.1.9) Since       dx dλ dy dλ −dy ds dx ds       = (dx) 2 + (dy) 2 ds dλ = 0, (5.1.10) it is possible to find ux and uy on L0 from the system of equations (5.1.6) and (5.1.9). Since ux and uy are known on L0, we find the higher derivatives by first differentiating ux and uy with respect to λ. Thus, we have ∂ 2u ∂x2 dx dλ + ∂ 2u ∂x ∂y dy dλ = d dλ  ∂u ∂x , (5.1.11) ∂ 2u ∂x ∂y dx dλ + ∂ 2u ∂y2 dy dλ = d dλ  ∂u ∂y  . (5.1.12) From equation (5.1.3), we have A ∂ 2u ∂x2 + B ∂ 2u ∂x ∂y + C ∂ 2u ∂y2 = F, (5.1.13) 120 5 The Cauchy Problem and Wave Equations where F is known since ux and uy have been found. The system of equations can be solved for uxx, uxy, and uyy, if           dx dλ dy dλ 0 0 dx dλ dy dλ ABC           = C  dx dλ2 − B  dx dλ dy dλ + A  dy dλ2 = 0. (5.1.14) The equation A  dy dx2 − B  dy dx + C = 0, (5.1.15) is called the characteristic equation. It is then evident that the necessary condition for obtaining the second derivatives is that the curve L0 must not be a characteristic curve. If the coefficients of equation (5.1.3) and the function (5.1.5) are analytic, then all the derivatives of higher orders can be computed by the above process. The solution can then be represented in the form of a Taylor series: u (x, y) = ∞ n=0 ∞ k=0 1 k! (n − k)! ∂ nu0 ∂xk 0 ∂yn−k 0 (x − x0) k (y − y0) n−k , (5.1.16) which can be shown to converge in the neighborhood of the curve L0. Thus, we may state the famous Cauchy–Kowalewskaya theorem. 5.2 The Cauchy–Kowalewskaya Theorem Let the partial differential equation be given in the form uyy = F (y, x1, x2,...,xn, u, uy, ux1 , ux2 ...,uxn , ux1y, ux2y,...,uxny, ux1x1 , ux2x2 ,...,uxnxn ),(5.2.1) and let the initial conditions u = f (x1, x2,...,xn), (5.2.2) uy = g (x1, x2,...,xn), (5.2.3) be given on the noncharacteristic manifold y = y0. If the function F is analytic in some neighborhood of the point  y 0 , x0 1 , x0 2 ,...,x0 n , u0 , u0 y ,... and if the functions f and g are analytic in some neighborhood of the point  x 0 1 , x0 2 ,...,x0 n , then the Cauchy problem has a unique analytic solution in some neighborhood of the point  y 0 , x0 1 , x0 2 ,...,x0 n . 5.3 Homogeneous Wave Equations 121 For the proof, see Petrovsky (1954). The preceding statement seems equally applicable to hyperbolic, parabolic, or elliptic equations. However, we shall see that difficulties arise in formulating the Cauchy problem for nonhyperbolic equations. Consider, for instance, the famous Hadamard (1952) example. The problem consists of the elliptic (or Laplace) equation uxx + uyy = 0, and the initial conditions on y = 0 u (x, 0) = 0, uy (x, 0) = n −1 sin nx. The solution of this problem is u (x, y) = n −2 sinh ny sin nx, which can be easily verified. It can be seen that, when n tends to infinity, the function n −1 sin nx tends uniformly to zero. But the solution n −2 sinh ny sin nx does not become small, as n increases for any nonzero y. Physically, the solution represents an oscillation with unbounded amplitude  n −2 sinh ny as y → ∞ for any fixed x. Even if n is a fixed number, this solution is unstable in the sense that u → ∞ as y → ∞ for any fixed x for which sin nx = 0. It is obvious then that the solution does not depend continuously on the data. Thus, it is not a properly posed problem. In addition to existence and uniqueness, the question of continuous dependence of the solution on the initial data arises in connection with the Cauchy–Kowalewskaya theorem. It is well known that any continuous function can accurately be approximated by polynomials. We can apply the Cauchy–Kowalewskaya theorem with continuous data by using polynomial approximations only if a small variation in the initial data leads to a small change in the solution. 5.3 Homogeneous Wave Equations To study Cauchy problems for hyperbolic partial differential equations, it is quite natural to begin investigating the simplest and yet most important equation, the one-dimensional wave equation, by the method of characteristics. The essential characteristic of the solution of the general wave equation is preserved in this simplified case. We shall consider the following Cauchy problem of an infinite string with the initial condition 122 5 The Cauchy Problem and Wave Equations utt − c 2uxx = 0, x ∈ R, t> 0, (5.3.1) u (x, 0) = f (x), x ∈ R, (5.3.2) ut (x, 0) = g (x), x ∈ R. (5.3.3) By the method of characteristics described in Chapter 4, the characteristic equation according to equation (4.2.4) is dx2 − c 2 dt2 = 0, which reduces to dx + c dt = 0, dx − c dt = 0. The integrals are the straight lines x + ct = c1, x − ct = c2. Introducing the characteristic coordinates ξ = x + ct, η = x − ct, we obtain uxx = uξξ + 2 uξη + uηη, utt = c 2 (uξξ − 2 uξη + uηη). Substitution of these in equation (5.3.1) yields −4c 2uξη = 0. Since c = 0, we have uξη = 0. Integrating with respect to ξ, we obtain uη = ψ ∗ (η), where ψ ∗ (η) is an arbitrary function of η. Integrating again with respect to η, we obtain u (ξ,η) =  ψ ∗ (η) dη + φ (ξ). If we set ψ (η) = * ψ ∗ (η) dη, we have u (ξ,η) = φ (ξ) + ψ (η), where φ and ψ are arbitrary functions. Transforming to the original variables x and t, we find the general solution of the wave equation 5.3 Homogeneous Wave Equations 123 u (x, t) = φ (x + ct) + ψ (x − ct), (5.3.4) provided φ and ψ are twice differentiable functions. Now applying the initial conditions (5.3.2) and (5.3.3), we obtain u (x, 0) = f (x) = φ (x) + ψ (x), (5.3.5) ut (x, 0) = g (x) = c φ′ (x) − c ψ′ (x). (5.3.6) Integration of equation (5.3.6) gives φ (x) − ψ (x) = 1 c  x x0 g (τ ) dτ + K, (5.3.7) where x0 and K are arbitrary constants. Solving for φ and ψ from equations (5.3.5) and (5.3.7), we obtain φ (x) = 1 2 f (x) + 1 2c  x x0 g (τ ) dτ + K 2 , ψ (x) = 1 2 f (x) − 1 2c  x x0 g (τ ) dτ − K 2 . The solution is thus given by u (x, t) = 1 2 [f (x + ct) + f (x − ct)] + 1 2c  x+ct x0 g (τ ) dτ −  x−ct x0 g (τ ) dτ = 1 2 [f (x + ct) + f (x − ct)] + 1 2c  x+ct x−ct g (τ ) dτ. (5.3.8) This is called the celebrated d’Alembert solution of the Cauchy problem for the one-dimensional wave equation. It is easy to verify by direct substitution that u (x, t), represented by (5.3.8), is the unique solution of the wave equation (5.3.1) provided f (x) is twice continuously differentiable and g (x) is continuously differentiable. This essentially proves the existence of the d’Alembert solution. By direct substitution, it can also be shown that the solution (5.3.8) is uniquely determined by the initial conditions (5.3.2) and (5.3.3). It is important to note that the solution u (x, t) depends only on the initial values of f at points x − ct and x + ct and values of g between these two points. In other words, the solution does not depend at all on initial values outside this interval, x − ct ≤ x ≤ x + ct. This interval is called the domain of dependence of the variables (x, t). Moreover, the solution depends continuously on the initial data, that is, the problem is well posed. In other words, a small change in either f or g results in a correspondingly small change in the solution u (x, t). Mathematically, this can be stated as follows: For every ε > 0 and for each time interval 0 ≤ t ≤ t0, there exists a number δ (ε, t0) such that 124 5 The Cauchy Problem and Wave Equations |u (x, t) − u ∗ (x, t)| < ε, whenever |f (x) − f ∗ (x)| < δ, |g (x) − g ∗ (x)| < δ. The proof follows immediately from equation (5.3.8). We have |u (x, t) − u ∗ (x, t)| ≤ 1 2 |f (x + ct) − f ∗ (x + ct)| + 1 2 |f (x − ct) − f ∗ (x − ct)| + 1 2c  x+ct x−ct |g (τ ) − g ∗ (τ )| dτ < ε, where ε = δ (1 + t0). For any finite time interval 0 <t<t0, a="" small="" change="" in="" the="" initial="" data="" only="" produces="" solution.="" this="" shows="" that="" problem="" is="" well="" posed.="" example="" 5.3.1.="" find="" solution="" of="" initial-value="" utt="c" 2uxx,="" x="" ∈="" r,="" t=""> 0, u (x, 0) = sin x, ut (x, 0) = cos x. From (5.3.8), we have u (x, t) = 1 2 [sin (x + ct) + sin (x − ct)] + 1 2c  x+ct x−ct cos τ dτ = sin x cos ct + 1 2c [sin (x + ct) − sin (x − ct)] = sin x cos ct + 1 c cos x sin ct. It follows from the d’Alembert solution that, if an initial displacement or an initial velocity is located in a small neighborhood of some point (x0, t0), it can influence only the area t>t0 bounded by two characteristics x−ct = constant and x+ct = constant with slope ± (1/c) passing through the point (x0, t0), as shown in Figure 5.3.1. This means that the initial displacement propagates with the speed dx dt = c, whereas the effect of the initial velocity propagates at all speeds up to c. This infinite sector R in this figure is called the range of influence of the point (x0, t0). According to (5.3.8), the value of u (x0, t0) depends on the initial data f and g in the interval [x0 − ct0, x0 + ct0] which is cut out of the initial line by the two characteristics x−ct = constant and x+ct = constant with slope ± (1/c) passing through the point (x0, t0). The interval [x0 − ct0, x0 + ct0] 5.3 Homogeneous Wave Equations 125 Figure 5.3.1 Range of influence on the line t = 0 is called the domain of dependence of the solution at the point (x0, t0), as shown in Figure 5.3.2. Figure 5.3.2 Domain of dependence 126 5 The Cauchy Problem and Wave Equations Since the solution u (x, t) at every point (x, t) inside the triangular region D in this figure is completely determined by the Cauchy data on the interval [x0 − ct0, x0 + ct0], the region D is called the region of determinancy of the solution. We will now investigate the physical significance of the d’Alembert solution (5.3.8) in greater detail. We rewrite the solution in the form u (x, t) = 1 2 f (x + ct) + 1 2c  x+ct 0 g (τ ) dτ + 1 2 f (x − ct) − 1 2c  x−ct 0 g (τ ) dτ. (5.3.9) Or, equivalently, u (x, t) = φ (x + ct) + ψ (x − ct), (5.3.10) where φ (ξ) = 1 2 f (ξ) + 1 2c  ξ 0 g (τ ) dτ, (5.3.11) ψ (η) = 1 2 f (η) − 1 2c  η 0 g (τ ) dτ. (5.3.12) Evidently, φ (x + ct) represents a progressive wave traveling in the negative x-direction with speed c without change of shape. Similarly, ψ (x − ct) is also a progressive wave propagating in the positive x-direction with the same speed c without change of shape. We shall examine this point in greater detail. Treat ψ (x − ct) as a function of x for a sequence of times t. At t = 0, the shape of this function of u = ψ (x). At a subsequent time, its shape is given by u = ψ (x − ct) or u = ψ (ξ), where ξ = x − ct is the new coordinate obtained by translating the origin a distance ct to the right. Thus, the shape of the curve remains the same as time progresses, but moves to the right with velocity c as shown in Figure 5.3.3. This shows that ψ (x − ct) represents a progressive wave traveling in the positive xdirection with velocity c without change of shape. Similarly, φ (x + ct) is also a progressive wave propagating in the negative x-direction with the same speed c without change of shape. For instance, u (x, t) = sin (x + ct) (5.3.13) represent sinusoidal waves traveling with speed c in the positive and negative directions respectively without change of shape. The propagation of waves without change of shape is common to all linear wave equations. To interpret the d’Alembert formula we consider two cases: Case 1. We first consider the case when the initial velocity is zero, that is, g (x)=0. 5.3 Homogeneous Wave Equations 127 Figure 5.3.3 Progressive Waves. Then, the d’Alembert solution has the form u (x, t) = 1 2 [f (x + ct) + f (x − ct)] . Now suppose that the initial displacement f (x) is different from zero in an interval (−b, b). Then, in this case the forward and the backward waves are represented by u = 1 2 f (x). The waves are initially superimposed, and then they separate and travel in opposite directions. We consider f (x) which has the form of a triangle. We draw a triangle with the ordinate x = 0 one-half that of the given function at that point, as shown in Figure 5.3.4. If we displace these graphs and then take the sum of the ordinates of the displaced graphs, we obtain the shape of the string at any time t. As can be seen from the figure, the waves travel in opposite directions away from each other. After both waves have passed the region of initial disturbance, the string returns to its rest position. Case 2. We consider the case when the initial displacement is zero, that is, f (x)=0, 128 5 The Cauchy Problem and Wave Equations Figure 5.3.4 Triangular Waves. and the d’Alembert solution assumes the form u (x, t) = 1 2  x+ct x−ct g (τ ) dτ = 1 2 [G (x + ct) − G (x − ct)] , where G (x) = 1 c  x x0 g (τ ) dτ. If we take for the initial velocity g (x) = ⎧ ⎨ ⎩ 0 |x| > b g0 |x| ≤ b, then, the function G (x) is equal to zero for values of x in the interval x ≤ −b, and G (x) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 1 c  x −b g0 dτ = g0 c (x + b) for −b ≤ x ≤ b, 1 c  x −b g0 dτ = 2bg0 c for x > b. 5.3 Homogeneous Wave Equations 129 Figure 5.3.5 Graph of u (x, t) at time t. As in the previous case, the two waves which differ in sign travel in opposite directions on the x-axis. After some time t the two functions (1/2) G (x) and − (1/2) G (x) move a distance ct. Thus, the graph of u at time t is obtained by summing the ordinates of the displaced graphs as shown in Figure 5.3.5. As t approaches infinity, the string will reach a state of rest, but it will not, in general, assume its original position. This displacement is known as the residual displacement. In the preceding examples, we note that f (x) is continuous, but not continuously differentiable and g (x) is discontinuous. To these initial data, there corresponds a generalized solution. By a generalized solution we mean the following: Let us suppose that the function u (x, t) satisfies the initial conditions (5.3.2) and (5.3.3). Let u (x, t) be the limit of a uniformly convergent sequence of solutions un (x, t) which satisfy the wave equation (5.3.1) and the initial conditions un (x, 0) = fn (x),  ∂un ∂t  (x, 0) = gn (x). Let fn (x) be a continuously differentiable function, and let the sequence converge uniformly to f (x); let gn (x) be a continuously differentiable function, and * x x0 gn (τ ) dτ approach uniformly to * x x0 g (τ ) dτ . Then, the function u (x, t) is called the generalized solution of the problem (5.3.1)–(5.3.3). In general, it is interesting to discuss the effect of discontinuity of the function f (x) at a point x = x0, assuming that g (x) is a smooth function. Clearly, it follows from (5.3.8) that u (x, t) will be discontinuous at each 130 5 The Cauchy Problem and Wave Equations point (x, t) such that x+ct = x0 or x−ct = x0, that is, at each point of the two characteristic lines intersecting at the point (x0, 0). This means that discontinuities are propagated along the characteristic lines. At each point of the characteristic lines, the partial derivatives of the function u (x, t) fail to exist, and hence, u can no longer be a solution of the Cauchy problem in the usual sense. However, such a function may be called a generalized solution of the Cauchy problem. Similarly, if f (x) is continuous, but either f ′ (x) or f ′′ (x) has a discontinuity at some point x = x0, the first- or second-order partial derivatives of the solution u (x, t) will be discontinuous along the characteristic lines through (x0, 0). Finally, a discontinuity in g (x) at x = x0 would lead to a discontinuity in the first- or second-order partial derivatives of u along the characteristic lines through (x0, 0), and a discontinuity in g ′ (x) at x0 will imply a discontinuity in the second-order partial derivatives of u along the characteristic lines through (x0, 0). The solution given by (5.3.8) with f, f ′ , f ′′ , g, and g ′ piecewise continuous on −∞ <x< ∞="" is="" usually="" called="" the="" generalized="" solution="" of="" cauchy="" problem.="" 5.4="" initial="" boundary-value="" problems="" we="" have="" just="" determined="" initial-value="" problem="" for="" infinite="" vibrating="" string.="" will="" now="" study="" effect="" a="" boundary="" on="" solution.="" (a)="" semi-infinite="" string="" with="" fixed="" end="" let="" us="" first="" consider="" end,="" that="" is,="" utt="c" 2uxx,="" 0="" <x<="" ∞,="" t=""> 0, u (x, 0) = f (x), 0 ≤ x < ∞, (5.4.1) ut (x, 0) = g (x), 0 ≤ x < ∞, u (0, t)=0, 0 ≤ t < ∞. It is evident here that the boundary condition at x = 0 produces a wave moving to the right with the velocity c. Thus, for x > ct, the solution is the same as that of the infinite string, and the displacement is influenced only by the initial data on the interval [x − ct, x + ct], as shown in Figure 5.4.1. When x < ct, the interval [x − ct, x + ct] extends onto the negative x-axis where f and g are not prescribed. But from the d’Alembert formula u (x, t) = φ (x + ct) + ψ (x − ct), (5.4.2) where 5.4 Initial Boundary-Value Problems 131 Figure 5.4.1 Displacement influenced by the initial data on [x − ct, x + ct]. φ (ξ) = 1 2 f (ξ) + 1 2c  ξ 0 g (τ ) dτ + K 2 , (5.4.3) ψ (η) = 1 2 f (η) − 1 2c  η 0 g (τ ) dτ − K 2 , (5.4.4) we see that u (0, t) = φ (ct) + ψ (−ct)=0. Hence, ψ (−ct) = −φ (ct). If we let α = −ct, then ψ (α) = −φ (−α). Replacing α by x − ct, we obtain for x < ct, ψ (x − ct) = −φ (ct − x), and hence, ψ (x − ct) = − 1 2 f (ct − x) − 1 2c  ct−x 0 g (τ ) dτ − K 2 . The solution of the initial boundary-value problem, therefore, is given by 132 5 The Cauchy Problem and Wave Equations u (x, t) = 1 2 [f (x + ct) + f (x − ct)] + 1 2c  x+ct x−ct g (τ ) dτ for x > ct, (5.4.5) u (x, t) = 1 2 [f (x + ct) − f (ct − x)] + 1 2c  x+ct ct−x g (τ ) dτ for x < ct. (5.4.6) In order for this solution to exist, f must be twice continuously differentiable and g must be continuously differentiable, and in addition f (0) = f ′′ (0) = g (0) = 0. Solution (5.4.6) has an interesting physical interpretation. If we draw the characteristics through the point (x0, t0) in the region x > ct, we see, as pointed out earlier, that the displacement at (x0, t0) is determined by the initial values on [x0 − ct0, x0 + ct0]. If the point (x0, t0) lies in the region x > ct as shown in Figure 5.4.1, we see that the characteristic x + ct = x0 + ct0 intersects the x-axis at (x0 + ct0, 0). However, the characteristic x − ct = x0 − ct0 intersects the t-axis at (0, t0 − x0/c), and the characteristic x + ct = ct0 − x0 intersects the x-axis at (ct0 − x0, 0). Thus, the disturbance at (ct0 − x0, 0) travels along the backward characteristic x + ct = ct0 − x0, and is reflected at (0, t0 − x0/c) as a forward moving wave represented by −φ (ct0 − x0). Example 5.4.1. Determine the solution of the initial boundary-value problem utt = 4 uxx, x > 0, t > 0, u (x, 0) = |sin x| , x > 0, ut (x, 0) = 0, x ≥ 0, u (x, 0) = 0, t ≥ 0. For x > 2t, u (x, t) = 1 2 [f (x + 2t) + f (x − 2t)] = 1 2 [|sin (x + 2t)|−|sin (x − 2t)|] , and for x < 2t, u (x, t) = 1 2 [f (x + 2t) − f (2t − x)] = 1 2 [|sin (x + 2t)|−|sin (2t − x)|] . Notice that u (0, t) = 0 is satisfied by u (x, t) for x < 2t (that is, t > 0). 5.4 Initial Boundary-Value Problems 133 (B) Semi-infinite String with a Free End We consider a semi-infinite string with a free end at x = 0. We will determine the solution of utt = c 2uxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = f (x), 0 ≤ x < ∞, (5.4.7) ut (x, 0) = g (x), 0 ≤ x < ∞, ux (0, t)=0, 0 ≤ t < ∞. As in the case of the fixed end, for x > ct the solution is the same as that of the infinite string. For x < ct, from the d’Alembert solution (5.4.2) u (x, t) = φ (x + ct) + ψ (x − ct), we have ux (x, t) = φ ′ (x + ct) + ψ ′ (x − ct). Thus, ux (0, t) = φ ′ (ct) + ψ ′ (−ct)=0. Integration yields φ (ct) − ψ (−ct) = K, where K is a constant. Now, if we let α = −ct, we obtain ψ (α) = φ (−α) − K. Replacing α by x − ct, we have ψ (x − ct) = φ (ct − x) − K, and hence, ψ (x − ct) = 1 2 f (ct − x) + 1 2c  ct−x 0 g (τ ) dτ − K 2 . The solution of the initial boundary-value problem, therefore, is given by u (x, t) = 1 2 [f (x + ct) + f (x − ct)] + 1 2c  x+ct x−ct g (τ ) dτ for x > ct. (5.4.8) u (x, t) = 1 2 [f (x + ct) + f (ct − x)] + 1 2c  x+ct 0 g (τ ) dτ +  ct−x 0 g (τ ) dτ for x < ct. (5.4.9) We note that for this solution to exist, f must be twice continuously differentiable and g must be continuously differentiable, and in addition, f ′ (0) = g ′ (0) = 0. 134 5 The Cauchy Problem and Wave Equations Example 5.4.2. Find the solution of the initial boundary-value problem utt = uxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = cos 4πx 2 5 , 0 ≤ x < ∞, ut (x, 0) = 0, 0 ≤ x < ∞, ux (x, 0) = 0, t ≥ 0. For x>t u (x, t) = 1 2 " cos π 2 (x + t) + cos π 2 (x − t) # = cos 4π 2 x 5 cos 4π 2 t 5 , and for x 0, t > 0, u (x, 0) = f (x), x ≥ 0, (5.5.1) ut (x, 0) = g (x), x ≥ 0, u (0, t) = p (t), t ≥ 0, we proceed in a manner similar to the case of homogeneous boundary conditions. Using equation (5.4.2), we apply the boundary condition to obtain u (0, t) = φ (ct) + ψ (−ct) = p (t). If we let α = −ct, we have ψ (α) = p 4 − α c 5 − φ (−α). Replacing α by x − ct, the preceding relation becomes ψ (x − ct) = p 4 t − x c 5 − φ (ct − x). 5.5 Equations with Nonhomogeneous Boundary Conditions 135 Thus, for 0 ≤ x < ct, u (x, t) = p 4 t − x c 5 + 1 2 [f (x + ct) − f (ct − x)] + 1 2c  x+ct ct−x g (τ ) dτ = p 4 t − x c 5 + φ (x + ct) − ψ (ct − x), (5.5.2) where φ (x + ct = ξ) is given by (5.3.11), and ψ (η) is given by ψ (η) = 1 2 f (η) + 1 2c  η 0 g (τ ) dτ. (5.5.3) The solution for x > ct is given by the solution (5.4.5) of the infinite string. In this case, in addition to the differentiability conditions satisfied by f and g, as in the case of the problem with the homogeneous boundary conditions, p must be twice continuously differentiable in t and p (0) = f (0), p′ (0) = g (0), p′′ (0) = c 2 f ′′ (0). We next consider the initial boundary-value problem utt = c 2uxx, x > 0, t > 0, u (x, 0) = f (x), x ≥ 0, ut (x, 0) = g (x), x ≥ 0, ux (0, t) = q (t), t ≥ 0. Using (5.4.2), we apply the boundary condition to obtain ux (0, t) = φ ′ (ct) + ψ ′ (−ct) = q (t). Then, integrating yields φ (ct) − ψ (−ct) = c  t 0 q (τ ) dτ + K. If we let α = −ct, then ψ (α) = φ (−α) − c  −α/c 0 q (τ ) dτ − K. Replacing α by x − ct, we obtain ψ (x − ct) = φ (ct − x) − c  t−x/c 0 q (τ ) dτ − K. The solution of the initial boundary-value problem for x < ct, therefore, is given by 136 5 The Cauchy Problem and Wave Equations u (x, t) = 1 2 [f (x + ct) + f (ct − x)] + 1 2c  x+ct 0 g (τ ) dτ +  ct−x 0 g (τ ) dτ −c  t−x/c 0 q (τ ) dτ. (5.5.4) Here f and g must satisfy the differentiability conditions, as in the case of the problem with the homogeneous boundary conditions. In addition f ′ (0) = q (0), g′ (0) = q ′ (0). The solution for the initial boundary-value problem involving the boundary condition ux (0, t) + h u (0, t)=0, h = constant can also be constructed in a similar manner from the d’Alembert solution. 5.6 Vibration of Finite String with Fixed Ends The problem of the finite string is more complicated than that of the infinite string due to the repeated reflection of waves from the boundaries We first consider the vibration of the string of length l fixed at both ends. The problem is that of finding the solution of utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, (5.6.1) u (0, t)=0, u (l, t)=0, t ≥ 0, From the previous results, we know that the solution of the wave equation is u (x, t) = φ (x + ct) + ψ (x − ct). Applying the initial conditions, we have u (x, 0) = φ (x) + ψ (x) = f (x), 0 ≤ x ≤ l, ut (x, 0) = c φ′ (x) − c ψ′ (x) = g (x), 0 ≤ x ≤ l. Solving for φ and ψ, we find φ (ξ) = 1 2 f (ξ) + 1 2c  ξ 0 g (τ ) dτ + K 2 , 0 ≤ ξ ≤ l, (5.6.2) ψ (η) = 1 2 f (η) − 1 2c  η 0 g (τ ) dτ − K 2 , 0 ≤ η ≤ l. (5.6.3) 5.6 Vibration of Finite String with Fixed Ends 137 Hence, u (x, t) = 1 2 [f (x + ct) + f (x − ct)] + 1 2c  x+ct x−ct g (τ ) dτ, (5.6.4) for 0 ≤ x + ct ≤ l and 0 ≤ x − ct ≤ l. The solution is thus uniquely determined by the initial data in the region t ≤ x c , t ≤ l − x c , t ≥ 0. For larger times, the solution depends on the boundary conditions. Applying the boundary conditions, we obtain u (0, t) = φ (ct) + ψ (−ct)=0, t ≥ 0, (5.6.5) u (l, t) = φ (l + ct) + ψ (l − ct)=0, t ≥ 0. (5.6.6) If we set α = −ct, equation (5.6.5) becomes ψ (α) = −φ (−α), α ≤ 0, (5.6.7) and if we set α = l + ct, equation (5.6.6) takes the form φ (α) = −ψ (2l − α), α ≥ l. (5.6.8) With ξ = −η, we may write equation (5.6.2) as φ (−η) = 1 2 f (−η) + 1 2c  −η 0 g (τ ) dτ + K 2 , 0 ≤ −η ≤ l. (5.6.9) Thus, from (5.6.7) and (5.6.9), we have ψ (η) = − 1 2 f (−η) − 1 2c  −η 0 g (τ ) dτ − K 2 , −l ≤ η ≤ 0. (5.6.10) We see that the range of ψ (η) is extended to −l ≤ η ≤ l. If we put α = ξ in equation (5.6.8), we obtain φ (ξ) = −ψ (2l − ξ), ξ ≥ l. (5.6.11) Then, by putting η = 2l − ξ in equation (5.6.3), we obtain ψ (2l − ξ) = 1 2 f (2l − ξ) − 1 2c  2l−ξ 0 g (τ ) dτ − K 2 , 0 ≤ 2l − ξ ≤ l. (5.6.12) Substitution of this in equation (5.6.11) yields 138 5 The Cauchy Problem and Wave Equations φ (ξ) = − 1 2 f (2l − ξ) + 1 2c  2l−ξ 0 g (τ ) dτ + K 2 , l ≤ ξ ≤ 2l. (5.6.13) The range of φ (ξ) is thus extended to 0 ≤ ξ ≤ 2l. Continuing in this manner, we obtain φ (ξ) for all ξ ≥ 0 and ψ (η) for all η ≤ l. Hence, the solution is determined for all 0 ≤ x ≤ l and t ≥ 0. In order to observe the effect of the boundaries on the propagation of waves, the characteristics are drawn through the end point until they meet the boundaries and then continue inward as shown in Figure 5.6.1. It can be seen from the figure that only direct waves propagate in region 1. In regions 2 and 3, both direct and reflected waves propagate. In regions, 4,5,6, ... , several waves propagate along the characteristics reflected from both of the boundaries x = 0 and x = l. Example 5.6.1. Determine the solution of the following problem utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = sin (πx/l), 0 ≤ x ≤ l, ut (x, 0) = 0, 0 ≤ x ≤ l, u (0, t)=0, u (l, t)=0, t ≥ 0. From equations (5.6.2) and (5.6.3), we have Figure 5.6.1 Regions of wave propagation. 5.7 Nonhomogeneous Wave Equations 139 φ (ξ) = 1 2 sin  πξ l  + K 2 , 0 ≤ ξ ≤ l. ψ (η) = 1 2 sin 4πη l 5 − K 2 , 0 ≤ η ≤ l. Using equation (5.6.10), we obtain ψ (η) = − 1 2 sin 4 − πη l 5 − K 2 , −l ≤ η ≤ 0 = 1 2 sin 4πη l 5 − K 2 . From equation (5.6.13), we find φ (ξ) = − 1 2 sin (π l (2l − ξ) ) + K 2 , l ≤ ξ ≤ 2l. Again by equation (5.6.7) and from the preceding φ (ξ), we have φ (η) = 1 2 sin 4πη l 5 − K 2 , −2l ≤ η ≤ −l. Proceeding in this manner, we determine the solution u (x, t) = φ (ξ) + ψ (η) = 1 2 " sin π l (x + ct) + sin π l (x − ct) # for all x in (0, l) and for all t > 0. Similarly, the solution of the finite initial boundary-value problem utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, u (0, t) = p (t), u (l, t) = q (t), t ≥ 0, can be determined by the same method. 5.7 Nonhomogeneous Wave Equations We shall consider next the Cauchy problem for the nonhomogeneous wave equation utt = c 2uxx + h ∗ (x, t), (5.7.1) with the initial conditions u (x, 0) = f (x), ut (x, 0) = g ∗ (x). (5.7.2) 140 5 The Cauchy Problem and Wave Equations By the coordinate transformation y = ct, (5.7.3) the problem is reduced to uxx − uyy = h (x, y), (5.7.4) u (x, 0) = f (x), (5.7.5) uy (x, 0) = g (x), (5.7.6) where h (x, y) = −h ∗/c2 and g (x) = g ∗/c. Let P0 (x0, y0) be a point of the plane, and let Q0 be the point (x0, 0) on the initial line y = 0. Then the characteristics, x + y = constant, of equation (5.7.4) are two straight lines drawn through the point P0 with slopes + 1. Obviously, they intersect the x-axis at the points P1 (x0 − y0, 0) and P2 (x0 + y0, 0), as shown in Figure 5.7.1. Let the sides of the triangle P0P1P2 be designated by B0, B1, and B2, and let D be the region representing the interior of the triangle and its boundaries B. Integrating both sides of equation (5.7.4), we obtain  R (uxx − uyy) dR =  R h (x, y) dR. (5.7.7) Now we apply Green’s theorem to obtain Figure 5.7.1 Triangular Region. 5.7 Nonhomogeneous Wave Equations 141  R (uxx − uyy) dR =  B (uxdy + uydx). (5.7.8) Since B is composed of B0, B1, and B2, we note that  B0 (ux dy + uy dx) =  x0+y0 x0−y0 uy dx,  B1 (ux dy + uy dx) =  B1 (−ux dx − uy dy), = u (x0 + y0, 0) − u (x0, y0),  B2 (ux dy + uy dx) =  B2 (ux dx + uy dy), = u (x0 − y0, 0) − u (x0, y0). Hence,  B (ux dy + uy dx) = −2 u (x0, y0) + u (x0 − y0, 0) +u (x0 + y0, 0) +  x0+y0 x0−y0 uy dx. (5.7.9) Combining equations (5.7.7), (5.7.8) and (5.7.9), we obtain u (x0, y0) = 1 2 [u (x0 + y0, 0) + u (x0 − y0, 0)] + 1 2  x0+y0 x0−y0 uy dx − 1 2  R h (x, y) dR. (5.7.10) We have chosen x0, y0 arbitrarily, and as a consequence, we replace x0 by x and y0 by y. Equation (5.7.10) thus becomes u (x, y) = 1 2 [f (x + y) + f (x − y)] + 1 2  x+y x−y g (τ ) dτ − 1 2  R h (x, y) dR. In terms of the original variables u (x, t) = 1 2 [f (x + ct) + f (x − ct)] + 1 2c  x+ct x−ct g ∗ (τ ) dτ − 1 2  R h (x, t) dR. (5.7.11) Example 5.7.1. Determine the solution of uxx − uyy = 1, u (x, 0) = sin x, uy (x, 0) = x. 142 5 The Cauchy Problem and Wave Equations Figure 5.7.2 Triangular Region. It is easy to see that the characteristics are x + y = constant = x0 + y0 and x − y = constant = x0 − y0, as shown in Figure 5.7.2. Thus, u (x0, y0) = 1 2 [sin (x0 + y0) + sin (x0 − y0)] + 1 2  x0+y0 x0−y0 τ dτ − 1 2  y0 0  −y+x0+y0 y+x0−y0 dx dy = 1 2 [sin (x0 + y0) + sin (x0 − y0)] + x0y0 − 1 2 y 2 0 . Now dropping the subscript zero, we obtain the solution u (x, y) = 1 2 [sin (x + y) + sin (x − y)] + xy − 1 2 y 2 . 5.8 The Riemann Method We shall discuss Riemann’s method of integrating the linear hyperbolic equation L[u] ≡ uxy + aux + buy + cu = f (x, y), (5.8.1) 5.8 The Riemann Method 143 where L denotes the linear operator, and a (x, y), b (x, y), c (x, y), and f (x, y) are differentiable functions in some domain D∗ . The method consists essentially of the derivation of an integral formula which represents the solution of the Cauchy problem. Let v (x, y) be a function having continuous second-order partial derivatives. Then, we may write vuxy − uvxy = (vux)y − (vuy)x , vaux = (avu)x − u (av)x , (5.8.2) vbuy = (bvu)y − u (bv)y , so that vL[u] − uM [v] = Ux + Vy, (5.8.3) where M is the operator represented by M [v] = vxy − (av)x − (bv)y + cv, (5.8.4) and U = auv − uvy, V = buv + vux. (5.8.5) The operator M is called the adjoint operator of L. If M = L, then the operator L is said to be self-adjoint. Now applying Green’s theorem, we have  D (Ux + Vy) dx dy =  C (U dy − V dx), (5.8.6) where C is the closed curve bounding the region of integration D which is in D∗ . Let Λ be a smooth initial curve which is continuous, as shown in Figure 5.8.1. Since equation (5.8.1) is in first canonical form, x and y are the characteristic coordinates. We assume that the tangent to Λ is nowhere parallel to the x or y axis. Let P (α, β) be a point at which the solution to the Cauchy problem is sought. Line P Q parallel to the x axis intersects the initial curve Λ at Q, and line P R parallel to the y axis intersects the curve Λ at R. We suppose that u and ux or uy are prescribed along Λ. Let C be the closed contour P QRP bounding D. Since dy = 0 on P Q and dx = 0 on P R, it follows immediately from equations (5.8.3) and (5.8.6) that  D (vL[u] − uM [v]) dx dy =  R Q (U dy − V dx) +  P R U dy −  Q P V dx. (5.8.7) 144 5 The Cauchy Problem and Wave Equations Figure 5.8.1 Smooth initial curve. From equation (5.8.5), we find  Q P V dx =  Q P bvu dx +  Q P vux dx. Integrating by parts, we obtain  Q P vuxdx = [uv] Q P −  Q P uvxdx. Hence, we may write  Q P V dx = [uv] Q P +  Q P u (bv − vx) dx. Substitution of this integral in equation (5.8.7) yields [uv]P = [uv]Q +  Q P u (bv − vx) dx −  P R u (av − vy) dy −  R Q (U dy − V dx) +  D (vL[u] − uM [v]) dx dy. (5.8.8) Suppose we can choose the function v (x, y; α, β) to be the solution of the adjoint equation 5.8 The Riemann Method 145 M [v]=0, (5.8.9) satisfying the conditions vx = bv when y = β, vy = av when x = α, (5.8.10) v = 1 when x = α and y = β. The function v (x, y; α, β) is called the Riemann function. Since L[u] = f, equation (5.8.8) reduces to, [u]P = [uv]Q −  R Q uv (a dy − b dx) +  R Q (uvydy + vuxdx) +  D vf dx dy. (5.8.11) This gives us the value of u at the point P when u and ux are prescribed along the curve Λ. When u and uy are prescribed, the identity [uv]R − [uv]Q =  R Q ( (uv)x dx + (uv)y dy) , may be used to put equation (5.8.8) in the form [u]P = [uv]R −  R Q uv (a dy − b dx) −  R Q (uvxdx + vuydy) +  D vf dx dy. (5.8.12) By adding equations (5.8.11) and (5.8.12), the value of u at P is given by [u]P = 1 2 4 [uv]Q + [uv]R 5 −  R Q uv (a dy − b dx) − 1 2  R Q u (vxdx − vydy) + 1 2  R Q v (uxdx − uydy) +  D vf dx dy (5.8.13) which is the solution of the Cauchy problem in terms of the Cauchy data given along the curve Λ. It is easy to see that the solution at the point (α, β) depends only on the Cauchy data along the arc QR on Λ. If the initial data were to change outside this arc QR, the solution would change only outside the triangle P QR. Thus, from Figure 5.8.2, we can see that each characteristic separates the region in which the solution remains unchanged from the region in which it varies. Because of this fact, the unique continuation of the solution across any characteristic is not possible. This is evident from Figure 5.8.2. The solution on the right of the characteristic P1R1 is determined by the initial data given in Q1R2, whereas the solution 146 5 The Cauchy Problem and Wave Equations Figure 5.8.2 Solution on the right and left of the characteristic. on the left is determined by the initial data given on Q1R1. If the initial data on R1R2 were changed, the solution on the right of P1R1 only will be affected. It should be remarked here that the initial curve can intersect each characteristic at only one point. Suppose, for example, the initial curve Λ intersects the characteristic at two points, as shown in Figure 5.8.3. Then, the solution at P obtained from the initial data on QR will be different from the solution obtained from the initial data on RS. Hence, the Cauchy problem, in this case, is not solvable. Figure 5.8.3 Initial curve intersects the characteristic at two points. 5.8 The Riemann Method 147 Example 5.8.1. The telegraph equation wtt + a ∗wt + b ∗w = c 2wxx, may be transformed into canonical form L[u] = uξη + ku = 0, by the successive transformations w = u e−a ∗ t/2 , and ξ = x + ct, η = x − ct, where k =  a ∗2 − 4b ∗ /16c 2 . We apply Riemann’s method to determine the solution satisfying the initial conditions u (x, 0) = f (x), ut (x, 0) = g (x). Since t = 1 2c (ξ − η), the line t = 0 corresponds to the straight line ξ = η in the ξ − η plane. The initial conditions may thus be transformed into [u] ξ=η = f (ξ), (5.8.14) [uξ − uη] ξ=η = c −1 g (ξ). (5.8.15) We next determine the Riemann function v (ξ,η; α, β) which satisfies vξη + kv = 0, (5.8.16) vξ (ξ, β; α, β)=0, (5.8.17) vη (α, η; α, β)=0, (5.8.18) v (α, β; α, β)=1. (5.8.19) The differential equation (5.8.16) is self-adjoint, that is, L[v] = M [v] = vξη + kv. We assume that the Riemann function is of the form v (ξ,η; α, β) = F (s), with the argument s = (ξ − α) (η − β). Substituting this value in equation (5.8.16), we obtain 148 5 The Cauchy Problem and Wave Equations sFss + Fs + kF = 0. If we let λ = √ 4ks, the above equation becomes F ′′ (λ) + 1 λ F ′ (λ) + F (λ)=0. This is the Bessel equation of order zero, and the solution is F (λ) = J0 (λ), disregarding Y0 (λ) which is unbounded at λ = 0. Thus, the Riemann function is v (ξ,η; α, β) = J0 4 4k (ξ − α) (η − β) 5 which satisfies equation (5.8.16) and is equal to one on the characteristics ξ = α and η = β. Since J ′ 0 (0) = 0, equations (5.8.17) and (5.8.18) are satisfied. From this, it immediately follows that [vξ] ξ=η = √ k (ξ − β)  (ξ − α) (η − β) [J ′ 0 (λ)]ξ=η , [vη] ξ=η = √ k (ξ − α)  (ξ − α) (η − β) [J ′ 0 (λ)]ξ=η . Thus, we have [vξ − uη] ξ=η = √ k (α − β)  (ξ − α) (ξ − β) [J ′ 0 (λ)]ξ=η . (5.8.20) From the initial condition u (Q) = f (β) and u (R) = f (α), (5.8.21) and substituting equations (5.8.15), (5.8.19), and (5.8.20) into equation (5.8.13), we obtain u (α, β) = 1 2 [f (α) + f (β)] − 1 2  α β √ k (α − β)  (τ − α) (τ − β) J ′ 0 4 4k (τ − α) (τ − β) 5 f (τ ) dτ + 1 2c  α β J0 4 4k (τ − α) (τ − β) 5 g (τ ) dτ. (5.8.22) Replacing α and β by ξ and η, and substituting the original variables x and t, we obtain 5.9 Solution of the Goursat Problem 149 u (x, t) = 1 2 [f (x + ct) + f (x − ct)] + 1 2  x+ct x−ct G (x, t, τ ) dτ, (5.8.23) where G (x, t, τ ) = / −2 √ k ctf (τ ) J0 42 4k " (τ − x) 2 − c 2t 2 # 501% (τ − x) 2 − c 2t 2 + c −1 g (τ ) J0 42 4k " (τ − x) 2 − c 2t 2 # 5 . If we set k = 0, we arrive at the d’Alembert solution for the wave equation u (x, t) = 1 2 [f (x + ct) + f (x − ct)] + 1 2c  x+ct x−ct g (τ ) dτ. 5.9 Solution of the Goursat Problem The Goursat problem is that of finding the solution of a linear hyperbolic equation uxy = a1 (x, y) ux + a2 (x, y) uy + a3 (x, y) u + h (x, y), (5.9.1) satisfying the prescribed conditions u (x, y) = f (x), (5.9.2) on a characteristic, say, y = 0, and u (x, y) = g (x) (5.9.3) on a monotonic increasing curve y = y (x) which, for simplicity, is assumed to intersect the characteristic at the origin. The solution in the region between the x-axis and the monotonic curve in the first quadrant can be determined by the method of successive approximations. The proof is given in Garabedian (1964). Example 5.9.1. Determine the solution of the Goursat problem utt = c 2uxx, (5.9.4) u (x, t) = f (x), on x − ct = 0, (5.9.5) u (x, t) = g (x), on t = t(x), (5.9.6) where f (0) = g (0). 150 5 The Cauchy Problem and Wave Equations The general solution of the wave equation is u (x, t) = φ (x + ct) + ψ (x − ct). Applying the prescribed conditions, we obtain f (x) = φ (2x) + ψ (0), (5.9.7) g (x) = φ (x + c t(x)) + ψ (x − c t(x)). (5.9.8) It is evident that f (0) = φ (0) + ψ (0) = g (0). Now, if s = x − c t(x), the inverse of it is x = α (s). Thus, equation (5.9.8) may be written as g (α (s)) = φ (x + c t(x)) + ψ (s). (5.9.9) Replacing x by (x + c t(x)) /2 in equation (5.9.7), we obtain f  x + c t(x) 2  = φ (x + c t(x)) + ψ (0). (5.9.10) Thus, using (5.9.10), equation (5.9.9) becomes ψ (s) = g (α (s)) − f  α (s) + c t(α (s)) 2  + ψ (0). Replacing s by x − ct , we have ψ (x − ct) = g (α (x − ct)) − f  α (x − c t) + c t(α (x − c t)) 2  + ψ (0). Hence, the solution is given by u (x, t) = f  x + c t 2  − f  α (x − c t) + c t(α (x − c t)) 2  + g (α (x − c t)). (5.9.11) Let us consider a special case when the curve t = t(x) is a straight line represented by t − kx = 0 with a constant k > 0. Then s = x − ckx and hence x = s/ (1 − ck). Using these values in (5.9.11), we obtain u (x, t) = f  x + c t 2  − f  (1 + c k) (x − c t) 2 (1 − c k)  + g  x − c t 1 − c k  . (5.9.12) When the values of u are prescribed on both characteristics, the problem of finding u of a linear hyperbolic equation is called a characteristic initialvalue problem. This is a degenerate case of the Goursat problem. 5.9 Solution of the Goursat Problem 151 Consider the characteristic initial-value problem uxy = h (x, y), (5.9.13) u (x, 0) = f (x), (5.9.14) u (0, y) = g (y), (5.9.15) where f and g are continuously differentiable, and f (0) = g (0). Integrating equation (5.9.13), we obtain u (x, y) =  x 0  y 0 h (ξ,η) dη dξ + φ (x) + ψ (y), (5.9.16) where φ and ψ are arbitrary functions. Applying the prescribed conditions (5.9.14) and (5.9.15), we have u (x, 0) = φ (x) + ψ (0) = f (x), (5.9.17) u (0, y) = φ (0) + ψ (y) = g (y). (5.9.18) Thus, φ (x) + ψ (y) = f (x) + g (y) − φ (0) − ψ (0). (5.9.19) But from (5.9.17), we have φ (0) + ψ (0) = f (0). (5.9.20) Hence, from (5.9.16), (5.9.19) and (5.9.20), we obtain u (x, y) = f (x) + g (y) − f (0) +  x 0  y 0 h (ξ,η) dη dξ. (5.9.21) Example 5.9.2. Determine the solution of the characteristic initial-value problem utt = c 2uxx, u (x, t) = f (x) on x + ct = 0, u (x, t) = g (x) on x − ct = 0, where f (0) = g (0). Here it is not necessary to reduce the given equation to canonical form. The general solution of the wave equation is u (x, t) = φ (x + ct) + ψ (x − ct). The characteristics are x + ct = 0, x − ct = 0. 152 5 The Cauchy Problem and Wave Equations Applying the prescribed conditions, we have u (x, t) = φ (2x) + ψ (0) = f (x) on x + ct = 0, (5.9.22) u (x, t) = φ (0) + ψ (2x) = g (x) on x − ct = 0. (5.9.23) We observe that these equations are compatible, since f (0) = g (0). Now, replacing x by (x + ct) /2 in equation (5.9.22) and replacing x by (x − ct) /2 in equation (5.9.23), we have φ (x + ct) = f  x + ct 2  − ψ (0), φ (x − ct) = g  x − ct 2  − φ (0). Hence, the solution is given by u (x, t) = f  x + ct 2  + g  x − ct 2  − f (0). (5.9.24) We note that this solution can be obtained by substituting k = −1/c into (5.9.12). Example 5.9.3. Find the solution of the characteristic initial-value problem y 3uxx − yuyy + uy = 0, (5.9.25) u (x, y) = f (x) on x + y 2 2 = 4 for 2 ≤ x ≤ 4, u (x, y) = g (x) on x − y 2 2 = 0 for 0 ≤ x ≤ 2, with f (2) = g (2). Since the equation is hyperbolic except for y = 0, we reduce it to the canonical form uξη = 0, where ξ = x +  y 2/2 and η = x −  y 2/2 . Thus, the general solution is u (x, y) = φ  x + y 2 2  + ψ  x − y 2 2  . (5.9.26) Applying the prescribed conditions, we have f (x) = φ (4) + ψ (2x − 4), (5.9.27) g (x) = φ (2x) + ψ (0). (5.9.28) Now, if we replace (2x − 4) by  x − y 2/2 in (5.9.27) and (2x) by  x + y 2/2 in (5.9.28), we obtain 5.10 Spherical Wave Equation 153 ψ  x − y 2 2  = f  x 2 − y 2 4 + 2 − φ (4), φ  x + y 2 2  = g  x 2 + y 2 4  − ψ (0). Thus, u (x, y) = f  x 2 − y 2 4 + 2 + g  x 2 + y 2 4  − φ (4) − ψ (0). But from (5.9.27) and (5.9.28), we see that f (2) = φ (4) + ψ (0) = g (2). Hence, u (x, y) = f  x 2 − y 2 4 + 2 + g  x 2 + y 2 4  − f (2). 5.10 Spherical Wave Equation In spherical polar coordinates (r, θ, φ), the wave equation (3.1.1) takes the form 1 r 2 ∂ ∂r  r 2 ∂u ∂r  + 1 r 2 sin θ ∂ ∂θ  sin θ ∂u ∂θ  + 1 r 2 sin2 θ ∂ 2u ∂φ2 = 1 c 2 ∂ 2u ∂t2 .(5.10.1) Solutions of this equation are called spherical symmetric waves if u depends on r and t only. Thus, the solution u = u (r, t) which satisfies the wave equation with spherical symmetry in three-dimensional space is 1 r 2 ∂ ∂r  r 2 ∂u ∂r  = 1 c 2 ∂ 2u ∂t2 . (5.10.2) Introducing a new dependent variable U = ru (r, t), this equation reduces to a simple form Utt = c 2Urr. (5.10.3) This is identical with the one-dimensional wave equation (5.3.1) and has the general solution in the form U (r, t) = φ (r + ct) + ψ (r − ct), (5.10.4) or, equivalently, u (r, t) = 1 r [φ (r + ct) + ψ (r − ct)] . (5.10.5) 154 5 The Cauchy Problem and Wave Equations This solution consists of two progressive spherical waves traveling with constant velocity c. The terms involving φ and ψ represent the incoming waves to the origin and the outgoing waves from the origin respectively. Physically, the solution for only outgoing waves generated by a source is of most interest, and has the form u (r, t) = 1 r ψ (r − ct), (5.10.6) where the explicit form of ψ is to be determined from the properties of the source. In the context of fluid flows, u represents the velocity potential so that the limiting total flux through a sphere of center at the origin and radius r is Q (t) = limr→0 4πr2ur (r, t) = −4π ψ (−ct). (5.10.7) In physical terms, we say that there is a simple (or monopole) point source of strength Q (t) located at the origin. Thus, the solution (5.10.6) can be expressed in terms of Q as u (r, t) = − 1 4πr Q 4 t − r c 5 . (5.10.8) This represents the velocity potential of the point source, and ur is called the radial velocity. In fluid flows, the difference between the pressure at any time t and the equilibrium value is given by p − p0 = ρ ut = − ρ 4πr Q˙ 4 t − r c 5 , (5.10.9) where ρ is the density of the fluid. Following an analysis similar to Section 5.3, the solution of the initialvalue problem with the initial data u (r, 0) = f (r), ut (r, 0) = g (r), r ≥ 0, (5.10.10) where f and g are continuously differentiable, is given by u (r, t) = 1 2r (r + ct) f (r + ct)+(r − ct) f (r − ct) + 1 c  r+ct r−ct τg (τ ) dτ , (5.10.11) provided r ≥ ct. However, when r < ct, this solution fails because f and g are not defined for r < 0. This initial data at t = 0, r ≥ 0 determine the solution u (r, t) only up to the characteristic r = ct in the r-t plane. To find u for r < ct, we require u to be finite at r = 0 for all t ≥ 0, that is, U = 0 at r = 0. Thus, the solution for U (r, t) is 5.11 Cylindrical Wave Equation 155 U (r, t) = 1 2 (r + ct) f (r + ct)+(r − ct) f (r − ct) + 1 c  r+ct r−ct τg (τ ) dτ , (5.10.12) provided r ≥ ct ≥ 0, and U (r, t) = 1 2 [φ (ct + r) + ψ (ct − r)] , ct ≥ r ≥ 0, (5.10.13) where φ (ct) + ψ (ct)=0, for ct ≥ 0. (5.10.14) In view of the fact that Ur + 1 c Ut is constant on each characteristic r + ct = constant, it turns out that φ ′ (ct + r)=(r + ct) f ′ (r + ct) + f (r + ct) + 1 c (r + ct) g (r + ct), or φ ′ (ct) = ctf′ (ct) + f (ct) + t g (ct). Integration gives φ (t) = tf (t) + 1 c  t 0 τg (τ ) dτ + φ (0), so that ψ (t) = −tf (t) − 1 c  t 0 τg (τ ) dτ − φ (0). Substituting these values into (5.10.13) and using U (r, t) = ru (r, t), we obtain, for ct > r, u (r, t) = 1 2r (ct + r) f (ct + r) − (ct − r) f (ct − r) + 1 c  ct+r ct−r τg (τ ) dτ . (5.10.15) 5.11 Cylindrical Wave Equation In cylindrical polar coordinates (R, θ, z), the wave equation (3.1.1) assumes the form uRR + 1 R uR + 1 R2 uθθ + uzz = 1 c 2 utt. (5.11.1) If u depends only on R and t, this equation becomes 156 5 The Cauchy Problem and Wave Equations uRR + 1 R uR = 1 c 2 utt. (5.11.2) Solutions of (5.11.2) are called cylindrical waves. In general, it is not easy to find the solution of (5.11.1). However, we shall solve this equation by using the method of separation of variables in Chapter 7. Here we derive the solution for outgoing cylindrical waves from the spherical wave solution (5.10.8). We assume that sources of constant strength Q (t) per unit length are distributed uniformly on the z-axis. The solution for the cylindrical waves produced by the line source is given by the total disturbance u (R, t) = − 1 4π  ∞ −∞ 1 r Q 4 t − r c 5 dz = − 1 2π  ∞ 0 1 r Q 4 t − r c 5 dz, (5.11.3) where R is the distance from the z-axis so that R2 =  r 2 − z 2 . Substitution of z = R sinh ξ and r = R cosh ξ in (5.11.3) gives u (R, t) = − 1 2π  ∞ 0 Q  t − R c cosh ξ  dξ. (5.11.4) This is usually considered as the cylindrical wave function due to a source of strength Q (t) at R = 0. It follows from (5.11.4) that utt = − 1 2π  ∞ 0 Q ′′  t − R c cosh ξ  dξ, (5.11.5) uR = 1 2πc  ∞ 0 cosh ξ Q′  t − R c cosh ξ  dξ, (5.11.6) uRR = − 1 2πc2  ∞ 0 cosh2 ξ Q′′  t − R c cosh ξ  dξ, (5.11.7) which give c 2  uRR + 1 R uR  − utt = 1 2π  ∞ 0 d dξ c R Q ′  t − R c cosh ξ  sinh ξ dξ = lim ξ→∞ c 2πR Q ′  t − R c cosh ξ  sinh ξ = 0, provided the differentiation under the sign of integration is justified and the above limit is zero. This means that u (R, t) satisfies the cylindrical wave equation (5.11.2). In order to find the asymptotic behavior of the solution as R → 0, we substitute cosh ξ = c(t−ζ) R into (5.11.4) and (5.11.6) to obtain u = − 1 2π  t−R/c −∞ Q (ζ) dζ " (t − ζ) 2 − R2 c 2 # 1 2 , (5.11.8) uR = 1 2π  t−R/c −∞  t − ζ R  Q′ (ζ) dζ " (t − ζ) 2 − R2 c 2 # 1 2 , (5.11.9) 5.11 Cylindrical Wave Equation 157 which, in the limit R → 0, give uR ∼ 1 2πR  t −∞ Q ′ (ζ) dζ = 1 2πR Q (t). (5.11.10) This leads to the result lim R→0 2πR uR = Q (t), (5.11.11) or u (R, t) ∼ 1 2π Q (t) log R as R → 0. (5.11.12) We next investigate the nature of the cylindrical wave solution near the waterfront (R = ct) and in the far field (R → ∞). We assume Q (t) = 0 for t < 0 so that the lower limit of integration in (5.11.8) may be taken to be zero, and the solution is non-zero for τ = t − R c > 0, where τ is the time passed after the arrival of the wavefront. Consequently, (5.11.8) becomes u (R, t) = − 1 2π  τ 0 Q (ζ) dζ (t − ζ)  t − ζ + 2R c ! 1 2 . (5.11.13) Since 0 <ζ<τ , 2R c > R c >τ>τ − ζ > 0, so that the second factor under the radical is approximately equal to 2R c when R ≫ cτ , and hence, u (R, t) ∼ − 1 2π 4 c 2R 5 1 2  τ 0 Q (ζ) dζ (t − ζ) 1 2 = − 4 c 2R 5 1 2 q (τ ) = − 4 c 2R 5 1 2 q  t − R c  , R ≫ ct 2 , (5.11.14) where q (τ ) = 1 2π  τ 0 Q (ζ) dζ √ τ − ζ . (5.11.15) Evidently, the amplitude involved in the solution (5.11.14) decays like R− 1 2 for large R (R → ∞). Example 5.11.1. Determine the asymptotic form of the solution (5.11.4) for a harmonically oscillating source of frequency ω. We take the source in the form Q (t) = q0 exp [−i(ω + iε)t], where ε is positive and small so that Q (t) → 0 as t → −∞. The small imaginary part ε of ω will make insignificant contributions to the solution at finite time as ε → 0. Thus, the solution (5.11.4) becomes u (R, t) = − 4 q0 2π 5 e −iωt  ∞ 0 exp  iωR c cosh ξ  dξ = −  iq0 4  e −iωtH (1) 0  ωR c  , (5.11.16) 158 5 The Cauchy Problem and Wave Equations where H (1) 0 (z) is the Hankel function given by H (1) 0 (z) = 2 πi  ∞ 0 exp (iz cosh ξ) dξ. (5.11.17) In view of the asymptotic expansion of H (1) 0 (z) in the form H (1) 0 (z) ∼  2 πz 1 2 exp " i 4 z − π 4 5# , z → ∞, (5.11.18) the asymptotic solution for u (R, t) in the limit  ωR c → ∞ is u (R, t) ∼ −  iq0 4  2c πωR1 2 exp −i  ωt − ωR c − π 4  . This represents the cylindrical wave propagating with constant velocity c. The amplitude of the wave decays like R− 1 2 as R → ∞. Example 5.11.2. For a supersonic flow (M > 1) past a solid body of revolution, the perturbation potential Φ satisfies the cylindrical wave equation ΦRR + 1 R ΦR = N 2Φxx, N2 = M2 − 1, where R is the distance from the path of the moving body and x is the distance from the nose of the body. It follows from problem 12 in 3.9 Exercises that Φ satisfies the equation Φyy + Φzz = N 2 Φxx. This represents a two-dimensional wave equation with x ↔ t and N2 ↔ 1 c 2 . For a body of revolution with (y, z) ↔ (R, θ), ∂ ∂θ ≡ 0, the above equation reduces to the cylindrical wave equation ΦRR + 1 R ΦR = 1 c 2 Φtt. 5.12 Exercises 1. Determine the solution of each of the following initial-value problems: (a) utt − c 2uxx = 0, u (x, 0) = 0, ut (x, 0) = 1. (b) utt − c 2uxx = 0, u (x, 0) = sin x, ut (x, 0) = x 2 . (c) utt − c 2uxx = 0, u (x, 0) = x 3 , ut (x, 0) = x. 5.12 Exercises 159 (d) utt − c 2uxx = 0, u (x, 0) = cos x, ut (x, 0) = e −1 . (e) utt − c 2uxx = 0, u (x, 0) = log  1 + x 2 , ut (x, 0) = 2. (f) utt − c 2uxx = 0, u (x, 0) = x, ut (x, 0) = sin x. 2. Determine the solution of each of the following initial-value problems: (a) utt − c 2uxx = x, u (x, 0) = 0, ut (x, 0) = 3. (b) utt − c 2uxx = x + ct, u (x, 0) = x, ut (x, 0) = sin x. (c) utt − c 2uxx = e x , u (x, 0) = 5, ut (x, 0) = x 2 . (d) utt − c 2uxx = sin x, u (x, 0) = cos x, ut (x, 0) = 1 + x. (e) utt − c 2uxx = xet , u (x, 0) = sin x, ut (x, 0) = 0. (f) utt − c 2uxx = 2, u (x, 0) = x 2 , ut (x, 0) = cos x. 3. A gas which is contained in a sphere of radius R is at rest initially, and the initial condensation is given by s0 inside the sphere and zero outside the sphere. The condensation is related to the velocity potential by s (t) =  1/c2 ut, at all times, and the velocity potential satisfies the wave equation utt = ∇2u. Determine the condensation s (t) for all t > 0. 4. Solve the initial-value problem uxx + 2uxy − 3uyy = 0, u (x, 0) = sin x, uy (x, 0) = x. 5. Find the longitudinal oscillation of a rod subject to the initial conditions u (x, 0) = sin x, ut (x, 0) = x. 6. By using the Riemann method, solve the following problems: (a) sin2 µ φxx − cos2 µ φyy −  λ 2 sin2 µ cos2 µ φ = 0, φ (0, y) = f1 (y), φ (x, 0) = g1 (x), φx (0, y) = f2 (y), φy (x, 0) = g2 (x). 160 5 The Cauchy Problem and Wave Equations (b) x 2uxx − t 2utt = 0, u (x, t1) = f (x), ut (x, t2) = g (x). 7. Determine the solution of the initial boundary-value problem utt = 4 uxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = x 4 , 0 ≤ x < ∞, ut (x, 0) = 0, 0 ≤ x < ∞, u (0, t)=0, t ≥ 0. 8. Determine the solution of the initial boundary-value problem utt = 9 uxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = 0, 0 ≤ x < ∞, ut (x, 0) = x 3 , 0 ≤ x < ∞, ux (0, t)=0, t ≥ 0. 9. Determine the solution of the initial boundary-value problem utt = 16 uxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = sin x, 0 ≤ x < ∞, ut (x, 0) = x 2 , 0 ≤ x < ∞, u (0, t)=0, t ≥ 0. 10. In the initial boundary-value problem utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, u (0, t)=0, t ≥ 0, if f and g are extended as odd functions, show that u (x, t) is given by the solution (5.4.5) for x > ct and solution (5.4.6) for x < ct. 11. In the initial boundary-value problem utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, ux (0, t)=0, t ≥ 0, if f and g are extended as even functions, show that u (x, t) is given by solution (5.4.8) for x > ct, and solution (5.4.9) for x < ct. 5.12 Exercises 161 12. Determine the solution of the initial boundary-value problem utt = c 2uxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = f (x), 0 ≤ x < ∞, ut (x, 0) = 0, 0 ≤ x < ∞, ux (0, t) + h u (0, t)=0, t ≥ 0, h = constant. State the compatibility condition of f. 13. Find the solution of the problem utt = c 2uxx, at < x < ∞, t> 0, u (x, 0) = f (x), 0 <x< ∞,="" ut="" (x,="" 0)="0," 0="" <x<="" u="" (at,="" t)="0," t=""> 0, where f (0) = 0 and a is constant. 14. Find the solution of the initial boundary-value problem utt = uxx, 0 <x< 2,="" t=""> 0, u (x, 0) = sin (πx/2), 0 ≤ x ≤ 2, ut (x, 0) = 0, 0 ≤ x ≤ 2, u (0, t)=0, u (2, t)=0, t ≥ 0. 15. Find the solution of the initial boundary-value problem utt = 4 uxx, 0 <x< 1,="" t=""> 0, u (x, 0) = 0, 0 ≤ x ≤ 1, ut (x, 0) = x (1 − x), 0 ≤ x ≤ 1, u (0, t)=0, u (1, t)=0, t ≥ 0. 16. Determine the solution of the initial boundary-value problem utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, ux (0, t)=0, ux (l, t)=0, t ≥ 0, by extending f and g as even functions about x = 0 and x = l. 17. Determine the solution of the initial boundary-value problem utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, u (0, t) = p (t), u (l, t) = q (t), t ≥ 0. 162 5 The Cauchy Problem and Wave Equations 18. Determine the solution of the initial boundary-value problem utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, ux (0, t) = p (t), ux (l, t) = q (t), t ≥ 0. 19. Solve the characteristic initial-value problem xy3uxx − x 3 y uyy − y 3ux + x 3uy = 0, u (x, y) = f (x) on y 2 − x 2 = 8 for 0 ≤ x ≤ 2, u (x, y) = g (x) on y 2 + x 2 = 16 for 2 ≤ x ≤ 4, with f (2) = g (2). 20. Solve the Goursat problem xy3uxx − x 3 y uyy − y 3ux + x 3uy = 0, u (x, y) = f (x) on y 2 + x 2 = 16 for 0 ≤ x ≤ 4, u (x, y) = g (y) on x = 0 for 0 ≤ y ≤ 4, where f (0) = g (4). 21. Solve utt = c 2uxx, u (x, t) = f (x) on t = t(x), u (x, t) = g (x) on x + ct = 0, where f (0) = g (0). 22. Solve the characteristic initial-value problem xuxx − x 3uyy − ux = 0, x = 0, u (x, y) = f (y) on y − x 2 2 = 0 for 0 ≤ y ≤ 2, u (x, y) = g (y) on y + x 2 2 = 4 for 2 ≤ y ≤ 4, where f (2) = g (2). 23. Solve uxx + 10 uxy + 9 uyy = 0, u (x, 0) = f (x), uy (x, 0) = g (x). 5.12 Exercises 163 24. Solve 4 uxx + 5 uxy + uyy + ux + uy = 2, u (x, 0) = f (x), uy (x, 0) = g (x). 25. Solve 3 uxx + 10 uxy + 3 uyy = 0, u (x, 0) = f (x), uy (x, 0) = g (x). 26. Solve uxx − 3 uxy + 2 uyy = 0, u (x, 0) = f (x), uy (x, 0) = g (x). 27. Solve x 2uxx − t 2utt = 0 x > 0, t> 0, u (x, 1) = f (x), ut (x, 1) = g (x). 28. Consider the initial boundary-value problem for a string of length l under the action of an external force q (x, t) per unit length. The displacement u (x, t) satisfies the wave equation ρ utt = T uxx + ρ q (x, t), where ρ is the line density of the string and T is the constant tension of the string. The initial and boundary conditions of the problem are u (x, 0) = f (x), ut (x, 0) = g (x), 0 ≤ x ≤ l, u (0, t) = u (l, t)=0, t > 0. Show that the energy equation is dE dt = [T uxut] l 0 +  l 0 ρqut dx, where E represents the energy integral E (t) = 1 2  l 0  ρ u2 t + T u2 x dx. Explain the physical significance of the energy equation. Hence or otherwise, derive the principle of conservation of energy, that is, that the total energy is constant for all t ≥ 0 provided that the string has free or fixed ends and there are no external forces. 164 5 The Cauchy Problem and Wave Equations 29. Show that the solution of the signaling problem governed by the wave equation utt = c 2uxx, x > 0, t > 0, u (x, 0) = ut (x, 0) = 0, x > 0, u (0, t) = U (t), t > 0, is u (x, t) = U 4 t − x c 5 H 4 t − x c 5 , where H is the Heaviside unit step function. 30. Obtain the solution of the initial-value problem of the homogeneous wave equation utt − c 2uxx = sin (kx − ωt), −∞ <x< ∞,="" t=""> 0, u (x, 0) = 0 = ut (x, 0), for all x ∈ R, where c, k and ω are constants. Discuss the non-resonance case, ω = ck and the resonance case, ω = ck. 31. In each of the following Cauchy problems, obtain the solution of the system utt − c 2uxx = 0, x ∈ R, t> 0, u (x, 0) = f (x) and ut (x, 0) = g (x) for x ∈ R, for the given c, f (x) and g (x): (a) c = 3, f (x) = cos x, g (x) = sin 2x. (b) c = 1, f (x) = sin 3x, g (x) = cos 3x. (c) c = 7, f (x) = cos 3x, g (x) = x. (d) c = 2, f (x) = cosh x, g (x)=2x. (e) c = 3, f (x) = x 3 , g (x) = x cos x. (f) c = 4, f (x) = cos x, g (x) = xe−x . 32. If u (x, t) is the solution of the nonhomogeneous Cauchy problem utt − c 2uxx = p (x, t), for x ∈ R, t> 0, u (x, 0) = 0 = ut (x, 0), for x ∈ R, 5.12 Exercises 165 and if v (x, t, τ ) is the solution of the nonhomogeneous Cauchy problem vtt − c 2 vxx = 0, for x ∈ R, t> 0, v (x, 0; τ )=0, vt (x, 0; τ ) = p (x, τ ), x ∈ R, show that u (x, t) =  t 0 v (x, t; τ ) dτ. This is known as the Duhamel principle for the wave equation. 33. Show that the solution of the nonhomogeneous diffusion equation with homogeneous boundary and initial data ut = κuxx + p (x, t), 0 < x < l, t > 0, u (0, t)=0= u (l, t), t > 0, u (x, 0) = 0, 0 < x < l, is u (x, t) =  t 0 v (x, t; τ ) dτ, where v = v (x, t; τ ) satisfies the homogeneous diffusion equation with nonhomogeneous boundary and initial data vtt = κvxx + p (x, t), 0 < x < l, t > 0, v (0, t; τ )=0= v (l, t; τ ), t > 0, v (x, τ ; τ ) = p (x, τ ). This is known as the Duhamel principle for the diffusion equation. 34. Use the Duhamel principle to solve the nonhomogeneous diffusion equation ut = κuxx + e −t sin πx, 0 < x < l, t > 0, with the homogeneous boundary and initial data u (0, t)=0, u (1, t)=0, t > 0, u (x, 0) = 0, 0 ≤ x ≤ 1. 35. (a) Verify that un (x, y) = exp  ny − √ n sin nx, is the solution of the Laplace equation 166 5 The Cauchy Problem and Wave Equations uxx + uyy = 0, x ∈ R, y > 0, u (x, 0) = 0, uy (x, 0) = n exp  − √ n sin nx, where n is a positive integer. (b) Show that this Cauchy problem is not well posed. 36. Show that the following Cauchy problems are not well posed: (a) ut = uxx, x ∈ R, t > 0, u (0, t) =  2 n sin  2n 2 t , ux (0, t)=0, t > 0. (b) uxx + uyy = 0, x ∈ R, t > 0, un (x, 0) → 0, (un)y (x, 0) → 0, as n → ∞. 6 Fourier Series and Integrals with Applications “The thorough study of nature is the most ground for mathematical discoveries.” Joseph Fourier “Nearly fifty years had passed without any progress on the question of analytic representation of an arbitrary function, when an assertion of Fourier threw new light on the subject. Thus a new era began for the development of this part of Mathematics and this was heralded in a stunning way by major developments in mathematical Physics.” Bernhard Riemann “Fourier created a coherent method by which the different components of an equation and its solution in series were neatly identified with different aspects of physical solution being analyzed. He also had a uniquely sure instinct for interpreting the asymptotic properties of the solutions of his equations for their physical meaning. So powerful was his approach that a full century passed before non-linear equations regained prominence in mathematical physics.” Ioan James 6.1 Introduction This chapter is devoted to the theory of Fourier series and integrals. Although the treatment can be extensive, the exposition of the theory here will be concise, but sufficient for its application to many problems of applied mathematics and mathematical physics. 168 6 Fourier Series and Integrals with Applications The Fourier theory of trigonometric series is of great practical importance because certain types of discontinuous functions which cannot be expanded in power series can be expanded in Fourier series. More importantly, a wide class of problems in physics and engineering possesses periodic phenomena and, as a consequence, Fourier’s trigonometric series become an indispensable tool in the analysis of these problems. We shall begin our study with the basic concepts and definitions of some properties of real-valued functions. 6.2 Piecewise Continuous Functions and Periodic Functions A single-valued function f is said to be piecewise continuous in an interval [a, b] if there exist finitely many points a = x1 < x2 < ... < xn = b, such that f is continuous in the intervals xj <x<xj+1 and="" the="" one-sided="" limits="" f="" (xj+)="" (xj+1−)="" exist="" for="" all="" j="1," 2,="" 3,...,n="" −="" 1.="" a="" piecewise="" continuous="" function="" is="" shown="" in="" figure="" 6.2.1.="" functions="" such="" as="" 1="" x="" sin="" (1="" x)="" fail="" to="" be="" closed="" interval="" [0,="" 1]="" because="" limit="" (0+)="" does="" not="" either="" case.="" if="" an="" [a,="" b],="" then="" it="" necessarily="" bounded="" integrable="" over="" that="" interval.="" also,="" follows="" immediately="" product="" of="" two="" on="" common="" b]="" if,="" addition,="" first="" derivative="" ′="" each="" intervals="" xj="" <x<xj+1,="" (xj−)="" exist,="" said="" smooth;="" second="" ′′="" 6.2.1="" graph="" function.="" 6.2="" periodic="" 169="" very="" smooth.="" (x)="" there="" exists="" real="" positive="" number="" p="" (x="" +="" p)="f" (x),="" (6.2.1)="" x,="" called="" period="" f,="" smallest="" value="" termed="" fundamental="" period.="" sample="" given="" 6.2.2.="" with="" p,="" 2p)="f" p),="" 3p)="f" 2p="" 2p),="" np)="f" (n="" 1)="" any="" integer="" n.="" hence,="" integral="" values="" n="" (x).="" (6.2.2)="" can="" readily="" f1,="" f2,="" ...,="" fk="" have="" ck="" are="" constants,="" c2f2="" ...="" ckfk,="" (6.2.3)="" has="" p.="" well="" known="" examples="" sine="" cosine="" functions.="" special="" case,="" constant="" also="" arbitrary="" thus,="" by="" relation="" (6.2.3),="" series="" a0="" a1="" cos="" a2="" 2x="" b1="" b2="" converges,="" obviously="" 2π.="" types="" series,="" which="" occur="" frequently="" problems="" applied="" mathematics="" mathematical="" physics,="" will="" treated="" later.="" 6.2.2="" 170="" 6="" fourier="" integrals="" applications="" 6.3="" systems="" orthogonal="" sequence="" {φn="" (x)}="" respect="" weight="" q="" ="" b="" φm="" φn="" dx="0," m="n." (6.3.1)="" we="" φn="1" φ="" 2="" dx3="" (6.3.2)="" norm="" system="" (x)}.="" example="" 6.3.1.="" {sin="" mx},="" 2,...,="" form="" [−π,="" π],="" π="" −π="" mx="" nx="" ⎨="" ⎩="" 0,="" π,="" this="" notice="" equal="" unity,="" √="" π.="" φ1,="" φ2,="" φn,="" where="" may="" finite="" or="" infinite,="" satisfies="" relations="" 1,="" (6.3.3)="" orthonormal="" b].="" evident="" obtained="" from="" dividing="" its="" 6.3.2.="" x,sin="" .="" ,="" nx,sin="" forms="" π]="" since="" m,="" n,="" (6.3.4)="" 6.4="" 171="" integers="" normalize="" system,="" divide="" elements="" original="" their="" norms.="" 2π="" ,...,="" system.="" 2x,sin="" 2x,="" mutually="" other="" linearly="" independent.="" formally="" associate="" trigonometric="" write="" ∼="" ∞="" k="1" (ak="" kx="" bk="" kx),="" (6.4.1)="" symbol="" indicates="" association="" a0,="" ak,="" some="" unique="" manner.="" coefficients="" ak="" determined="" soon.="" coefficient="" (a0="" 2)="" instead="" used="" convenience="" representation.="" however,="" easy="" say="" right="" hand="" side="" itself="" converges="" represents="" indeed,="" converge="" diverge.="" let="" riemann="" defined="" π].="" suppose="" define="" nth="" partial="" sum="" sn="" n="" (6.4.2)="" represent="" shall="" seek="" best="" approximation="" sense="" least="" squares,="" is,="" minimize="" i="" (a0,="" bk)="" [f="" (x)]2="" dx.="" (6.4.3)="" extremal="" problem.="" necessary="" condition="" bk,="" so="" minimum,="" derivatives="" these="" vanish.="" substituting="" equation="" into="" differentiating="" obtain="" 172="" ∂i="" ∂a0="−" ⎡="" ⎣f="" (aj="" jx="" bj="" jx)="" ⎤="" ⎦dx.="" (6.4.4)="" ∂ak="−2" ⎦cos="" dx.(6.4.5)="" ∂bk="−2" ⎦sin="" dx.(6.4.6)="" using="" orthogonality="" noting="" (6.4.7)="" integers,="" equations="" (6.4.4),="" (6.4.5),="" (6.4.6)="" become="" dx,="" (6.4.8)="" (6.4.9)="" (6.4.10)="" must="" vanish="" value.="" (6.4.11)="" (6.4.12)="" (6.4.13)="" note="" case="" reason="" writing="" rather="" than="" (6.4.1).="" (6.4.8),="" (6.4.9),="" ∂="" ∂a2="" 0="π," (6.4.14)="" ∂b2="" (6.4.15)="" mixed="" order="" remaining="" higher="" now="" expand="" taylor="" about="" a1,...,an,="" b1,...,bn),="" 6.5="" convergence="" 173="" ∆a0,...,bn="" ∆bn)="I" (a0,...,bn)="" ∆i,="" (6.4.16)="" ∆i="" stands="" terms.="" derivatives,="" vanish,="" 2!="" 1="" ∆a2="" ="" ∆b2="" 3="" (6.4.17)="" virtue="" (6.4.15),="" positive.="" minimum="" value,="" (6.4.11),="" (6.4.12),="" respectively.="" corresponding="" correspondence="" asserts="" nothing="" divergence="" constructed="" series.="" question="" arises="" whether="" possible="" investigation="" sufficient="" conditions="" representation="" turns="" out="" difficult="" remark="" possibility="" representing="" imply="" uniformly,="" matter="" fact,="" convergent="" need="" instance,="" log="" no="" introduce="" three="" kinds="" fouriers="" series:="" (i)="" pointwise="" convergence,="" (ii)="" uniform="" (iii)="" mean-square="" convergence.="" definition="" 6.5.1.="" (pointwise="" convergence).="" infinite="" 2∞="" fn="" a<x<b="" a<x<b.="" words,="" a<x<b,="" |f="" (x)|="" →="" ∞,="" 174="" 6.5.2.="" (="" uniformly="" ≤="" max="" a≤x≤b="" ∞.="" evidently,="" implies="" but="" converse="" true.="" 6.5.3.="" (or="" l="" )="" noted="" stronger="" both="" study="" long="" complex="" history.="" f.="" answer="" certainly="" obvious.="" 2π-="" function,="" leads="" questions="" local="" behavior="" near="" point="" global="" overall="" entire="" another="" deals="" (−π,="" π),="" theorem="" provide="" insight="" problem="" guarantee="" x.="" hand,="" 2π-periodic="" smooth="" r,="" every="" been="" 1876="" whose="" diverge="" at="" certain="" points.="" was="" open="" century="" point.="" 1966,="" lennart="" carleson="" (1966)="" provided="" affirmative="" deep="" states="" square="" almost="" obvious="" 175="" ≥="" (6.5.1)="" expanding="" gives="" [sn="" but,="" definitions="" (6.3.4),="" kx)="" 3="" ="" (6.5.2)="" s="" 32="" (6.5.3)="" consequently,="" πa2="" 0.="" (6.5.4)="" (6.5.5)="" independent="" (6.5.6)="" bessel’s="" inequality.="" 176="" see="" left="" nondecreasing="" above,="" therefore,="" (6.5.7)="" converges.="" lim="" k→∞="" (6.5.8)="" mean="" when="" limn→∞="" 4="" kx532="" (6.5.9)="" (6.5.10)="" parseval’s="" one="" central="" results="" theory="" derive="" many="" important="" numerical="" furthermore,="" holds="" true,="" set="" complete.="" parseval="" derived="" (6.5.11)="" respectively,="" multiply="" 6.4.11)="" integrate="" resulting="" expression="" .(6.5.12)="" replacing="" (6.5.12)="" (6.5.10).="" 6.6="" 177="" section="" different="" way.="" expansion="" kx).="" (6.6.1)="" assume="" term-by-term="" (we="" later="" this),="" (6.6.2)="" again,="" sides="" (6.6.3)="" similar="" manner,="" find="" (6.6.4)="" just="" found="" exactly="" same="" those="" 6.4.="" 6.6.1.="" <="" here="" 178="" 6.6.1="" 3="" ="" kπ="" (−1)k="" 3,....="" similarly,="" 179="" 4="" ....="" (6.6.5)="" 6.6.2.="" consider="" −π,="" −π<x<="" −πdx="" 6.6.2="" 180="" (cos="" "="" #="" kπ)="1" ="" kx0="" (6.6.6)="" 6.6.3.="" sawtooth="" wave="" −π<x<π,="" 2kπ)="" 2,....="" 6.6.3="" continuous.="" 181="" (−1)k+1="" 3x="" 4x="" ...="" (6.6.7)="" agree="" endpoints="" zero,="" endpoint.="" (π="" 0)="" 0)]="1" π)="" uniform.="" kx.="" (6.6.8)="" neighborhood="" difference="" between="" (x)and="" seems="" smaller="" increases,="" size="" region="" occurs="" decreases="" indicating="" nonuniform="" oscillatory="" nature="" close="" discontinuities="" gibbs="" phenomenon.="" what="" happens="" π?="" (6.6.8),="" put="" xn="π" approximate="" (xn)="n" 4="" 5="n" ="" rewritten="" πk="" ·="" 4π="" (6.6.9)="" 182="" identified="" definite="" subintervals="" (k−1)π="" obviously,="" subinterval="" length="" evaluate="" right-hand="" endpoint="" ≈="" 1.18π.="" xn,="" approaches="" left.="" tends="" jump="" (π−)−f="" (π+)="2π," and,="" sufficiently="" large="" (π−)="" 1.18π="" next="" draw="" graphs="" s7="" s10="" exhibit="" oscillations="" figures="" 6.6.4="" (a)="" (b).="" show="" so-called="" overshooting="" 3π,="" several="" 5="" 7="" well-known="" slowly="" (6.6.10)="" putting="" 8="1" 9="" 10="" 11√="" 13√="" 14="" 11="" view="" (6.6.10),="" (6.6.11)="" 6.7="" 183="" even="" odd="" (6.7.1)="" 184="" written="" kx,="" (6.7.2)="" formula="" (6.7.1).="" consequence,="" (6.7.3)="" (6.7.4)="" (6.7.5)="" (6.7.4).="" 6.7.1.="" −1,="" +1,="" (x+="" clearly,="" 3,...="" b2k="0" b2k−1="[(4/π)" (2k="" 1)].="" (6.7.6)="" consists="" only="" harmonics.="" loss="" harmonics="" due="" fact="" sgn="" (6.6.10).="" 185="" 6.7.1="" (6.7.7)="" examine="" manner="" terms="" tend="" 3,="" 6.7.2.="" investigate="" locate="" peak="" origin="" calculate="" height="" 6.7.2="" 186="" overshoot="" maximum="" [sin="" 2kx="" x]="2" (6.7.8)="" points="" 2n="" ,...,(2n="" so,="" becomes="" (1.852)="1.179." 1.179="" approximately.="" compared="" (1.179="" ×="" 100%="" 9%.="" onset="" phenomenon="" 6.7.3.="" historically,="" observed="" physicist="" a.="" michelson="" (1852–1931)="" end="" nineteenth="" century.="" graphically,="" he="" developed="" equipment="" harmonic="" analyser="" synthesizer.="" calculated="" sums,="" sn,="" graphically.="" graphical="" that,="" functions,="" were="" sums="" error="" origin,="" (the="" function)="" sums.="" j.w.="" (1839–1903)="" who="" explanation="" strikingly="" new="" showed="" errors="" associated="" computations.="" further="" 187="" 6.7.3="" discontinuity="" properties="" discontinuity.="" |sin="" x|="" 6.7.4,="" 2,...="" k)="" 1+(−1)k="" (2kx)="" 4k="" triangular="" 188="" 6.7.4="" rectified="" =="" ⎧="" −x,="" (6.7.9)="" 2nπ),="" 6.7.5.="" bn="0" 6.7.5="" 189="" |x|="" x="" dx="" integrating="" parts="2" πn2="" [cos="" nx]="" [(−1)n="" even,="" odd.="" 5x="" (6.7.10)="" yields="" following="" (2n="" 1)2="" (6.7.11)="" reciprocals="" squares="" n2="π" (6.7.12)="" vise="" versa.="" (2n)="" s.="" (6.7.12).="" zeta="" ζ="" (s)="∞" ns="" (6.7.13)="" 190="" complex.="" extended="" natural="" way="" extension="" analytic="" continuation="" include="" numbers="" except="" introduced="" bernhard="" 1841.="" proved="" made="" conjectures,="" still="" mathematics.="" zeros="" axis="" negative="" integers.="" conjectured="" lie="" line="" re="" hypothesis;="" unsolved="" line.="" fall="" 2002,="" fifty="" billion="" —="" them="" stated="" 6.7.4.="" extensions="" 6.7.6.="" ≡="" 6.7.6="" 191="" nxπ="" nπ="" "="" nx#π="" (−1)n="" 4∞="" (6.7.14)="" ···="π" 12="" (6.7.15)="" vice="" s,="" adding="" (6.7.16)="" then,="" subtracting="" 24="" (6.7.17)="" 192="" 6.7.7="" preceding="" sections,="" prescribed="" assumed="" (−∞,∞).="" practice,="" encounter="" π).="" simply="" extend="" periodically="" 2π,="" 6.7.7.="" way,="" able="" expansion,="" although="" interested="" (0,="" ways.="" denoted="" (see="" 6.7.8)="" fe="" (−x),="" while="" 6.7.9)="" f0="" −f="" 193="" 6.7.8="" expansions="" 6.7.9="" 194="" 6.8="" sometimes="" convenient="" form.="" easily="" euler’s="" formulas="" ix="" e="" −ix="" 2i="" ikx="" −ikx="" ak="" ibk="" c−k="" c0="a0" isin="" −ikxdx="" ikxdx.="" 195="" (6.8.1)="" −ikxdx.="" (6.8.2)="" |ck|="" (6.8.3)="" (6.8.1)–(6.8.2).="" multiplying="" −π<x<π="" ikxdx="∞" c¯k="∞" 6.8.1.="" ik)="" sinh="" eikx="" (6.8.4)="" apply="" |1="" ik|="" sinh2="" 4π="" −2π="1" simplifying="" result="" coth="" (6.8.5)="" 196="" 6.8.2.="" −1="" <a<="" 2a="" (6.8.6)="" (6.8.7)="" denote="" c="" nx)="∞" eix="" a="" eix="" ia="" sin2="" (6.8.8)="" equating="" imaginary="" part="" desired="" results.="" 6.9="" far="" concerned="" applications,="" restrictive,="" interest="" arbitrary,="" variable="" t="" transformation="" (b="" a)="" t,="" (6.9.1)="" [(b="" ((b="" 2π)t]="F" (t)="" kt="" kt),="" (6.9.2)="" dt,="" 197="" changing="" (2x="" ,(6.9.3)="" (6.9.4)="" (6.9.5)="" take="" [−l,l].="" once="" letting="" [−l,l]="" takes="" kπx="" (6.9.6)="" −l="" (6.9.7)="" (6.9.8)="" 2l,="" (6.9.6),="" determine="" (6.9.9)="" (6.9.10)="" (6.9.11)="" 198="" (6.9.12)="" finally,="" make="" change="" −l<x<l.="" 2l.="" lt="" (t).="" (6.9.13)="" smooth,="" expanded="" cke="" ikt,="" −iktdt.="" (6.9.14)="" exp="" ixπk="" 2l="" dx.(6.9.15)="" particular,="" time="" ω="" frequency,="" [an="" (nωt)="" (nωt)]="" (6.9.16)="" anand="" replaced="" (a1="" ωt="" ωt),="" (a2="" 2ωt="" 2ωt),="" (an="" nωt="" nωt),="" first,="" second,="" 6.9.1.="" −2="" <x<="" odd,="" 199="" 6.9.1="" 2.="" 6.9.2.="" (kπx)="" (−1)k−1="" πx.="" 6.9.3.="" (−l,l)="" 200="" 6.9.2="" directly="" πkx="" l.="" 6.9.4.="" l)="" know="" 2∞="" πt="" πkt="" 4l="" 6.10="" riemann–lebesgue="" lemma="" 201="" 6.9.5.="" solution="" 25="" 6.14="" exercises,="" −l<t<l.="" use="" exercise="" ⎫="" ⎬="" ⎭="1" ="" n="0" inlt="" earlier="" section,="" discuss="" proof="" lemma.="" 6.10.1.="" (riemann–lebesgue="" lemma)="" g="" λ→∞="" λx="" (6.10.1)="" proof.="" (λ)="" (6.10.2)="" λ,="" λ="" (t="" λ)="−" λt,="" b−π="" a−π="" λt="" dt.="" (6.10.3)="" dummy="" variable,="" above="" (6.10.4)="" addition="" 202="" [g="" λ)]="" (6.10.5)="" bounded,="" |g="" m.="" a+π="" πm="" |i="" (λ)|="" λ)|="" (6.10.6)="" ε="" a),="" (6.10.7)="">Λ and all x in [a, b]. We now choose λ such that πM/λ < ε/2, whenever λ>Λ. Then |I (λ)| < ε 2 + ε 2 = ε. If g (x) is piecewise continuous in [a, b], then the proof consists of a repeated application of the preceding argument to every subinterval of [a, b] in which g (x) is continuous. Theorem 6.10.1. (Pointwise Convergence Theorem). If f (x) is piecewise smooth and periodic function with period 2π in [−π, π], then for any x a0 2 + ∞ k=1 (ak cos kx + bk sin kx) = 1 2 [f (x+) + f (x−)] , (6.10.8) where ak = 1 π  π −π f (t) cos kt dt, k = 0, 1, 2,..., (6.10.9) bk = 1 π  π −π f (t) sin kt dt, k = 1, 2, 3,.... (6.10.10) 6.10 The Riemann–Lebesgue Lemma and Pointwise Convergence Theorem 203 Proof. The nth partial sum sn (x) of the series (6.10.8) is sn (x) = 1 2 a0 + n k=1 (ak cos kx + bk sin kx). (6.10.11) We use integrals in (6.10.9)–(6.10.10) to replace ak and bk in (6.10.11) so that sn (x) = 1 2π  π −π 1 1+2n k=1 (cos kt cos kx + sin ktsin kx) 3 f (t) dt = 1 2π  π −π 1 1+2n k=1 cos k (x − t) 3 f (t) dt = 1 2π  π −π Dn (x − t) f (t) dt, (6.10.12) where Dn (θ) is called the Dirichlet kernel defined by Dn (θ)=1+2n k=1 cos kθ. (6.10.13) The next step is to study the properties of this kernel Dn (θ) which is an even function with period 2π and satisfies the condition 1 2π  π −π Dn (θ) dθ =1+0+0+ ... +0=1. (6.10.14) We find the value of the sum in (6.10.13) by Euler’s formula so that Dn (θ)=1+n k=1  e ikθ + e −ikθ = n k=−n e ikθ = e −inθ + ... +1+ ... + e inθ . This is a finite geometric series with the first term e −inθ, the ratio e iθ, and the last term e inθ, and hence, its sum is given by Dn (θ) = e −inθ − e i(n+1)θ 1 − e iθ = exp −  n + 1 2 iθ! − exp n + 1 2 iθ! exp  − 1 2 iθ − exp  + 1 2 iθ = sin  n + 1 2 θ sin 1 2 θ . (6.10.15) The graph of Dn (θ) is shown in Figure 6.10.1. It looks similar to that of the diffusion kernel as drawn in Figure 12.4.1 in Chapter 12 except for its symmetric oscillatory trail. 204 6 Fourier Series and Integrals with Applications Figure 6.10.1 Graph of Dn (θ) against θ. We next put t − x = θ in (6.10.12) to obtain sn (x) = 1 2π  x+π x−π Dn (θ) f (x + θ) dθ. (6.10.16) Since both Dn and f have period 2π, the limits of the integral can be taken from −π to π, and hence, (6.10.16) assumes the form sn (x) = 1 2π  π −π Dn (θ) f (x + θ) dθ. (6.10.17) We next use (6.10.14) to express the difference of sn (x) and 1 2 [f (x+) + f (x−)] in the form sn (x) − 1 2 [f (x+) + f (x−)] = 1 2π  0 −π Dn (θ) [f (x + θ) − f (x−)] dθ + 1 2π  π 0 Dn (θ) [f (x + θ) − f (x+)] dθ which is, by (6.10.15), = 1 2π  0 −π g− (θ) sin  n + 1 2  θ dθ + 1 2π  π 0 g+ (θ) sin  n + 1 2  θ dθ, (6.10.18) 6.10 The Riemann–Lebesgue Lemma and Pointwise Convergence Theorem 205 where g + (θ) =  sin θ 2 −1 [f (x + θ) − f (x + )] . (6.10.19) Since the denominators of the functions g + (θ) vanish at θ = 0, integrals in (6.10.18) may diverge at this point. However, by assumption, f is piecewise smooth, and hence, lim θ→0 + g + (θ) = lim θ→0 + f (x + θ) − f (x) θ 2 · 4 θ 2 sin θ 2 5 = 2f ′ (x +).(6.10.20) Evidently, the above limits exist, and g + (θ) are piecewise continuous elsewhere in the interval (−π, π). Therefore, by the Riemann–Lebesgue Lemma 6.10.1, both integrals in (6.10.18) vanish as n → ∞. Thus, limn→∞ sn (x) = 1 2 [f (x+) + f (x−)] . This proves that the Fourier series converges for each x in (−π, π). Remark 3. At a point of continuity the series converges to the function f (x). Remark 4. At a point of discontinuity, the series is equal to the arithmetic mean of the limits of the function on both sides of the discontinuity. Remark 5. The condition of piecewise smoothness under which the Fourier series converges pointwise is a sufficient condition. A large number of examples of applications is covered by this case. However, the pointwise convergence Theorem 6.10.1 can be proved under weaker conditions. Example 6.10.1. In Example 6.6.1, we obtained that the Fourier series expansion for  x + x 2 in [−π, π], as shown in Figure 6.10.2, is f (x) ∼ π 2 3 + ∞ k=1 4 k 2 (−1)k cos kx − 2 k (−1)k sin kx . Since f (x) = x + x 2 is piecewise smooth, the series converges, and hence, we write x + x 2 = π 2 3 + ∞ k=1 4 k 2 (−1)k cos kx − 2 k (−1)k sin kx , at points of continuity. At points of discontinuity, such as x = π, by virtue of the Pointwise Convergence Theorem, 206 6 Fourier Series and Integrals with Applications Figure 6.10.2 Graph of f (x). 1 2 π + π 2 +  −π + π 2 ! = π 2 3 + ∞ k=1 4 k 2 (−1)k cos kπ, (6.10.21) since f (π−) = π + π 2 and f (π+) = f (−π+) = −π + π 2 . Simplification of equation (6.10.21) gives π 2 = π 2 3 + ∞ k=1 4 k 2 (−1)2k , or π 2 6 = ∞ k=1 1 k 2 . The series can be used to obtain the sum of reciprocals of squares of odd positive integers, that is, ∞ n=1 1 (2n − 1)2 . We have π 2 6 = ∞ n=1 1 n2 = ∞ n=1 1 (2n) 2 + ∞ n=1 1 (2n − 1)2 = 1 4 · π 2 6 + ∞ n=1 1 (2n − 1)2 , 6.10 The Riemann–Lebesgue Lemma and Pointwise Convergence Theorem 207 or, ∞ n=1 1 (2n − 1)2 = π 2 6  1 − 1 4  = π 2 8 . Conversely, this series can be used to find the sum of reciprocals of squares of all positive integers. Example 6.10.2. Find the Fourier series of the following function f (x) = ⎧ ⎨ ⎩ 0, −2 ≤ x < 0 2 − x, 0 < x ≤ 2. This function is defined over the interval −2 ≤ x ≤ 2, where it is piecewise smooth with a finite discontinuity at x = 0. We use (6.9.7) and (6.9.8) to calculate the Fourier coefficients a0 = 1 2  2 0 (2 − x) dx = 1 ak = 1 2  2 0 (2 − x) cos  πkx 2  dx = 2 π 2k 2 " 1 − (−1)k # , k = 1, 2, 3,.... bk = 1 2  2 0 (2 − x) sin  πkx 2  dx = 2 πk , k = 1, 2, 3,.... Consequently, the Fourier series (6.9.6) becomes f (x) = 1 2 + 2 π ∞ k=1 ⎡ ⎣ ( 1 − (−1)k ) πk2 cos  πkx 2  + 1 k sin  πkx 2  ⎤ ⎦. (6.10.22) The function f (x) is continuous at x = 1 where f (1) = 1, so that the Fourier series (6.10.22) gives 1 = 1 2 + 2 π ∞ n=1 {1 − (−1)n } πn2 cos 4nπ 2 5 + 1 n sin 4nπ 2 5 . Since the factor 1 − (−1)n = 0 for even n, and cos nπ 2 = 0 when n is odd, every term of the cosine series vanishes for all n. Consequently, π 4 = ∞ n=1 1 n sin 4nπ 2 5 = ∞ n=1 (−1)n (2n + 1). (6.10.23) On the other hand, f (x) is discontinuous at x = 0 and the Fourier series must converge to 1 2 (0 + 2) = 1. Thus, 208 6 Fourier Series and Integrals with Applications π 2 4 = ∞ n=1 [1 − (−1)n ] n2 = 2∞ n=1 1 (2n − 1)2 , or ∞ n=1 1 (2n − 1)2 = π 2 8 . (6.10.24) 6.11 Uniform Convergence, Differentiation, and Integration In the preceding section, we have proved the pointwise convergence of the Fourier series for a piecewise smooth function. Here, we shall consider several theorems without proof concerning uniform convergence, term-by-term differentiation, and integration of Fourier series. Theorem 6.11.1. (Uniform and Absolute Convergence Theorem) Let f (x) be a continuous function with period 2π, and let f ′ (x) be piecewise continuous in the interval [−π, π]. If, in addition, f (−π) = f (π), then the Fourier series expansion for f (x) is uniformly and absolutely convergent. In the preceding theorem, we have assumed that f (x) is continuous and f ′ (x) is piecewise continuous. With less stringent conditions on f, the following theorem can be proved. Theorem 6.11.2. Let f (x) be piecewise smooth in the interval [−π, π]. If f (x) is periodic with period 2π, then the Fourier series for f converges uniformly to f in every closed interval containing no discontinuity. We note that the partial sums sn (x) of a Fourier series cannot approach the function f (x) uniformly over any interval containing a point of discontinuity of f. The behavior of the deviation of sn (x) from f (x) in such an interval is known as the Gibbs phenomenon. For instance, in the Example 6.7.1, the Fourier series of the function is given by f (x) = 4 π ∞ k=1 sin (2k − 1) x (2k − 1) . (6.11.1) From graphs of the partial sums sn (x) against the x-axis, as shown in Figures 6.7.2 and 6.7.3, we find that sn (x) oscillate above and below the value of f. It can be observed that, near the discontinuous points x = 0 and x = π, sn deviate from the function rather significantly. Although the magnitude of oscillation decreases at all points in the interval for large n, very near the points of discontinuity the amplitude remains practically independent of n as n increases. This illustrates the fact that the Fourier 6.11 Uniform Convergence, Differentiation, and Integration 209 series of a function f does not converge uniformly on any interval which contains a discontinuity. Termwise differentiation of Fourier series is, in general, not permissible. From Example 6.6.3, the Fourier series for f (x) = x is given by x = 2 sin x − sin 2x 2 + sin 3x 3 − ... , (6.11.2) which converges for all x, whereas the series after formal term-by-term differentiation, 1 ∼ 2 [cos x − cos 2x + cos 3x − ...] . This series is not the Fourier series of f ′ (x) = 1, since the Fourier series of f ′ (x) = 1 is the function 1. In fact, this series is not a Fourier series of any piecewise continuous function defined in [−π, π] as the coefficients do not tend to zero which contradicts the Riemann–Lebesque lemma. In fact, the series of f ′ (x) = 1 diverges for all x since the nth term, cos nx does not tend to zero as n → ∞. The difficulty arises from the fact that the given function f (x) = x in [−π, π] when extended periodically is discontinuous at the points + π, + 3π, .... We shall see below that the continuity of the periodic function is one of the conditions that must be met for the termwise differentiation of a Fourier series. Theorem 6.11.3. (Differentiation Theorem) Let f (x) be a continuous function in the interval [−π, π] with f (−π) = f (π), and let f ′ (x) be piecewise smooth in that interval. Then Fourier series for f ′ can be obtained by termwise differentiation of the series for f, and the differentiated series converges pointwise to f ′ at points of continuity and to [f ′ (x) + f ′ (−x)] /2 at discontinuous points. The termwise integration of Fourier series is possible under more general conditions than termwise differentiation. We recall that in calculus, the series of functions to be integrated must converge uniformly in order to assure the convergence of a termwise integrated series. However, in the case of Fourier series, this condition is not necessary. Theorem 6.11.4. (Integration Theorem) Let f (x) be piecewise continuous in [−π, π], and periodic with period 2π. Then the Fourier series of f (x) a0 2 + ∞ k=1 (ak cos kx + bk sin kx), whether convergent or not, can be integrated term by term between any limits. 210 6 Fourier Series and Integrals with Applications Example 6.11.1. In Example 6.7.2, we have found that f (x) = |sin x| is represented by the Fourier series sin x = 2 π + 4 π ∞ k=1 cos (2kx) (1 − 4k 2) , −π < x < π. (6.11.3) Since f (x) = |sin x| is continuous in the interval [−π, π] and f (−π) = f (π), we differentiate the series term by term, obtaining cos x = − 8 π ∞ k=1 k sin (2kx) (1 − 4k 2) , (6.11.4) by use of Theorem 6.11.3, since f ′ (x) is piecewise smooth in [−π, π]. In this way, we obtain the Fourier sine series expansion of the cosine function in (−π, π). Note that the reverse process is not permissible. Example 6.11.2. Consider the function f (x) = x in the interval −π 0, lim λ→∞  b 0 f (x) sin λx x dx = π 2 f (0+). Proof.  b 0 f (x) sin λx x dx =  b 0 f (0+) sin λx x dx +  b 0 f (x) − f (0+) x sin λx dx = f (0+)  λb 0 sin t t dt +  b 0 f (x) − f (0+) x sin λx dx. Since f is piecewise smooth, the integrand of the last integral is bounded as λ → ∞, and thus, by the Riemann–Lebesgue lemma 6.10.1, the last integral tends to zero as λ → ∞. Hence, lim λ→∞  b 0 f (x) sin λx x dx = π 2 f (0+), (6.13.6) since  ∞ 0 sin t t dt = π 2 . Theorem 6.13.1. (Fourier Integral Theorem) If f is piecewise smooth in every finite interval, and absolutely integrable on (−∞,∞), then 1 π  ∞ 0  ∞ −∞ f (t) cos k (t − x) dt dk = 1 2 [f (x+) + f (x−)] . Proof. Noting that |cos k (t − x)| ≤ 1 and that by hypothesis  ∞ −∞ f (t) dt < ∞, we see that the integral  ∞ −∞ f (t) cos k (t − x) dt converges independently of k and x. It therefore follows that in the double integral I =  λ 0  ∞ −∞ f (t) cos k (t − x) dt dk, the order of integration may be interchanged. We then have 218 6 Fourier Series and Integrals with Applications I =  ∞ −∞ f (t) 1 λ 0 cos k (t − x) dk3 dt =  ∞ −∞ f (t) sin λ (t − x) (t − x) dt = 1 −M −∞ +  x −M +  M x +  ∞ M 3 f (t) sin λ (t − x) (t − x) dt. If we substitute u = t − x, we have  M x f (t) sin λ (t − x) (t − x) dt =  M−x 0 f (u + x)  sin λu u  du which is equal to πf (x+) /2 in the limit λ → ∞, by Lemma 6.13.1. Similarly, the second integral tends to πf (x−) /2 when λ → ∞. If we make M sufficiently large, the absolute values of the first and the last integrals are each less than ε/2. Consequently, as λ → ∞  ∞ 0  ∞ −∞ f (t) cos k (t − x) dt dk = π 2 [f (x+) + f (x−)] . (6.13.7) If f is continuous at the point x, then f (x+) = f (x−) = f (x) so that integral (6.13.7) reduces to the Fourier integral representation for f as f (x) = 1 π  ∞ 0  ∞ −∞ f (t) cos k (t − x) dt dk. (6.13.8) We may express the Fourier integral representation (6.13.8) in complex form. In this case, we substitute cos k (t − x) = cos k (x − t) = 1 2 " e ik(x−t) + e −ik(x−t) # into equation (6.13.8) and write it as the sum of two integrals f (x) = 1 2π  ∞ 0  ∞ −∞ f (t) e ik(x−t) dt dk + 1 2π  ∞ 0  ∞ −∞ f (t) e −ik(x−t) dt dk. Changing the integration variable from k to −k in the second integral, we obtain f (x) = 1 2π  ∞ 0  ∞ −∞ f (t) e ik(x−t) dt dk −  −∞ 0  ∞ −∞ f (t) e ik(x−t) dt dk = 1 2π  ∞ 0  ∞ −∞ f (t) e ik(x−t) dt dk +  0 −∞  ∞ −∞ f (t) e ik(x−t) dt dk = 1 2π  ∞ −∞  ∞ −∞ f (t) e ik(x−t) dt dk. (6.13.9) 6.13 Fourier Integrals 219 Or, equivalently, f (x) = 1 √ 2π  ∞ −∞ e ikxdk 1 √ 2π  ∞ −∞ e −iktf (t) dt = 1 √ 2π  ∞ −∞ F (k) e ikxdk, (6.13.10) where F (k) = 1 √ 2π  ∞ −∞ e −iktf (t) dt. (6.13.11) Either (6.13.9) or (6.13.10) with coefficient F (k) is called the complex form of the Fourier integral representation for f (x). Now we assume that f (x) is either an even or an odd function. Any function that is not even or odd can be expressed as a sum of two such functions. Expanding the cosine function in (6.13.8), we obtain the Fourier cosine formula f (x) = f (−x) = 2 π  ∞ 0 cos kx dk  ∞ 0 cos kt f (t) dt. (6.13.12) Similarly, for an odd function, we obtain the Fourier sine formula f (x) = −f (−x) = 2 π  ∞ 0 sin kx dk  ∞ 0 sin kt f (t) dt. (6.13.13) Example 6.13.1. The rectangular pulse can be expressed as a sum of Heaviside functions f (x) = H (x + 1) − H (x − 1). Find its Fourier integral representation. From (6.13.5) we find f (x) = 1 π  ∞ 0  1 −1 cos [k (t − x)] dt dk = 1 π  ∞ 0 cos kx  1 −1 cos kt dt + sin kx  1 −1 sin kt dt dk = 2 π  ∞ 0  sin k k  cos kx dk. Example 6.13.2. Find the Fourier cosine integral representation of the function f (x) = ⎧ ⎨ ⎩ 1, 0 <x< 1,="" 0,="" x="" ≥="" 1.="" 220="" 6="" fourier="" series="" and="" integrals="" with="" applications="" we="" have,="" from="" (6.13.12),="" f="" (x)="2" π="" ="" ∞="" 0="" cos="" kx="" dk="" 1="" kt="" dt="2" ="" sin="" k="" ="" dk,="" or,="" dk.="" 6.14="" exercises="" find="" the="" of="" following="" functions:="" (a)="" ⎨="" ⎩="" h="" −π<x<="" <="" π,="" is="" a="" constant="" (b)="" 2="" (c)="" +="" −π<x<π,="" (d)="" (e)="" (f)="" −π<x<π.="" 2.="" determine="" sine="" −="" <x<π,="" 3="" <x<π.="" 221="" 3.="" obtain="" cosine="" representation="" for="" 3x="" 4.="" expand="" functions="" in="" series:="" −1="" <x<="" 6,="" (πx="" l)="" <x<l,="" −2="" 2,="" −x="" 5.="" complex="" 2x="" 2)="" 222="" 6.="" expansion="" function="" π.="" use="" 6(a),="" show="" that="" 8="1+" 5="" 7="" ....="" 7.="" ,="" −l="" l.="" 7(a),="" 12="1" 4="" 8.="" each="" by="" performing="" differentiation="" appropriate="" sin2="" cos2="" 9.="" represented="" new="" which="" are="" obtained="" termwise="" integration="" to="" x:="" ∞="" (−1)k+1="" 1−(−1)k="" ="" 223="" sin(2k+1)x="" (2k+1)3="π" 2x−πx2="" 2π,="" sin(2k−1)x="" (2k−1)="⎧" 10.="" double="" (x,="" y)="1" <x<π="" <y<π,="" 2y="" y="" −π<x<π="" −π<y<π,="" x+y="" (g)="" <y<="" (h)="" <x<a="" <y<b,="" (i)="" (j)="" (k)="" −π<y<π.="" 11.="" deduce="" general="" formula="" rectangle="" −a<x<a,="" −b<y<b.="" 12.="" prove="" weierstrass="" approximation="" theorem:="" if="" continuous="" on="" interval="" −π="" ≤="" (−π)="f" (π),="" then,="" any="" ε=""> 0, there exists a trigonometric polynomial T (x) = a0 2 + n k=1 (ak cos kx + bk sin kx) such that |f (x) − T (x)| < ε for all x in [−π, π]. 224 6 Fourier Series and Integrals with Applications 13. Use the Fourier cosine or sine integral formula to show that (a) e −αx = 2 π  ∞ 0 α α2+β2 cos βx dβ, x ≥ 0, α > 0, (b) e −αx = 2 π  ∞ 0 β α2+β2 sin βx dβ, x > 0, α > 0. 14. Show that the Fourier integral representation of the function f (x) = ⎧ ⎨ ⎩ x 2 , 0 <xa is f (x) = 2 π  ∞ 0 a 2 − 2 k 2  sin ak + 2a k cos ak cos kx k dk. 15. Apply the Parseval relation (6.5.10) to Example 6.7.3 or Example 6.7.4 to show that (a) ∞ n=1 1 (2n − 1)4 = π 4 96 and (b) ∞ n=1 1 n4 = π 4 90 . 16. (a) Obtain the Fourier series for the 2π-periodic odd function f (x) = x (π − x) on [0, π]. (b) Use the Parseval relation (6.5.10) to show that ∞ n=1 1 (2n − 1)6 = π 6 960 and ∞ n=1 1 n6 = π 6 945 . 17. If the 2π-periodic even function is given by f (x) = |x| for −π ≤ x ≤ π, show that f (x) = π 2 − 4 π ∞ n=1 cos (2n − 1) x (2n − 1)2 . 18. Consider the sawtooth function defined by f (x) = π − x, 0 <x< 2π,="" and="" f="" (x="" +="" 2nπ)="f" (x)="" with="" (0)="0." (a)="" show="" that="" the="" fourier="" series="" for="" this="" function="" is="" k="0" 2="" sin="" kx,="" has="" a="" jump="" discontinuity="" at="" origin="" 6.14="" exercises="" 225="" (0+)="π" ,="" (0−)="−" π="" −="" (b)="" max="" 0≤x≤="" n="" sn="" 0="" θ="" dθ="" .="" (c)="" result="" manifestation="" of="" gibbs="" phenomenon,="" is,="" near="" discontinuity,="" overshoots="" (or="" undershoots)="" it="" by="" approximately="" 9%="" jump.="" (d)="" if="" dn="" in="" ≤="" x="" d="" ′="" (x),="" where="" given="" (6.10.15).="" (e)="" using="" e="" ikx,="" first="" critical="" point="" to="" right="" occurs="" xn="π/" ="" 1="" limn→∞="" (xn)="2" ="" π.="" (f)="" draw="" graph="" fortieth="" partial="" sum="" s40="" 40="" kx="" −2π<x<="" then="" examine="" phenomenon="" (x).="" 19.="" consider="" characteristic="" interval="" [a,="" b]="" ⊂="" [−π,="" π]="" defined="" ⎨="" ⎩="" 1,="" b="" 0,="" otherwise.="" ∼="" 2π="" (b="" a)="" +="" k="0" exp="" (−ika)="" (−ikb)="" 2πik="" ·="" ikx="" 20.="" obtain="" (π="" x),="" derive="" following="" numerical="" 3="" 5="" 7="" ...="π" 12="" 9="" 113="" √="" 128="" 226="" 6="" integrals="" applications="" 21.="" triangular="" vertices="" (0,="" 0),="" 2,="" 1)="" (π,="" 0)="" 2x="" ="" 3x="" 5x="" 7x="" ...0="" ∞="" (2n="" 1)2="π" 8="" 22.="" sine="" cosine="" functions:="" <x<a,="" <x<a.="" 23.="" find="" full="" functions="" −1="" x,="" <x<="" +1="" 1.="" π,="" 2π.="" 24.="" cos="" <="" 25.="" 2i="" ="" n="0" inx="" complex="" 2π-periodic="" sawtooth="" 227="" −π<x<="" 26.="" suppose="" g="" have="" expansion="" −π="" π:="" a0="" (ak="" bk="" kx),="" α0="" (αk="" βk="" together="" their="" two="" derivatives="" are="" continuous="" on="" (−π)="f" (π),="" (π)="" hold.="" prove="" general="" parseval="" relation="" holds:="" dx="1" a0α0="" (akαk="" bkβk).="" when="" (6.5.10)="" special="" case="" above="" result.="" 27.="" integral="" representation="" (a="" |x|)="⎧" |x|=""> a, (b) f (x) = ⎧ ⎨ ⎩ sin x, |x| < π 0, |x| > π. 28. If f (x), x ∈ R, is defined by f (x) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ −1, −a<x< 0,="" +1,="" 0="" ≤="" x="" <="" a,="" otherwise,="" show="" that="" f="" (x)="" has="" the="" fourier="" sine="" integral="" representation="" π="" ="" ∞="" 1="" k="" (1="" −="" cos="" ka)="" sin="" kx="" dk.="" 228="" 6="" series="" and="" integrals="" with="" applications="" 29.="" if="" −x="" ,="" <x<="" ∞,="" (a)="" is="" +="" 2="" dk,="" (b)="" cosine="" 30.="" defined="" by="" ⎨="" ⎩="" x<="" e=""> 0, show that (a) the Fourier integral representation of f (x) is f (x) = 1 π  ∞ 0 (cos kx + k sin kx) (1 + k 2) dk, (b) the Fourier cosine integral representation of f (x) is f (x) = 1 2π  ∞ −∞  1 − ik 1 + k 2  e ikxdk. 31. (a) Obtain both the complex Fourier series and the usual Fourier series of f (x) = exp [x (1 + 2πi)] on the interval [−1, 1]. (b) Find the sum of each of the series ∞ k=1 1 (1 + π 2k 2) and ∞ k=1 (−1)k (1 + π 2k 2) . 32. Use Example 6.7.2 to calculate the value of the following series: (a) ∞ k=1 1 (4k 2 − 1), (b) ∞ k=1 (−1)k (4k 2 − 1), (c) ∞ k=1 1 (4k 2 − 1)2 , and (d) ∞ k=1 k 2 (4k 2 − 1)2 . 33. Show that the complex Fourier series of f (x) = x is given by x ∼ ∞ k=1 (−1)k k  i eikx + −∞ k=−1 (−1)k k  i eikx . 6.14 Exercises 229 34. (a) Show that the Fourier series for f (x) is defined by f (x) = ⎧ ⎨ ⎩ sin 2x, 0 ≤ x ≤ π 2 0, π 2 ≤ x ≤ π, is f (x) = 1 π + 1 2 sin 2x −  2 π ∞ k=1 cos 4kx (4k 2 − 1). (b) Show that ∞ k=1 1 (4k 2 − 1)2 = 1 16  π 2 − 8 . (c) Find the sum of the infinite series sin (4x) 1.2.3 + sin (2.4x) 3.4.5 + sin (3.4x) 5.6.7 + ..., 0 ≤ x ≤ π. 35. (a) Obtain the complex Fourier series of f (x) = cos (ax), −π ≤ x ≤ π, where a is real but not an integer. (b) Hence, show that π cot πx = 1 x − ∞ k=1 2x (k 2 − x 2) . (c) Derive the product formula sin πx = πx 3∞ n=1  1 − x 2 n2  . (d) Show that π 2 = 3∞ n=1 2n (2n − 1) · 2n (2n + 1) =  2 1 · 2 3  ·  4 3 · 4 5  ·  6 5 · 6 7  ·  8 7 · 8 9  .... 36. Obtain the Fourier series of the following functions: (a) f (x) = e x , 0 ≤ x ≤ 2π, f (x + 2π) = f (x). (b) f (x) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ +1, −π<x< −="" π="" 2="" ,="" 0="" <x<="" −1,="" 0,="" <="" x="" π,="" n="0," +="" 1,="" 2,...="" 230="" 6="" fourier="" series="" and="" integrals="" with="" applications="" draw="" the="" graph="" of="" this="" function.="" (c)="" f="" (x)="x" [x],="" where="" [x]="" is="" greatest="" integer="" not="" exceeding="" x.="" 37.="" find="" for="" each="" functions="" in="" −l<x<l="" defined="" outside="" interval="" so="" that="" (x="" 2l)="f" all="" x:="" (a)="" ⎨="" ⎩="" −l<x<="" l,="" l.="" (b)="" −x,="" −l="" ≤="" x,="" l="" (d)="" examine="" gibbs="" phenomenon="" at="" points="" discontinuity="" function="" (a).="" 38.="" prove="" following="" identities:="" 1="" n="" k="1" cos="" kx="sin" ="" sin="" .="" sn="" ="" −π="" (ξ="" x)="" ξ="" dξ,="" nth="" partial="" sum="" a="" (−π,="" π).="" 7="" method="" separation="" variables="" “however,="" emphasis="" should="" be="" somewhat="" more="" on="" how="" to="" do="" mathematics="" quickly="" easily,="" what="" formulas="" are="" true,="" rather="" than="" mathematicians’="" interest="" methods="" rigorous="" proof.”="" richard="" feynman="" “as="" science,="" has="" been="" adapted="" description="" natural="" phenomena,="" great="" practitioners="" field,="" such="" as="" von="" k´arm´an,="" taylor="" lighthill,="" have="" never="" concerned="" themselves="" logical="" foundations="" mathematics,="" but="" boldly="" taken="" pragmatic="" view="" an="" intellectual="" machine="" which="" works="" successfully.="" verified="" by="" further="" observation,="" still="" strikingly="" prediction,="" ....="" ”="" george="" temple="" 7.1="" introduction="" combined="" principle="" superposition="" widely="" used="" solve="" initial="" boundary-value="" problems="" involving="" linear="" differential="" equations.="" usually,="" dependent="" variable="" u="" (x,="" y)="" expressed="" separable="" form="" y="" (y),="" respectively.="" many="" cases,="" equation="" reduces="" two="" ordinary="" equations="" similar="" treatment="" can="" applied="" three="" or="" independent="" variables.="" however,="" question="" separability="" into="" no="" means="" trivial="" one.="" spite="" question,="" finding="" solutions="" large="" class="" problems.="" 232="" solution="" also="" known="" (or="" eigenfunction="" expansion).="" thus,="" procedure="" outlined="" above="" leads="" important="" ideas="" eigenvalues,="" eigenfunctions,="" orthogonality,="" very="" general="" powerful="" dealing="" examples="" illustrate="" nature="" solution.="" 7.2="" section,="" we="" shall="" introduce="" one="" most="" common="" elementary="" methods,="" called="" variables,="" solving="" applicable="" contains="" wide="" range="" mathematical="" physics,="" engineering="" science.="" now="" describe="" conditions="" applicability="" involve="" second-order="" consider="" homogeneous="" ∗ux∗x∗="" b="" ∗ux∗y∗="" c="" ∗uy∗y∗="" d="" ∗ux∗="" e="" ∗uy∗="" ∗u="0" (7.2.1)="" ∗="" stated="" chapter="" 4="" transformation="" y∗="" ),="" (7.2.2)="" ∂="" ∗,="" y∗)="0," always="" transform="" canonical="" uxx="" uyy="" ux="" uy="" when="" (i)="" hyperbolic,="" (ii)="" parabolic,="" (iii)="" elliptic.="" assume="" (7.2.3)="" (y)="0," (7.2.4)="" are,="" respectively,="" alone,="" twice="" continuously="" differentiable.="" substituting="" (7.2.3),="" obtain="" 233="" x′′y="" cxy="" ′′="" x′y="" exy="" ′="" xy="0," (7.2.5)="" primes="" denote="" differentiation="" respect="" appropriate="" let="" there="" exist="" p="" y),="" that,="" if="" divide="" a1="" b1="" a2="" b2="" [a3="" b3="" (y)]="" (7.2.6)="" dividing="" again="" x′′="" x′="" a3="−" (7.2.7)="" left="" side="" only.="" right="" depends="" only="" upon="" y.="" differentiate="" dx="" (7.2.8)="" integration="" yields="" (7.2.9)="" λ="" constant.="" from="" (7.2.9),="" (7.2.10)="" may="" rewrite="" a1x′′="" a2x′="" (a3="" λ)="" (7.2.11)="" b1y="" b2y="" (b3="" (7.2.12)="" coefficients="" constant,="" then="" reduction="" longer="" necessary.="" this,="" auxx="" buxy="" cuyy="" dux="" euy="" (7.2.13)="" 234="" a,="" b,="" c,="" d,="" e,="" constants="" zero.="" before,="" (7.2.13),="" ax′′y="" bx′y="" dx′y="" fxy="0." (7.2.14)="" division="" axy="" (7.2.15)="" ="" ′="" (7.2.16)="" 4="" 5′="" (7.2.17)="" obviously="" separable,="" both="" sides="" must="" equal="" constant="" λ.="" therefore,="" λy="0," (7.2.18)="" ="" (7.2.19)="" integrating="" (7.2.20)="" β="" determined.="" original="" (7.2.15),="" (7.2.21)="" comparing="" (7.2.21),="" clearly="" satisfy="" 7.3="" vibrating="" string="" problem="" 235="" just="" described="" given="" equation.="" now,="" take="" look="" boundary="" involved.="" several="" types="" conditions.="" ones="" appear="" frequently="" physics="" include="" dirichlet="" condition:="" prescribed="" neumann="" (∂u="" ∂n)="" mixed="" hu="" boundary,="" directional="" derivative="" along="" outward="" normal="" h="" continuous="" boundary.="" details,="" see="" 9="" besides="" these="" conditions,="" as,="" first,="" second,="" third="" other="" robin="" condition;="" condition="" portion="" another="" remainder="" variety="" treat="" later.="" separate="" listed="" above,="" it="" best="" choose="" coordinate="" system="" suitable="" instance,="" cartesian="" rectangular="" region="" lines="" polar="" (r,="" θ)="" circular="" r="constant" θ="constant." imposed="" say="" contain="" derivatives="" only,="" their="" depend="" example,="" [u="" uy]x="x0" =="" cannot="" separated.="" needless="" say,="" condition,="" uy,="" axis.="" first="" tension="" t="" density="" ρ="" stretched="" xaxis="" fixed="" its="" end="" points.="" seen="" 5="" utt="" 2uxx="0,"> 0, (7.3.1) u (x, 0) = f (x), 0 ≤ x ≤ l, (7.3.2) ut (x, 0) = g (x), 0 ≤ x ≤ l, (7.3.3) u (0, t)=0, t ≥ 0, (7.3.4) u (l, t)=0, t ≥ 0, (7.3.5) 236 7 Method of Separation of Variables where f and g are the initial displacement and initial velocity respectively. By the method of separation of variables, we assume a solution in the form u (x, t) = X (x) T (t) = 0. (7.3.6) If we substitute equation (7.3.6) into equation (7.3.1), we obtain XT′′ = c 2X′′T, and hence, X′′ X = 1 c 2 T ′′ T , (7.3.7) whenever XT = 0. Since the left side of equation (7.3.7) is independent of t and the right side is independent of x, we must have X′′ X = 1 c 2 T ′′ T = λ, where λ is a separation constant. Thus, X′′ − λX = 0, (7.3.8) T ′′ − λc2T = 0. (7.3.9) We now separate the boundary conditions. From equations (7.3.4) and (7.3.6), we obtain u (0, t) = X (0) T (t)=0. We know that T (t) = 0 for all values of t, therefore, X (0) = 0. (7.3.10) In a similar manner, boundary condition (7.3.5) implies X (l)=0. (7.3.11) To determine X (x) we first solve the eigenvalue problem (eigenvalue problems are also treated in Chapter 8) X′′ − λX = 0, X (0) = 0, X (l)=0. (7.3.12) We look for values of λ which gives us nontrivial solutions. We consider three possible cases λ > 0, λ = 0, λ < 0. Case 1. λ > 0. The general solution in this case is of the form 7.3 The Vibrating String Problem 237 X (x) = Ae− √ λ x + Be √ λ x where A and B are arbitrary constants. To satisfy the boundary conditions, we must have A + B = 0, Ae − √ λ l + Be √ λ l = 0. (7.3.13) We see that the determinant of the system (7.3.13) is different from zero. Consequently, A and B must both be zero, and hence, the general solution X (x) is identically zero. The solution is trivial and hence, is no interest. Case 2. λ = 0. Here, the general solution is X (x) = A + Bx. Applying the boundary conditions, we have A = 0, A + Bl = 0. Hence A = B = 0. The solution is thus identically zero. Case 3. λ < 0. In this case, the general solution assumes the form X (x) = A cos √ −λ x + B sin √ −λ x. From the condition X (0) = 0, we obtain A = 0. The condition X (l)=0 gives B sin √ −λ l = 0. If B = 0, the solution is trivial. For nontrivial solutions, B = 0, hence, sin √ −λ l = 0. This equation is satisfied when √ −λ l = nπ for n = 1, 2, 3,..., or −λn = (nπ/l) 2 . (7.3.14) For this infinite set of discrete values of λ, the problem has a nontrivial solution. These values of λn are called the eigenvalues of the problem, and the functions sin (nπ/l) x, n = 1, 2, 3,... are the corresponding eigenfunctions. We note that it is not necessary to consider negative values of n since sin (−n) πx/l = − sin nπx/l. 238 7 Method of Separation of Variables No new solution is obtained in this way. The solutions of problems (7.3.12) are, therefore, Xn (x) = Bn sin (nπx/l). (7.3.15) For λ = λn, the general solution of equation (7.3.9) may be written in the form Tn (t) = Cn cos 4nπc l 5 t + Dn sin 4nπc l 5 t, (7.3.16) where Cn and Dn are arbitrary constants. Thus, the functions un (x, t) = Xn (x) Tn (t) = 4 an cos nπc l t + bn sin nπc l t 5 sin 4nπx l 5 (7.3.17) satisfy equation (7.3.1) and the boundary conditions (7.3.4) and (7.3.5), where an = BnCn and bn = BnDn. Since equation (7.3.1) is linear and homogeneous, by the superposition principle, the infinite series u (x, t) = ∞ n=1 4 an cos nπc l t + bn sin nπc l t 5 sin 4nπx l 5 (7.3.18) is also a solution, provided it converges and is twice continuously differentiable with respect to x and t. Since each term of the series satisfies the boundary conditions (7.3.4) and (7.3.5), the series satisfies these conditions. There remain two more initial conditions to be satisfied. From these conditions, we shall determine the constants an and bn. First we differentiate the series (7.3.18) with respect to t. We have ut = ∞ n=1 nπc l 4 −an sin nπc l t + bn cos nπc l t 5 sin 4nπx l 5 . (7.3.19) Then applying the initial conditions (7.3.2) and (7.3.3), we obtain u (x, 0) = f (x) = ∞ n=1 an sin 4nπx l 5 , (7.3.20) ut (x, 0) = g (x) = ∞ n=1 bn 4nπc l 5 sin 4nπx l 5 . (7.3.21) These equations will be satisfied if f (x) and g (x) can be represented by Fourier sine series. The coefficients are given by an = 2 l  l 0 f (x) sin 4nπx l 5 dx, bn = 2 nπc  l 0 g (x) sin 4nπx l 5 dx, (7.3.22ab) 7.3 The Vibrating String Problem 239 The solution of the vibrating string problem is therefore given by the series (7.3.18) where the coefficients an and bn are determined by the formulae (7.3.22ab). We examine the physical significance of the solution (7.3.17) in the context of the free vibration of a string of length l. The eigenfunctions un (x, t)=(an cos ωnt + bn sin ωnt) sin 4nπx l 5 , ωn = nπc l , (7.3.23) are called the nth normal modes of vibration or the nth harmonic, and ωn represent the discrete spectrum of circular (or radian) frequencies or νn = ωn 2π = nc 2l , which are called the angular frequencies. The first harmonic (n = 1) is called the fundamental harmonic and all other harmonics (n > 1) are called overtones. The frequency of the fundamental mode is given by ω1 = πc l , ν1 = 1 2l $ T ∗ ρ . (7.3.24) Result (7.3.24) is considered the fundamental law (or Mersenne law) of a stringed musical instrument. The angular frequency of the fundamental mode of transverse vibration of a string varies as the square root of the tension, inversely as length, and inversely as the square root of the density. The period of the fundamental mode is T1 = 2c ω1 = 2l c , which is called the fundamental period. Finally, the solution (7.3.18) describes the motion of a plucked string as a superposition of all normal modes of vibration with frequencies which are all integral multiples (ωn = nω1 or νn = nν1) of the fundamental frequency. This is the main reason that stringed instruments produce sweeter musical sounds (or tones) than drum instruments. In order to describe waves produced in the plucked string with zero initial velocity (ut (x, 0) = 0), we write the solution (7.3.23) in the form un (x, t) = an sin 4nπx l 5 cos  nπct l  , n = 1, 2, 3,.... (7.3.25) These solutions are called standing waves with amplitude an sin  nπx l , which vanishes at x = 0, l n , 2l n , . . . ,l. These are called the nodes of the nth harmonic. The string displays n loops separated by the nodes as shown in Figure 7.3.1. It follows from elementary trigonometry that (7.3.25) takes the form un (x, t) = 1 2 an " sin nπ l (x − ct) + sin nπ l (x + ct) # . (7.3.26) This shows that a standing wave is expressed as a sum of two progressive waves of equal amplitude traveling in opposite directions. This result is in agreement with the d’Alembert solution. 240 7 Method of Separation of Variables Figure 7.3.1 Several modes of vibration in a string. Finally, we can rewrite the solution (7.3.23) of the nth normal modes in the form un (x, t) = cn sin 4nπx l 5 cos  nπct l − εn  , (7.3.27) where cn =  a 2 n + b 2 n 1 2 and tan εn = 4 bn an 5 . This solution represents transverse vibrations of the string at any point x and at any time t with amplitude cn sin  nπx l and circular frequency ωn = nπc l . This form of the solution enables us to calculate the kinetic and potential energies of the transverse vibrations. The total kinetic energy (K.E.) is obtained by integrating with respect to x from 0 to l, that is, Kn = K.E. =  l 0 1 2 ρ  ∂un ∂t 2 dx, (7.3.28) where ρ is the line density of the string. Similarly, the total potential energy (P.E.) is given by Vn = P.E. = 1 2 T ∗  l 0  ∂un ∂x 2 dx. (7.3.29) Substituting (7.3.27) in (7.3.28) and (7.3.29) gives Kn = 1 2 ρ 4nπc l cn 52 sin2  nπct l − εn   l 0 sin2 4nπx l 5 dx = ρc2π 2 4l (n cn) 2 sin2  nπct l − εn  = 1 4 ρlω2 n c 2 n sin2 (ωnt − εn),(7.3.30) where ωn = nπc l . 7.3 The Vibrating String Problem 241 Similarly, Vn = 1 2 T ∗ 4nπcn l 52 cos2  nπct l − εn   l 0 cos2 4nπx l 5 dx = π 2T ∗ 4l (n cn) 2 cos2  nπct l − εn  = 1 4 ρlω2 n c 2 n cos2 (ωnt − εn). (7.3.31) Thus, the total energy of the nth normal mode of vibrations is given by En = Kn + Vn = 1 4 ρl(ωncn) 2 = constant. (7.3.32) For a given string oscillating in a normal mode, the total energy is proportional to the square of the circular frequency and to the square of the amplitude. Finally, the total energy of the system is given by E = ∞ n=1 En = 1 4 ρl∞ n=1 ω 2 n c 2 n , (7.3.33) which is constant because En = constant. Example 7.3.1. The Plucked String of length l As a special case of the problem just treated, consider a stretched string fixed at both ends. Suppose the string is raised to a height h at x = a and then released. The string will oscillate freely. The initial conditions, as shown in Figure 7.3.2, may be written u (x, 0) = f (x) = ⎧ ⎨ ⎩ hx/a, 0 ≤ x ≤ a h (l − x) / (l − a), a ≤ x ≤ l. Since g (x) = 0, the coefficients bn are identically equal to zero. The coeffi- cients an, according to equation (7.3.22a), are given by an = 2 l  l 0 f (x) sin 4nπx l 5 dx = 2 l  a 0 hx a sin 4nπx l 5 dx + 2 l  l a h (l − x) (l − a) sin 4nπx l 5 dx. Integration and simplification yields an = 2hl2 π 2a (l − a) 1 n2 sin 4nπa l 5 . Thus, the displacement of the plucked string is u (x, t) = 2hl2 π 2a (l − a) ∞ n=1 1 n2 sin 4nπa l 5 sin 4nπx l 5 cos 4nπc l 5 t. 242 7 Method of Separation of Variables Figure 7.3.2 Plucked String Example 7.3.2. The struck string of length l Here, we consider the string with no initial displacement. Let the string be struck at x = a so that the initial velocity is given by ut (x, 0) = ⎧ ⎨ ⎩ v0 a x, 0 ≤ x ≤ a v0 (l − x) / (l − a), a ≤ x ≤ l . Since u (x, 0) = 0, we have an = 0. By applying equation (7.3.22b), we find that bn = 2 nπc  a 0 v0 a x sin 4nπx l 5 dx + 2 nπc  l a v0 (l − x) (l − a) sin 4nπx l 5 dx = 2v0l 3 π 3ca (l − a) 1 n3 sin 4nπa l 5 . Hence, the displacement of the struck string is u (x, t) = 2v0l 3 π 3ca (l − a) ∞ n=1 1 n3 sin 4nπa l 5 sin 4nπx l 5 cos 4nπc l 5 t. 7.4 Existence and Uniqueness of Solution of the Vibrating String Problem 243 7.4 Existence and Uniqueness of Solution of the Vibrating String Problem In the preceding section we found that the initial boundary-value problem (7.3.1)–(7.3.5) has a formal solution given by (7.3.18). We shall now show that the expression (7.3.18) is the solution of the problem under certain conditions. First we see that u1 (x, t) = ∞ n=1 an cos 4nπc l t 5 sin 4nπx l 5 (7.4.1) is the formal solution of the problem (7.3.1)–(7.3.5) with g (x) ≡ 0, and u2 (x, t) = ∞ n=1 bn sin 4nπc l t 5 sin 4nπx l 5 (7.4.2) is the formal solution of the above problem with f (x) ≡ 0. By linearity of the problem, the solution (7.3.18) may be considered as the sum of the two formal solutions (7.4.1) and (7.4.2). We first assume that f (x) and f ′ (x) are continuous on [0, l], and f (0) = f (l) = 0. Then by Theorem 6.10.1, the series for the function f (x) given by (7.3.20) converges absolutely and uniformly on the interval [0, l]. Using the trigonometric identity sin 4nπx l 5 cos 4nπc l t 5 = 1 2 sin nπ l (x − ct) + 1 2 sin nπ l (x + ct), (7.4.3) u1 (x, t) may be written as u1 (x, t) = 1 2 ∞ n=1 an sin nπ l (x − ct) + 1 2 ∞ n=1 an sin nπ l (x + ct). Define F (x) = ∞ n=1 an sin 4nπx l 5 (7.4.4) and assume that F (x) is the odd periodic extension of f (x), that is, F (x) = f (x) 0 ≤ x ≤ l F (−x) = −F (x) for all x F (x + 2l) = F (x). We can now rewrite u1 (x, t) in the form u1 (x, t) = 1 2 [F (x − ct) + F (x + ct)] . (7.4.5) 244 7 Method of Separation of Variables To show that the boundary conditions are satisfied, we note that u1 (0, t) = 1 2 [F (−ct) + F (ct)] = 1 2 [−F (ct) + F (ct)] = 0 u1 (l, t) = 1 2 [F (l − ct) + F (l + ct)] = 1 2 [F (−l − ct) + F (l + ct)] = 1 2 [−F (l + ct) + F (l + ct)] = 0. Since u1 (x, 0) = 1 2 [F (x) + F (x)] = F (x) = f (x), 0 ≤ x ≤ l, we see that the initial condition u1 (x, 0) = f (x) is satisfied. Thus, equation (7.3.1) and conditions (7.3.2)–(7.3.3) with g (x) ≡ 0 are satisfied. Since f ′ is continuous in [0, l], F ′ exists and is continuous for all x. Thus, if we differentiate u1 (x, t) with respect to t, we obtain ∂u1 ∂t = 1 2 [−c F′ (x − ct) + c F′ (x + ct)] , and ∂u1 ∂t (x, 0) = 1 2 [−c F′ (x) + c F′ (x)] = 0. We therefore see that initial condition (7.3.3) is also satisfied. In order to show that u1 (x, t) satisfies the differential equation (7.3.1), we impose additional restrictions on f. Let f ′′ be continuous on [0, l] and f ′′ (0) = f ′′ (l) = 0. Then, F ′′ exists and is continuous everywhere, and therefore, ∂ 2u1 ∂t2 = 1 2 c 2 [F ′′ (x − ct) + F ′′ (x + ct)] , ∂ 2u1 ∂x2 = 1 2 [F ′′ (x − ct) + F ′′ (x + ct)] . We find therefore that ∂ 2u1 ∂t2 = c 2 ∂ 2u1 ∂x2 . Next, we shall state the assumptions which must be imposed on g to make u2 (x, t) the solution of problem (7.3.1)–(7.3.5) with f (x) ≡ 0. Let g 7.4 Existence and Uniqueness of Solution of the Vibrating String Problem 245 and g ′ be continuous on [0, l] and let g (0) = g (l) = 0. Then the series for the function g (x) given by (7.3.21) converges absolutely and uniformly in the interval [0, l]. Introducing the new coefficients cn = (nπc/l) bn, we have u2 (x, t) =  l πc∞ n=1 cn n sin 4nπc l t 5 sin 4nπx l 5 . (7.4.6) We shall see that term-by-term differentiation with respect to t is permitted, and hence, ∂u2 ∂t = ∞ n=1 cn cos 4nπc l t 5 sin 4nπx l 5 . (7.4.7) Using the trigonometric identity (7.4.3), we obtain ∂u2 ∂t = 1 2 ∞ n=1 cn sin nπ l (x − ct) + 1 2 ∞ n=1 cn sin nπ l (x + ct). (7.4.8) These series are absolutely and uniformly convergent because of the assumptions on g, and hence, the series (7.4.6) and (7.4.7) converge absolutely and uniformly on [0, l]. Thus, the term-by-term differentiation is justified. Let G (x) = ∞ n=1 cn sin 4nπx l 5 be the odd periodic extension of the function g (x). Then, equation (7.4.8) can be written in the form ∂u2 ∂t = 1 2 [G (x − ct) + G (x + ct)] . Integration yields u2 (x, t) = 1 2  t 0 G (x − ct′ ) dt′ + 1 2  t 0 G (x + ct′ ) dt′ = 1 2c  x+ct x−ct G (τ ) dτ. (7.4.9) It immediately follows that u2 (x, 0) = 0, and ∂u2 ∂t (x, 0) = G (x) = g (x), 0 ≤ x ≤ l. Moreover, u2 (0, t) = 1 2  t 0 G (−ct′ ) dt′ + 1 2  t 0 G (ct′ ) dt′ = − 1 2  t 0 G (ct′ ) dt′ + 1 2  t 0 G (ct′ ) dt′ = 0 246 7 Method of Separation of Variables and u2 (l, t) = 1 2  t 0 G (l − ct′ ) dt′ + 1 2  t 0 G (l + ct′ ) dt′ = 1 2  t 0 G (−l − ct′ ) dt′ + 1 2  t 0 G (l + ct′ ) dt′ = − 1 2  t 0 G (l + ct′ ) dt′ + 1 2  t 0 G (l + ct′ ) dt′ = 0. Finally, u2 (x, t) must satisfy the differential equation. Since g ′ is continuous on [0, l], G′ exists so that ∂ 2u2 ∂t2 = c 2 [−G ′ (x − ct) + G ′ (x + ct)] . Differentiating u2 (x, t) represented by equation (7.4.6) with respect to x, we obtain ∂u2 ∂x = 1 c ∞ n=1 cn sin 4nπc l t 5 cos 4nπx l 5 = 1 2c ∞ n=1 cn " − sin nπ l (x − ct) + sin nπ l (x + ct) # = 1 2c [−G (x − ct) + G (x + ct)] . Differentiating again with respect to x, we obtain ∂ 2u2 ∂x2 = 1 2c [−G ′ (x − ct) + G ′ (x + ct)] . It is quite evident that ∂ 2u2 ∂t2 = c 2 ∂ 2u2 ∂x2 . Thus, the solution of the initial boundary-value problem (7.3.1)–(7.3.5) is established. Theorem 7.4.2. (Uniqueness Theorem) There exists at most one solution of the wave equation utt = c 2uxx, 0 < x < l, t > 0, satisfying the initial conditions u (x, 0) = f (x), ut (x, 0) = g (x), 0 ≤ x ≤ l, and the boundary conditions u (0, t)=0, u (l, t)=0, t ≥ 0, where u (x, t) is a twice continuously differentiable function with respect to both x and t. 7.4 Existence and Uniqueness of Solution of the Vibrating String Problem 247 Proof. Suppose that there are two solutions u1 and u2 and let v = u1−u2. It can readily be seen that v (x, t) is the solution of the problem vtt = c 2 vxx, 0 < x < l, t > 0, v (0, t)=0, t ≥ 0, v (l, t)=0, t ≥ 0, v (x, 0) = 0, 0 ≤ x ≤ l, vt (x, 0) = 0, 0 ≤ x ≤ l. We shall prove that the function v (x, t) is identically zero. To do so, consider the energy integral E (t) = 1 2  l 0  c 2 v 2 x + v 2 t dx (7.4.10) which physically represents the total energy of the vibrating string at time t. Since the function v (x, t) is twice continuously differentiable, we differentiate E (t) with respect to t. Thus, dE dt =  l 0  c 2 vxvxt + vtvtt dx. (7.4.11) Integrating the first integral in (7.4.11) by parts, we have  l 0 c 2 vxvxtdx = c 2 vxvt !l 0 −  l 0 c 2 vtvxxdx. But from the condition v (0, t) = 0 we have vt (0, t) = 0, and similarly, vt (l, t) = 0 for x = l. Hence, the expression in the square brackets vanishes, and equation (7.4.11) becomes dE dt =  l 0 vt  vtt − c 2 vxx dx. (7.4.12) Since vtt − c 2vxx = 0, equation (7.4.12) reduces to dE dt = 0 which means E (t) = constant = C. Since v (x, 0) = 0 we have vx (x, 0) = 0. Taking into account the condition vt (x, 0) = 0, we evaluate C to obtain 248 7 Method of Separation of Variables E (0) = C = 1 2  l 0 c 2 v 2 x + v 2 t ! t=0 dx = 0. This implies that E (t) = 0 which can happen only when vx = 0 and vt = 0 for t > 0. To satisfy both of these conditions, we must have v (x, t) = constant. Employing the condition v (x, 0) = 0, we then find v (x, t) = 0. Therefore, u1 (x, t) = u2 (x, t) and the solution u (x, t) is unique. 7.5 The Heat Conduction Problem We consider a homogeneous rod of length l. The rod is sufficiently thin so that the heat is distributed equally over the cross section at time t. The surface of the rod is insulated, and therefore, there is no heat loss through the boundary. The temperature distribution of the rod is given by the solution of the initial boundary-value problem ut = kuxx, 0 < x < l, t > 0, u (0, t)=0, t ≥ 0, u (l, t)=0, t ≥ 0, (7.5.1) u (x, 0) = f (x), 0 ≤ x ≤ l. If we assume a solution in the form u (x, t) = X (x) T (t) = 0. Equation (7.5.1) yields XT′ = kX′′T. Thus, we have X′′ X = T ′ kT = −α 2 , where α is a positive constant. Hence, X and T must satisfy X′′ + α 2X = 0, (7.5.2) T ′ + α 2 kT = 0. (7.5.3) From the boundary conditions, we have u (0, t) = X (0) T (t)=0, u (l, t) = X (l) T (t)=0. Thus, X (0) = 0, X (l)=0, 7.5 The Heat Conduction Problem 249 for an arbitrary function T (t). Hence, we must solve the eigenvalue problem X′′ + α 2X = 0, X (0) = 0, X (l)=0. The solution of equation (7.5.2) is X (x) = A cos αx + B sin αx. Since X (0) = 0, A = 0. To satisfy the second condition, we have X (l) = B sin αl = 0. Since B = 0 yields a trivial solution, we must have B = 0 and hence, sin αl = 0. Thus, α = nπ l for n = 1, 2, 3 .... Substituting these eigenvalues, we have Xn (x) = Bn sin 4nπx l 5 . Next, we consider equation (7.5.3), namely, T ′ + α 2 kT = 0, the solution of which is T (t) = Ce−α 2kt . Substituting α = (nπ/l), we have Tn (t) = Cne −(nπ/l) 2kt . Hence, the nontrivial solution of the heat equation which satisfies the two boundary conditions is un (x, t) = Xn (x) Tn (t) = an e −(nπ/l) 2kt sin 4nπx l 5 , n = 1, 2, 3 ..., where an = BnCn is an arbitrary constant. By the principle of superposition, we obtain a formal series solution as u (x, t) = ∞ n=1 un (x, t), = ∞ n=1 an e −(nπ/l) 2kt sin 4nπx l 5 , (7.5.4) 250 7 Method of Separation of Variables which satisfies the initial condition if u (x, 0) = f (x) = ∞ n=1 an sin 4nπx l 5 . This holds true if f (x) can be represented by a Fourier sine series with Fourier coefficients an = 2 l  l 0 f (x) sin 4nπx l 5 dx. (7.5.5) Hence, u (x, t) = ∞ n=1 1 2 l  l 0 f (τ ) sin 4nπτ l 5 dτ3 e −(nπ/l) 2kt sin 4nπx l 5 (7.5.6) is the formal series solution of the heat conduction problem. Example 7.5.1. (a) Suppose the initial temperature distribution is f (x) = x (l − x). Then, from equation (7.5.5), we have an = 8l 2 n3π 3 , n = 1, 3, 5,.... Thus, the solution is u (x, t) =  8l 2 π 3  ∞ n=1,3,5,... 1 n3 e −(nπ/l) 2kt sin 4nπx l 5 . (b) Suppose the temperature at one end of the rod is held constant, that is, u (l, t) = u0, t ≥ 0. The problem here is ut = k uxx, 0 < x < l, t > 0, u (0, t)=0, u (l, t) = u0, (7.5.7) u (x, 0) = f (x), 0 < x < l. Let u (x, t) = v (x, t) + u0x l . Substitution of u (x, t) in equations (7.5.7) yields vt = k vxx, 0 < x < l, t > 0, v (0, t)=0, v (l, t)=0, v (x, 0) = f (x) − u0x l , 0 < x < l. 7.6 Existence and Uniqueness of Solution of the Heat Conduction Problem 251 Hence, with the knowledge of solution (7.5.6), we obtain the solution u (x, t) = ∞ n=1 1 2 l  l 0 4 f (τ ) − u0τ l 5 sin 4nπτ l 5 dτ3 e −(nπ/l) 2kt sin 4nπx l 5 + 4u0x l 5 . (7.5.8) 7.6 Existence and Uniqueness of Solution of the Heat Conduction Problem In the preceding section, we found that (7.5.4) is the formal solution of the heat conduction problem (7.5.1), where an is given by (7.5.5). We shall prove the existence of this formal solution if f (x) is continuous in [0, l] and f (0) = f (l) = 0, and f ′ (x) is piecewise continuous in (0, l). Since f (x) is bounded, we have |an| = 2 l       l 0 f (x) sin 4nπx l 5 dx      ≤ 2 l  l 0 |f (x)| dx ≤ C, where C is a positive constant. Thus, for any finite t0 > 0,    an e −(nπ/l) 2kt sin 4nπx l 5   ≤ C e−(nπ/l) 2kt0 when t ≥ t0. According to the ratio test, the series of terms exp " − (nπ/l) 2 kt0 # converges. Hence, by the Weierstrass M-test, the series (7.5.4) converges uniformly with respect to x and t whenever t ≥ t0 and 0 ≤ x ≤ l. Differentiating equation (7.5.4) termwise with respect to t, we obtain ut = − ∞ n=1 an 4nπ l 52 k e−(nπ/l) 2kt sin 4nπx l 5 . (7.6.1) We note that    −an 4nπ l 52 k e−(nπ/l) 2kt sin 4nπx l 5     ≤ C 4nπ l 52 k e−(nπ/l) 2kt0 when t ≥ t0, and the series of terms C (nπ/l) 2 k exp " − (nπ/l) 2 kt0 # converges by the ratio test. Hence, equation (7.6.1) is uniformly convergent in the region 0 ≤ x ≤ l, t ≥ t0. In a similar manner, the series (7.5.4) can be differentiated twice with respect to x, and as a result uxx = − ∞ n=1 an 4nπ l 52 e −(nπ/l) 2kt sin 4nπx l 5 . (7.6.2) 252 7 Method of Separation of Variables Evidently, from equations (7.6.1) and (7.6.2), ut = k uxx. Hence, equation (7.5.4) is a solution of the one-dimensional heat equation in the region 0 ≤ x ≤ l, t ≥ 0. Next, we show that the boundary conditions are satisfied. Here, we note that the series (7.5.4) representing the function u (x, t) converges uniformly in the region 0 ≤ x ≤ l, t ≥ 0. Since the function represented by a uniformly convergent series of continuous functions is continuous, u (x, t) is continuous at x = 0 and x = l. As a consequence, when x = 0 and x = l, solution (7.5.4) satisfies u (0, t)=0, u (l, t)=0, for all t > 0. It remains to show that u (x, t) satisfies the initial condition u (x, 0) = f (x), 0 ≤ x ≤ l. Under the assumptions stated earlier, the series for f (x) given by f (x) = ∞ n=1 an sin 4nπx l 5 is uniformly and absolutely convergent. By Abel’s test of convergence the series formed by the product of the terms of a uniformly convergent series ∞ n=1 an sin 4nπx l 5 and a uniformly bounded and monotone sequence exp " − (nπ/l) 2 kt# converges uniformly with respect to t. Hence, u (x, t) = ∞ n=1 an e −(nπ/l) 2kt sin 4nπx l 5 converges uniformly for 0 ≤ x ≤ l, t ≥ 0, and by the same reasoning as before, u (x, t) is continuous for 0 ≤ x ≤ l, t ≥ 0. Thus, the initial condition u (x, 0) = f (x), 0 ≤ x ≤ l is satisfied. The existence of solution is therefore established. In the above discussion the condition imposed on f (x) is stronger than necessary. The solution can be obtained with a less stringent condition on f (x) (see Weinberger (1965)). 7.6 Existence and Uniqueness of Solution of the Heat Conduction Problem 253 Theorem 7.6.1. (Uniqueness Theorem) Let u (x, t) be a continuously differentiable function. If u (x, t) satisfies the differential equation ut = k uxx, 0 < x < l, t > 0, the initial conditions u (x, 0) = f (x), 0 ≤ x ≤ l, and the boundary conditions u (0, t)=0, u (l, t)=0, t ≥ 0, then, the solution is unique. Proof. Suppose that there are two distinct solutions u1 (x, t) and u2 (x, t). Let v (x, t) = u1 (x, t) − u2 (x, t). Then, vt = k vxx, 0 < x < l, t > 0, v (0, t)=0, v (l, t)=0, t ≥ 0, (7.6.3) v (x, 0) = 0, 0 ≤ x ≤ l, Consider the function defined by the integral J (t) = 1 2k  l 0 v 2 dx. Differentiating with respect to t, we have J ′ (t) = 1 k  l 0 vvtdx =  l 0 vvxxdx, by virtue of equation (7.6.3). Integrating by parts, we have  l 0 vvxxdx = [vvx] l 0 −  l 0 v 2 xdx. Since v (0, t) = v (l, t) = 0, J ′ (t) = −  l 0 v 2 x dx ≤ 0. From the condition v (x, 0) = 0, we have J (0) = 0. This condition and J ′ (t) ≤ 0 implies that J (t) is a nonincreasing function of t. Thus, 254 7 Method of Separation of Variables J (t) ≤ 0. But by definition of J (t), J (t) ≥ 0. Hence, J (t)=0, for t ≥ 0. Since v (x, t) is continuous, J (t) = 0 implies v (x, t)=0 in 0 ≤ x ≤ l, t ≥ 0. Therefore, u1 = u2 and the solution is unique. 7.7 The Laplace and Beam Equations Example 7.7.1. Consider the steady state temperature distribution in a thin rectangular slab. Two sides are insulated, one side is maintained at zero temperature, and the temperature of the remaining side is prescribed to be f (x). Thus, we are required to solve ∇2u = 0, 0 < x < a, 0 < y < b, u (x, 0) = f (x), 0 ≤ x ≤ a, u (x, b)=0, 0 ≤ x ≤ a, ux (0, y)=0, ux (a, y)=0. Let u (x, y) = X (x) Y (y). Substitution of this into the Laplace equation yields X′′ − λX = 0, Y ′′ + λX = 0. Since the boundary conditions are homogeneous on x = 0 and x = a, we have λ = −α 2 with α ≥ 0 for nontrivial solutions of the eigenvalue problem X′′ + α 2X = 0, X′ (0) = X′ (a)=0. The solution is X (x) = A cos αx + B sin αx. Application of the boundary conditions then yields B = 0 and α = (nπ/a) with n = 0, 1, 2,.... Hence, Xn (x) = A cos 4nπx a 5 . 7.7 The Laplace and Beam Equations 255 The solution of the Y equation is clearly Y (y) = C cosh αy + D sinh αy which can be written in the form Y (y) = E sinh α (y + F), where E =  D2 − C 2 1 2 and F = tanh−1 (C/D) ! /α. Applying the homogeneous boundary condition Y (b) = 0, we obtain Y (b) = E sinh α (b + F)=0 which implies F = −b, E = 0 for nontrivial solutions. Hence, we have u (x, y) = (b − y) b a0 2 + ∞ n=1 an cos 4nπx a 5 sinh(nπ a (y − b) ) . Now we apply the remaining nonhomogeneous condition to obtain u (x, 0) = f (x) = a0 2 + ∞ n=1 an cos 4nπx a 5 sinh  − nπb a  . Since this is a Fourier cosine series, the coefficients are given by a0 = 2 a  a 0 f (x) dx, an = −2 a sinh  nπb a  a 0 f (x) cos 4nπx a 5 dx, n = 1, 2,.... Thus, the solution is u (x, y) =  b − y b  a0 2 + ∞ n=1 a ∗ n sinh nπ a (b − y) sinh nπb a cos 4nπx a 5 , where a ∗ n = 2 a  a 0 f (x) cos 4nπx a 5 dx. If, for example f (x) = x in 0 <x<π, 0="" <y<π,="" then="" we="" find="" (note="" that="" a="π)" a0="π," a∗="" n="2" πn2="" [(−1)n="" −="" 1]="" ,="" 2,...="" and="" hence,="" the="" solution="" has="" final="" form="" u="" (x,="" y)="1" 2="" (π="" +="" ∞="" sinh="" nπ="" cos="" nx.="" 256="" 7="" method="" of="" separation="" variables="" example="" 7.7.2.="" as="" another="" example,="" consider="" transverse="" vibration="" beam.="" equation="" motion="" is="" governed="" by="" utt="" 2uxxxx="0," <="" x="" l,="" t=""> 0, where u (x, t) is the displacement and a is the physical constant. Note that the equation is of the fourth order in x. Let the initial and boundary conditions be u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, u (0, t) = u (l, t)=0, t > 0, (7.7.1) uxx (0, t) = uxx (l, t)=0, t> 0. The boundary conditions represent the beam being simple supported, that is, the displacements and the bending moments at the ends are zero. Assume a nontrivial solution in the form u (x, t) = X (x) T (t), which transforms the equation of motion into the forms T ′′ + a 2α 4T = 0, X(iv) − α 4X = 0, α > 0. The equation for X (x) has the general solution X (x) = A cosh αx + B sinh αx + C cos αx + D sin αx. The boundary conditions require that X (0) = X (l)=0, X′′ (0) = X′′ (l)=0. Differentiating X twice with respect to x, we obtain X′′ (x) = Aα2 cosh αx + Bα2 sinh αx − Cα2 cos αx − Dα2 sin αx. Now applying the conditions X (0) = X′′ (0) = 0, we obtain A + C = 0, α2 (A − C)=0, and hence, A = C = 0. The conditions X (l) = X′′ (l) = 0 yield B sinh αl + D sin αl = 0, B sinh αl − D sin αl = 0. 7.7 The Laplace and Beam Equations 257 These equations are satisfied if B sinh αl = 0, D sin αl = 0. Since sinh αl = 0, B must vanish. For nontrivial solutions, D = 0, sin αl = 0, and hence, α = 4nπ l 5 , n = 1, 2, 3,.... We then obtain Xn (x) = Dn sin 4nπx l 5 . The general solution for T (t) is T (t) = E cos  aα2 t + F sin  aα2 t . Inserting the values of α 2 , we obtain Tn (t) = En cos  a 4nπ l 52 t 0 + Fn sin  a 4nπ l 52 t 0 . Thus, the general solution for the transverse vibrations of a beam is u (x, t) = ∞ n=1 an cos  a 4nπ l 52 t 0 + bn sin  a 4nπ l 52 t 0 sin 4nπx l 5 . (7.7.2) To satisfy the initial condition u (x, 0) = f (x), we must have u (x, 0) = f (x) = ∞ n=1 an sin 4nπx l 5 from which we find an = 2 l  l 0 f (x) sin 4nπx l 5 dx. (7.7.3) Now the application of the second initial condition gives ut (x, 0) = g (x) = ∞ n=1 bna 4nπ l 52 sin 4nπx l 5 and hence, bn = 2 al  l nπ 2  l 0 g (x) sin 4nπx l 5 dx. (7.7.4) Thus, the solution of the initial boundary-value problem is given by equations (7.7.2)–(7.7.4). 258 7 Method of Separation of Variables 7.8 Nonhomogeneous Problems The partial differential equations considered so far in this chapter are homogeneous. In practice, there is a very important class of problems involving nonhomogeneous equations. First, we shall illustrate a problem involving a time-independent nonhomogeneous equations. Example 7.8.1. Consider the initial boundary-value problem utt = c 2uxx + F (x), 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, (7.8.1) u (0, t) = A, u (l, t) = B, t > 0. We assume a solution in the form u (x, t) = v (x, t) + U (x). Substitution of u (x, t) in equation (7.8.1) yields vtt = c 2 (vxx + Uxx) + F (x), and if U (x) satisfies the equation c 2Uxx + F (x)=0, then v (x, t) satisfies the wave equation vtt = c 2 vxx. In a similar manner, if u (x, t) is inserted in the initial and boundary conditions, we obtain u (x, 0) = v (x, 0) + U (x) = f (x), ut (x, 0) = vt (x, 0) = g (x), u (0, t) = v (0, t) + U (0) = A, u (l, t) = v (l, t) + U (l) = B . Thus, if U (x) is the solution of the problem c 2Uxx + F = 0, U (0) = A, U (l) = B, then v (x, t) must satisfy vtt = c 2 vxx, v (x, 0) = f (x) − U (x), vt (x, 0) = g (x), (7.8.2) v (0, t)=0, v (l, t)=0. 7.8 Nonhomogeneous Problems 259 Now v (x, t) can be solved easily since U (x) is known. It can be seen that U (x) = A + (B − A) x l + x l  l 0 1 c 2  η 0 F (ξ) dξ dη −  x 0 1 c 2  η 0 F (ξ) dξ dη. As a specific example, consider the problem utt = c 2uxx + h, h is a constant u (x, 0) = 0, ut (x, 0) = 0, (7.8.3) u (0, t)=0, u (l, t)=0. Then, the solution of the system c 2Uxx + h = 0, U (0) = 0, U (l)=0, is U (x) = h 2c 2  lx − x 2 . The function v (x, t) must satisfy vtt = c 2 vxx, v (x, 0) = − h 2c 2  lx − x 2 , vt (x, 0) = 0, v (0, t)=0, v (l, t)=0. The solution is given (see Section 7.3 with g (x) = 0) by v (x, t) = ∞ n=1 an cos 4nπc l t 5 sin 4nπx l 5 , and the coefficient is an = 2 l  l 0 − h 2c 2  lx − x 2 sin 4nπx l 5 dx an = − 4l 2h n3π 3c 2 for n odd an = 0 for n even. The solution of the given initial boundary-value problem is, therefore, given by 260 7 Method of Separation of Variables u (x, t) = v (x, t) + U (x) = hx 2c 2 (l − x) + ∞ n=1  − 4l 2h c 2π 3  cos (2n − 1) (πct/l) (2n − 1)3 × sin (2n − 1) (πx/l).(7.8.4) Let us now consider the problem of a finite string with an external force acting on it. If the ends are fixed, we have utt − c 2uxx = h (x, t), 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, (7.8.5) u (0, t)=0, u (l, t)=0, t ≥ 0. We assume a solution involving the eigenfunctions, sin (nπx/l), of the associated eigenvalue problem in the form u (x, t) = ∞ n=1 un (t) sin 4nπx l 5 , (7.8.6) where the functions un (t) are to be determined. It is evident that the boundary conditions are satisfied. Let us also assume that h (x, t) = ∞ n=1 hn (t) sin 4nπx l 5 . (7.8.7) Thus, hn (t) = 2 l  l 0 h (x, t) sin 4nπx l 5 dx. (7.8.8) We assume that the series (7.8.6) is convergent. We then find utt and uxx from (7.8.6) and substitution of these values into (7.8.5) yields ∞ n=1 u ′′ n (t) + λ 2 n un (t) ! sin 4nπx l 5 = ∞ n=1 hn (t) sin 4nπx l 5 , where λn = (nπc/l). Multiplying both sides of this equation by sin (mπx/l), where m = 1, 2, 3,..., and integrating from x = 0 to x = l, we obtain u ′′ n (t) + λ 2 n un (t) = hn (t) the solution of which is given by un (t) = an cos λnt + bn sin λnt + 1 λn  t 0 hn (τ ) sin [λn (t − τ )] dτ. (7.8.9) 7.8 Nonhomogeneous Problems 261 Hence, the formal solution (7.8.6) takes the final form u (x, t) = ∞ n=1  an cos λnt + bn sin λnt + 1 λn  t 0 hn (τ ) sin [λn (t − τ )] dτ0 · sin 4nπx l 5 . (7.8.10) Applying the initial conditions, we have u (x, 0) = f (x) = ∞ n=1 an sin 4nπx l 5 . Thus, an = 2 l  l 0 f (x) sin 4nπx l 5 dx. (7.8.11) Similarly, ut (x, 0) = g (x) = ∞ n=1 bnλn sin 4nπx l 5 . Thus, bn =  2 lλn   l 0 g (x) sin 4nπx l 5 dx. (7.8.12) Hence, the formal solution of the initial boundary-value problem (7.8.5) is given by (7.8.10) with an given by (7.8.11) and bn given by (7.8.12). Example 7.8.2. Determine the solution of the initial boundary-value problem utt − uxx = h, 0 <x< 1,="" t=""> 0, h = constant, u (x, 0) = x (1 − x), 0 ≤ x ≤ 1, ut (x, 0) = 0, 0 ≤ x ≤ 1, (7.8.13) u (0, t)=0, u (1, t)=0, t ≥ 0. In this case, c = 1, λn = nπ, bn = 0 and an is given by an = 2  1 0 x (1 − x) sin nπx dx = 4 (nπ) 3 [1 − (−1)n ] . We also have hn = 2  1 0 h sin 4nπx l 5 dx = 2h nπ [1 − (−1)n ] . 262 7 Method of Separation of Variables Hence, the integral term in (7.8.9) represents φn (t) given by φn (t) = 1 λn  t 0 hn (τ ) sin [λn (t − τ )] dτ = 2h nπλ2 n [1 − (−1)n ] (1 − cos λnt). The solution (7.8.10) is thus given by u (x, t) = ∞ n=1  4 n3π 3 [1 − (−1)n ] cos nπt + 2h n3π 3 [1 − (−1)n ] (1 − cos nπt) 0 · sin nπx. (7.8.14) We have treated the initial boundary-value problem with the fixed end conditions. Problems with other boundary conditions can also be solved in a similar manner. We will now consider the initial boundary-value problem with timedependent boundary conditions, namely, utt − uxx = h (x, t), 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, (7.8.15) u (0, t) = p (t), u (l, t) = q (t), t ≥ 0. We assume a solution in the form u (x, t) = v (x, t) + U (x, t). (7.8.16) Substituting this into equation (7.8.15), we obtain vtt − c 2 vxx = h − Utt + c 2Uxx. For the initial and boundary conditions, we have v (x, 0) = f (x) − U (x, 0), vt (x, 0) = g (x) − Ut (x, 0), v (0, t) = p (t) − U (0, t), v (l, t) = q (t) − U (l, t). In order to make the boundary conditions homogeneous, we set U (0, t) = p (t), U (l, t) = q (t). Thus, U (x, t) must take the form U (x, t) = p (t) + x l [q (t) − p (t)] . (7.8.17) 7.8 Nonhomogeneous Problems 263 The problem now is to find the function v (x, t) which satisfies vtt − c 2 vxx = h − Utt = H (x, t), v (x, 0) = f (x) − U (x, 0) = F (x), vt (x, 0) = g (x) − Ut (x, 0) = G (x), (7.8.18) v (0, t)=0, v (l, t)=0. This is the same type of problem as the one with homogeneous boundary condition that has previously been treated. Example 7.8.3. Find the solution of the problem utt − uxx = h, 0 <x< 1,="" t=""> 0, h = constant, u (x, 0) = x (1 − x), 0 ≤ x ≤ 1, ut (x, 0) = 0, 0 ≤ x ≤ 1, (7.8.19) u (0, t) = t, u (1, t) = sin t, t ≥ 0. In this case, we use (7.8.16) and (7.8.17) with c = 1 and λn = nπ so that u (x, t) = v (x, t) + U (x, t), U (x, t) = t + x (sin t − t). (7.8.20) Then, v must satisfy vtt − vxx = h + x sin t, v (x, 0) = x (1 − x), vt (x, 0) = −1, (7.8.21) v (0, t)=0, v (1, t)=0. It follows from (7.8.8) that hn (t)=2  1 0 (h + x sin t) sin nπx dx = 2h nπ [1 − (−1)n ] + 2 (−1)n+1 nπ sin t = a + b sin t(say). (7.8.22) We also find an = 2  1 0 x (1 − x) sin nπx dx = 4 (nπ) 3 [1 − (−1)n ] , and bn = 2 nπ  1 0 sin nπx dx = 2 (nπ) 2 [1 − (−1)n ] . 264 7 Method of Separation of Variables Then, we determine the integral term in (7.8.9) so that φn (t) = 1 nπ  t 0 (a + b sin τ ) sin [nπ (t − τ )] dτ = 1 nπ  a nπ (1 − cos nπt) + b 4 [(sin 2t − 2t) cos nπt − (cos 2t − 1) sin nπt] 0 . (7.8.23) Hence, the solution of the problem (7.8.21) is v (x, t) = ∞ n=1 [an cos nπt + bn sin nπt + φn (t)] sin nπx. (7.8.24) Thus, the solution of problem (7.8.19) is given by u (x, t) = v (x, t) + U (x, t), where v (x, t) is given by (7.8.24) and U (x, t) is given by (7.8.20) Example 7.8.4. Use the method of separation of variables to derive the Hermite equation from the Fokker–Planck equation of nonequilibrium statistical mechanics ut − uxx = (x u)x . (7.8.25) We seek a nontrivial separable solution u (x, t) = X (x) T (t) so that equation (7.8.25) reduces to a pair of ordinary differential equations X′′ + xX′ + (1 + n) X = 0 and T ′ + n T = 0, (7.8.26ab) where (−n) is a separation constant. We next use X (x) = exp  − 1 2 x 2  f (x) (7.8.27) and rescale the independent variable to obtain the Hermite equation for f in the form d 2f dξ2 − 2ξ df dξ + 2nf = 0. The solution of (7.8.26b) gives T (t) = cn exp (−nt), (7.8.28) where the coefficients cn are constants. 7.9 Exercises 265 Thus, the solution of the Fokker–Planck equation is given by u (x, t) = ∞ n=1 an exp  −nt − 1 2 x 2  Hn  x √ 2  , (7.8.29) where Hn is the Hermite function and an are arbitrary constants to be determined from the given initial condition u (x, 0) = f (x). (7.8.30) We make the change of variables ξ = x et and u = e t v, (7.8.31) in equation (7.8.25). Consequently, equation (7.8.25) becomes ∂v ∂t = e 2t ∂ 2v ∂ξ2 . (7.8.32) Making another change of variable t to τ (t), we transform (7.8.32) into the linear diffusion equation ∂v ∂τ = ∂ 2v ∂ξ2 . (7.8.33) Finally, we note that the asymptotic behavior of the solution u (x, t) as t → ∞ is of special interest. The reader is referred to Reif (1965) for such behavior. 7.9 Exercises 1. Solve the following initial boundary-value problems: (a) utt = c 2uxx, 0 <x< 1,="" t=""> 0, u (x, 0) = x (1 − x), ut (x, 0) = 0, 0 ≤ x ≤ 1, u (0, t) = u (1, t) = 0, t > 0. (b) utt = c 2uxx, 0 <x<π, t=""> 0, u (x, 0) = 3 sin x, ut (x, 0) = 0, 0 ≤ x ≤ π, u (0, t) = u (1, t) = 0, t > 0. 266 7 Method of Separation of Variables 2. Determine the solutions of the following initial boundary-value problems: (a) utt = c 2uxx, 0 <x<π, t=""> 0, u (x, 0) = 0, ut (x, 0) = 8 sin2 x, 0 ≤ x ≤ π, u (0, t) = u (π, t) = 0, t > 0. (b) utt = c 2uxx = 0, 0 <x< 1,="" t=""> 0, u (x, 0) = 0, ut (x, 0) = x sin πx, 0 ≤ x ≤ 1, u (0, t) = u (1, t) = 0, t > 0. 3. Find the solution of each of the following problems: (a) utt = c 2uxx = 0, 0 <x< 1,="" t=""> 0, u (x, 0) = x (1 − x), ut (x, 0) = x − tan πx 4 , 0 ≤ x ≤ 1, u (0, t) = u (π, t) = 0, t > 0. (b) utt = c 2uxx = 0, 0 <x<π, t=""> 0, u (x, 0) = sin x, ut (x, 0) = x 2 − πx, 0 ≤ x ≤ π, u (0, t) = u (π, t) = 0, t > 0. 4. Solve the following problems: (a) utt = c 2uxx = 0, 0 <x<π, t=""> 0, u (x, 0) = x + sin x, ut (x, 0) = 0, 0 ≤ x ≤ π, u (0, t) = ux (π, t) = 0, t > 0. (b) utt = c 2uxx = 0, 0 <x<π, t=""> 0, u (x, 0) = cos x, ut (x, 0) = 0, 0 ≤ x ≤ π, ux (0, t) = 0, ux (π, t) = 0, t > 0. 7.9 Exercises 267 5. By the method of separation of variables, solve the telegraph equation: utt + aut + bu = c 2uxx, 0 < x < l, t > 0, u (x, 0) = f (x), ut (x, 0) = 0, u (0, t) = u (l, t)=0, t > 0. 6. Obtain the solution of the damped wave motion problem: utt + aut = c 2uxx, 0 < x < l, t > 0, u (x, 0) = 0, ut (x, 0) = g (x), u (0, t) = u (l, t)=0. 7. The torsional oscillation of a shaft of circular cross section is governed by the partial differential equation θtt = a 2 θxx, where θ (x, t) is the angular displacement of the cross section and a is a physical constant. The ends of the shaft are fixed elastically, that is, θx (0, t) − h θ (0, t)=0, θx (l, t) + h θ (l, t)=0. Determine the angular displacement if the initial angular displacement is f (x). 8. Solve the initial boundary-value problem of the longitudinal vibration of a truncated cone of length l and base of radius a. The equation of motion is given by 4 1 − x h 52 ∂ 2u ∂t2 = c 2 ∂ ∂x 4 1 − x h 52 ∂u ∂x , 0 < x < l, t > 0, where c 2 = (E/ρ), E is the elastic modulus, ρ is the density of the material and h = la/ (a − l). The two ends are rigidly fixed. If the initial displacement is f (x), that is, u (x, 0) = f (x), find u (x, t). 9. Establish the validity of the formal solution of the initial boundaryvalue problems: utt = c 2uxx, 0 < x < π, t > 0, u (x, 0) = f (x), ut (x, 0) = g (x), 0 ≤ x ≤ π, ux (0, t)=0, ux (π, t)=0, t > 0. 10. Prove the uniqueness of the solution of the initial boundary-value problem: utt = c 2uxx, 0 < x < π, t > 0, u (x, 0) = f (x), ut (x, 0) = g (x), 0 ≤ x ≤ π, ux (0, t)=0, ux (π, t)=0, t > 0. 268 7 Method of Separation of Variables 11. Determine the solution of utt = c 2uxx + A sinh x, 0 < x < l, t > 0, u (x, 0) = 0, ut (x, 0) = 0, 0 ≤ x ≤ l, u (0, t) = h, u (l, t) = k, t > 0, where h, k, and A are constants. 12. Solve the problem: utt = c 2uxx + Ax, 0 <x< 1,="" t=""> 0, A = constant, u (x, 0) = 0, ut (x, 0) = 0, 0 ≤ x ≤ 1, u (0, t)=0, u (1, t)=0, t > 0. 13. Solve the problem: utt = c 2uxx + x 2 , 0 <x< 1,="" t=""> 0, u (x, 0) = x, ut (x, 0) = 0, 0 ≤ x ≤ 1, u (0, t)=0, u (1, t)=1, t ≥ 0. 14. Find the solution of the following problems: (a) ut = kuxx + h, 0 <x< 1,="" t=""> 0, h = constant, u (x, 0) = u0 (1 − cos πx), 0 ≤ x ≤ 1, u0 = constant, u (0, t)=0, u (l, t)=2u0, t ≥ 0. (b) ut = kuxx − hu, 0 < x < l, t > 0, h = constant, u (x, 0) = f (x), 0 ≤ x ≤ l, ux (0, t) = ux (l, t)=0, t > 0. 15. Obtain the solution of each of the following initial boundary-value problems: (a) ut = 4 uxx, 0 <x< 1,="" t=""> 0, u (x, 0) = x 2 (1 − x), 0 ≤ x ≤ 1, u (0, t) = 0, u (l, t) = 0, t ≥ 0. (b) ut = k uxx, 0 <x<π, t=""> 0, u (x, 0) = sin2 x, 0 ≤ x ≤ π, u (0, t) = 0, u (π, t) = 0, t ≥ 0. 7.9 Exercises 269 (c) ut = uxx, 0 <x< 2,="" t=""> 0, u (x, 0) = x, 0 ≤ x ≤ 2, u (0, t) = 0, ux (2, t) = 1, t ≥ 0. (d) ut = k uxx, 0 <x<l, t=""> 0, u (x, 0) = sin (πx/2l), 0 ≤ x ≤ l, u (0, t) = 0, u (l, t) = 1, t ≥ 0. 16. Find the temperature distribution in a rod of length l. The faces are insulated, and the initial temperature distribution is given by x (l − x). 17. Find the temperature distribution in a rod of length π, one end of which is kept at zero temperature and the other end of which loses heat at a rate proportional to the temperature at that end x = π. The initial temperature distribution is given by f (x) = x. 18. The voltage distribution in an electric transmission line is given by vt = k vxx, 0 < x < l, t > 0. A voltage equal to zero is maintained at x = l, while at the end x = 0, the voltage varies according to the law v (0, t) = Ct, t > 0, where C is a constant. Find v (x, t) if the initial voltage distribution is zero. 19. Establish the validity of the formal solution of the initial boundaryvalue problem: ut = k uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, u (0, t)=0, ux (l, t)=0, t ≥ 0. 20. Prove the uniqueness of the solution of the problem: ut = k uxx, 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ux (0, t)=0, ux (l, t)=0, t ≥ 0. 270 7 Method of Separation of Variables 21. Solve the radioactive decay problem: ut − k uxx = Ae−ax , 0 < x < π, t > 0, u (x, 0) = sin x, 0 ≤ x ≤ π, u (0, t)=0, u (π, t)=0, t ≥ 0. 22. Determine the solution of the initial boundary-value problem: ut − k uxx = h (x, t), 0 < x < l, t > 0, k = constant, u (x, 0) = f (x), 0 ≤ x ≤ l, u (0, t) = p (t), u (l, t) = q (t), t ≥ 0. 23. Determine the solution of the initial boundary-value problem: ut − k uxx = h (x, t), 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, u (0, t) = p (t), ux (l, t) = q (t), t ≥ 0. 24. Solve the problem: ut − k uxx = 0, 0 <x< 1,="" t=""> 0, u (x, 0) = x (1 − x), 0 ≤ x ≤ 1, u (0, t) = t, u (1, t) = sin t, t ≥ 0. 25. Solve the problem: ut − 4uxx = xt, 0 <x< 1,="" t="" ≥="" 0,="" u="" (x,="" 0)="sin" πx,="" 0="" ≤="" x="" (0,="" t)="t," (1,="" 2="" ,="" 0.="" 26.="" solve="" the="" problem:="" ut="" −="" k="" uxx="x" cost,="" <="" π,=""> 0, u (x, 0) = sin x, 0 ≤ x ≤ π, u (0, t) = t 2 , u (π, t)=2t, t ≥ 0. 27. Solve the problem: ut − uxx = 2x 2 t, 0 <x< 1,="" t=""> 0, u (x, 0) = cos (3πx/2), 0 ≤ x ≤ 1, u (0, t)=1, ux (1, t) = 3π 2 , t ≥ 0. 28. Solve the problem: ut − 2 uxx = h, 0 <x< 1,="" t=""> 0, h = constant, u (x, 0) = x, 0 ≤ x ≤ 1, u (0, t) = sin t, ux (1, t) + u (1, t)=2, t ≥ 0. 7.9 Exercises 271 29. Determine the solution of the initial boundary-value problem: utt − c 2uxx = h (x, t), 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, u (0, t) = p (t), ux (l, t) = q (t), t ≥ 0. 30. Determine the solution of the initial boundary-value problem: utt − c 2uxx = h (x, t), 0 < x < l, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ l, ut (x, 0) = g (x), 0 ≤ x ≤ l, ux (0, t) = p (t), ux (l, t) = q (t), t ≥ 0. 31. Solve the problem: utt − uxx = 0, 0 <x< 1,="" t=""> 0, u (x, 0) = x, ut (x, 0) = 0, 0 ≤ x ≤ 1, u (0, t) = t 2 , u (1, t) = cost, t ≥ 0. 32. Solve the problem: utt − 4 uxx = xt, 0 <x< 1,="" t=""> 0, u (x, 0) = x, ut (x, 0) = 0, 0 ≤ x ≤ 1, u (0, t)=0, ux (1, t)=1+ t, t ≥ 0. 33. Solve the problem: utt − 9 uxx = 0, 0 <x< 1,="" t=""> 0, u (x, 0) = sin 4πx 2 5 , ut (x, 0) = 1 + x, 0 ≤ x ≤ 1, ux (0, t) = π/2, ux (1, t)=0, t ≥ 0. 34. Find the solution of the problem: utt + 2k ut − c 2uxx = 0, 0 < x < l, t > 0, u (x, 0) = 0, ut (x, 0) = 0, 0 ≤ x ≤ l, ux (0, t)=0, u (l, t) = h, t ≥ 0, h = constant. 35. Solve the problem: ut − c 2uxx + hu = hu0, −π < x < π, t > 0, u (x, 0) = f (x), −π ≤ x ≤ π, u (−π, t) = u (π, t), ux (−π, t) = ux (π, t), t ≥ 0, where h and u0 are constants. 272 7 Method of Separation of Variables 36. Prove the uniqueness theorem for the boundary-value problem involving the Laplace equation: uxx + uyy = 0, 0 < x < a, 0 < y < b, u (x, 0) = f (x), u (x, b)=0, 0 ≤ x ≤ a, ux (0, y)=0= ux (a, y), 0 ≤ y ≤ b. 37. Consider the telegraph equation problem: utt − c 2uxx + aut + bu = 0, 0 < x < l, t > 0, u (x, 0) = f (x), ut (x, 0) = g (x) for 0 ≤ x ≤ l, u (0, t)=0= u (l, t) for t ≥ 0, where a and b are positive constants. (a) Show that, for any T > 0,  l 0  u 2 t + c 2u 2 x + bu2 t=T dx ≤  l 0  u 2 t + c 2u 2 x + bu2 t=0 dx. (b) Use the above integral inequality from (a) to show that the initial boundary-value problem for the telegraph equation can have only one solution. 8 Eigenvalue Problems and Special Functions “The tool which serves as intermediary between theory and practice, between thought and observation, is mathematics; it is mathematics which builds the linking bridges and gives the ever more reliable forms. From this it has come about that our entire contemporary culture, in as much as it is based on the intellectual penetration and the exploitation of nature, has its foundations in mathematics.” David Hilbert “In 1836/7, he published some important joint work with his friend Sturm on what became known as Sturm–Liouville theory, which became important in physics.” Ioan James 8.1 Sturm–Liouville Systems In the preceding chapter, we determined the solutions of partial differential equations by the method of separation of variables. In this chapter, we generalize the method of separation of variables and the associated eigenvalue problems. This generalization, usually known as the Sturm–Liouville theory, greatly extends the scope of the method of separation of variables. Under separable conditions we transformed the second-order homogeneous partial differential equation into two ordinary differential equations (7.2.11) and (7.2.12) which are of the form a1 (x) d 2y dx2 + a2 (x) dy dx + [a3 (x) + λ] y = 0. (8.1.1) If we introduce 274 8 Eigenvalue Problems and Special Functions p (x) = exp  x a2 (t) a1 (t) dt , q (x) = a3 (x) a1 (x) p (x), s (x) = p (x) a1 (x) , (8.1.2) into equation (8.1.1), we obtain d dx  p dy dx + (q + λ s) y = 0, (8.1.3) which is known as the Sturm–Liouville equation. In terms of the Sturm– Liouville operator L ≡ d dx  p d dx + q, equation (8.1.3) can be written as L[y] + λ s (x) y = 0, (8.1.4) where λ is a parameter independent of x, and p, q, and s are real-valued functions of x. To ensure the existence of solutions, we let q and s be continuous and p be continuously differentiable in a closed finite interval [a, b]. The Sturm–Liouville equation is called regular in the interval [a, b] if the functions p (x) and s (x) are positive in the interval [a, b]. Thus, for a given λ, there exist two linearly independent solutions of a regular Sturm– Liouville equation in the interval [a, b]. The Sturm–Liouville equation L[y] + λ s (x) y = 0, a ≤ x ≤ b, together with the separated end conditions a1y (a) + a2y ′ (a)=0, b1y (b) + b2y ′ (b)=0, (8.1.5) where a1 and a2, and likewise b1 and b2, are not both zero and are given real numbers, is called a regular Sturm–Liouville (RSL) system. The values of λ for which the Sturm–Liouville system has a nontrivial solution are called the eigenvalues, and the corresponding solutions are called the eigenfunctions. For a regular Sturm–Liouville problem, we denote the domain of L by D (L), that is, D (L) is the space of all complex-valued functions y defined on [a, b] for which y ′′ ∈ L 2 ([a, b]) and which satisfy boundary conditions (8.1.5). Example 8.1.1. Consider the regular Sturm–Liouville problem y ′′ + λy = 0, 0 ≤ x ≤ π, y (0) = 0, y′ (π)=0. 8.1 Sturm–Liouville Systems 275 When λ ≤ 0, it can be readily shown that λ is not an eigenvalue. However, when λ > 0, the solution of the Sturm–Liouville equation is y (x) = A cos √ λ x + B sin √ λ x. Applying the condition y (0) = 0, we obtain A = 0. The condition y ′ (π)=0 yields B √ λ cos √ λ π = 0. Since λ = 0 and B = 0 yields a trivial solution, we must have cos √ λ π = 0, B = 0. This equation is satisfied if √ λ = 2n − 1 2 , n = 1, 2, 3,..., and hence, the eigenvalues are λn = (2n − 1)2 /4, and the corresponding eigenfunctions are sin  2n − 1 2  x, n = 1, 2, 3,.... Example 8.1.2. Consider the Euler equation x 2 y ′′ + xy′ + λu = 0, 1 ≤ x ≤ e with the end conditions y (1) = 0, y (e)=0. By using the transformation (8.1.2), the Euler equation can be put into the Sturm–Liouville form: d dx  x dy dx + 1 x λ y = 0. The solution of the Euler equation is y (x) = c1 x i √ λ + c2 x −i √ λ . Noting that x ia = e ia ln x = cos (a ln x) + isin (a ln x), the solution y (x) becomes y (x) = A cos 4√ λ ln x 5 + B sin 4√ λ ln x 5 , where A and B are constants related to c1 and c2. The end condition y (1) = 0 gives A = 0, and the end condition y (e) = 0 gives 276 8 Eigenvalue Problems and Special Functions sin √ λ = 0, B = 0, which in turn yields the eigenvalues λn = n 2π 2 , n = 1, 2, 3,..., and the corresponding eigenfunctions sin (nπ ln x), n = 1, 2, 3,.... Another type of problem that often occurs in practice is the periodic Sturm–Liouville system. The Sturm–Liouville equation d dx  p (x) dy dx + [q (x) + λs (x)] y = 0, a ≤ x ≤ b, in which p (a) = p (b), together with the periodic end conditions y (a) = y (b), y′ (a) = y ′ (b) is called a periodic Sturm–Liouville system. Example 8.1.3. Consider the periodic Sturm–Liouville system y ′′ + λy = 0, −π ≤ x ≤ π, y (−π) = y (π), y′ (−π) = y ′ (π). Here we note that p (x) = 1, hence p (−π) = p (π). When λ > 0, we see that the solution of the Sturm–Liouville equation is y (x) = A cos √ λ x + B sin √ λ x. Application of the periodic end conditions yields 4 2 sin √ λ π5 B = 0, 4 2 √ λ sin √ λ π5 A = 0. Thus, to obtain a nontrivial solution, we must have sin 4√ λ 5 π = 0, A = 0, B = 0. Consequently, λn = n 2 , n = 1, 2, 3,.... Since sin √ λ π = 0 is satisfied for arbitrary A and B, we obtain two linearly independent eigenfunctions cos nx, and sin nx corresponding to the same eigenvalue n 2 . It can be readily shown that if λ < 0, the solution of the Sturm–Liouville equation does not satisfy the periodic end conditions. However, when λ = 0 the corresponding eigenfunction is 1. Thus, the eigenvalues of the periodic Sturm–Liouville system are 0, & n 2 ' , and the corresponding eigenfunctions are 1, {cos nx}, {sin nx}, where n is a positive integer. 8.2 Eigenvalues and Eigenfunctions 277 8.2 Eigenvalues and Eigenfunctions In Examples 8.1.1 and 8.1.2 of the regular Sturm–Liouville systems in the preceding section, we see that there exists only one linearly independent eigenfunction corresponding to the eigenvalue λ, which is called an eigenvalue of multiplicity one (or a simple eigenvalue). An eigenvalue is said to be of multiplicity k if there exist k linearly independent eigenfunctions corresponding to the same eigenvalue. In Example 8.1.3 of the periodic Sturm– Liouville system, the eigenfunctions cos nx, sin nx correspond to the same eigenvalue n 2 . Thus, this eigenvalue is of multiplicity two. In the preceding examples, we see that the eigenfunctions are cos nx and sin nx for n = 1, 2, 3,.... It can be easily shown by using trigonometric identities that  π −π cos mx cos nx dx = 0, m = n,  π −π cos mx sin nx dx = 0, for all integers m, n,  π −π sin mx sin nx dx = 0, m = n. We say that these functions are orthogonal to each other in the interval [−π, π]. The orthogonality relation holds in general for the eigenfunctions of Sturm–Liouville systems Let φ (x) and ψ (x) be any real-valued integrable functions on an interval I. Then φ and ψ are said to be orthogonal on I with respect to a weight function ρ (x) > 0, if and only if, φ, ψ =  I φ (x) ψ (x) ρ (x) dx = 0. (8.2.1) The interval I may be of infinite extent, or it may be either open or closed at one or both ends of the finite interval. When φ = ψ in (8.2.1) we define the norm of φ by φ =  I φ 2 (x) ρ (x) dx 1 2 . (8.2.2) Theorem 8.2.1. Let the coefficients p, q, and s in the Sturm–Liouville system be continuous in [a, b]. Let the eigenfunctions φj and φk, corresponding to λj and λk, be continuously differentiable. Then φj and φk are orthogonal with respect to the weight function s (x) in [a, b]. Proof. Since φj corresponding to λj satisfies the Sturm–Liouville equation, we have d dx  p φ′ j + (q + λj s) φj = 0 (8.2.3) 278 8 Eigenvalue Problems and Special Functions and for the same reason d dx (p φ′ k )+(q + λks) φk = 0. (8.2.4) Multiplying equation (8.2.3) by φk and equation (8.2.4) by φj , and subtracting, we obtain (λj − λk) s φjφk = φj d dx (p φ′ k ) − φk d dx  p φ′ j = d dx (p φ′ k ) φj −  p φ′ j φk ! and integrating yields (λj − λk)  b a s φjφkdx = p  φjφ ′ k − φ ′ jφk !b a = p (b) φj (b) φ ′ k (b) − φ ′ j (b) φk (b) ! −p (a) φj (a) φ ′ k (a) − φ ′ j (a) φk (a) ! (8.2.5) the right side of which is called the boundary term of the Sturm–Liouville system. The end conditions for the eigenfunctions φj and φk are b1φj (b) + b2φ ′ j (b)=0, b1φk (b) + b2φ ′ k (b)=0. If b2 = 0, we multiply the first condition by φk (b) and the second condition by φj (b), and subtract to obtain φj (b) φ ′ k (b) − φ ′ j (b) φk (b) ! = 0. (8.2.6) In a similar manner, if a2 = 0, we obtain φj (a) φ ′ k (a) − φ ′ j (a) φk (a) ! = 0. (8.2.7) We see by virtue of (8.2.6) and (8.2.7) that (λj − λk)  b a s φj φk dx = 0. (8.2.8) If λj and λk are distinct eigenvalues, then  b a s φj φk dx = 0. (8.2.9) Theorem 8.2.2. The eigenfunctions of a periodic Sturm–Liouville system in [a, b] are orthogonal with respect to the weight function s (x) in [a, b]. 8.2 Eigenvalues and Eigenfunctions 279 Proof. The periodic conditions for the eigenfunctions φj and φk are φj (a) = φj (b), φ′ j (a) = φ ′ j (b), φk (a) = φk (b), φ′ k (a) = φ ′ k (b). Substitution of these into equation (8.2.5) yields (λj − λk)  b a s φj φk dx = [p (b) − p (a)] φj (a) φ ′ k (a) − φ ′ j (a) φk (a) ! . Since p (a) = p (b), we have (λj − λk)  b a s φj φk dx = 0. (8.2.10) For distinct eigenvalues λj = λk, (λj − λk) = 0 and thus,  b a s φj φk dx = 0. (8.2.11) Theorem 8.2.3. For any y, z ∈ D (L), we have the Lagrange identity yL[z] − zL[y] = d dx [p (yz′ − zy′ )] . (8.2.12) Proof. Using the definition of the Sturm–Liouville operator, we have yL[z] − zL[y] = y d dx  p dz dx + qyz − z d dx  p dy dx − qyz = d dx [p (yz′ − zy′ )] . Theorem 8.2.4. The Sturm–Liouville operator L is self-adjoint. In other words, for any y, z ∈ D (L), we have L[y] , z = y,L[z], (8.2.13) where < , > is the inner product in L 2 ([a, b]) defined by f,g =  b a f (x) g (x) dx. (8.2.14) Proof. Since all constants involved in the boundary conditions of Sturm– Liouville system are real, if z ∈ D (L), then z ∈ D (L). Also since p, q and s are real-valued, L[z] = L[z]. Consequently, we have L[y] , z−y,L[z] =  b a (z L[y] − y L[z]) dx = [p (z y′ − y z ′ )]b a , by (8.2.12). (8.2.15) 280 8 Eigenvalue Problems and Special Functions We next show that the right hand side of this equality vanishes for a regular RSL system. If p (a) = 0, the result follows immediately. If p (a) > 0, then y and z satisfy the boundary conditions of the form (8.1.5) at x = a. That is, ⎡ ⎣ y (a) y ′ (a) z (a) z ′ (a) ⎤ ⎦ ⎡ ⎣ a1 a2 ⎤ ⎦ = 0. Since a1 and a2 are not both zero, we have z (a) y ′ (a) − y (a) z ′ (a)=0. A similar argument can be used to the other end point x = b, so that the right-hand side of (8.2.15) vanishes. This proves the theorem. Theorem 8.2.5. All the eigenvalues of a regular Sturm–Liouville system with s (x) > 0 are real. Proof. Let λ be an eigenvalue of a RSL system and let y (x) be the corresponding eigenfunction. This means that y = 0 and L[y] = −λsy. Then 0 = L[y] , y−y,L[y] =  λ − λ  b a s (x)|y (x)| 2 dx. Since s (x) > 0 in [a, b] and y = 0, the integrand is a positive number. Thus, λ = λ, and hence, the eigenvalues are real. This completes the proof. Theorem 8.2.6. If φ1 (x) and φ2 (x) are any two solutions of the equation L[y] + λsy = 0 on [a, b], then p (x) W (x; φ1, φ2) = constant, where W is the Wronskian. Proof. Since φ1 and φ2 are solutions of L[y] + λsy = 0, we have d dx  p dφ1 dx  + (q + λs) φ1 = 0, d dx  p dφ2 dx  + (q + λs) φ2 = 0. Multiplying the first equation by φ2 and the second equation by φ1, and subtracting, we obtain φ1 d dx  p dφ2 dx  − φ2 d dx  p dφ1 dx  = 0. Integrating this equation from a to x, we obtain p (x) [φ1 (x) φ ′ 2 (x) − φ ′ 1 (x) φ2 (x)] = p (a) [φ1 (a) φ ′ 2 (a) − φ ′ 1 (a) φ2 (a)] p (x) W (x; φ, φ2) = constant. (8.2.16) This is called Abel’s formula where W is the Wronskian. 8.2 Eigenvalues and Eigenfunctions 281 Theorem 8.2.7. An eigenfunction of a regular Sturm–Liouville system is unique except for a constant factor. Proof. Let φ1 (x) and φ2 (x) be eigenfunctions corresponding to an eigenvalue λ. Then, according to Abel’s formula (8.2.16), we have p (x) W (x; φ1, φ2) = constant, p (x) > 0, where W is the Wronskian. Thus, if W vanishes at a point in [a, b], it must vanish for all x ∈ [a, b]. Since φ1 and φ2 satisfy the end condition at x = a, we have a1φ1 (a) + a2φ ′ 1 (a)=0, a1φ2 (a) + a2φ ′ 2 (a)=0. Since a1 and a2 are not both zero, we have       φ1 (a) φ ′ 1 (a) φ1 (a) φ ′ 2 (a)       = W (a; φ1, φ2)=0. Therefore, W (x; φ1φ2) = 0 for all x ∈ [a, b], which is a sufficient condition for the linear dependence of two functions φ1 and φ2. Hence, φ1 (x) differs from φ2 (x) only by a constant factor. Theorem 8.2.5 states that all eigenvalues of a regular Sturm–Liouville system are real, but it does not guarantee that any eigenvalue exists. However, it can be proved that a self-adjoint regular Sturm–Liouville system has a denumerably infinite number of eigenvalues. To illustrate this, we consider the following example. Example 8.2.1. Consider the Sturm–Liouville system y ′′ + λy = 0, 0 ≤ x ≤ 1, y (0) = 0, y (1) + hy′ (1) = 0, h> 0 a constant. Here p = 1, q = 0, s = 1. The solution of the Sturm–Liouville equation is y (x) = A cos √ λ x + B sin √ λ x. Since y (0) = 0, gives A = 0, we have y (x) = B sin √ λ x. Applying the second end condition, we have sin √ λ + h √ λ cos √ λ = 0, B = 0 282 8 Eigenvalue Problems and Special Functions which can be rewritten as tan √ λ = −h √ λ. If α = √ λ is introduced in this equation, we have tan α = −h α. This equation does not possess an explicit solution. Thus, we determine the solution graphically by plotting the functions ξ = tan α and ξ = −hα against α, as shown in Figure 8.2.1. The roots are given by the intersection of two curves, and as is evident from the graph, there are infinitely many roots αn for n = 1, 2, 3,.... To each root αn, there corresponds an eigenvalue λn = α 2 n , n = 1, 2, 3,.... Thus, there exists an ordered sequence of eigenvalues λ0 < λ1 < λ2 < λ3 <... with limn→∞ λn = ∞. The corresponding eigenfunctions are sin √ λn x . Figure 8.2.1 Intersection of ξ = tan α and ξ = −h α. 8.3 Eigenfunction Expansions 283 Theorem 8.2.8. A self-adjoint regular Sturm–Liouville system has an infinite sequence of real eigenvalues λ1 < λ2 < λ3 <... with limn→∞ λn = ∞. For each n the corresponding eigenfunction φn (x), uniquely determined up to a constant factor, has exactly n zeros in the interval (a, b). Proof of this theorem can be found in the book by Myint-U (1978). 8.3 Eigenfunction Expansions A real-valued function φ (x) is said to be square-integrable with respect to a weight function ρ (x) > 0, if, on an interval I,  I φ 2 (x) ρ (x) dx < +∞. (8.3.1) An immediate consequence of this definition is the Schwarz inequality      I φ (x) ψ (x) ρ (x) dx     2 ≤  I φ 2 (x) ρ (x) dx  I ψ 2 (x) ρ (x) dx (8.3.2) for square-integrable functions φ (x) and ψ (x). Let {φn (x)}, for positive integers n, be an orthogonal set of squareintegrable functions with a positive weight function ρ (x) on an interval I. Let f (x) be a given function that can be represented by a uniformly convergent series of the form f (x) = ∞ n=1 cn φn (x), (8.3.3) where the coefficients cn are constants. Now multiplying both sides of (8.3.3) by φm (x) ρ (x) and integrating term by term over the interval I (uniform convergence of the series is a sufficient condition for this), we obtain  I f (x) φm (x) ρ (x) dx = ∞ n=1  I cnφn (x) φm (x) ρ (x) dx, and hence, for n = m,  I f (x) φn (x) ρ (x) dx = cn  I φ 2 n (x) ρ (x) dx. 284 8 Eigenvalue Problems and Special Functions Thus, cn = * I f φn ρ dx * I φ2 n ρ dx . (8.3.4) Hence, we have the following theorem: Theorem 8.3.1. If f is represented by a uniformly convergent series f (x) = ∞ n=1 cnφn (x) on an interval I, where φn are square-integrable functions orthogonal with respect to a positive weight function ρ (x), then cn are determined by cn = * I f φn ρ dx * I φ2 n ρ dx . Example 8.3.1. The Legendre polynomials Pn (x) are orthogonal with respect to the weight function ρ (x) = 1 on (−1, 1). If we assume that f (x) can be represented by the Fourier–Legendre series f (x) = ∞ n=1 cnPn (x) then, cn are given by cn = * 1 −1 f (x) Pn (x) dx * 1 −1 P2 n (x) dx =  2n + 1 2   1 −1 f (x) Pn (x) dx. In the above discussion, we assumed that the given function f (x) is represented by a uniformly convergent series. This is rather restrictive, and we will show in the following section that f (x) can be represented by a mean-square convergent series. 8.4 Convergence in the Mean Let {φn} be the set of square-integrable functions orthogonal with respect to a weight function ρ (x) on [a, b]. Let sn (x) = n k=1 ckφk (x) 8.4 Convergence in the Mean 285 be the nth partial sum of the series ∞ k=1 ckφk (x). Let f be a square-integrable function. The sequence {sn} is said to converge in the mean to f (x) on the interval I with respect to the weight function ρ (x) if lim n→+∞  I [f (x) − sn (x)]2 ρ (x) dx = 0. (8.4.1) We shall now seek the coefficients ck such that sn (x) represents the best approximation to f (x) in the sense of least squares, that is, we seek to minimize the integral E (ck) =  I [f (x) − sn (x)]2 ρ (x) dx =  I f 2 ρ dx − 2 n k=1 ck  I f φk ρ dx + n k=1 c 2 k  I φ 2 k ρ dx. (8.4.2) This is an extremal problem. A necessary condition on the ck for E to be minimum is that the first partial derivatives of E with respect to these coefficients vanish. Thus, differentiating (8.4.2) with respect to ck, we obtain ∂E ∂ck = −2  I f φk ρ dx + 2ck  I φ 2 k ρ dx = 0 (8.4.3) and hence, ck = * I f φk ρ dx * I φ 2 k ρ dx . (8.4.4) Now if we complete the square, the right side of (8.4.2) becomes E =  I f 2 ρ dx + n k=1  I φ 2 k ρ dx ck − * I f φk ρ dx * I φ 2 k ρ dx 2 − n k=1 * I f φk ρ dx 2 * I φ 2 k ρ dx . The right side shows that E is a minimum if and only if ck is given by (8.4.4). Therefore, this choice of ck yields the best approximation to f (x) in the sense of least squares. For series convergent in the mean to f (x), we conventionally write f (x) ∼ ∞ k=1 ckφk (x), where the coefficients ck are the generalized Fourier coefficients and the series is the generalized Fourier series. This series may or may not be pointwise or uniformly convergent. 286 8 Eigenvalue Problems and Special Functions 8.5 Completeness and Parseval’s Equality Substituting the Fourier coefficients (8.4.4) into (8.4.2), we obtain  I 4 f (x) − n k=1 ckφk (x) 52 ρ (x) dx =  I f 2 ρ dx − n k=1 c 2 k  I φ 2 k ρ dx. Since the left side is nonnegative, we have n k=1 c 2 k  I φ 2 k ρ dx ≤  I f 2 ρ dx. (8.5.1) The integral on the right side is finite, and hence, the series on the left side is bounded above for any n. Thus, as n → ∞, the inequality (8.5.1) may be written as ∞ k=1 c 2 k  I φ 2 k ρ dx ≤  I f 2 ρ dx. (8.5.2) This is called Bessel’s inequality. If the series converges in the mean to f (x), that is, limn→∞  I 4 f (x) − n k=1 ckφk (x) 52 ρ (x) dx = 0, then, it follows from the above derivation that ∞ k=1 c 2 k  I φ 2 kρ dx =  I f 2 ρ dx which is called Parseval’s equality. Sometimes it is known as the completeness relation. Thus, when every continuous square-integrable function f (x) can be expanded into an infinite series f (x) = ∞ k=1 ckφk (x), the sequence of continuous square-integrable functions {φk} orthogonal with respect to the weight function ρ is said to be complete. Next we state the following theorem: Theorem 8.5.1. The eigenfunctions of any regular Sturm–Liouville system are complete in the space of functions that are piecewise continuous on the interval [a, b] with respect to the weight function s (x). Moreover, any piecewise smooth function on [a, b] that satisfies the end conditions of 8.5 Completeness and Parseval’s Equality 287 the regular Sturm–Liouville system can be expanded in an absolutely and uniformly convergent series f (x) = ∞ k=1 ckφk (x), where ck are given by ck =  b a f φk s (x) dx4 b a φ 2 k s (x) dx. Proof of a more general theorem can be found in Coddington and Levinson (1955). Example 8.5.1. Consider a cylindrical wire of length l whose surface is perfectly insulated against the flow of heat. The end l = 0 is maintained at the zero degree temperature, while the other end radiates freely into the surrounding medium of zero degree temperature. Let the initial temperature distribution in the wire be f (x). Find the temperature distribution u (x, t). The initial boundary-value problem is ut = k uxx, 0 < x < l, t > 0, (8.5.3) u (x, 0) = f (x), 0 < x ≤ l, (8.5.4) u (0, t)=0, t > 0, (8.5.5) h u (l, t) + u ′ (l, t)=0, t > 0, h > 0. (8.5.6) By the method of separation of variables, we assume a nontrivial solution in the form u (x, t) = X (x) T (t) and substituting it in the heat equation, we obtain X′′ + λX = 0, T′ + kλT = 0, where λ > 0 is a separation constant. The solution of the latter equation is T (t) = Ce−kλt (8.5.7) where C is an arbitrary constant. The former equation has to be solved subject to the boundary conditions X (0) = 0, h X (l) + X′ (l)=0. This is a Sturm–Liouville system which gives the solution with X (0) = 0 X (x) = B sin √ λ x, (8.5.8) 288 8 Eigenvalue Problems and Special Functions where B is a constant to be determined. Application of the second end condition (8.5.6) yields h sin √ λ l + √ λ cos √ λ l = 0 for B = 0 which can be rewritten as tan √ λ l = − √ λ/h. If α = √ λ l is introduced in the preceding equation, we have tan α = −a α, where a = (1/hl). As in Example 8.2.1, there exists a sequence of eigenvalues λ1 < λ2 < λ3 <... with limn→∞ λn = ∞. The corresponding eigenfunctions are sin √ λn x, and hence, Xn (x) = Bn sin  λn x. (8.5.9) Therefore, combining (8.5.7) with C = Cn and (8.5.9), the solution takes the form un (x, t) = an e −kλnt sin  λn x, an = BnCn which satisfies the heat equation and the boundary conditions. Since the heat equation is linear and homogeneous, we form the series solution u (x, t) = ∞ n=1 an e −kλnt sin  λn x, (8.5.10) which is also a solution, provided it converges and is twice differentiable with respect to x and once differentiable with respect to t. According to Theorem 8.2.1, the eigenfunctions sin √ λn x form an orthogonal system over the interval (0, l). Application of the initial condition yields u (x, 0) = f (x) ∼ ∞ n=1 an sin  λn x. If we assume that f is a piecewise smooth function on [a, b], then, by Theorem 8.5.1, we can expand f (x) in terms of the eigenfunctions, and formally write f (x) = ∞ n=1 an sin  λn x, where the coefficient an is given by an =  l 0 f (x) sin λn x dx4 l 0 sin2  λn x dx. With this value of an, the temperature distribution is given by (8.5.10). 8.6 Bessel’s Equation and Bessel’s Function 289 8.6 Bessel’s Equation and Bessel’s Function Bessel’s equation frequently occurs in problems of applied mathematics and mathematical physics involving cylindrical symmetry. The standard form of Bessel’s equation is given by x 2 y ′′ + xy′ +  x 2 − ν 2 y = 0, (8.6.1) where ν is a nonnegative real number. We shall first restrict our attention to x > 0. Since x = 0 is the regular singular point, a solution is taken in accordance with the Frobenius method to be y (x) = ∞ n=0 an x s+n , (8.6.2) where the index s is to be determined. Substitution of this series into equation (8.6.1) then yields  s 2 − ν 2 a0 x s + " (s + 1)2 − ν 2 # a1x s+1 + ∞ n=2 ("(s + n) 2 − ν 2 # an + an−2 ) x s+n = 0. (8.6.3) The requirement that the coefficient of x s vanish leads to the initial equation  s 2 − ν 2 a0 = 0, (8.6.4) from which it follows that s = + ν for arbitrary a0 = 0. Since the leading term in the series (8.6.2) is a0 x s , it is clear that for ν > 0 the solution of Bessel’s equation corresponding to the choice s = ν vanishes at the origin, whereas the solution corresponding to s = −ν is infinite at that point. We consider first the regular solution of Bessel’s equation, that is, the solution corresponding to the choice s = ν. The vanishing of the coefficient of x s+1 in equation (8.6.3) requires that (2ν + 1) a1 = 0, (8.6.5) which in turn implies that a1 = 0 (since ν ≥ 0). From the requirement that the coefficient of x s+n in equation (8.6.3) be zero, we obtain the two-term recurrence relation an = − an−2 n (2ν + n) . (8.6.6) Since a1 = 0, it is obvious that an = 0 for n = 3, 5, 7,.... The remaining coefficients are given by a2k = (−1)k a0 2 2kk! (ν + k) (ν + k − 1)...(ν + 1) (8.6.7) 290 8 Eigenvalue Problems and Special Functions for k = 1, 2, 3,.... This relation may also be written as a2k = (−1)k 2 νΓ (ν + 1) a0 2 2k+νk!Γ (ν + k + 1), k = 1, 2,..., (8.6.8) where Γ (α) is the gamma function, whose properties are described in the Appendix. Hence, the regular solution of Bessel’s equation takes the form y (x) = a0 ∞ k=0 (−1)k 2 νΓ (ν + 1) 2 2k+νk!Γ (ν + k + 1) x 2k+ν . (8.6.9) It is customary to choose a0 = 1 2 νΓ (ν + 1) (8.6.10) and to denote the corresponding solution by Jν (x). This solution, called the Bessel function of the first kind of order ν, is therefore given by Jν (x) = ∞ k=0 (−1)k x 2k+ν 2 2k+νk! Γ (ν + k + 1). (8.6.11) To determine the irregular solution of the Bessel equation for s = −ν, we proceed as above. In this way, we obtain as the analogue of equation (8.6.5) the relation (−2ν + 1) a1 = 0 from which it follows, without loss of generality, that a1 = 0. Using the recurrence relation an = − an−2 n (n − 2ν) , n ≥ 2 (8.6.12) we obtain the irregular solution of the Bessel function of the first kind of order −ν as J−ν (x) = ∞ k=0 (−1)k x 2k−ν 2 2k−νk! Γ (−ν + k + 1). (8.6.13) It can be easily proved that, if ν is not an integer, Jν and J−ν converge for all values of x, and are linearly independent. Thus, the general solution of the Bessel equation for nonintegral ν is y (x) = c1Jν (x) + c2J−ν (x). (8.6.14) If ν is an integer, say ν = n, then from equation (8.6.13), noting that, when gamma functions in the coefficients of the first n terms become infi- nite, the coefficients become zero, hence we have 8.6 Bessel’s Equation and Bessel’s Function 291 J−n (x) = ∞ k=n (−1)k x 2k−n 2 2k−nk! Γ (−n + k + 1). = (−1)n∞ k=0 (−1)k x 2k+n 2 2k+nk! Γ (n + k + 1). = (−1)n Jn (x). (8.6.15) This shows that J−n is not independent of Jn, and therefore, a second linearly independent solution is required. A number of distinct irregular solutions are discussed in the literature, but the one most commonly used, as defined by Watson (1966), is Yν (x) = (cos νπ) Jν (x) − J−ν (x) sin νπ . (8.6.16) For nonintegral ν, it is obvious that Yν (x), being a linear combination of Jν (x) and J−ν (x), is linearly independent of Jν (x). When ν is a nonnegative integer n, Yν (x) is indeterminate. But Yn (x) = limν→n Yν (x) exists and is a solution of the Bessel equation. Moreover, it is linearly independent of Jn (x). (For an extended treatment, see Watson (1966)). The function Yν (x) is called the Bessel function of the second kind of order ν. Thus, the general solution of the Bessel equation is y (x) = c1Jν (x) + c2Yν (x), for ν ≥ 0. (8.6.17) Like elementary functions, the Bessel functions are tabulated (see Jahnke et al. (1960)). For illustration, the functions J0, J1, Y0 and Y1 are plotted for small values of x in Figure 8.6.1. It should be noted that Jν (x) for ν ≥ 0 and J−ν (x) for a positive integer ν are finite at the origin, but J−ν (x) for nonintegral ν and Yν (x) for ν ≥ 0 approach infinity as x tends to zero. Some of the useful recurrence relations are Jν−1 (x) + Jν+1 (x) = 2ν x Jν (x), (8.6.18) νJν (x) + xJ′ ν (x) = xJν−1 (x), (8.6.19) Jν−1 (x) − Jν+1 (x)=2J ′ ν (x), (8.6.20) νJν (x) − xJ′ ν (x) = xJν+1 (x), (8.6.21) d dx [x νJν (x)] = x νJν−1 (x), (8.6.22) d dx x −νJν (x) ! = −x −νJν+1 (x). (8.6.23) All of these relations also hold true for Yν (x). 292 8 Eigenvalue Problems and Special Functions Figure 8.6.1 Graphs of Jν (x) and Yν (x). For |x| ≫ 1 and |x| ≫ ν, the asymptotic expansion of Jν (x) is Jν−1 (x) ∼ 2 2 πx 1/1 −  4ν 2 − 1 2 4ν 2 − 3 2 2! (8x) 2 +  4ν 2 − 1 2 4ν 2 − 3 2 4ν 2 − 5 2 4ν 2 − 7 2 4! (8x) 4 − ...0 cos φ − / 4ν 2 − 1 2 8x −  4ν 2 − 1 2 4ν 2 − 3 2 4ν 2 − 5 2 3! (8x) 3 + ...0 sin φ 3 (8.6.24) where φ = x −  ν + 1 2  π 2 . For |x| ≫ 1 and |x| ≫ ν, the asymptotic expansion of Yν (x) is 8.6 Bessel’s Equation and Bessel’s Function 293 Yν (x) ∼ 2 2 πx 1/1 −  4ν 2 − 1 2 4ν 2 − 3 2 2! (8x) 2 +  4ν 2 − 1 2 4ν 2 − 3 2 4ν 2 − 5 2 4ν 2 − 7 2 4! (8x) 4 − ...0 sin φ + / 4ν 2 − 1 2 8x −  4ν 2 − 1 2 4ν 2 − 3 2 4ν 2 − 5 2 3! (8x) 3 + ...0 cos φ 3 . (8.6.25) When ν = + (1/2), Bessel’s function may be expressed in the form J 1 2 (x) = 2 2 πx sin x, (8.6.26) J− 1 2 (x) = 2 2 πx cos x. (8.6.27) The Bessel functions which satisfy the condition Jν (akm) + hJ′ ν (akm)=0, h, a = constant, (8.6.28) are orthogonal to each other with respect to the weight function x, that is, for the nonnegative integer ν, the orthogonal relation is  a 0 xJν (xkn) Jν (xkm) dx = 0, n = m. (8.6.29) When n = m, we have the norm Jν (xkm) 2 =  a 0 x [Jν (xkm)]2 dx = 1 2k 2 m ( a 2 k 2 m [J ′ ν (akm)]2 +  a 2 k 2 m − ν 2 [Jν (akm)]2 ) , (8.6.30) where km are the roots of (8.6.28). We now give a particular example of the eigenfunction expansion theorem discussed in Sections 8.4 and 8.5. Assume a formal expansion of the function f (x) defined in 0 ≤ x ≤ a in the form f (x) = ∞ m=1 amJν (xkm), (8.6.31) where the summation is taken over all the positive roots k1, k2, k3, ..., of equation (8.6.28). Multiplying (8.6.31) by xJν (xkn), integrating, and utilizing (8.6.30), we obtain 294 8 Eigenvalue Problems and Special Functions  a 0 xf (x) Jν (xkm) dx = am  a 0 x [Jν (xkm)]2 dx = am 2k 2 m ( a 2 k 2 m [J ′ ν (akm)]2 +  a 2 k 2 m − ν 2 [Jν (akm)]2 ) . (8.6.32) Thus, we have the following theorem: Theorem 8.6.1. If bm =  a 0 xf (x) Jν (xkm) dx (8.6.33) then the expansion (8.6.31) of f (x) takes the form f (x) = ∞ m=1 2k 2 mbmJν (xkm) k 2 m [J ′ ν (akm)]2 + (a 2k 2 m − ν 2) [Jν (akm)]2 . (8.6.34) In particular, when h = 0 in (8.6.28), that is, when km are the positive roots of Jν (akm)=0, then (8.6.34) becomes f (x) = 2 a 2 ∞ m=1 bmJν (xkm) k 2 m [J ′ ν (akm)]2 = 2 a 2 ∞ m=1 bmJν (xkm) [Jν+1 (akm)]2 . (8.6.35) These expansions are known as the Bessel–Fourier series for f (x). They are generated by Sturm–Liouville problems involving the Bessel equation, and arise from problems associated with partial differential equations. Closely related to Bessel’s functions are Hankel’s functions of the first and second kind, defined by H(1) ν (x) = Jν (x) + iYν (x), H(2) ν (x) = Jν (x) − iYν (x), (8.6.36) respectively, where i = √ −1. Other closely related functions are the modified Bessel functions. Consider Bessel’s equation containing a parameter λ, namely, x 2 y ′′ + xy′ +  λ 2x 2 − ν 2 y = 0. (8.6.37) The general solution of this equation is y (x) = c1Jν (λx) + c2Yν (λx). If λ = i, then y (x) = c1Jν (ix) + c2Yν (ix). We write 8.7 Adjoint Forms and Lagrange Identity 295 Jν (ix) = ∞ k=0 (−1)k (ix) 2k+ν 2 2k+νk! Γ (ν + k + 1) = i ν Iν (x), where Iν (x) = ∞ k=0 x 2k+ν 2 2k+νk! Γ (ν + k + 1), (8.6.38) Iν (x) is called the modified Bessel function of the first kind of order ν. As in the case of Jν and J−ν, Iν and I−ν (which is defined in a similar manner) are linearly independent solutions except when ν is an integer. Consequently, we define the modified Bessel function of the second kind of order ν by Kν (x) = π 2  I−ν (x) − Iν (x) sin νπ  . (8.6.39) Thus, we obtain the general solution of the modified Bessel equation x 2 y ′′ + xy′ −  x 2 + ν 2 y = 0 (8.6.40) in the form y (x) = c1Iν (x) + c2Kν (x). (8.6.41) We should note that Iν (0) = ⎧ ⎨ ⎩ 1, ν = 0 0, ν> 0 (8.6.42) and that Kν approaches infinity as x → 0. For a detailed treatment of Bessel and related functions, refer to Watson’s (1966) Theory of Bessel Functions. The eigenvalue problems which involve Bessel’s functions will be described in Section 8.8 on singular Sturm–Liouville systems. 8.7 Adjoint Forms and Lagrange Identity Self-adjoint equations play a very important role in many areas of applied mathematics and mathematical physics. Here we will give a brief account of self-adjoint operators and the Lagrange identity. We consider the equation L[y] = a0 (x) y ′′ + a1 (x) y ′ + a2 (x) y = 0 296 8 Eigenvalue Problems and Special Functions defined on an interval I. Integrating z (x)L[y] by parts from a to x, we have  x a zL[y] dx = (za0) y ′ − (za0) ′ y + (za1) y !x a +  x a (za0) ′′ − (za1) ′ + (za2) ! y dx. (8.7.1) Now, if we define the second-order operator L ∗ by L ∗ [z]=(za0) ′′ − (za1) ′ + (za2) = a0 z ′′ (2a ′ 0 − a1) z ′ + (a ′′ 0 − a ′ 1 + a2) z the relation (8.7.1) takes the form  x a (zL[y] − yL∗ [z]) dx = [a0 (y ′ z − yz′ )+(a1 − a ′ 0 ) yz] x a . (8.7.2) The operator L ∗ is called the adjoint operator corresponding to the operator L. It can be readily verified that the adjoint of L ∗ is L itself. If L and L ∗ are the same, L is said to be self-adjoint. The necessary and sufficient condition for this is that a1 = 2a ′ 0 − a1, a2 = a ′′ 0 − a ′ 1 + a2, which is satisfied if a1 = a ′ 0 . Thus, if L is self-adjoint, we have L(y) = a0y ′′ + a ′ 0 y ′ + a2y = (a0y ′ ) ′ + a2 (x) y. (8.7.3) In general, L[y] is not self-adjoint. But if we let h (x) = 1 a0 exp  x a1 (t) a0 (t) dt0 (8.7.4) then h (x)L[y] is self-adjoint. Thus, any second-order linear differential equation a0 (x) y ′′ + a1 (x) y ′ + a2 (x) y = 0 (8.7.5) can be made self-adjoint. Multiplying by h (x) given by equation (8.7.4), equation (8.7.5) is transformed into the self-adjoint form d dx p (x) dy dx + q (x) y = 0, (8.7.6) 8.8 Singular Sturm–Liouville Systems 297 where p (x) = exp  x a1 (t) a0 (t) dt0 , q (x) =  a2 a0  exp  x a1 (t) a0 (t) dt0 . (8.7.7) For example, the self-adjoint form of the Legendre equation  1 − x 2 y ′′ − 2xy′ + n (n + 1) y = 0 is d dx  1 − x 2 dy dx + n (n + 1) y = 0, (8.7.8) and the self-adjoint form of the Bessel equation x 2 y ′′ + xy′ +  x 2 − ν 2 y = 0 is d dx  x dy dx +  x − ν 2 x  y = 0. (8.7.9) Now, if we differentiate both sides of equation (8.7.2), we obtain zL[y] − yL∗ [z] = d dx [a0 (y ′ z − yz′ )+(a1 − a ′ 0 ) yz] (8.7.10) which is known as the Lagrange identity for the operator L. If we consider the integral from a to b of equation (8.7.2), we obtain Green’s identity  b a (zL[y] − yL∗ [z]) dx = [a0 (y ′ z − yz′ )+(a1 − a ′ 0 ) yz] b a . (8.7.11) When L is self-adjoint, this relation becomes  b a (zL[y] − yL[z]) dx = [a0 (y ′ z − yz′ )]b a . (8.7.12) 8.8 Singular Sturm–Liouville Systems A Sturm–Liouville equation is called singular when it is given on a semiinfinite or infinite interval, or when the coefficient p (x) or s (x) vanishes, or when one of the coefficients becomes infinite at one end or both ends of a finite interval. A singular Sturm–Liouville equation together with appropriate linear homogeneous end conditions is called a singular Sturm–Liouville (SSL) system. The conditions imposed in this case are not like the separated boundary end conditions in the regular Sturm–Liouville system. The 298 8 Eigenvalue Problems and Special Functions condition that is often necessary to prescribe is the boundedness of the function y (x) at the singular end point. To exhibit this, let us consider a problem with a singularity at the end point x = a. By the relation (8.7.12), for any twice continuously differentiable functions y (x) and z (x), we have on (a, b)  b a+ε {zL[y] − yL[z]} dx = p (b) [y ′ (b) z (b) − y (b) z ′ (b)] −p (a + ε) [y ′ (a + ε) z (a + ε) − y (a + ε) z ′ (a + ε)] , where ε is a small positive number. If the conditions lim x→a+ p (x) [y ′ (x) z (x) − y (x) z ′ (x)] = 0, (8.8.1) p (b) [y ′ (b) z (b) − y (b) z ′ (b)] = 0, (8.8.2) are imposed on y and z, it follows that  b a {zL[y] − yL[z]} dx = 0. (8.8.3) For example, when p (a) = 0, the relations (8.8.1) and (8.8.2) are replaced by the conditions 1. y (x) and y ′ (x) are finite as x → a 2. b1y (b) + b2y ′ (b) = 0. Thus, we say that the singular Sturm–Liouville system is self-adjoint, if any functions y (x) and z (x) that satisfy the end conditions satisfy  b a {zL[y] − yL[z]} dx = 0. Example 8.8.1. Consider the singular Sturm–Liouville system involving Legendre’s equation d dx  1 − x 2 dy dx + λy = 0, −1 <x< 1,="" with="" the="" conditions="" that="" y="" and="" ′="" are="" finite="" as="" x="" →="" +="" 1.="" in="" this="" case,="" p="" (x)="1" −="" 2="" s="" vanishes="" at="" legendre="" functions="" of="" first="" kind,="" pn="" (x),="" n="0," 2,...,="" eigenfunctions="" which="" corresponding="" eigenvalues="" λn="n" (n="" 1)="" for="" 2,....="" we="" observe="" here="" singular="" sturm–liouville="" system="" has="" infinitely="" many="" real="" eigenvalues,="" orthogonal="" to="" each="" other.="" example="" 8.8.2.="" another="" a="" is="" bessel="" equation="" fixed="" ν="" 8.8="" systems="" 299="" d="" dx="" ="" dy="" dx="" λ="" ="" 0="" <="" a,="" end="" (a)="0" 0+.="" q="" ,="" now="" (0)="0," becomes="" infinite="" 0+,="" therefore,="" singular.="" if="" kind="" order="" ν,="" namely="" jν="" (knx),="" 2,="" 3,...,="" where="" kna="" nth="" zero="" jν.="" function="" its="" derivative="" both="" .="" thus,="" other="" respect="" weight="" preceding="" examples,="" have="" seen="" (x).="" general,="" they="" square-integrable="" theorem="" 8.8.1.="" distinct="" proof.="" proceeding="" 8.2.1,="" arrive="" (λj="" λk)="" ="" b="" φj="" φk="" (b)="" φ="" k="" j="" !="" suppose="" boundary="" term="" vanishes,="" case="" mentioned="" earlier,="" b1y="" b2y="" then,="" integral="" exists="" by="" virtue="" (8.3.2).="" λj="λk," 8.8.3.="" consider="" involving="" hermite="" u="" ′′="" 2xu′="" λu="0," −∞="" <x<="" ∞,="" (8.8.4)="" not="" self-adjoint.="" let="" −x="" 2u="" takes="" self-adjoint="" form="" 1="" ∞.="" 300="" 8="" eigenvalue="" problems="" special="" nonnegative="" integers="" n,="" φn="" 2hn="" hn="" polynomials="" solutions="" (see="" magnus="" oberhettinger="" (1949)).="" now,="" impose="" tends="" +∞.="" satisfied="" because="" fact="" ne="" since="" square-integrable,="" ∞="" hm="" e="" m="n." 8.8.4.="" problem="" transverse="" vibration="" thin="" elastic="" circular="" membrane="" utt="c" urr="" 1="" r="" ur="" t=""> 0, u (r, 0) = f (r), ut (r, 0) = 0, 0 ≤ r ≤ 1, (8.8.5) u (1, t)=0, limr→0 u (r, t) < ∞, t ≥ 0. We seek a nontrivial separable solution in the form u (r, t) = R (r) T (t). Substituting this in the wave equation yields R′′ + (1/r) R′ R = 1 c 2 T ′′ T = −α 2 , where α is a positive constant. The negative sign in front of α 2 is chosen to obtain the solution periodic in time. Thus, we have rR′′ + R ′ + α 2 rR = 0, T′′ + α 2 c 2T = 0. The solution T (t) is therefore given by T (t) = A cos (αct) + B sin (αct). Next, it is required to determine the solution R (r) of the following singular Sturm–Liouville system d dr r dR dr + α 2 rR = 0, (8.8.6) R (1) = 0, limr→0 R (r) < ∞. (8.8.7) We note that in this case, p = r which vanishes at r = 0. The condition on the boundedness of the function R (r) is obtained from the fact that limr→0 u (r, t) = limr→0 R (r) T (t) < ∞ 8.8 Singular Sturm–Liouville Systems 301 which implies that limr→0 R (r) < ∞ (8.8.8) for arbitrary T (t). Equation (8.8.6) is Bessel’s equation of order zero, the solution of which is given by R (r) = CJ0 (αr) + DY0 (αr), (8.8.9) where J0 and Y0 are Bessel’s functions of the first and second kinds respectively of order zero. The condition (8.8.7) requires that D = 0 since Y0 (αr) → −∞ as r → 0. Hence, R (r) = CJ0 (αr). The remaining condition R (1) = 0 yields J0 (α) = 0. This transcendental equation has infinitely many positive zeros α1 < α2 < α3 <.... Thus, the solution of problem (8.8.5) is given by un (r, t) = J0 (αnr) (An cos αnct + Bn sin αnct), n = 1, 2, 3,.... Since the Bessel equation is linear and homogeneous, the linear superposition principle gives u (r, t) = ∞ n=1 J0 (αnr) (An cos αnct + Bn sin αnct), (8.8.10) is also a solution, provided the series converges and is sufficiently differentiable with respect to r and t. Differentiating (8.8.10) formally with respect to t, we obtain ut (r, t) = ∞ n=1 J0 (αnr) (−An αn c sin αnct + Bn αnc cos αnct). Application of the initial condition ut (r, 0) = 0 yields Bn = 0. Consequently, we have u (r, t) = ∞ n=1 AnJ0 (αnr) cos (αnct). (8.8.11) It now remains to show that u (r, t) satisfies the initial condition u (r, 0) = f (r). For this, we have u (r, 0) = f (r) ∼ ∞ n=1 AnJ0 (αnr). 302 8 Eigenvalue Problems and Special Functions If f (r) is piecewise smooth on [0, 1], then the eigenfunctions J0 (αnr) form a complete orthogonal system with respect to the weight function r over the interval (0, 1). Hence, we can formally expand f (r) in terms of the eigenfunctions. Thus, f (r) = ∞ n=1 AnJ0 (αnr), (8.8.12) where the coefficient An is represented by An =  1 0 rf (r) J0 (αnr) dr1 1 0 r [J0 (αnr)]2 dr. (8.8.13) The solution of the problem (8.8.5) is therefore given by (8.8.11) with the coefficients An given by (8.8.13). 8.9 Legendre’s Equation and Legendre’s Function The Legendre equation is  1 − x 2 y ′′ − 2xy′ + ν (ν + 1) y = 0, (8.9.1) where ν is a real number. This equation arises in problems with spherical symmetry in mathematical physics. Its coefficients are analytic at x = 0. Thus, if we expand near the point x = 0, the coefficients are p (x) = − 2x 1 − x 2 = −2x ∞ m=0 x 2m = ∞ m=0 (−2) x 2m+1 , and q (x) = ν (ν + 1) 1 − x 2 = ν (ν + 1) ∞ m=0 x 2m = ∞ m=0 ν (ν + 1) x 2m. We see that these series converge for |x| < 1. Thus, the Legendre equation on |x| < 1 has convergent power series solution at x = 0. Now to find the solution near the ordinary point x = 0, we assume y (x) = ∞ m=0 am x m. Substituting y, y ′ , and y ′′ in the Legendre equation, we obtain  1 − x 2 ∞ m=0 m (m − 1) am x m−2 − 2x ∞ m=0 mam x m−1 +ν (ν + 1) ∞ m=0 am x m = 0. 8.9 Legendre’s Equation and Legendre’s Function 303 Simplification gives ∞ m=0 [(m + 1) (m + 2) am+2 + (ν − m) (ν + m + 1) am] x m = 0. Therefore the coefficients in the power series must satisfy the recurrence relation am+2 = − (ν − m) (ν + m + 1) (m + 1) (m + 2) am, m ≥ 0. (8.9.2) This relation determines a2, a4, a6, ... in terms of a0, and a3, a5, a7, ... in terms of a1. It can easily be verified that a2k and a2k+1 can be expressed in terms of a0 and a1 respectively as a2k = (−1)k ν (ν − 2)...(ν − 2k + 2) (ν + 1) (ν + 3)...(ν + 2k − 1) (2k)! a0 and a2k+1 = (−1)k (ν − 1) (ν − 3)...(ν − 2k + 1) (ν + 2) (ν + 4)...(ν + 2k) (2k + 1)! a1. Hence, the solution of the Legendre equation is y (x) = a0 1 + ∞ k=1 (−1)k ν (ν − 2)...(ν − 2k + 2) (ν + 1) (ν + 3)...(ν + 2k − 1) x 2k (2k)! 3 +a1 x + ∞ k=1 (−1)k (ν − 1) (ν − 3)...(ν − 2k + 1) (ν + 2) (ν + 4)...(ν + 2k) x 2k+1 (2k + 1)! 3 = a0φν (x) + a1ψν (x). (8.9.3) It can easily be proved that the functions φν (x) and ψν (x) converge for x < 1 and are linearly independent. Now consider the case in which ν = n, with n a nonnegative integer. It is then evident from the recurrence relation (8.9.2) that, when m = n, an+2 = an+4 = ... = 0. Consequently, when n is even, the series φn (x) terminates with x n, whereas the series for ψn (x) does not terminate. When n is odd, it is the series for ψn (x) which terminates with x n, while that for φn (x) does not terminate. 304 8 Eigenvalue Problems and Special Functions In the first case (n even), φn (x) is a polynomial of degree n; the same is true for ψn (x) in the second case (n odd). Thus, for any nonnegative integer n, either φn (x) or ψn (x), but not both, is a polynomial of degree n. Consequently, the general solution of the Legendre equation contains a polynomial solution Pn (x) and an infinite series solution Qn (x) for a nonnegative integer n. To find the polynomial solution Pn (x), it is convenient to choose an so that Pn (1) = 1. Let this an be an = (2n)! 2 n (n!)2 . (8.9.4) Rewriting the recurrence relation (8.9.2), we have an−2 = − (n − 1) n 2 (2n − 1) an. Substituting an from (8.9.4) into this relation, we obtain an−2 = − (2n − 2)! 2 n (n − 1)! (n − 2)!, and an−4 = (2n − 4)! 2 n2! (n − 2)! (n − 4)!. It follows by induction that an−2k = (−1)k (2n − 2k)! 2 nk! (n − k)! (n − 2k)!. Hence, we may write Pn (x) in the form Pn (x) =  N k=0 (−1)k (2n − 2k)! 2 nk! (n − k)! (n − 2k)! x n−2k , (8.9.5) where N = (n/2) when n is even, and N = (n − 1) /2 when n is odd. This polynomial Pn (x) is called the Legendre function of the first kind of order n. It is also known as the Legendre polynomial of degree n. The first few Legendre polynomials are P0 (x)=1, P1 (x) = x, P2 (x) = 1 2  3x 2 − 1 , P3 (x) = 1 2  5x 3 − 3x , P4 (x) = 1 8  35x 4 − 30x 2 + 3 . 8.9 Legendre’s Equation and Legendre’s Function 305 Figure 8.9.1 The first four Legendre’s polynomials. These polynomials are plotted in Figure 8.9.1 for small values of x. Recall that for a given nonnegative integer n, only one of the two solutions φn (x) and ψn (x) of Legendre’s equation is a polynomial, while the other in an infinite series. This infinite series, when appropriately normalized, is called the Legendre function of the second kind. It is defined for |x| < 1 by Qn (x) = ⎧ ⎨ ⎩ φn (1) ψn (x) for n even −ψn (1) φn (x) for n odd . (8.9.6) Thus, when n is a nonnegative integer, the general solution of the Legendre equation is given by y (x) = c1Pn (x) + c2Qn (x). (8.9.7) The Legendre polynomial may also be expressed in the form Pn (x) = 1 2 nn! d n dxn  x 2 − 1 n . (8.9.8) This expression is known as the Rodriguez formula. Like Bessel’s functions, Legendre polynomials satisfy certain recurrence relations. Some of the important relations are 306 8 Eigenvalue Problems and Special Functions (n + 1) Pn+1 (x) − (2n + 1) xPn (x) + nPn−1 (x)=0, n ≥ 1, (8.9.9)  x 2 − 1 P ′ n (x) = nxPn (x) − nPn−1 (x), n ≥ 1, (8.9.10) nPn (x) + P ′ n−1 (x) − xP′ n (x)=0, n ≥ 1, (8.9.11) P ′ n+1 (x) = xP′ n (x)+(n + 1) Pn (x), n ≥ 0. (8.9.12) In addition, P2n (−x) = P2n (x), (8.9.13) P2n+1 (−x) = −P2n+1 (x). (8.9.14) These indicate that Pn (x) is an even function for even n, and an odd function for odd n. It can easily be shown that the Legendre polynomials form a sequence of orthogonal functions on the interval [−1, 1]. Thus, we have  1 −1 Pn (x) Pm (x) dx = 0, for n = m. (8.9.15) The norm of the function Pn (x) is given by Pn (x) 2 =  1 −1 P 2 n (x) dx = 2 2n + 1 . (8.9.16) Another important equation in mathematical physics, one which is closely related to the Legendre equation (8.9.1), is Legendre’s associated equation:  1 − x 2 y ′′ − 2xy′ + n (n + 1) − m2 1 − x 2 y = 0, (8.9.17) where m is an integer. Although this equation is independent of the algebraic sign of the integer m, it is often convenient to have the solutions for negative m differ somewhat from those for positive m. We consider first the case for a nonnegative integer m. Introducing the change of variable y =  1 − x 2 m/2 u, |x| < 1, Legendre’s associated equation becomes  1 − x 2 u ′′ − 2 (m + 1) xu′ + (n − m) (n + m + 1) u = 0. But this is the same as the equation obtained by differentiating the Legendre equation (8.9.1) m times. Thus, the general solution of (8.9.17) is given by y (x) =  1 − x 2 m/2 d mY (x) dxm , (8.9.18) 8.9 Legendre’s Equation and Legendre’s Function 307 and Y (x) = c1Pn (x) + c2Qn (x) (8.9.19) is the general solution of (8.9.1). Hence, we have the linearly independent solutions of (8.9.17), known as the associated Legendre functions of the first and second kind, respectively given by P m n (x) =  1 − x 2 m/2 d mPn (x) dxm , (8.9.20) and Q m n (x) =  1 − x 2 m/2 d mQn (x) dxm . (8.9.21) We observe that P 0 n (x) = Pn (x), Q0 n (x) = Qn (x), and that P m n (x) vanishes for m>n. The functions P −m n (x) and Q−m n (x) are defined by P −m n (x)=(−1)m (n − m)! (n + m)! P m n (x), m = 0, 1, 2, . . . , n, (8.9.22) Q −m n (x)=(−1)m (n − m)! (n + m)! Q m n (x), m = 0, 1, 2, . . . , n. (8.9.23) The first few associated Legendre functions are P 1 1 (x) =  1 − x 2 1 2 , P 1 2 (x)=3x  1 − x 2 1 2 , P 2 2 (x)=3  1 − x 2 . The associated Legendre functions of the first kind also form a sequence of orthogonal functions in the interval [−1, 1]. Their orthogonality, as well as their norm, is expressed by the equation  1 −1 P m n (x) P m k (x) dx = 2 (n − m)! (2n + 1) (n + m)! δnk. (8.9.24) Note that (8.9.15) and (8.9.16) are special cases of (8.9.24), corresponding to the choice m = 0. We finally observe that P m n (x) is bounded everywhere in the interval [−1, 1], whereas Qm n (x) is unbounded at the end points x = + 1. Problems in which Legendre’s polynomials arise will be treated in Chapter 10. 308 8 Eigenvalue Problems and Special Functions 8.10 Boundary-Value Problems Involving Ordinary Differential Equations A boundary-value problem consists in finding an unknown solution which satisfies an ordinary differential equation and appropriate boundary conditions at two or more points. This is in contrast to an initial-value problem for which a unique solution exists for an equation satisfying prescribed initial conditions at one point. The linear two-point boundary-value problem, in general, may be written in the form L[y] = f (x), a < x < b, (8.10.1) Ui [y] = αi , 1 ≤ i ≤ n, where L is a linear operator of order n and Ui is the boundary operator defined by Ui [y] = n j=1 aij y (j−1) (a) +n j=1 bij y (j−1) (b). (8.10.2) Here aij , bij , and αi are constants. The treatment of this problem can be found in Coddington and Levinson (1955). More complicated boundary conditions occur in practice. Treating a general differential system is rather complicated and difficult. A large class of boundary-value problems that occur often in the physical sciences consists of the second-order equations of the type y ′′ = f (x, y, y′ ), a<xξ . (8.12.1) Since G (x, ξ) is continuous at x = ξ, we have 316 8 Eigenvalue Problems and Special Functions c2 (ξ) y2 (ξ) − c1 (ξ) y1 (ξ)=0. (8.12.2) The discontinuity in the derivative of G at the point requires that dG dx (x, ξ)     x=ξ+ x=ξ− = c2 (ξ) y ′ 2 (ξ) − c1 (ξ) y ′ 1 (ξ) = − 1 p (ξ) . (8.12.3) Solving equations (8.12.2) and (8.12.3) for c1 and c2, we find c1 (ξ) = −y2 (ξ) p (ξ) W (y1, y2; ξ) , c2 (ξ) = −y1 (ξ) p (ξ) W (y1, y2; ξ) , (8.12.4) where W (y1, y2; ξ) is the Wronskian given by W (y1, y2; ξ) = y1 (ξ) y ′ 2 (ξ) − y2 (ξ) y ′ 1 (ξ). Since the two solutions are linearly independent, the Wronskian differs from zero. From Theorem 8.2.6, with λ = 0, we have p W = constant = C. (8.12.5) Hence, Green’s function is given by G (x, ξ) = ⎧ ⎨ ⎩ −y1 (x) y2 (ξ) /C, for x ≤ ξ −y2 (x) y1 (ξ) /C, for x ≥ ξ. (8.12.6) Thus, we state the following theorem: Theorem 8.12.1. If the associated homogeneous boundary-value problem of (8.11.1)–(8.11.3) has the trivial solution only, then Green’s function exists and is unique. Proof. The proof for uniqueness of Green’s function is left as an exercise for the reader. Example 8.12.1. Consider the problem y ′′ + y = −1, y (0) = 0, y 4π 2 5 = 0. (8.12.7) The solution of L[y]=(dy′/dx) + y = 0 satisfying y (0) = 0 is y1 (x) = sin x, 0 ≤ x<ξ and the solution of L[y] = 0 satisfying y (π/2) = 0 is y2 (x) = cos x, ξ < x ≤ π 2 . The Wronskian of y1 and y2 is then given by W (ξ) = y1 (ξ) y ′ 2 (ξ) − y2 (ξ) y ′ 1 (ξ) = −1. 8.13 The Schr¨odinger Equation and Linear Harmonic Oscillator 317 Since in this case p = 1, (8.12.6) becomes G (x, ξ) = ⎧ ⎨ ⎩ sin x cos ξ, for x ≤ ξ cos x sin ξ, for x ≥ ξ. Therefore, the solution of (8.12.7) is y (x) =  x 0 G (x, ξ) f (ξ) dξ +  π/2 x G (x, ξ) f (ξ) dξ =  x 0 cos x sin ξ dξ +  π/2 x sin x cos ξ dξ = −1 + sin x + cos x. It can be seen in the formula (8.12.6) that Green’s function is symmetric in x and ξ. Example 8.12.2. Construct the Green’s function for the two-point boundaryvalue problem y ′′ (x) + ω 2 y = f (x), y (a) = y (b)=0. This describes the forced oscillation of an elastic string with fixed ends at x = a and x = b. It is easy to check that sin ωx and cos ωx are two functions which satisfy the homogeneous equation y ′′ + ω 2y = 0. These are used to construct two functions y1 (x) and y2 (x) which satisfy the boundary conditions y1 (a) = y2 (b) = 0. Accordingly, y1 (x) = A sin ωx+B cos ωx and y2 (x) = C sin ωx+ D cos ωx, and the resulting functions are y1 (x) = sin ω (x − a), y2 (x) = sin ω (x − b). The corresponding Wronskian is W = −ω sin ω (a − b). Substituting these results into (8.11.10) yields G (x, ξ) = ⎧ ⎪⎨ ⎪⎩ sin ω(ξ−a) sin ω(x−b) −ω sin ω(a−b) , a ≤ ξ 0, the equation (8.13.1) takes the form d 2ψ dx2 +  β − α 2x 2 ψ = 0. (8.13.3) For small β and large x, β − α 2x 2 ∼ −α 2x 2 so that the equation becomes d 2ψ dx2 − α 2x 2ψ = 0. As |x|→∞, ψ (x) = x n exp 4 + αx2 2 5 satisfies (8.13.3) for a finite n so far as leading terms  ∼ −α 2x 2 are concerned. The positive exponential factor is unacceptable because of the boundary conditions, so the asymptotic solution ψ (x) = x n exp 4 − αx2 2 5 suggests the possibility of the exact solution in the form ψ (x) = v (x) exp 4 − αx2 2 5 where v (x) is to be determined. Substituting this result into (8.13.3), we obtain d 2v dx2 − 2αx dv dx + (β − α) v = 0. (8.13.4) In terms of a new independent variable ζ = x √ α, this equation reduces to the form d 2v dζ2 − 2ζ dv dζ +  β α − 1  v = 0. (8.13.5) We seek a power series solution v (ζ) = ∞ n=0 anζ n . (8.13.6) 8.13 The Schr¨odinger Equation and Linear Harmonic Oscillator 319 Substituting this series into equation (8.13.5) and equating the coefficients of ζ n to zero, we obtain the recurrence relation an+2 = (2n + 1 − β/α) (n + 1) (n + 2) an (8.13.7) which gives an+2 an ∼ 2 n as n → ∞. (8.13.8) This ratio is the same as that of the series for ζ n exp  ζ 2 4 ∼ x ne αx2 5 with finite n. This leads to the fact that ψ (x) = v (x) e −αx2/2 ∼ x ne αx2/2 which does not satisfy the basic requirement for |x|→∞. This unacceptable result can only be avoided if n is an integer and the series terminates so that it becomes a polynomial of degree n. This means that an+2 = 0 but an = 0 so that 2n + 1 − β α = 0, (8.13.9) or β α = (2n + 1). Substituting the values for α and β, it turns out that E ≡ En =  n + 1 2  ω, n = 0, 1, 2,.... (8.13.10) This represents a discrete set of energies. Thus, in quantum mechanics, a stationary state of the harmonic oscillator can assume only one of the values from the set En. The energy is thus quantized, and forms a discrete spectrum. According to the classical theory, the energy forms a continuous spectrum, that is, all non-negative numbers are allowed for the energy of a harmonic oscillator. This shows a remarkable contrast between the results of the classical and quantum theory. The number n which characterizes the energy eigenvalues and eigenfunctions is called the quantum number. The value of n = 0 corresponds to the minimum value of the quantum number with the energy E0 = 1 2 ω. (8.13.11) This is called the lowest (or ground) state energy which never vanishes as the lowest possible classical energy would. E0 is proportional to , representing a quantum phenomenon. The discrete energy spectrum is in perfect agreement with the quantization rules of the quantum theory. 320 8 Eigenvalue Problems and Special Functions To determine the eigenfunctions for the harmonic oscillator associated with the eigenvalues En, we obtain the solution of equation (8.13.5) which has the form d 2v dζ2 − 2ζ dv dζ + 2nv = 0. (8.13.12) This is a well-known differential equation for the Hermite polynomials Hn (ζ) of degree n. Thus, the complete eigenfunctions can be expressed in terms of Hn (ζ) as ψn (x) = AnHn  x √ α exp  − αx2 2  , (8.13.13) where An are arbitrary constants. The Hermite polynomials Hn (x) are usually defined by Hn (x)=(−1)n e x 2 Dn 4 e −x 2 5 , D ≡ d dx. (8.13.14) They form an orthogonal system in (−∞,∞) with the weight function exp  −x 2 . The orthogonal relation for these polynomials is  ∞ −∞ e −x 2 Hm (x) Hn (x) dx = ⎧ ⎨ ⎩ 0, n = m 2 nn! √ π, n = m. The Hermite polynomials Hn (x) for n = 0, 1, 2, 3, 4 are H0 (x)=1 H1 (x)=2x H2 (x) = −2+4x 2 H3 (x) = −12x + 8x 3 H4 (x) = 12 − 48x 2 + 16x 4 . Finally, the eigenfunctions ψn of the linear harmonic oscillator for the quantum number n = 0, 1, 2, 3 are given in Figure 8.13.1. 8.14 Exercises 321 Figure 8.13.1 Eigenfunctions ψn for n = 0, 1, 2, 3. 8.14 Exercises 1. Determine the eigenvalues and eigenfunctions of the following regular Sturm–Liouville systems: (a) y ′′ + λy = 0, y (0) = 0, y (π)=0. (b) y ′′ + λy = 0, y (0) = 0, y′ (1) = 0. (c) y ′′ + λy = 0, y ′ (0) = 0, y′ (π)=0. (d) y ′′ + λy = 0, y (1) = 0, y (0) + y ′ (0) = 0. 322 8 Eigenvalue Problems and Special Functions 2. Find the eigenvalues and eigenfunctions of the following periodic Sturm– Liouville systems: (a) y ′′ + λy = 0, y (−1) = y (−1), y′ (−1) = y ′ (1). (b) y ′′ + λy = 0, y (0) = y (2π), y′ (0) = y ′ (2π). (c) y ′′ + λy = 0, y (0) = y (π), y′ (0) = y ′ (π). 3. Obtain the eigenvalues and eigenfunctions of the following Sturm– Liouville systems: (a) y ′′ + y ′ + (1 + λ) y = 0, y (0) = 0, y (1) = 0. (b) y ′′ + 2y ′ + (1 − λ) y = 0, y (0) = 0, y′ (1) = 0. (c) y ′′ − 3y ′ + 3 (1 + λ) y = 0, y ′ (0) = 0, y′ (π) = 0. 4. Find the eigenvalues and eigenfunctions of the following regular Sturm– Liouville systems: (a) x 2y ′′ + 3xy′ + λy = 0, 1 ≤ x ≤ e, y (1) = 0, y (e) = 0. (b) d dx " (2 + x) 2 y ′ # + λy = 0, −1 ≤ x ≤ 1, y (−1) = 0, y (1) = 0. (c) (1 + x) 2 y ′′ + 2 (1 + x) y ′ + 3λy = 0, 0 ≤ x ≤ 1, y (0) = 0, y (1) = 0. 8.14 Exercises 323 5. Determine all eigenvalues and eigenfunctions of the Sturm–Liouville systems: (a) x 2y ′′ + xy′ + λy = 0, y (1) = 0, y, y′ are bounded at x = 0. (b) y ′′ + λy = 0, y (0) = 0, y, y′ are bounded at infinity. 6. Expand the function f (x) = sin x, 0 ≤ x ≤ π in terms of the eigenfunctions of the Sturm–Liouville problem y ′′ + λy = 0, y (0) = 0, y (π) + y ′ (π)=0. 7. Find the expansion of f (x) = x, 0 ≤ x ≤ π in a series of eigenfunctions of the Sturm–Liouville system y ′′ + λy = 0, y ′ (0) = 0, y′ (π)=0. 8. Transform each of the following equations into the equivalent selfadjoint form: (a) The Laguerre equation xy′′ + (1 − x) y ′ + ny = 0, n = 0, 1, 2,.... (b) The Hermite equation y ′′ − 2xy′ + 2ny = 0, n = 0, 1, 2,.... (c) The Tchebycheff equation  1 − x 2 y ′′ − xy′ + n 2 y = 0, n = 0, 1, 2,.... 9. If q (x) and s (x) are continuous and p (x) is twice continuously differentiable in [a, b], show that the solutions of the fourth-order Sturm– Liouville system 324 8 Eigenvalue Problems and Special Functions [p (x) y ′′] ′′ + [q (x) + λs (x)] y = 0, " a1y + a2 (py′′) ′ # x=a = 0, " b1y + b2 (py′′) ′ # x=b = 0, [c1y ′ + c2 (py′′)]x=a = 0, [d1y ′ + d2 (py′′)]x=b = 0, where a 2 1 + a 2 2 = 0, b 2 1 + b 2 2 = 0, c 2 1 + c 2 2 = 0, d 2 1 + d 2 2 = 0, are orthogonal with respect to s (x) in [a, b]. 10. If the eigenfunctions of the problem 1 r d dr (ry′ ) + λy = 0, 0 < r < a, c1y (a) + c2y ′ (a)=0, limr→0+ y (r) < ∞, satisfy limr→0+ ry′ (r)=0, show that all the eigenvalues are real for real c1 and c2. 11. Find the Green’s function for each of the following problems: (a) L[y] = y ′′ = 0, y (0) = 0, y′ (1) = 0. (b) L[y] =  1 − x 2 y ′′ − 2xy′ = 0, y (0) = 0, y′ (1) = 0. (c) L[y] = y ′′ + a 2y = 0, a = constant, y (0) = 0, y (1) = 0. 12. Determine the solution of each of the following boundary-value problems: (a) y ′′ + y = 1, y (0) = 0, y (1) = 0. 8.14 Exercises 325 (b) y ′′ + 4y = e x , y (0) = 0, y′ (1) = 0. (c) y ′′ = sin x, y (0) = 0, y (1) + 2y ′ (1) = 0. (d) y ′′ + 4y = −2, y (0) = 0, y  π 4 = 0. (e) y ′′ = −x, y (0) = 2, y (1) + y ′ (1) = 4. (f) y ′′ = −x 2 , y (0) + y ′ (0) = 4, y′ (1) = 2. (g) y ′′ = −x, y (0) = 1, y′ (1) = 2. 13. Determine the solution of the following boundary-value problems: (a) y ′′ = −f (x), y (0) = 0, y′ (1) = 0. (b) y ′′ = −f (x), y (−1) = 0, y (1) = 0. 14. Find the solution of the following boundary-value problems: (a) y ′′ − y = −f (x), y (0) = y (1) = 0. (b) y ′′ − y = −f (x), y′ (0) = y ′ (1) = 0. 15. Show that the Green’s function G (t, ξ) for the forced harmonic oscillator described by initial-value problem x¨ + ω 2x =  F m  sin Ωt, x (0) = a, x˙ (0) = 0, is 326 8 Eigenvalue Problems and Special Functions G (t, ξ) = 1 ω sin ω (t − ξ). Hence, the particular solution is xp (t) = F mω  t 0 sin ω (t − ξ) sin (Ωξ) dξ. 16. Determine the Green’s function for the boundary-value problem xy′′ + y ′ = −f (x), y (1) = 0, limx→0 |y (x)| < ∞. 17. Determine the Green’s function for the boundary-value problem xy′′ + y ′ − n 2 x y = −f (x), y (1) = 0, limx→0 |y (x)| < ∞. 18. Determine the Green’s function for the boundary-value problem 1 − x 2 y ′ !′ − h 2 (1 − x 2) y = −f (x), h = 1, 2, 3,..., lim r→+ 1 |y (x)| < ∞. 19. Prove the uniqueness of the Green’s function for the boundary-value problem L[y] = −f (x), a1y (a) + a2y ′ (a)=0, b1y (b) + b2y ′ (b)=0. 20. Find the Green’s function for the boundary-value problem L[y] = y (iv) = −f (x), y (0) = y (1) = y ′ (0) = y ′ (1) = 0. Prove that the homogeneous problem has a trivial solution only, and prove that the nonhomogeneous problem has a unique solution. 8.14 Exercises 327 21. Determine the Green’s function for the boundary-value problem y ′′ = −f (x), y (−1) = y (1), y′ (−1) = y ′ (1). 22. Consider the nonself-adjoint boundary-value problem L[y] = y ′′ + 3y ′ + 2y = −f (x), 2 y (0) − y (1) = 0, y′ (1) = 2. By direct integration of GL[y] from 0 to 1, show that y (x) = −2 G (1, x) −  1 0 G (x, ξ) f (ξ) dξ is the solution of the boundary-value problem, if G satisfies the system Gξξ − 3Gξ + 2G = 0, ξ = x, G (0, x)=0, 6 G (1, x) − 2 Gξ (1, x) + Gξ (0, x)=0. Find the Green’s function G (x, ξ). 23. Show that dG (x, ξ) dx     ξ=x+ ξ=x− = 1 p (x) is equivalent to dG (x, ξ) dx     x=ξ+ x=ξ− = − 1 p (ξ) . 24. (a) Apply the Pr¨ufer transformation R 2 = y 2 + p 2 (y ′ ) 2 , θ = tan−1  y py′  to transform the Sturm–Liouville equation (8.1.3) into the first order nonlinear equation in the form dR dx = 1 2 r  1 p − q − λr sin 2θ, dθ dx = (q + λr) sin2 θ + 1 p cos2 θ, where a<x 0. (9.2.2) But for v to be a maximum in D, vxx ≤ 0, vyy ≤ 0. Thus, vxx + vyy ≤ 0 which contradicts equation (9.2.2). Hence, the maximum of u must be attained on B. Theorem 9.2.2. (The Minimum Principle) If u (x, y) is harmonic in a bounded domain D and continuous in D = D ∪ B, then u attains its minimum on the boundary B of D. 9.3 Uniqueness and Continuity Theorems 333 Proof. The proof follows directly by applying the preceding theorem to the harmonic function −u (x, y). As a result of the above theorems, we see that u =constant which is evidently harmonic attains the same value in the domain D as on the boundary B. 9.3 Uniqueness and Continuity Theorems Theorem 9.3.1. (Uniqueness Theorem) The solution of the Dirichlet problem, if it exists, is unique. Proof. Let u1 (x, y) and u2 (x, y) be two solutions of the Dirichlet problem. Then u1 and u2 satisfy ▽2u1 = 0, ▽2u2 = 0 in D, u1 = f, u2 = f on B. Since u1 and u2 are harmonic in D, (u1 − u2) is also harmonic in D. But u1 − u2 = 0 on B. The maximum-minimum principle gives u1 − u2 = 0 at all interior points of D. Thus, we have u1 = u2. Therefore, the solution is unique. Theorem 9.3.2. (Continuity Theorem) The solution of the Dirichlet problem depends continuously on the boundary data. Proof. Let u1 and u2 be the solutions of ▽2u1 = 0 in D, u1 = f1 on B, and ▽2u2 = 0 in D, u2 = f2 on B. If v = u1 − u2, then v satisfies 334 9 Boundary-Value Problems and Applications ▽2 v = 0 in D, v = f1 − f2 on B. By maximum and minimum principles, f1 − f2 attains the maximum and minimum of v on B . Thus, if |f1 − f2| < ε, then −ε 0, there exists an integer N such that everywhere on B |fn − fm| < ε for n, m > N. It follows from the continuity theorem that for all n, m > N |un − um| < ε in D, and hence, the theorem is proved. 9.4 Dirichlet Problem for a Circle 1. Interior Problem We shall now establish the existence of the solution of the Dirichlet problem for a circle. The Dirichlet problem is ▽2u = urr + 1 r ur + 1 r 2 uθθ = 0, 0 ≤ r < a, 0 < θ ≤ 2π,(9.4.1) u (a, θ) = f (θ) for all θ in [0, 2π] . (9.4.2) By the method of separation of variables, we seek a solution in the form u (r, θ) = R (r) Θ (θ) = 0. Substitution of this in equation (9.4.1) yields 9.4 Dirichlet Problem for a Circle 335 r 2 R ′′ R + r R ′ R = − Θ ′′ Θ = λ. Hence, r 2R ′′ + rR ′ − λR = 0, (9.4.3) Θ ′′ + λΘ = 0. (9.4.4) Because of the periodicity conditions Θ (0) = Θ (2π) and Θ ′ (0) = Θ ′ (2π) which ensure that the function Θ is single-valued, the case λ < 0 does not yield an acceptable solution. When λ = 0, we have u (r, θ)=(A + B log r) (Cθ + D). Since log r → −∞ as r → 0+ (note that r = 0 is a singular point of equation (9.4.1)), B must vanish in order for u to be finite at r = 0. C must also vanish in order for u to be periodic with period 2π. Hence, the solution for λ = 0 is u = constant. When λ > 0, the solution of equation (9.4.4) is Θ (θ) = A cos √ λ θ + B sin √ λ θ. The periodicity conditions imply √ λ = n for n = 1, 2, 3,.... Equation (9.4.3) is the Euler equation and therefore, the general solution is R (r) = Crn + Dr−n . Since r −n → ∞ as r → 0, D must vanish for u to be continuous at r = 0. Thus, the solution is u (r, 0) = Crn (A cos n θ + B sin n θ) for n = 1, 2,.... Hence, the general solution of equation (9.4.1) may be written in the form u (r, θ) = a0 2 + ∞ n=1 4r a 5n (an cos nθ + bn sin nθ), (9.4.5) where the constant term (a0/2) represents the solution for λ = 0, and an and bn are constants. Letting ρ = r/a, we have u (ρ, θ) = a0 2 + ∞ n=1 ρ n (an cos nθ + bn sin nθ). (9.4.6) 336 9 Boundary-Value Problems and Applications Our next task is to show that u (r, θ) is harmonic in 0 ≤ r 0 such that |a0| < M, |an| < M, |bn| < M, n = 1, 2, 3,.... Thus, if we consider the sequence of functions {un} defined by un (ρ, θ) = ρ n (an cos nθ + bn sin nθ), (9.4.8) we see that |un| < 2ρ n 0M, 0 ≤ ρ ≤ ρ0 < 1. Hence, in any closed circular region, series (9.4.6) converges uniformly. Next, differentiate un with respect to r. Then, for 0 ≤ ρ ≤ ρ0 < 1,     ∂un ∂r     =    n a ρ n−1 (an cos nθ + bn sin nθ)    < 2 4n a 5 ρ n−1 0 M. Thus, the series obtained by differentiating series (9.4.6) term by term with respect to r converges uniformly. In a similar manner, we can prove that the series obtained by twice differentiating series (9.4.6) term by term with respect to r and θ converge uniformly. Consequently, ▽2u = urr + 1 r ur + 1 r 2 uθθ = ∞ n=1 ρ n−2 a 2 (an cos nθ + bn sin nθ) n (n − 1) + n − n 2 ! = 0, 0 ≤ ρ ≤ ρ0 < 1. Since each term of series (9.4.6) is a harmonic function, and since the series converges uniformly, u (r, θ) is harmonic at any interior point of the region 0 ≤ ρ < 1. It now remains to show that u satisfies the boundary data f (θ). Substitution of the Fourier coefficients an and bn into equation (9.4.6) yields 9.4 Dirichlet Problem for a Circle 337 u (ρ, θ) = 1 2π  2π 0 f (θ) dθ + 1 π ∞ n=1 ρ n  2π 0 f (τ ) × (cos nτ cos nθ + sin nτ sin nθ) dτ = 1 2π  2π 0 1 1+2∞ n=1 ρ n cos n (θ − τ ) 3 f (τ ) dτ. (9.4.9) The interchange of summation and integration is permitted due to the uniform convergence of the series. For 0 ≤ ρ ≤ 1 1+2∞ n=1 [ρ n cos n (θ − τ )] = 1 + ∞ n=1 " ρ n e in(θ−τ) + ρ n e −in(θ−τ) # =1+ ρ ei(θ−τ) 1 − ρ ei(θ−τ) + ρ e−i(θ−τ) 1 − ρ e−i(θ−τ) = 1 − ρ 2 1 − ρ ei(θ−τ) − ρ e−i(θ−τ) + ρ 2 = 1 − ρ 2 1 − 2ρ cos (θ − τ ) + ρ 2 . Hence, u (ρ, θ) = 1 2π  2π 0 1 − ρ 2 1 − 2ρ cos (θ − τ ) + ρ 2 f (τ ) dτ. (9.4.10) The integral on the right side of (9.4.10) is called the Poisson integral formula for a circle. Now if f (θ) = 1, then, according to series (9.4.9), u (r, θ) = 1 for 0 ≤ ρ ≤ 1. Thus, equation (9.4.10) gives 1 = 1 2π  2π 0 1 − ρ 2 1 − 2ρ cos (θ − τ ) + ρ 2 dτ. Hence, f (θ) = 1 2π  2π 0 1 − ρ 2 1 − 2ρ cos (θ − τ ) + ρ 2 f (θ) dτ, 0 ≤ ρ < 1. Therefore, u (ρ, θ) − f (θ) = 1 2π  2π 0  1 − ρ 2 [f (τ ) − f (θ)] 1 − 2ρ cos (θ − τ ) + ρ 2 dτ. (9.4.11) Since f (θ) is uniformly continuous on [0, 2π], for given ε > 0, there exists a positive number δ (ε) such that |θ − τ | < δ implies |f (θ) − f (τ )| < ε. If |θ − τ | ≥ δ so that θ − τ = 2nπ for n = 0, 1, 2,..., then 338 9 Boundary-Value Problems and Applications lim ρ→1− 1 − ρ 2 1 − 2ρ cos (θ − τ ) + ρ 2 = 0. In other words, there exists ρ0 such that if |θ − τ | ≥ δ, then 1 − ρ 2 1 − 2ρ cos (θ − τ ) + ρ 2 < ε, for 0 ≤ ρ ≤ ρ0 < 1. Hence, equation (9.4.10) yields |u (r, θ)| − f (θ)| ≤ 1 2π  2π |0−τ|≥δ  1 − ρ 2 |f (τ ) − f (θ)| 1 − 2ρ cos (θ − τ ) + ρ 2 dτ + 1 2π  2π |θ−τ|<δ  1 − ρ 2 |f (θ) − f (τ )| 1 − 2ρ cos (θ − τ ) + ρ 2 dτ ≤ 1 2π (2πε) 2 max 0≤θ≤2π |f (θ)| + ε 2π · 2π = ε 1+2  max 0≤θ≤2π |f (θ)|  which implies that lim ρ→1− u (r, θ) = f (θ) uniformly in θ. Therefore, we state the following theorem: Theorem 9.4.1. There exists one and only one harmonic function u (r, θ) which satisfies the continuous boundary data f (θ). This function is either given by u (r, θ) = 1 2π  2π 0 a 2 − r 2 a 2 − 2ar cos (θ − τ ) + r 2 f (τ ) dτ, (9.4.12) or u (r, θ) = a0 2 + ∞ n=1 r n a n (an cos nθ + bn sin nθ), (9.4.13) where an and bn are the Fourier coefficients of f (θ). For ρ = 0, the Poisson integral formula (9.4.10) becomes u (0, θ) = u (0) = 1 2π  2π 0 f (τ ) dτ. (9.4.14) This result may be stated as follows: Theorem 9.4.2. (Mean Value Theorem) If u is harmonic in a circle, then the value of u at the center is equal to the mean value of u on the boundary of the circle. 9.4 Dirichlet Problem for a Circle 339 Several comments are in order. First, the Continuity Theorem 9.3.2 for the Dirichlet problem for the Laplace equation is a special example of the general result that the Dirichlet problems for all elliptic equations are well-posed. Second, the formula (9.4.12) represents the unique continuous solution of the Laplace equation in 0 ≤ r 1.(9.4.18) 9.5 Dirichlet Problem for a Circular Annulus The natural extension of the Dirichlet problem for a circle is the Dirichlet problem for a circular annulus, that is ∇2u = 0, r2 <r<r1, (9.5.1)="" u="" (r1,="" θ)="f" (θ),="" (r2,="" (θ).="" (9.5.2)="" in="" addition,="" (r,="" must="" satisfy="" the="" periodicity="" condition.="" accordingly,="" f="" (θ)="" and="" g="" also="" be="" periodic="" with="" period="" 2π.="" proceeding="" as="" case="" of="" dirichlet="" problem="" for="" a="" circle,="" we="" obtain="" λ="0" +="" b="" log="" r)="" (cθ="" d).="" condition="" on="" requires="" that="" c="0." then,="" becomes="" 2="" b0="" r,="" where="" a0="2AD" solution=""> 0 is u (r, θ) = 4 Cr √ λ + Dr− √ λ 54A cos √ λ θ + B sin √ λ θ5 , where √ λ = n = 1, 2, 3,.... Thus, the general solution is u (r, θ) = 1 2 (a0 + b0 log r) + ∞ n=1 anr n + bnr −n cos nθ +  cnr n + dnr −n sin nθ! , (9.5.3) where an, bn, cn, and dn are constants. Applying the boundary conditions (9.5.2), we find that the coefficients are given by 9.6 Neumann Problem for a Circle 341 a0 + b0 log r1 = 1 π  2π 0 f (τ ) dτ, anr n 1 + bnr −n 1 = 1 π  2π 0 f (τ ) cos nτ dτ, cnr n 1 + dnr −n 1 = 1 π  2π 0 f (τ ) sin nτ dτ, and a0 + b0 log r2 = 1 π  2π 0 g (τ ) dτ, anr n 2 + bnr −n 2 = 1 π  2π 0 g (τ ) cos nτ dτ, cnr n 2 + dnr −n 2 = 1 π  2π 0 g (τ ) sin nτ dτ. The constants a0, b0, an, bn, cn, dn for n = 1, 2, 3,... can then be determined. Hence, the solution of the Dirichlet problem for an annulus is given by (9.5.3). 9.6 Neumann Problem for a Circle Let u be a solution of the Neumann problem ∇2u = 0 in D, ∂u ∂n = f on B. It is evident that u + constant is also a solution. Thus, we see that the solution of the Neumann problem is not unique, and it differs from another by a constant. Consider the interior Neumann problem ∇2u = 0, r < R, (9.6.1) ∂u ∂n = ∂u ∂r = f (θ), r = R. (9.6.2) Before we determine a solution of the Neumann problem, a necessary condition for the existence of a solution will be established. In Green’s second formula  D   v∇2u − u∇2 v dS =  B  v ∂u ∂n − u ∂v ∂n ds, (9.6.3) we put v = 1, so that ∇2v = 0 in D and ∂v/∂n = 0 on B. Then, the result is 342 9 Boundary-Value Problems and Applications  D  ∇2u dS =  B ∂u ∂n ds. (9.6.4) Substituting of (9.6.1) and (9.6.2) into equation (9.6.4) yields  B f ds = 0 (9.6.5) which may also be written in the form R  2π 0 f (θ) dθ = 0. (9.6.6) As in the case of the interior Dirichlet problem for a circle, the solution of the Laplace equation is u (r, θ) = a0 2 + ∞ k=1 r k (ak cos kθ + bk sin kθ). (9.6.7) Differentiating this with respect to r and applying the boundary condition (9.6.2), we obtain ∂u ∂r (R, θ) = ∞ k=1 kRk−1 (ak cos kθ + bk sin kθ) = f (θ). (9.6.8) Hence, the coefficients are given by ak = 1 kπRk−1  2π 0 f (τ ) cos kτ dτ, k = 1, 2, 3,..., (9.6.9) bk = 1 kπRk−1  2π 0 f (τ ) sin kτ dτ, k = 1, 2, 3,.... Note that the expansion of f (θ) in a series of the form (9.6.8) is possible only by virtue of the compatibility condition (9.6.6) since a0 = 1 π  2π 0 f (τ ) dτ = 0. Inserting ak and bk in equation (9.6.7), we obtain u (r, θ) = a0 2 + R π  2π 0 1∞ k=1 4 r R 5k cos k (θ − τ ) 3 f (τ ) dτ. Using the identity − 1 2 log 1 + ρ 2 − 2ρ cos (θ − τ ) ! = ∞ k=1 1 k ρ k cos {k (θ − τ )} , 9.7 Dirichlet Problem for a Rectangle 343 with ρ = (r/R), we find that u (r, θ) = a0 2 − R 2π  2π 0 log R 2 − 2rR cos (θ − τ ) + r 2 ! f (τ ) dτ. (9.6.10) in which a constant factor R2 in the argument of the logarithm was eliminated by virtue of equation (9.6.6). In a similar manner, for the exterior Neumann problem, we can readily find the solution in the form u (r, θ) = a0 2 + R 2π  2π 0 log R 2 − 2rR cos (θ − τ ) + r 2 ! f (τ ) dτ. (9.6.11) 9.7 Dirichlet Problem for a Rectangle We first consider the boundary-value problem ∇2u = uxx + uyy = 0, 0 < x < a, 0 < y < b, (9.7.1) u (x, 0) = f (x), u (x, b)=0, 0 ≤ x ≤ a, (9.7.2) u (0, y)=0, u (a, y)=0, 0 ≤ y ≤ b. (9.7.3) We seek a nontrivial separable solution in the form u (x, y) = X (x) Y (y) Substituting u (x, y) in the Laplace equation, we obtain X′′ − λX = 0, (9.7.4) Y ′′ + λY = 0, (9.7.5) where λ is a separation constant. Since the boundary conditions are homogeneous for x = 0 and x = a, we choose λ = −α 2 with α > 0 in order to obtain nontrivial solutions of the eigenvalue problem X′′ + α 2X = 0, X (0) = X (a)=0. It is easily found that the eigenvalues are α = nπ a , n = 1, 2, 3,.... and the corresponding eigenfunctions are sin (nπx/a). Hence Xn (x) = Bn sin 4nπx a 5 . 344 9 Boundary-Value Problems and Applications The solution of equation (9.7.5) is Y (y) = C cosh αy +D sinh αy, which may also be written in the form Y (y) = E sinh α (y + F), where E =  D2 − C 2 1 2 and F = (1/α) tanh−1 (C/D). Applying the remaining homogeneous boundary condition u (x, b) = X (x) Y (b)=0, we obtain Y (b) = E sinh α (b + F)=0, and hence, F = −b, E = 0 for a nontrivial solution u (x, y). Thus, we have Yn (y) = En sinh(nπ a (y − b) ) . Because of linearity, the solution becomes u (x, y) = ∞ n=1 an sin 4nπx a 5 sinh(nπ a (y − b) ) , where an = BnEn. Now, we apply the nonhomogeneous boundary condition to obtain u (x, 0) = f (x) = ∞ n=1 an sinh  −nπb a  sin 4nπx a 5 . This is a Fourier sine series and hence, an = −2 a sinh  nπb a  a 0 f (x) sin 4nπx a 5 dx. Thus, the formal solution is given by u (x, y) = ∞ n=1 a ∗ n sinh & nπ a (b − y) ' sinh  nπb a sin 4nπx a 5 , (9.7.6) where a ∗ n = 2 a  a 0 f (x) sin 4nπx a 5 dx. 9.7 Dirichlet Problem for a Rectangle 345 To prove the existence of solution (9.7.6), we first note that sinh nπ a (b − y) sinh nπb a = e −nπy/a 1 − e −(2nπ/a)(b−y) 1 − e−2nπb/a ≤ C1e −nπy/a , where C1 is a constant. Since f (x) is bounded, we have |a ∗ n | ≤ 2 a  a 0 |f (x)| dx = C2. Thus, the series for u (x, y) is dominated by the series ∞ n=1 Me−nπy0/a for y ≥ y0 > 0, M = constant, and hence, u (x, y) converges uniformly in x and y whenever 0 ≤ x ≤ a, y ≥ y0 > 0. Consequently, u (x, y) is continuous in this region and satisfies the boundary values u (0, y) = u (a, y) = u (x, b) = 0. Now differentiating u twice with respect to x, we obtain uxx (x, y) = ∞ n=1 −a ∗ n 4nπ a 52 sinh nπ a (b − y) sinh nπb a sin 4nπx a 5 and differentiating u twice with respect to y, we obtain uyy (x, y) = ∞ n=1 a ∗ n 4nπ a 52 sinh nπ a (b − y) sinh nπb a sin 4nπx a 5 . It is evident that the series for uxx and uyy are both dominated by ∞ n=1 M∗n 2 e −nπy0/a and hence, converge uniformly for any 0 < y0 < b. It follows that uxxand uyy exist, and hence, u satisfies the Laplace equation. It now remains to show that u (x, 0) = f (x). Let f (x) be a continuous function and let f ′ (x) be piecewise continuous on [0, a]. If, in addition, f (0) = f (a) = 0, then, the Fourier series for f (x) converges uniformly. Putting y = 0 in the series for u (x, y), we obtain u (x, 0) = ∞ n=1 a ∗ n sin 4nπx a 5 . Since u (x, 0) converges uniformly to f (x), we write, for ε > 0, |sm (x, 0) − sn (x, 0)| < ε for m, n > Nε, 346 9 Boundary-Value Problems and Applications where sm (x, y) = ∞ n=1 a ∗ n sin 4nπx a 5 . We also know that sm (x, y) − sn (x, y) satisfies the Laplace equation and the boundary conditions at x = 0, x = a and y = b. Then, by the maximum principle, |sm (x, y) − sn (x, y)| < ε for m, n > Nε in the region 0 ≤ x ≤ a, 0 ≤ y ≤ b. Thus, the series for u (x, y) converges uniformly, and as a consequence, u (x, y) is continuous in the region 0 ≤ x ≤ a, 0 ≤ y ≤ b. Hence, we obtain u (x, 0) = ∞ n=1 a ∗ n sin 4nπx a 5 = f (x). Thus, the solution (9.7.6) is established. The general Dirichlet problem ∇2u = 0, 0 < x < a, 0 < y < b, u (x, 0) = f1 (x), u (x, a) = f2 (x), 0 ≤ x ≤ a, u (0, y) = f3 (y), u (b, y) = f4 (y), 0 ≤ y ≤ b can be solved by separating it into four problems, each of which has one nonhomogeneous boundary condition and the rest zero. Thus, determining each solution as in the preceding problem and then adding the four solutions, the solution of the Dirichlet problem for a rectangle can be obtained. 9.8 Dirichlet Problem Involving the Poisson Equation The solution of the Dirichlet problem involving the Poisson equation can be obtained for simple regions when the solution of the corresponding Dirichlet problem for the Laplace equation is known. Consider the Poisson equation ∇2u = uxx + uuu = f (x, y) in D, with the condition u = g (x, y) on B. Assume that the solution can be written in the form u = v + w, 9.8 Dirichlet Problem Involving the Poisson Equation 347 where v is a particular solution of the Poisson equation and w is the solution of the associated homogeneous equation, that is, ∇2w = 0, ∇2 v = f. As soon as v is ascertained, the solution of the Dirichlet problem ∇2w = 0 in D, w = −v + g (x, y) on B can be determined. The usual method of finding a particular solution for the case in which f (x, y) is a polynomial of degree n is to seek a solution in the form of a polynomial of degree (n + 2) with undetermined coefficients. As an example, we consider the torsion problem ∇2u = −2, 0 < x < a, 0 < y < b, u (0, y)=0, u (a, y) = 0; u (x, 0) = 0, u (x, b)=0. We let u = v + w. Now assume v to be the form v (x, y) = A + Bx + Cy + Dx2 + Exy + F y2 . Substituting this in the Poisson equation, we obtain 2D + 2F = −2. The simplest way of satisfying this equation is to choose D = −1 and F = 0. The remaining coefficients are arbitrary. Thus, we take v (x, y) = ax − x 2 so that v reduces to zero on the sides x = 0 and x = a. Next, we find w from ∇2w = 0, 0 < x < a, 0 < y < b, w (0, y) = −v (0, y)=0, w (a, y) = −v (a, 0) = 0, w (x, 0) = −v (x, 0) = −  ax − x 2 , w (x, b) = −v (x, b) = −  ax − x 2 . As in the Dirichlet problem (Section 9.7), the solution is found to be w (x, y) = ∞ n=1 4 an cosh nπy a + bn sinh nπy a 5 sin 4nπx a 5 . 348 9 Boundary-Value Problems and Applications Application of the nonhomogeneous boundary conditions yields w (x, 0) = −  ax − x 2 = ∞ n=1 an sin 4nπx a 5 , w (x, b) = −  ax − x 2 = ∞ n=1  an cosh nπb a + bn sinh nπb a  sin 4nπx a 5 , from which we find an = 2 a  a 0  x 2 − ax sin 4nπx a 5 dx =  0, if n is even −8a 2 π3n3 if n is odd and  an cosh nπb a + bn sinh nπb a  = 2 a  a 0  x 2 − ax sin 4nπx a 5 dx. Thus, we have bn =  1 − cosh nπb a an sinh  nπb a . Hence, the solution of the Dirichlet problem for the Poisson equation is given by u (x, y)=(a − x) x − 8a 2 π 3 ∞ n=1 " sinh (2n − 1) π(b−y) a + sinh (2n − 1) πy a # sinh (2n − 1) πb a sin (2n − 1) πx a (2n − 1)3 . 9.9 The Neumann Problem for a Rectangle Consider the Neumann problem ∇2u = 0, 0 < x < a, 0 < y < b, (9.9.1) ux (0, y) = f1 (y), ux (a, y) = f2 (y), 0 ≤ y ≤ b, (9.9.2) uy (x, 0) = g1 (x), uy (x, b) = g2 (x), 0 ≤ x ≤ a. (9.9.3) The compatibility condition that must be fulfilled in this case is  a 0 [g1 (x) − g2 (x)] dx +  b 0 [f1 (y) − f2 (y)] dy = 0. (9.9.4) We assume a solution in the form 9.9 The Neumann Problem for a Rectangle 349 u (x, y) = u1 (x, y) + u2 (x, y), (9.9.5) where u1 (x, y) is a solution of ∇2u1 = 0, ∂u1 ∂x (0, y)=0, ∂u1 ∂x (a, y)=0, (9.9.6) ∂u1 ∂x (x, 0) = g1 (x), ∂u1 ∂x (x, b) = g2 (x), and where g1 and g2 satisfy the compatibility condition  a 0 [g1 (x) − g2 (x)] dx = 0. (9.9.7) The function u2 (x, y) is a solution of ∇2u2 = 0, ∂u2 ∂x (0, y) = f1 (y) ∂u2 ∂x (a, y) = f2 (y) (9.9.8) ∂u2 ∂y (x, 0) = 0, ∂u2 ∂y (x, b)=0, where f1 and f2 satisfy the compatibility condition  b 0 [f1 (y) − f2 (y)] dy = 0. (9.9.9) Hence, u1 (x, y) and u2 (x, y) can be determined. Conditions (9.9.7) and (9.9.9) ensure that condition (9.9.4) is fulfilled. Thus, the problem is solved. However, the solution obtained in this manner is rather restrictive. In general, condition (9.9.4) does not imply conditions (9.9.7) and (9.9.9). Thus, generally speaking, it is not possible to obtain a solution of the Neumann problem for a rectangle by the method described above. To obtain a general solution, Grunberg (1946) proposed the following method. Suppose we assume a solution in the form u (x, y) = Y0 2 (y) + ∞ n=1 Xn (x) Yn (y), (9.9.10) 350 9 Boundary-Value Problems and Applications where Xn (x) = cos (nπx/a) is an eigenfunction of the eigenvalue problem X′′ + λX = 0, X′ (0) = X′ (a)=0, corresponding to the eigenvalue λn = (nπ/a) 2 . Then, from equation (9.9.10), we see that Yn (y) = 2 a  a 0 u (x, y) Xn (x) dx, = 2 a  a 0 u (x, y) cos 4nπx a 5 dx. (9.9.11) Multiplying both sides of equation (9.9.1) by 2 cos (nπx/a) and integrating with respect to x from 0 to a, we obtain 2 a  a 0 (uxx + uyy) cos 4nπx a 5 dx = 0, or, Y ′′ n + 2 a  a 0 uxx cos 4nπx a 5 dx = 0. Integrating the second term by parts and applying the boundary conditions (9.9.2), we obtain Y ′′ n (y) − 4nπ a 52 Yn (y) = Fn (y), (9.9.12) where Fn (y) = 2[f1 (y) − (−1)n f2 (y)] /a. This is an ordinary differential equation whose solution may be written in the form Yn (y) = An cosh 4nπy a 5 + Bn sinh 4nπy a 5 + 2 πn  y 0 Fn (τ ) sinh(nπ a (y − τ ) ) dτ. (9.9.13) The coefficients Anand Bn are determined from the boundary conditions Y ′ n (0) = 2 a  a 0 uy (x, 0) cos 4nπx a 5 dx = 2 a  a 0 g1 (x) cos 4nπx a 5 dx (9.9.14) and Y ′ n (b) = 2 a  a 0 g2 (x) cos 4nπx a 5 dx. (9.9.15) 9.10 Exercises 351 For n = 0, equation (9.9.12) takes the form Y ′′ 0 (y) = 2 a [f1 (y) − f2 (y)] and hence, Y ′ 0 (y) = 2 a  y 0 [f1 (τ ) − f2 (τ )] dτ + C, where C is an integration constant. Employing the condition (9.9.14) for n = 0, we find C = 2 a  a 0 g1 (x) dx. Thus, we have Y ′ 0 (y) = 2 a  y 0 [f1 (τ ) − f2 (τ )] dτ +  a 0 g1 (x) dx0 . Consequently, Y ′ 0 (b) = 2 a / b 0 [f1 (τ ) − f2 (τ )] dτ +  a 0 g1 (x) dx0 . Also from equation (9.9.14), we have Y ′ 0 (b) = 2 a  a 0 g2 (x) dx. It follows from these two expressions for Y ′ 0 (b) that  b 0 [f1 (y) − f2 (y)] dy +  a 0 [g1 (x) − g2 (x)] dx = 0. which is the necessary condition for the existence of a solution to the Neumann problem for a rectangle. 9.10 Exercises 1. Reduce the Neumann problem to the Dirichlet problem in the twodimensional case. 2. Reduce the wave equation un = c 2 (uxx + uyy + uzz) 352 9 Boundary-Value Problems and Applications to the Laplace equation uxx + uyy + uzz + uτ τ = 0 by letting τ = ict where i = √ −1. Obtain the solution of the wave equation in cylindrical coordinates via the solution of the Laplace equation. Assume that u (r, θ, z, τ ) is independent of z. 3. Prove that, if u (x, t) satisfies ut = k uxx for 0 ≤ x ≤ 1, 0 ≤ t ≤ t0, then the maximum value of u is attained either at t = 0 or at the end points x = 0 or x = 1 for 0 ≤ t ≤ t0. This is called the maximum principle for the heat equation. 4. Prove that a function which is harmonic everywhere on a plane and is bounded either above or below is a constant. This is called the Liouville theorem. 5. Show that the compatibility condition for the Neumann problem ∇2u = f in D ∂u ∂n = g on B is  D f dS +  B g ds = 0, where B is the boundary of domain D. 6. Show that the second degree polynomial P = Ax2 + Bxy + Cy2 + Dyz + F z2 + F xz is harmonic if E = − (A + C) and obtain P = A  x 2 − z 2 + Bxy + C  y 2 − z 2 + Dyz + F xz. 7. Prove that a solution of the Neumann problem ∇2u = f in D u = g on B differs from another solution by at most a constant. 9.10 Exercises 353 8. Determine the solution of each of the following problems: (a) ∇2u = 0, 1 <r< 2,="" 0="" <="" θ="" π,="" u="" (1,="" θ)="sin" θ,="" (2,="" ≤="" (r,="" 0)="0," π)="0," 1="" r="" 2.="" (b)="" ∇2u="0," <r<="" (θ="" −="" π),="" (c)="" 3,="" π="" (3,="" 1)="" (r="" 3),="" ="" r,="" 2="0," 3.="" (d)="" (r),="" 9.="" solve="" the="" boundary-value="" problem="" a="" b,="" α,="" (a,="" (θ),="" (b,="" α)="0," b.="" 10.="" verify="" directly="" that="" poisson="" integral="" is="" solution="" of="" laplace="" equation.="" 11.="" a,="" (π="" θ),="" (0,="" bounded.="" 354="" 9="" problems="" and="" applications="" 12.="" +="" 13.="" find="" dirichlet="" <θ<="" 2π,="" 14.="" following="" problems:="" (a)="" ur="" 2π.="" 15.="" where="" ="" f="" ds="" g="" 16.="" robin="" for="" semicircular="" disk="" 9.10="" exercises="" 355="" 17.="" 18.="" determine="" mixed="" r<r,="" hu="" h="constant." 19.="" (θ).="" 20.="" neumann="" sin="" 2θ,="" r1="" <r<r2,="" (r1,="" (r2,="" 21.="" 22.="" <x<="" 1,="" <y<="" (x,="" (x="" 1),="" x="" y)="0," y="" 1.="" (πx),="" 356="" cos="" πy="" ,="" πy,="" 23.="" ux="" (π,="" uy="" π.="" y,="" x,="" 24.="" steady-state="" temperature="" distribution="" in="" rectangular="" plate="" length="" width="" b="" described="" by="" at="" kept="" zero="" degrees,="" while="" insulated.="" prescribed="" 357="" heat="" allowed="" to="" radiate="" freely="" into="" surrounding="" medium="" degree="" temperature.="" is,="" boundary="" conditions="" are="" (x),="" b)="" a.="" distribution.="" 25.="" 26.="" harmonic="" function="" which="" vanishes="" on="" hypotenuse="" has="" values="" other="" two="" sides="" an="" isosceles="" right-angled="" triangle="" formed="" constant.="" 27.="" a)="0," 28.="" third="" 358="" 29.="" 30.="" obtain="" representation="" d,="" ∂u="" ∂n="g" d.="" 31.="" terms="" green’s="" 32.="" inside="" circular="" annular="" region="" governed="" urr="" uθθ="0," −π="" 33.="" consider="" radially="" symmetric="" solid="" homogeneous="" cylinder="" radius="" unity="" height="" h.="" z)="" satisfies="" equation="" uzz="0," z="" with="" conditions:="" (z),="" h)="0," 3πz="" =="" h).="" 359="" 34.="" show="" 33(c)="" given="" n="1" anj0="" (knr)="" sinh="" knz="" knh="" j0="" (kn)="0," 3,....="" 35.="" problem:="" <z<="" 0),="" <z<π,="" constant,="" uz="" <z<h,="" h),="" (h="" z),="" each="" above="" (a)–(d),="" uzz.="" 10="" higher-dimensional="" “as="" long="" as="" branch="" knowledge="" offers="" abundance="" problems,="" it="" full="" vitality.”="" david="" hilbert="" 10.1="" introduction="" treatment="" more="" than="" space="" variables="" much="" involved="" variables.="" here="" number="" multidimensional="" involving="" equation,="" wave="" equations="" various="" will="" be="" presented.="" included="" cube,="" sphere,="" three="" dimensional="" rectangular,="" cylindrical="" polar="" spherical="" coordinates.="" three-dimensional="" schr¨odinger="" central="" field="" hydrogen="" helium="" atoms="" discussed.="" we="" also="" forced="" vibration="" membrane="" three-dimensional,="" nonhomogeneous="" moving="" boundaries.="" 10.2="" cube="" uyy="" (10.2.1)="" <x<π,="" <y<π,="" <z<π.="" faces="" except="" face="" 362="" 0,="" y),="" assume="" nontrivial="" separable="" form="" (x)="" (y)z="" (z).="" substituting="" this="" x′′y="" xy="" ′′z="" z′′="0." division="" yields="" x′′="" ′′="" .="" since="" right="" side="" depends="" only="" left="" independent="" z,="" both="" must="" equal="" thus,="" have="" same="" reasoning,="" hence,="" ordinary="" differential="" µx="0," (λ="" µ)="" λz="0." using="" conditions,="" eigenvalue="" (0)="X" (π)="0," eigenvalues="" µ="−m2" m="1," 3,...="" corresponding="" eigenfunctions="" mx.="" similarly,="" gives="" λ−µ="−n" ny.="" 10.3="" 363="" λ="" m2="" follows="" ′′+λz="0" satisfying="" condition="" (z)="C" "="" n2="" #="" takes="" ∞="" amn="" 4="" 5="" mx="" applying="" condition,="" formally="" coefficient="" double="" fourier="" series="" thus="" ny="" dx="" dy.="" therefore,="" formal="" may="" written="" bmn="" √="" ny,="" (10.2.2)="" example="" 10.3.1.="" determining="" electric="" potential="" charge-free="" cylinder.="" coordinates="" (10.3.1)="" r<a,="" <z<l.="" let="" lateral="" surface="" top="" grounded,="" potential.="" base="" (10.3.2)="" 364="" (r)="" Θ="" (θ)z="" r′′="" rr′="" Θ′′="" 2r′′="" 2λ="−" 2r="" r2="" (10.3.3)="" µΘ="0," (10.3.4)="" (10.3.5)="" periodicity="" (2π),="" Θ′="" ′="" 2,...="" nθ,="" nθ.="" (θ)="A" nθ="" (10.3.6)="" suppose="" real="" negative="" β=""> 0. If the condition Z (l) = 0 is imposed, then the solution of equation (10.3.5) can be written in the form Z (z) = C sinh β (l − z). (10.3.7) Next we introduce the new independent variable ξ = βr. Equation (10.3.3) transforms into ξ 2 d 2R dξ2 + ξ dR dξ +  ξ 2 − n 2 R = 0 which is the Bessel equation of order n. The general solution is Rn (ξ) = DJn (ξ) + E Yn (ξ) where Jn and Yn are the Bessel functions of the first and second kind respectively. In terms of the original variable, we have 10.3 Dirichlet Problem for a Cylinder 365 Rn (r) = DJn (βr) + E Yn (βr). Since Yn (βr) is unbounded at r = 0, we choose E = 0. The condition R (a) = 0 requires that Jn (βa)=0. For each n ≥ 0, there exist positive zeros. Arranging these in an infinite increasing sequence, we have 0 < αn1 < αn2 <...<αnm <.... Thus, we obtain βnm = (αnm/a). Consequently, Rn (r) = DJn (αnmr/a). The solution u then finally takes the form u (r, θ, z) = ∞ n=0 ∞ m=1 Jn 4r a αnm5 (anm cos nθ + bnm sin nθ) × sinh (l − z) a αnm . To satisfy the nonhomogeneous boundary condition, it is required that f (r, θ) = ∞ n=0 ∞ m=1 Jn 4r a αnm5 (anm cos nθ + bnm sin nθ) sinh  l a αnm . The coefficients anm and bnm are given by a0m = 1 πa2 sinh  1 a α0m [J1 (α0m)]2  a 0  2π 0 f (r, θ) J0 4r a α0m 5 r dr dθ, anm = 2 πa2 sinh  1 a αnm [Jn+1 (αnm)]2  a 0  2π 0 f (r, θ) Jn 4r a αnm5 × cos nθ r dr dθ, bnm = 2 πa2 sinh  1 a αnm [Jn+1 (αnm)]2  a 0  2π 0 f (r, θ) Jn 4r a αnm5 × sin nθ r dr dθ. Example 10.3.2. We shall illustrate the same problem with different boundary conditions. Consider the problem 366 10 Higher-Dimensional Boundary-Value Problems ∇2u = 0, 0 ≤ r < a, 0 < z < π, u (r, θ, 0) = 0, u (r, θ, π)=0, u (a, θ, z) = f (θ, z). As before, by the separation of variables, we obtain r 2R ′′ + rR′ −  λr2 + µ R = 0, Θ ′′ + µΘ = 0, Z ′′ + λZ = 0. By the periodicity conditions, again as in the previous example, the Θ equation yields the eigenvalues µ = n 2 with n = 0, 1, 2,...; the corresponding eigenfunctions are sin nθ, cos nθ. Thus, we have Θ (θ) = An cos n cos θ + Bn sin nθ. Now let λ = β 2 with β > 0. Then, the boundary value problem Z ′′ + β 2Z = 0 Z (0) = 0, Z (π)=0, has the solution Z (z) = Cm sin mz, m = 1, 2, 3,.... Finally, we have r 2R ′′ + rR′ −  m2 r 2 + n 2 R = 0, or R ′′ + 1 r R ′ −  m2 + n 2 r 2  R = 0, the general solution of which is R (r) = DIn (mr) + EKn (mr), where In and Kn are the modified Bessel functions of the first and second kind, respectively. Since R must remain finite at r = 0, we set E = 0. Then R takes the form R (r) = DIn (mr). Applying the nonhomogeneous condition, we find the solution 10.4 Dirichlet Problem for a Sphere 367 u (r, θ, z) = ∞ m=1 4am0 2 5 I0 (mr) I0 (ma) sin mz + ∞ m=1 ∞ n=1 (amn cos nθ + bmn sin nθ) In (mr) In (ma) sin mz, where amn = 2 π 2  π 0  2π 0 f (θ, z) sin mz cos nθ dθ dz, bmn = 2 π 2  π 0  2π 0 f (θ, z) sin mz sin nθ dθ dz. 10.4 Dirichlet Problem for a Sphere Example 10.4.3. To determine the potential in a sphere, we transform the Laplace equation into spherical coordinates. It has the form ∇2u = urr + 2 r ur + 1 r 2 uθθ + cot θ r 2 uθ + 1 r 2 sin2 θ uϕϕ, (10.4.1) where 0 ≤ r<a, 0="" <θ<π,="" and="" <ϕ<="" 2π.="" let="" the="" prescribed="" potential="" on="" sphere="" be="" u="" (a,="" θ,="" ϕ)="f" (θ,="" ϕ).="" (10.4.2)="" we="" assume="" a="" nontrivial="" separable="" solution="" in="" form="" (r,="" (r)="" Θ="" (θ)Φ(ϕ).="" substitution="" of="" laplace="" equation="" yields="" r="" 2r="" ′′="" +="" 2rr′="" −="" λr="0," (10.4.3)="" sin2="" θ="" Θ′′="" sin="" cos="" Θ′="" ="" λ="" µ="" (10.4.4)="" Φ="" µΦ="0." (10.4.5)="" general="" is="" Φ(ϕ)="A" √="" ϕ="" bn="" ϕ.="" (10.4.6)="" periodicity="" condition="" requires="" that="" m="0," 1,="" 2,....="" since="" euler="" type,="" β="" .="" 368="" 10="" higher-dimensional="" boundary-value="" problems="" inserting="" this="" (10.4.3),="" obtain="" 2="" roots="" are="" −1="" 1+4λ="" (1="" β).="" hence,="" rβ="" d="" r−(1+β)="" (10.4.7)="" variable="" ξ="cos" transforms="" into="" 1="" 2ξΘ′="" (β="" 1)="" m2="" (10.4.8)="" which="" legendre’s="" associated="" equation.="" with="" for="" n="0," 2,...="" (θ)="E" p="" (cos="" θ)="" f="" qm="" θ).="" continuity="" at="" π="" corresponds="" to="" (ξ)="" 1.="" has="" logarithmic="" singularity="" choose="" thus,="" becomes="" consequently,="" spherical="" coordinates="" n="" np="" (anm="" mϕ="" bnm="" mϕ).="" order="" satisfy="" function="" boundary,="" it="" necessary="" mϕ)="" ≤="" π,="" by="" orthogonal="" properties="" functions="" mϕ,="" coefficients="" given="" anm="(2n" 2πan="" (n="" m)!="" ="" 2π="" dθ="" dϕ,="" 2,...,="" an0="(2n" 4πan="" pn="" 10.4="" dirichlet="" problem="" 369="" example="" 10.4.4.="" determine="" grounded="" conducting="" uniform="" field="" satisfies="" ∇2u="0," <="" a,="" <φ<="" 2π,="" →="" −e0="" as="" ∞.="" z="" direction="" so="" will="" independent="" φ.="" then,="" takes="" urr="" ur="" uθθ="" cot="" uθ="0." (θ).="" if="" set="" then="" second="" legendre="" qn="" θ),="" where="" first="" kind="" respectively.="" not="" singular="" r-equation="" obtained="" dn="" −(n+1)="" 4="" an="" −(n+1)5="" infinity,="" must="" have="" a1="−E0," ≥="" ∞="" n+1="" 370="" using="" orthogonality="" functions,="" find="" e0="" n+2="" −π="" 3="" δn1,="" integral="" vanishes="" all="" except="" θ.="" 10.4.5.="" dielectric="" radius="" placed="" electric="" e0.="" potentials="" inside="" outside="" sphere.="" u1="" u2="" ∇2u1="∇2u2" =="" 0,="" k="" ∂u1="" ∂r="∂u2" ,="" −e0r="" ∞,="" sphere,="" respectively,="" constant.="" preceeding="" example,="" (10.4.9)="" finite="" origin,="" take="" anr="" npn="" a.="" (10.4.10)="" u2,="" approach="" infinity="" manner,="" −(n+1)pn="" (10.4.11)="" from="" two="" conditions="" b1="" ka1="−E0" 2b1="" 2.="" 371="" found="" 3e0="" (k="" 2).="" 2)="" −2="" 10.4.6.="" between="" concentric="" spheres="" held="" different="" constant="" potentials.="" here="" need="" solve="" b,="" case,="" depends="" only="" radial="" distance.="" ∂="" ="" ∂u="" ="0." elementary="" integration,="" c2="" c1="" arbitrary="" constants.="" applying="" boundary="" conditions,="" aa="" b="" b)="" ab="" (b="" a)="" (a="" bb="" r="" b="" 372="" 10.5="" three-dimensional="" wave="" heat="" equations="" three="" space="" variables="" may="" written="" utt="c" 2∇2u,="" (10.5.1)="" ∇2="" operator.="" (x,="" y,="" z,="" t)="U" z)="" t="" (t).="" substituting="" (10.5.1),="" λc2t="0," (10.5.2)="" λu="0," (10.5.3)="" −λ="" separation="" separated="" solutions="" determined.="" next="" consider="" ut="k∇2u." (10.5.4)="" before,="" seek="" (10.5.4),="" ′="" λkt="0," see="" here,="" previous="" essentially="" solving="" helmholtz="" 10.6="" vibrating="" membrane="" specific="" equation,="" us="" length="" width="" b.="" initial="" displacement="" (uxx="" uyy),="" x="" y=""> 0, (10.6.1) u (x, y, 0) = f (x, y), 0 ≤ x ≤ a, 0 ≤ y ≤ b, (10.6.2) ut (x, y, 0) = g (x, y), 0 ≤ x ≤ a, 0 ≤ y ≤ b, (10.6.3) u (0, y, t)=0, u (a, y, t)=0, (10.6.4) u (x, 0, t)=0, u (x, b, t)=0. (10.6.5) 10.6 Vibrating Membrane 373 We have just shown that the separated equations for the wave equation are T ′′ + λc2T = 0, (10.6.6) ∇2U + λU = 0, (10.6.7) where, in this case, ∇2U = Uxx + Uyy. Let λ = α 2 . Then the solution of equation (10.6.6) is T (t) = A cos αct + B sin αct. Now we look for a nontrivial solution of equation (10.6.7) in the form U (x, y) = X (x) Y (y). Substituting this into equation (10.6.7) yields X′′ − µX = 0, Y ′′ + (λ + µ) Y = 0. If we let µ = −β 2 , then the solutions of these equations take the form X (x) = C cos βx + D sin βx. Y (y) = E cos γy + F sin γy, where γ 2 = (λ + µ) = α 2 − β 2 . The homogeneous boundary conditions in x require that C = 0 and D sin βa = 0 which implies that β = (mπ/a) with D = 0. Similarly, the homogeneous boundary conditions in y require that E = 0 and F sin γb = 0 which implies that γ = (nπ/b) with F = 0. Noting that m and n are independent integers, we obtain the displacement function in the form u (x, y, t) = ∞ m=1 ∞ n=1 (amn cos αmn ct + bmn sin αmn ct) sin 4mπx a 5 sin 4nπy b 5 , (10.6.8) where αmn =  m2π 2/a2 +  n 2π 2/b2 , amn and bmn are constants. Now applying the nonhomogeneous initial conditions, we have 374 10 Higher-Dimensional Boundary-Value Problems u (x, y, 0) = f (x, y) = ∞ m=1 ∞ n=1 amn sin 4mπx a 5 sin 4nπy b 5 , and thus, amn = 4 ab  a 0  b 0 f (x, y) sin 4mπx a 5 sin 4nπy b 5 dx dy. (10.6.9) In a similar manner, the initial condition on ut implies ut (x, y, 0) = g (x, y) = ∞ m=1 ∞ n=1 bmn αmn c sin 4mπx a 5 sin 4nπy b 5 , from which it follows that bmn = 4 αmn abc  a 0  b 0 g (x, y) sin 4mπx a 5 sin 4nπy b 5 dx dy. (10.6.10) The solution of the rectangular membrane problem is, therefore, given by equation (10.6.8). Example 10.6.1. (Vibration of a Circular Membrane). For a circular elastic membrane that is stretched over a circular frame of radius a, the motion of the membrane can be described by a function u (r, θ, t) that satisfies the partial differential equation 1 c 2 utt = urr + 1 r ur + 1 r 2 uθθ, (10.6.11) where c 2 = (T /ρ), T is the tension in the membrane and ρ is its mass density. We consider the synchronous vibrations of the vibration of the membrane defined by the separable solution u (r, θ, t) = v (r, θ, t) cos (ωct). (10.6.12) Substituting (10.6.12) into (10.6.11) gives vrr + 1 r vr + 1 r 2 uθθ + ω 2 v = 0. (10.6.13) We seek a nontrivial separable solution v (r, θ) = R (r) Θ (θ) of equation (10.6.13) so that r 2R′′ + r R′ R + ω 2 r 2 = − Θ′′ Θ = λ 2 . (10.6.14) 10.7 Heat Flow in a Rectangular Plate 375 This must hold for all points of the membrane, 0 <r 0, (10.7.1) u (x, y, 0) = f (x, y), 0 ≤ x ≤ a, 0 ≤ y ≤ b, (10.7.2) 376 10 Higher-Dimensional Boundary-Value Problems ux (0, y, t)=0, ux (a, y, t)=0, (10.7.3ab) u (x, 0, t)=0, u (x, b, t)=0. (10.7.4ab) As shown earlier, the separated equations for this problem are found to be T ′ + λkT = 0, (10.7.5) ∇2U + λU = 0. (10.7.6) We assume a nontrivial separable solution in the form U (x, y) = X (x) Y (y). Inserting this in equation (10.7.6), we obtain X′′ − µX = 0, (10.7.7) Y ′′ + (λ + µ) Y = 0. (10.7.8) Because the conditions in x are homogeneous, we choose µ = −α 2 so that X (x) = A cos αx + B sin αx. Since X′ (0) = 0, B = 0 and since X′ (a) = 0, sin αa = 0, A = 0 which gives α = (mπ/a), m = 1, 2, 3,.... We note that µ = 0 is also an eigenvalue. Consequently, Xm (x) = Am cos (mπx/a), m = 0, 1, 2,.... Similarly, for nontrivial solution Y , we select β 2 = λ + µ = λ − α 2 so that the solution of equation (10.7.8) is Y (y) = C cos βy + D sin βy. Applying the homogeneous conditions, we find C = 0 and sin βb = 0, D = 0. Thus, we obtain β = (nπ/b) ; n = 1, 2, 3,..., and 10.7 Heat Flow in a Rectangular Plate 377 Yn (y) = Dn sin (nπy/b). Recalling that λ = α 2 + β 2 , the solution of equation (10.7.5) may be written in the form Tmn (t) = Emn e −(m2/a2+n 2/b2 )π 2kt . Thus, the solution of the heat equation satisfying the prescribed boundary conditions may be written as u (x, y, t) = ∞ m=0 ∞ n=1 amn e −(m2/a2+n 2/b2 )π 2kt cos 4mπx a 5 sin 4nπy b 5 , (10.7.9) where amn = AmDmEmn are arbitrary constants. Applying the initial condition, we obtain u (x, y, 0) = f (x, y) = ∞ m=0 ∞ n=1 amn cos 4mπx a 5 sin 4nπy b 5 . (10.7.10) This is a double Fourier series, and the coefficients are given by a0n =  2 ab  a 0  b 0 f (x, y) sin 4nπy b 5 dx dy, and for m ≥ 1 amn =  4 ab  a 0  b 0 f (x, y) cos 4mπx a 5 sin 4nπy b 5 dx dy. The solution of the heat equation is thus given by equation (10.7.9). Example 10.7.1. (Steady-state temperature in a Circular Disk). We next consider the steady-state temperature distribution u (r, θ) in a circular disk of radius r = a that satisfies the Laplace equation urr + 1 r ur + 1 r 2 uθθ = 0, 0 ≤ r ≤ a, 0 ≤ θ ≤ 2π, (10.7.11) u (r, θ) = f (θ), on r = a for all θ, (10.7.12) where f (θ) is a given function of θ. This is exactly the Dirichlet problem for a circle that was already solved in Section 9.4. We also consider the steady-state temperature distribution u (r, θ, φ) in a sphere of radius a where 0 ≤ r<a, 0="" <θ<π="" and="" <φ<="" 2π.="" for="" simplicity,="" we="" assume="" only="" steady="" temperature="" distribution="" which="" depends="" on="" r="" θ.="" thus,="" u="" is="" independent="" of="" the="" longitudinal="" coordinate="" 378="" 10="" higher-dimensional="" boundary-value="" problems="" φ,="" hence,="" steady-state="" (r,="" θ)="" satisfies="" laplace="" equation="" in="" spherical="" polar="" coordinates="" form="" ∂="" ∂r="" ="" 2="" ∂u="" ="" +="" 1="" sin="" θ="" ∂θ="" (10.7.13)="" seek="" a="" separable="" solution="" (r)="" Θ="" (θ)="" so="" that="" leads="" to="" d="" dr="" dθ="" dΘ="" (10.7.14)="" this="" must="" hold="" <r<a="" <θ<π.="" consequently,="" (10.7.15)="" or="" −="" λr="0," <="" a,="" (10.7.16)="" λ="" π.="" (10.7.17)="" can="" also="" be="" written="" as="" ′′="" ′="" (10.7.18)="" simplify="" by="" change="" variable.="" x="cos" θ,="" y="" (x)="Θ" (θ).="" using="" chain="" rule="" obtain="" dx="" (sin="" dy="" sin2="" dx="−" ="" .="" combining="" result="" with="" legendre="" 10.8="" waves="" three="" dimensions="" 379="" λy="0," −1="" ≤="" 1,="" (10.7.19)="" or,="" equivalently,="" 2y="" dx2="" 2x="" (10.7.20)="" was="" completely="" solved="" section="" 8.9.="" well-known="" sturm–liouville="" (−1)="" (+1)="" finite.="" results="" are="" =="" n="" (n="" 1),="" (x),="" 2,="" 3,...,="" where="" pn="" polynomial="" degree="" n.="" propagation="" due="" an="" initial="" disturbance="" rectangular="" volume="" best="" described="" problem="" utt="c" 2∇2u,="" b,="" z="" d,="" t=""> 0, (10.8.1) u (x, y, z, 0) = f (x, y, z), 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ d, (10.8.2) ut (x, y, z, 0) = g (x, y, z), 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ d, (10.8.3) u (0, y, z, t)=0, u (a, y, z, t)=0, (10.8.4) u (x, 0, z, t)=0, u (x, b, z, t)=0, (10.8.5) u (x, y, 0, t)=0, u (x, y, d, t)=0. (10.8.6) We assume a nontrivial separable solution in the form u (x, y, z, t) = U (x, y, z) T (t). The separated equations are given by T ′′ + λc2T = 0, (10.8.7) ∇2U + λU = 0. (10.8.8) We assume that U has the nontrivial separable solution in the form U (x, y, z) = X (x) Y (y)Z (z). Substitution of this into equation (10.8.8) yields X′′ − µX = 0, (10.8.9) Y ′′ − νY = 0, (10.8.10) Z ′′ + (λ + µ + ν)Z = 0. (10.8.11) 380 10 Higher-Dimensional Boundary-Value Problems Because of the homogeneous conditions in x, we let µ = −α 2 so that X (x) = A cos αx + B sin αx. As in the preceding examples, we obtain Xl (x) = Bl sin  lπx a  , l = 1, 2, 3,.... In a similar manner, we let ν = −β 2 to obtain Y (y) = C cos βy + D sin βy and accordingly, Ym (y) = Dm sin 4mπy b 5 , m = 1, 2, 3,.... We again choose γ 2 = λ + µ + ν = λ − α 2 − β 2 so that Z (z) = E cos (γz) + F sin (γz). Applying the homogeneous conditions in z, we obtain Zn (z) = Fn sin 4nπz d 5 . Since the solution of equation (10.8.7) is T (t) = G cos 4√ λ ct5 + H sin 4√ λ ct5 , the solution of the wave equation has the form u (x, y, z, t) = ∞ l=1 ∞ m=1 ∞ n=1 4 almn cos √ λ ct + blmn sin √ λ ct5 × sin  lπx a  sin 4mπy b 5 sin 4nπz d 5 where almn and blmn are arbitrary constants. The coefficients almn are determined from the initial condition u (x, y, z, 0) = f (x, y, z) and are found to be almn = 8 abd  a 0  b 0  d 0 f (x, y, z) sin  lπx a  sin 4mπy b 5 sin 4nπz d 5 dx dy dz. Similarly the coefficients blmn are determined from the initial condition u (x, y, z, 0) = g (x, y, z) and are found to be blmn = 8 √ λ acbd  a 0  b 0  d 0 g (x, y, z) sin  lπx a  sin 4mπy b 5 sin 4nπz d 5 dx dy dz, where λ =  l 2 a 2 + m2 b 2 + n 2 d 2  π 2 . 10.9 Heat Conduction in a Rectangular Volume 381 10.9 Heat Conduction in a Rectangular Volume As in the case of the wave equation, the solution of the heat equation in three spaces variables can be determined. Consider the problem of heat distribution in a rectangular volume. The faces are maintained at zero degree temperature. The solid is initially heated so that the problem may be written as ut = k ∇2u, 0 < x < a, 0 < y < b, 0 < z < d, t > 0, u (x, y, z, 0) = f (x, y, z), 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ d, u (0, y, z, t)=0, u (a, y, z, t)=0, u (x, 0, z, t)=0, u (x, b, z, t)=0, u (x, y, 0, t)=0, u (x, y, d, t)=0. As before, the separable equations are T ′ + λkT = 0, (10.9.1) ∇2U + λU = 0. (10.9.2) If we assume the solution U to be of the form U (x, y, z) = X (x) Y (y)Z (z), then the solution of the Helmholtz equation is Ulmn (x, y, z) = BlDmFn sin  lπx a  sin 4mπy b 5 sin 4nπz d 5 . Since the solution of equation (10.9.1) is T (t) = G e−λkt , the solution of the heat equation takes the form u (x, y, z, t) = ∞ l=1 ∞ m=1 ∞ n=1 almn e −λkt sin  lπx a  sin 4mπy b 5 sin 4nπz d 5 , where λ = l 2/a2 +  m2/b2 +  n 2/d2 ! π 2 and almn are constants. Application of the initial condition yields almn =  8 abd  a 0  b 0  d 0 f (x, y, z) sin  lπx a  sin 4mπy b 5 sin 4nπz d 5 dx dy dz. 382 10 Higher-Dimensional Boundary-Value Problems 10.10 The Schr¨odinger Equation and the Hydrogen Atom In quantum mechanics, the Hamiltonian (or energy operator) is usually denoted by H and is defined by H = p 2 2M + V (r) (10.10.1) where p = (/i) ∇ = −i∇ is the momentum of a particle of mass M, h = 2π is the Planck constant, and V (r) is the potential energy. The physical state of a particle at time t is described as fully as possible by the wave function Ψ (r, t). The probability of finding the particle at position r = (x, y, z) within a finite volume dV = dx dy dz is  |Ψ| 2 dx dy dz. The particle must always be somewhere in the space, so the probability of finding the particle within the whole space is one, that is,  ∞ −∞  ∞ −∞  ∞ −∞ |Ψ| 2 dx dy dz = 1. The time dependent Schr¨odinger equation for the function Ψ (r, t) is i Ψt = HΨ, (10.10.2) where H is explicitly given by H = −  2 2M ∇2 + V (r). (10.10.3) Given the potential V (r), the fundamental problem of quantum mechanics is to obtain a solution of (10.10.2) which agrees with a given initial state Ψ (r, 0). For the stationary state solutions, we seek a solution of the form Ψ (r, t) = f (t) ψ (r). Substituting this into (10.10.2) gives df dt + iE  f = 0, (10.10.4) Hψ (r) = Eψ (r), (10.10.5) where E is a separation constant and has the dimension of energy. Integration of (10.10.4) gives 10.10 The Schr¨odinger Equation and the Hydrogen Atom 383 f (t) = A exp  − iEt   , (10.10.6) where A is an arbitrary constant. Equation (10.10.5) is called the time independent Schr¨odinger equation. The great importance of this equation follows from the fact that the separation of variables gives not just some particular solution of (10.10.5), but generally yields all solutions of physical interest. If ψE (r) represents one particular solution of (10.10.5), then most general solutions of (10.10.2) can be obtained by the principle of superposition of such particular solutions. In fact, the general solution is given by ψ (r, t) =  E AE exp  − iEt   ψE (r), (10.10.7) where the summation is taken over all admissible values of E, and AE is an arbitrary constant to be determined from the initial conditions. We now solve the eigenvalue problem for the Schr¨odinger equation for the spherically symmetric potential so that V (r) = V (r). The equation for the wave function ψ (r) is ∇2ψ + 2M  2 [E − V (r)] ψ = 0, (10.10.8) where ∇2 is the three-dimensional Laplacian. To determine the wave function ψ, it is convenient to introduce spherical polar coordinates (r, θ, φ) so that equation (10.10.8) takes the form 1 r 2 ∂ ∂r  r 2 ∂ψ ∂r  + 1 r 2 sin θ ∂ ∂θ  sin θ ∂ψ ∂θ  + 1 r 2 sin2 θ ∂ 2ψ ∂φ2 +K [E − V (r)] ψ = 0, (10.10.9) where K =  2M/ 2 , ψ ≡ ψ (r, θ, φ), 0 ≤ r < ∞, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π. We seek a nontrivial separable solution of the form ψ = R (r) Y (θ, φ) and then substitute into (10.10.9) to obtain the following equations d dr  r 2 dR dr  + K (E − V ) r 2 − λ ! R = 0, (10.10.10) 1 sin θ ∂ ∂θ  sin θ ∂ ∂θ  + 1 sin2 θ ∂ 2 ∂φ2 Y + λY = 0, (10.10.11) where λ is a separation constant. 384 10 Higher-Dimensional Boundary-Value Problems We first solve (10.10.11) by separation of variables through Y = Θ (θ)Φ(φ) so that the equation becomes sin θ d dθ  sin θ dΘ dθ  +  λ sin2 θ − m2 Θ = 0, (10.10.12) d 2Φ dφ2 + m2Φ = 0, (10.10.13) where m2 is a separation constant. The general solution of (10.10.13) is Φ = A eimφ + B e−imφ , where A and B are arbitrary constants to be determined by the boundary conditions on ψ (r, θ, φ) = R (r) Θ (θ)Φ(φ) which will now be formulated. According to the fundamental postulate of quantum mechanics, the wave function for a particle without spin must have a definite value at every point in space. Hence, we assume that ψ is a single-valued function of position. In particular, ψ must have the same value whether the azimuthal coordinate φ is given by φ or φ + 2π, that is, Φ(φ) = Φ(φ + 2π). Consequently, the solution for Φ has the form Φ = C eimΦ, m = 0, + 1, + 2,..., (10.10.14) where C is an arbitrary constant. In order to solve (10.10.12), it is convenient to change the variable x = cos θ, Θ (θ) = u (x), −1 ≤ x ≤ 1 so that this equation becomes d dx  1 − x 2 du dx +  λ − m2 1 − x 2  u = 0. (10.10.15) For the particular case m = 0, this equation becomes d dx  1 − x 2 du dx + λ u = 0. (10.10.16) This is known as the Legendre equation, which gives the Legendre polynomials Pl (x) of degree l as solutions provided λ = l(l + 1) where l is a positive integer or zero. When m = 0, equation (10.10.15) with λ = l(l + 1) admits solutions which are well known as associated Legendre functions, P m l (x) of degree l and order m defined by P m l (x) =  1 − x 2 m/2 d m dxm P m l (x), x = cos θ. Clearly, P m l (x) vanishes when m>l. As for the negative integral values of m, it can be readily shown that 10.10 The Schr¨odinger Equation and the Hydrogen Atom 385 P −m l (x)=(−1)m (l − m)! (l + m)! P m l (x). Hence, the functions P −m l (x) differ from P m l (x) by a constant factor, and as a consequence, m is restricted to a positive integer or zero. Thus, the associated Legendre functions P m l (x) with |m| ≤ l are the only nonsingular and physically acceptable solutions of (10.10.15). Since |m| ≤ l, when l = 0, m = 0; when l = 1, m = −1, 0, +1; when l = 2, m = −2, −1, 0, 1, 2, etc. This means that, given l, there are exactly (2l + 1) different values of m = −l, ..., −1, 0, 1,..., l. The numbers l and m are called the orbital quantum member and the magnetic quantum number respectively. It is convenient to write down the solutions of (10.10.11) as functions which are normalized with respect to an integration over the whole solid angle. They are called spherical harmonics and are given by, for m ≥ 0, Y m l (θ, φ) = (2l + 1) 4π (l − m)! (l + m)! 1 2 (−1)m e imφP m l (cos θ). (10.10.17) Spherical harmonics with negative m and with |m| ≤ l are defined by Y m l (θ, φ)=(−1)m Y −m l (θ, φ). (10.10.18) We now return to a general discussion of the radial equation (10.10.10) which becomes, under the transformation R (r) = P (r) /r, d 2P dr2 + K (E − V ) − λ r 2 P (r)=0. (10.10.19) Almost all cases of physical interest require V (r) to be finite everywhere except at the origin r = 0. Also, V (r) → 0 as r → ∞. The Coulomb and square well potentials are typical examples of this kind. In the neighborhood of r = 0, V (r) can be neglected compared to the centrifugal term  ∼ 1/r2 so that equation (10.10.19) takes the form d 2P dr2 − l(l + 1) r 2 P (r) = 0 (10.10.20) for all states with l = 0. The general solution of this equation is P (r) = A rl+1 + B r−l , (10.10.21) where A and B are arbitrary constants. With the boundary condition P (0) = 0, B = 0 so that the solution is proportional to r l+1 . On the other hand, in view of the assumption that V (r) → 0 as r → ∞, the radial equation (10.10.19) reduces to d 2P dr2 + KE P (r)=0. (10.10.22) 386 10 Higher-Dimensional Boundary-Value Problems The general solution of this equation is P (r) = C eir√ KE + D e−ir√ KE. (10.10.23) The solution is oscillatory for E > 0, and exponential in nature for E < 0. The oscillatory solutions are not physically acceptable because the wave function does not tend to zero as r → ∞. When E < 0, the second term in (10.10.23) tends to infinity as r → ∞. Consequently, the only physically acceptable solutions for E > 0, have the asymptotic form P (r) = C e−αr/2 , (10.10.24) where KE = −  α 2/4 . Thus, the general solution of (10.10.19) can be written as P (r) = f (r) e −(α/2)r , so that f (r) satisfies the ordinary differential equation d 2f dr2 − α df dr − KV + l(l + 1) r 2 f = 0. (10.10.25) Note that this general solution is physically acceptable because the wave function tends to zero as r → 0 and as r → ∞. We now specify the form of the potential V (r). One of the most common potentials is the Coulomb potential V (r) = −Ze2/r representing the attraction between an atomic nucleus of charge +Ze and a moving electron of charge −e. For the hydrogen atom Z = 1. It is a two particle system consisting of a negatively charged electron interacting with a positively charged proton. On the other hand, a helium atom consists of two protons and two neutrons. There are two electrons in orbit around the nucleus of a helium atom. For the singly charged helium ion Z = 2, where Z represents the number of unit charges of the nucleus. Consequently, equation (10.10.25) reduces to d 2f dr2 − α df dr + KZe2 r − l(l + 1) r 2 f (r)=0. (10.10.26) We seek a power series solution of this equation in the form f (r) = r k∞ s=1 asr s , k = 0. (10.10.27) Substituting this series into (10.10.26), we obtain r k∞ s=1 [(s + k) (s + k − 1) − l(l + 1)] asr s+k−1 + ∞ s=1 Zke2 − α (s + k) ! asr s+k−1 = 0. 10.10 The Schr¨odinger Equation and the Hydrogen Atom 387 Clearly, the lowest power of r is (k − 1), so that [k (k + 1) − l(l + 1)] a1 = 0. This implies that k = l or − (l + 1) provided a1 = 0. The negative root of k is not acceptable because it leads to an unbounded solution. Equating the coefficient of r s+k−1 , we get the recurrence relation for the coefficients as as+1 = α (s + l) − ZKe2 s (s + 2l + 1) as, s = 1, 2, 3,.... (10.10.28) The asymptotic nature of this result is as+1 as ∼ α s as s → ∞. This ratio is the same as that of the series for e αr. This means that R (r) is unbounded as r → ∞, which is physically unacceptable. Hence, the series for f (r) must terminate, and f (r) must be a polynomial so that as+1 = 0, but as = 0. Hence α (s + l) − ZKe2 = 0, s = 1, 2, 3,..., or, α 2 4 = Z 2K2 e 4 4 (s + l) 2 = −KE. (10.10.29) Putting K =  2M/ 2 , the energy levels are given by E = En = − Z 2K2 e 4 4n2K = − MZ2 e 4 2 2n2 , (10.10.30) where n = (s + l) is called the principal quantum number and n = 1, 2, 3,.... Thus, it turns out that the complete solution of the Schr¨odinger equation is given by ψn,l,m (r, θ, φ) = Rn,l (r) Y m l (θ, φ), where the radial part is the solution of the radial equation (10.10.10), and it depends on the principle quantum number n (energy levels) and the orbital quantum number l. However, it does not depend on the magnetic quantum number m. Of course, there are (2l + 1) states with the same l value but with different m values. Each of these states has the same energy, and therefore, such systems have a (2l + 1)-fold degeneracy, as a result of rotational symmetry. For the hydrogen atom, Z = 1, the discrete energy spectrum is En = − Me4 2 2n2 = − e 2 2an2 , (10.10.31) 388 10 Higher-Dimensional Boundary-Value Problems where a =   2/e2M is called the Bohr radius of the hydrogen atom of mass M and charge of the electron, −e. This discrete energy spectrum depends only on the principle quantum number n (but not on m) and has an excellent agreement with experimental prediction of spectral lines. For a given n, there are n sets of l and s n = 1, {l = 0, s = 1} ; n = 2, ⎧ ⎨ ⎩ l = 0, s = 2 l = 1, s = 1 ⎫ ⎬ ⎭ ; n = 3, ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ l = 0, s = 3 l = 1, s = 2 l = 2, s = 1 ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ ; etc. Given n, there are exactly n values of l(l = 0, 1, 2,...,n − 1) and the highest value of l is n − 1. Thus, the three numbers n, l, m, determine a unique eigenfunction, ψn,l,m (r, θ, φ) = Rn,l (r) Y m l (θ, φ). Since the energy levels depend only on the principle quantum number n, there are, in general, several linearly independent eigenfunctions of the Schr¨odinger equation for the hydrogen atom corresponding to each energy level, so the energy levels are said to be degenerate. There are (2l + 1) different eigenfunctions of the same energy obtained by varying the magnetic quantum number m from −l to l. In general, the total number of degenerate energy states En for the hydrogen atom is then n−1 l=0 (2l + 1) = 2 n (n − 1) 2 + n = n 2 . (10.10.32) The energy levels of the hydrogen atom (10.10.31) can be expressed in terms the Rydberg, Ry, as En = − Ry n2 , (10.10.33) where Ry represents the Rydberg given by Ry = Me4 2 2 = M c2 e 4 2 (c) 2 = M c2 2 ×  e 2 c 2 ≃ 5 × 105 2 eV ×  1 1372 ≃ 13.3 eV. Consequently, En = − 13.3 n2 eV. (10.10.34) 10.10 The Schr¨odinger Equation and the Hydrogen Atom 389 Thus, the ground state of the hydrogen atom, which is the most tightly bound, has an energy −13.3 eV (more accurately −13.6 eV ) and therefore, it would take 13.6 eV to release the electron from its ground state. Therefore, this is called the binding energy of the hydrogen atom. Finally, the electron is treated here as a nonrelativistic particle. However, in reality, small relativistic effects can be calculated. These are known as fine structure corrections. Thus, the nonrelativistic Schr¨odinger equation describes the hydrogen atom extremely well. Example 10.10.1. (Infinite Square well potential V0 → ∞). We consider the one-dimensional Schr¨odinger equation (10.10.8) in the form −   2 2M  d 2 dx2 + V (x) ψ (x) = Eψ (x), (10.10.35) where the potential V (x) is given by V (x) = ⎧ ⎨ ⎩ V0, x ≤ −a, x ≥ a, 0, −a ≤ x ≤ a (10.10.36) and take the limit V0 → +∞ as shown in Figure 10.10.1. It is noted that the potential is zero inside the square well and E is the kinetic energy of the particle in this region (−a ≤ x ≤ a) which must be positive, E > 0. It is convenient to fix the origin at the center of the well so that V (x) is an even function of x. A case of special interest is that V0 > E ≥ 0 and eventually, V0 → ∞. Figure 10.10.1 Square well with potential V0 → ∞. 390 10 Higher-Dimensional Boundary-Value Problems The given potential is different in different regions, we solve (10.10.35) separately in three regions. Region 1. V = V0 in this region x ≤ −a. The Schr¨odinger equation (10.10.35) in this region is −  2 2M ψxx + V0ψ = Eψ. Or, equivalently, ψxx =  2M  2  (V0 − E) ψ, V0 >E> 0. (10.10.37) The general solution of (10.10.37) is ψ1 (x) = A ekx + B e−kx , (10.10.38) where A and B are constants and k = 2M  2 (V0 − E) 1 2 . (10.10.39) The wave function ψ1 (x) must be bounded as x → −∞ to retain its probabilistic interpretation, hence B = 0, and the solution in x ≤ −a is ψ1 (x) = A ekx . As V0 → ∞, k → ∞, and, in this limit, the solution must vanish, that is, ψ1 (x)=0, for x ≤ −a. (10.10.40) Region 2. V = 0 in −a ≤ x ≤ a. In this case, the equation takes the form ψxx + k 2ψ = 0, (10.10.41) where k 2 = 2ME  2 . (10.10.42) The general solution of (10.10.41) is given by ψ2 (x) = C sin kx + D cos kx, (10.10.43) where C and D are arbitrary constants. Region 3. V = V0 in this region x ≥ a. An argument similar to region 1 leads to zero solution, that is, 10.10 The Schr¨odinger Equation and the Hydrogen Atom 391 ψ3 (x)=0, for x ≥ a. (10.10.44) From a physical point of view, the solution of the Schr¨odinger equation must be continuous everywhere including at the boundaries. Thus, matching of solutions at x = + a is required so that ψ2 (a) = C sin ak + D cos ak =0= ψ3 (a), (10.10.45) ψ2 (−a) = −C sin ak + D cos ak =0= ψ1 (−a). (10.10.46) This system of linear homogeneous equations has nontrivial solutions for C and D only if the determinant of the coefficient matrix vanishes. This means that sin ak cos ak = 0. (10.10.47) There are two possible nontrivial solutions for the set of conditions (10.10.47). Case 1. Even solution: cos ak = 0. In this case, it follows from (10.10.45)–(10.10.46) that C = 0. Hence, ak = (2n + 1) π 2 , n is an integer, or, k 2 = k 2 n = " (2n + 1) π 2a #2 . (10.10.48) Consequently, (10.10.42) gives the energy levels E = En as En = (2n + 1)2 π 2 2 8M a2 . (10.10.49) In this case, the nontrivial solution in region 2 takes the form ψ2 (x) = D cos kx, −a ≤ x ≤ a. (10.10.50) Case 2. Odd solution: sin ak = 0. It follows from (10.10.45)–(10.10.46) that D = 0 and sin ak = 0 holds. Consequently, ak = nπ, n is an integer, n = 0, or, k 2 n = 4nπ a 52 . (10.10.51) 392 10 Higher-Dimensional Boundary-Value Problems Thus, the energy levels are given by En =  2k 2 n 2M = (nπ) 2 2M a2 . (10.10.52) The nontrivial solution in region 2 is ψ2 (x) = C sin kx, −a ≤ x ≤ a. (10.10.53) Thus, it turns out that, corresponding to every value of En given by (10.10.49) or (10.10.52), there exists a physically acceptable solution. Hence, the general solution of the Schr¨odinger equation is obtained from (10.10.7) in the form ψ (x, t) =  n Cnψn (x) exp  − itEn   , (10.10.54) where Cn are constants. In classical mechanics, the motion of the particle is allowed for E > 0. In quantum mechanics, it follows from (10.10.49) or (10.10.52) that particle motion is allowed for discrete values of energy, that is, the energy for this system is quantized. This is a remarkable contrast between the results of the classical mechanics and quantum mechanics. Finally, it follows from the above analysis is that lim |x|→∞ ψ (x)=0. (10.10.55) Such a system, where the wave function vanishes beyond range or asymptotically, is called a bound state, and energy is quantized. A very common example is the hydrogen atom which was discussed in this section. In the present system ψ (x) = 0 for x 2 ≥ a 2 . Therefore, this system is also referred to as a particle in a box of length 2a. The probability for finding the particle outside this region is zero. 10.11 Method of Eigenfunctions and Vibration of Membrane Consider the nonhomogeneous initial boundary-value problem L[u] = ρ utt − G in D (10.11.1) with prescribed homogeneous boundary conditions on the boundary B of D, and the initial conditions u (x1, x2,...,xn, 0) = f (x1, x2,...,xn), (10.11.2) ut (x1, x2,...,xn, 0) = g (x1, x2,...,xn). (10.11.3) 10.11 Method of Eigenfunctions and Vibration of Membrane 393 Here ρ ≡ ρ (x1, x2,...,xn) is a real-valued positive continuous function and G ≡ G (x1, x2,...,xn) is a real-valued continuous function. We assume that the only solution of the associated homogeneous problem L[u] = ρutt (10.11.4) with the prescribed boundary conditions is the trivial solution. Then, if there exists a solution of the given problem (10.11.1)–(10.11.3), it can be represented by a series of eigenfunctions of the associated eigenvalue problem L[ϕ] + λρϕ = 0 (10.11.5) with ϕ satisfying the boundary conditions given for u. For problems with one space variable, see Section 7.8. As a specific example, we shall determine the solution of the problem of forced vibration of a rectangular membrane of length a and width b. The problem is utt − c 2∇2u = F (x, y, t) in D (10.11.6) u (x, y, 0) = f (x, y), 0 ≤ x ≤ a, 0 ≤ y ≤ b, (10.11.7) ut (x, y, 0) = g (x, y), 0 ≤ x ≤ a, 0 ≤ y ≤ b, (10.11.8) u (0, y, t)=0, u (a, y, t)=0, (10.11.9) u (x, 0, t)=0, u (x, b, t)=0. (10.11.10) The associated eigenvalue problem is ∇2ϕ + λϕ = 0 in D, ϕ = 0 on the boundary B of D. The eigenvalues for this problem according to Section 10.6 are given by αmn =  m2π 2 a 2 + n 2π 2 b 2  , m, n = 1, 2, 3 ... and the corresponding eigenfunctions are ϕmn (x, y) = sin 4mπx a 5 sin 4nπy b 5 . Thus, we assume the solution u (x, y, t) = ∞ m=1 ∞ n=1 umn (t) sin 4mπx a 5 sin 4nπy b 5 and the forcing function 394 10 Higher-Dimensional Boundary-Value Problems F (x, y, t) = ∞ m=1 ∞ n=1 Fmn (t) sin 4mπx a 5 sin 4nπy b 5 . Here Fmn (t) are given by Fmn (t) = 4 ab  a 0  b 0 F (x, y, t) sin 4mπx a 5 sin 4nπy b 5 dx dy. Note that u automatically satisfies the homogeneous boundary conditions. Now inserting u (x, y, t) and F (x, y, t) in equation (10.11.6), we obtain u¨mn + c 2α 2 mnumn = Fmn, where α 2 mn = (mπ/a) 2 + (nπ/b) 2 . We have assumed that u is twice continuously differentiable with respect to t. Thus, the solution of the preceding ordinary differential equation takes the form umn (t) = Amn cos (αmnct) + Bmn sin (αmnct) + 1 (αmn c)  t 0 Fmn (τ ) sin [αmnc (t − τ )] dτ. The first initial condition gives u (x, y, 0) = f (x, y) = ∞ m=1 ∞ n=1 Amn sin 4mπx a 5 sin 4nπy b 5 . Assuming that f (x, y) is continuous in x and y, the coefficient Amn of the double Fourier series is given by Amn = 4 ab  a 0  b 0 f (x, y) sin 4mπx a 5 sin 4nπy b 5 dx dy. Similarly, from the remaining initial condition, we have ut (x, y, 0) = g (x, y) = ∞ m=1 ∞ n=1 Bmn (αmn c) sin 4mπx a 5 sin 4nπy b 5 , and hence, for continuous g (x, y), Bmn = 4 (ab αmnc)  a 0  b 0 g (x, y) sin 4mπx a 5 sin 4nπy b 5 dx dy. The solution of the given initial boundary-value problem is therefore given by u (x, y, t) = ∞ m=1 ∞ n=1 umn (t) sin 4mπx a 5 sin 4nπy b 5 , 10.12 Time-Dependent Boundary-Value Problems 395 provided the series for u and its first and second derivatives converge uniformly. If F (x, y, t) = e x+y cos ωt, then Fmn (t) = 4mnπ2 (m2π 2 + a 2) (n2π 2 + b 2) " 1+(−1)m+1 e a # × " 1+(−1)n+1 e b # cos ωt = Cmn cos ωt. Hence, we have umn (t) = 1 (αmnc)  t 0 Cmn cos ωtsin [αmnc (t − τ )] dτ = Cmn (α2 mnc 2 − ω2) (cos ωt − cos αmnct) provided ω = (αmnc). Thus, the solution may be written in the form u (x, y, t) = ∞ m=1 ∞ n=1 Cmn (α2 mnc 2 − ω2) (cos ωt − cos αmnct) × sin 4mπx a 5 sin 4nπy b 5 . 10.12 Time-Dependent Boundary-Value Problems The preceding chapters have been devoted to problems with homogeneous boundary conditions. Due to the frequent occurrence of problems with time dependent boundary conditions in practice, we consider the forced vibration of a rectangular membrane with moving boundaries. The problem here is to determine the displacement function u which satisfies utt − c 2∇2u = F (x, y, t), 0 < x < a, 0 < y < b, (10.12.1) u (x, y, 0) = f (x, y), 0 ≤ x ≤ a, 0 ≤ y ≤ b, (10.12.2) ut (x, y, 0) = g (x, y), 0 ≤ x ≤ a, 0 ≤ y ≤ b, (10.12.3) u (0, y, t) = p1 (y, t), 0 ≤ y ≤ b, t ≥ 0, (10.12.4) u (a, y, t) = p2 (y, t), 0 ≤ y ≤ b, t ≥ 0, (10.12.5) u (x, 0, t) = q1 (x, t), 0 ≤ x ≤ a, t ≥ 0, (10.12.6) u (x, b, t) = q2 (x, t), 0 ≤ x ≤ a, t ≥ 0. (10.12.7) For such problems, we seek a solution in the form u (x, y, t) = U (x, y, t) + v (x, y, t), (10.12.8) 396 10 Higher-Dimensional Boundary-Value Problems where v is the new dependent variable to be determined. Before finding v, we must first determine U. If we substitute equation (10.12.8) into equations (10.12.1)–(10.12.7), we respectively obtain vtt − c 2 (vxx + vyy) = F − Utt + c 2 (Uxx + Uyy) = F5 (x, y, t) and v (x, y, 0) = f (x, y) − U (x, y, 0) = f5(x, y), vt (x, y, 0) = g (x, y) − Ut (x, y, 0) = g5(x, y), v (0, y, t) = p1 (y, t) − U (0, y, t) = p51 (y, t), v (a, y, t) = p2 (y, t) − U (a, y, t) = p52 (y, t), v (x, 0, t) = q1 (x, t) − U (x, 0, t) = q51 (x, t), v (x, b, t) = q2 (x, t) − U (x, b, t) = q52 (x, t). In order to make the conditions on v homogeneous, we set p51 = p52 = q51 = q52 = 0, so that U (0, y, t) = p1 (y, t), U (a, y, t) = p2 (y, t), (10.12.9) U (x, 0, t) = q1 (x, t), U (x, b, t) = q2 (x, t). (10.12.10) In order that the boundary conditions be compatible, we assume that the prescribed functions take the forms p1 (y, t) = ϕ (y) p ∗ 1 (y, t), p2 (y, t) = ϕ (y) p ∗ 2 (y, t), q1 (x, t) = ψ (x) q ∗ 1 (x, t), q2 (x, t) = ψ (x) q ∗ 2 (x, t), where the function ϕ must vanish at the end points y = 0, y = b and the function ψ must vanish at x = 0, x = a. Thus, U (x, y, t) which satisfies equations (10.12.9)–(10.12.10) takes the form U (x, y, t) = ϕ (y) " p ∗ 1 + x a (p ∗ 2 + p ∗ 1 ) # + ψ (x) " q ∗ 1 + y b (q ∗ 2 + q ∗ 1 ) # . The problem then is to find the function v (x, y, t) which satisfies vtt − c 2 (vxx + vyy) = F5 (x, y, t), v (x, y, 0) = f5(x, y), vt (x, y, 0) = g5(x, y), v (0, y, t)=0, v (a, y, t)=0, v (x, 0, t)=0, v (x, b, t)=0. This is an initial boundary-value problem with homogeneous boundary conditions, which has already been solved. 10.12 Time-Dependent Boundary-Value Problems 397 As a particular case, consider the following problem utt − c 2 (uxx + uyy)=0, u (x, y, 0) = 0, ut (x, y, 0) = y b sin 4πx a 5 , u (0, y, t)=0, u (a, y, t)=0, u (x, 0, t)=0, u (x, b, t) = sin 4πx a 5 sin t. We assume a solution in the form u (x, y, t) = v (x, y, t) + U (x, y, t). The function U (x, y, t) which satisfies U (0, y, t)=0, U (a, y, t)=0, U (x, 0, t)=0, U (x, b, t) = sin 4πx a 5 sin t is U (x, y, t) = sin 4πx a 54y b sin t 5 . Thus, the new problem to be solved is vtt − c 2 (vxx + vyy) =  1 − c 2π 2 a 2  y b sin 4πx a 5 sin t, v (x, y, 0) = 0, vt (x, y, 0) = 0, v (0, y, t)=0, v (a, y, t)=0, v (x, 0, t)=0, v (x, b, t)=0. Then, we find Fmn from Fmn (t) = 4 ab  a 0  b 0 F (x, y, t) sin 4mπx a 5 sin 4nπy b 5 dx dy, where F (x, y, t) =  1 − c 2π 2 a 2  y b sin 4πx a 5 sin t, and obtain Fmn (t) = 2 (−1)n+1 an  1 − c 2π 2 a 2  sin t. Now we determine vmn (t) which are given by 398 10 Higher-Dimensional Boundary-Value Problems vmn (t) = Amn cos (αmnct) + Bmn sin (αmnct) + 1 αmnc  t 0 Fmn (τ ) sin [αmnc (t − τ )] dτ. Since v (x, y, 0) = 0, Amn = 0, but Bmn = 4 ab αmnc  a 0  b 0 4 − y b sin πx a 5 sin 4mπx a 5 sin 4nπy b 5 dx dy = 2 (−1)n αmnnac . Thus, we have vmn (t) = 2 (−1)n αmnnac sin (αmnct) + 2 (−1)n αmnca3n (1 − α2c 2)  a 2 − c 2π 2 (sin αmnct − αc sin t). The solution is therefore given by u (x, y, t) = y b sin 4πx a 5 sin t + ∞ m=1 ∞ n=1 vmn (t) sin 4mπx a 5 sin 4nπy b 5 . 10.13 Exercises 1. Solve the Dirichlet problem ∇2u = 0, 0 < x < a, 0 < y < b, 0 < z < c, u (0, y, z) = sin 4πy b 5 sin 4πz c 5 , u (a, y, z)=0, u (x, 0, z)=0, u (x, b, z)=0, u (x, y, 0) = 0, u (x, y, c)=0. 2. Solve the Neumann problem ∇2u = 0, 0 <x< 1,="" 0="" <y<="" <z<="" ux="" (0,="" y,="" z)="0," (1,="" uy="" (x,="" 0,="" uz="" 0)="cos" πx="" cos="" πy,="" 1)="0." 3.="" solve="" the="" robin="" boundary-value="" problem="" ∇2u="0," <="" x="" π,="" y="" z="" u="" (y,="" z),="" (π,="" +="" h="" π)="" ⎫="" ⎬="" ⎭="" 10.13="" exercises="" 399="" 4.="" determine="" solution="" of="" each="" following="" problems="" for="" a="" cylinder:="" (a)="" r<a,="" <θ<="" 2π,="" l,="" (a,="" θ,="" (r,="" l)="0," θ).="" (b)="" (θ,="" 5.="" find="" dirichlet="" sphere="" θ="" <ϕ<="" ϕ)="cos2" θ.="" 6.="" in="" region="" bounded="" by="" two="" concentric="" spheres="" r="" b,="" <φ<="" φ)="f" φ),="" (b,="" φ).="" 7.="" steady-state="" temperature="" distribution="" cylinder="" radius="" if="" constant="" flow="" heat="" t="" is="" supplied="" at="" end="" and="" surface="" are="" maintained="" zero="" temperature.="" 8.="" potential="" electrostatic="" field="" inside="" length="" l="" a,="" grounded,="" charged="" to="" u0.="" 9.="" electric="" upper="" half="" u1="" lower="" u2.="" 10.="" r<="" 11.="" neumann="" 400="" 10="" higher-dimensional="" ur="" ϕ),="" where="" ="" 2π="" π="" f="" sin="" dθ="" dϕ="0." 12.="" initial="" utt="c" 2∇2u,="" <x<=""> 0, u (x, y, 0) = sin2 πx sin πy, ut (x, y, 0) = 0, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, u (0, y, t)=0, u (1, y, t)=0, 0 ≤ y ≤ 1, t> 0, u (x, 0, t)=0, u (x, 1, t)=0, 0 ≤ x ≤ 1, t> 0. 13. Obtain the solution of the problem utt = c 2∇2u, r < a, 0 <θ< 2π, t > 0, u (r, θ, 0) = f (r, θ), ut (r, θ, 0) = g (r, θ), u (a, θ, t)=0. 14. Determine the temperature distribution in a rectangular plate with radiation from its surface. The temperature distribution is described by ut = k (uxx + uyy) − h (u − u0), 0 < x < a, 0 < y < b, t > 0, u (x, y, 0) = f (x, y), u (0, y, t)=0, u (a, y, t)=0, u (x, 0, t)=0, u (x, b, t)=0, where k, h and u0 are constants. 15. Solve the heat conduction problem in a circular plate ut = k  urr + 1 r ur + 1 r 2 uθθ , r< 1, 0 <θ< 2π, t > 0, u (r, θ, 0) = f (r, θ), u (1, θ, t)=0. 16. Solve the initial boundary-value problem utt = c 2∇2u, 0 <x< 1,="" 0="" <y<="" <z<="" t=""> 0, u (x, y, z, 0) = sin πx sin πy sin πz, 10.13 Exercises 401 ut (x, y, z, 0) = 0, u (0, y, z, t) = u (1, y, z, t)=0, u (x, 0, z, t) = u (x, 1, z, t)=0, u (x, y, 0, t) = u (x, y, 1, t)=0. 17. Solve utt + k ut = c 2∇2u, 0 < x < a, 0 < y < b, 0 < z < d, t > 0, u (x, y, z, 0) = f (x, y, z), ut (x, y, z, 0) = g (x, y, z), u (0, y, z, t) = u (a, y, z, t)=0, u (x, 0, z, t) = u (x, b, z, t)=0, u (x, y, 0, t) = u (x, y, d, t)=0. 18. Obtain the solution of the problem for t > 0, utt = c 2  urr + 1 r ur + 1 r 2 uθθ + uzz , r < a, 0 <θ< 2π, 0 < z < l, u (r, θ, z, 0) = f (r, θ, z), ut (r, θ, z, 0) = g (r, θ, z), u (a, θ, z, t)=0, u (r, θ, 0, t) = u (r, θ, l, t)=0. 19. Determine the solution of the heat conduction problem ut = k ∇2u, 0 < x < a, 0 < y < b, 0 < z < c, t > 0, u (x, y, z, 0) = f (x, y, z), ux (0, y, z, t) = ux (a, y, z, t)=0, uy (x, 0, z, t) = uy (x, b, z, t)=0, uz (x, y, 0, t) = uz (x, y, c, t)=0. 402 10 Higher-Dimensional Boundary-Value Problems 20. Solve the problem ut = k ∇2u, r < a, 0 <θ< 2π, 0 < z < l, t > 0, u (r, θ, z, 0) = f (r, θ, z), ur (a, θ, z, t)=0, u (r, θ, 0, t) = u (r, θ, l, t)=0. 21. Find the temperature distribution in the section of a sphere cut out by the cone θ = θ0. The surface temperature is zero while the initial temperature is given by f (r, θ, ϕ). 22. Solve the initial boundary-value problem utt = c 2∇2u + F (x, y, t), 0 < x < a, 0 < y < b, t > 0, u (x, y, 0) = f (x, y), ut (x, y, 0) = g (x, y), ux (0, y, t) = ux (a, y, t) = 0 for all t > 0, uy (x, 0, t) = uy (x, b, t) = 0 for all t > 0. 23. Solve the problem utt = c 2∇2u + xy sin t, 0 < x < π, 0 < y < π, t > 0, u (x, y, 0) = 0, ut (x, y, 0) = 0, u (0, y, t) = u (π, y, t)=0, u (x, 0, t) = u (x, π, t)=0. 24. Solve ut = k∇2u + F (x, y, z, t), 0 < x < a, 0 < y < b, 0 < z < c, t > 0, u (x, y, z, 0) = f (x, y, z), u (0, y, z, t) = u (a, y, z, t)=0, 10.13 Exercises 403 u (x, 0, z, t) = u (x, b, z, t)=0, uz (x, y, 0, t) = uz (x, y, c, t)=0. 25. Solve the nonhomogeneous diffusion problem ut = k ∇2u + A, 0 < x < π, 0 < y < π, t > 0, u (x, y, 0) = 0, u (0, y, t) = u (π, y, t)=0, uy (x, 0, t) + u (x, 0, t)=0, uy (x, π, t) + u (x, π, t)=0, where k and A are constants. 26. Find the temperature distribution of the composite cylinder consisting of an inner cylinder 0 ≤ r ≤ r0 and an outer cylindrical tube r0 ≤ r ≤ a. The surface temperature is maintained at zero degrees, and the initial temperature distribution is given by f (r, θ, z). 27. Solve the initial boundary-value problem ut − c 2∇2u = 0, 0 < x < π, 0 < y < π, t > 0, u (x, y, 0) = 0, u (0, y, t) = u (π, y, t)=0, u (x, 0, t) = x (x − π) sin t, u (x, π, t)=0, 0 ≤ x ≤ π, t ≥ 0. 28. Solve the problem utt = c 2∇2u, r < a, 0 <θ< 2π, t > 0, u (r, θ, 0) = f (r, θ), ut (r, θ, 0) = g (r, θ), u (a, θ, t) = p (θ, t). 404 10 Higher-Dimensional Boundary-Value Problems 29. Solve ut = c 2∇2u, r < a, t > 0, u (r, θ, 0) = f (r, θ), ut (a, θ, t) = g (θ, t), 0 < θ < π. 30. Determine the solution of the biharmonic equation ∇4u = q/D with the boundary conditions u (x, 0) = u (x, b)=0, u 4 − a 2 , y5 = u 4a 2 , y5 = 0, uxx 4 − a 2 , y5 = uxx 4a 2 , y5 = 0, uyy (x, 0) = uyy (x, b)=0, where q is the load per unit area and D is the flexural rigidity of the plate. This is the problem of the deflection of a uniformly loaded plate, the sides of which are simply supported. 31. (a) Show that the solution of the one-dimensional Schr¨odinger equation for a free particle of mass M ψt =  i 2M  ψxx is ψ (x, t) =  N b  exp  − x 2 2b 2  , b =  a 2 + it M 1 2 , where a is an integrating constant that can be determined from the initial value of the wave function ψ (x, t), and N is also a constant that can be determined from the normalization of the probability (wave function) of finding the particle. (b) Show that the Gaussian probability density is |ψ| 2 = ψψ∗ = |N| 2 ac exp  − x 2 c 2  , and its mean width is δ = c √ 2 , C =  a 2 +  2 t 2 M2a 2 1 2 . 10.13 Exercises 405 32. Analogous to Example 10.10.1, solve the problem for a finite square well potential (see Figure 10.10.1) with a finite value for the height of the potential given as V (x) = ⎧ ⎨ ⎩ 0, for −a ≤ x ≤ a V0, for x ≤ −a, x ≥ a. 33. Consider the quantum mechanical problem described by the one-dimensional Schr¨odinger equation ψxx + k 2ψ = 0 where the wavenumber k = 1   2M (E − V ) in the rectangular potential barrier of height V0 and width 2a, and V (x) = V0H (a − |x|), where H is the Heaviside unit step function. The particle is free for x < −a and x>a, and V (x) is an even function; the case V0 > E is of great interest here. Show that the general solution of the Schr¨odinger equation for V0 > E is ψ (x) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ A eikx + B e−ikx, x ≤ −a, C e−κx + D e+κx , −a ≤ x ≤ a, F eikx + G e−ikx, x ≥ a, where k = √ 2ME and κ =  2M (V0 − E). Matching the boundary conditions at x = −a, show that ⎡ ⎣ A B ⎤ ⎦ = 1 2 ⎡ ⎣  1 + iκ k exp (κa + ika)  1 − iκ k exp (−κa + ika)  1 − iκ k exp (κa − ika)  1 + iκ k exp (−κa − ika) ⎤ ⎦ ⎡ ⎣ C D ⎤ ⎦ where [ ] denotes a matrix. Using the matching conditions at x = a, show that ⎡ ⎣ C D ⎤ ⎦ = 1 2 ⎡ ⎣  1 − ik κ exp (aκ + iak)  1 + ik κ exp (aκ − iak)  1 + ik κ exp (−aκ + iak)  1 − ik κ exp (−aκ − ika) ⎤ ⎦ ⎡ ⎣ F G ⎤ ⎦. Hence, deduce ⎡ ⎣ A B ⎤ ⎦ = ⎡ ⎣  cosh 2aκ + iε 2 sinh 2aκ e 2iak 1 2 (iη) sinh 2aκ − 1 2 (iη) sinh 2aκ  cosh 2aκ − 1 2 iε sinh 2aκ e −2ika ⎤ ⎦ ⎡ ⎣ F G ⎤ ⎦ where ε =  κ k − k κ and η =  κ k + k κ . 11 Green’s Functions and Boundary-Value Problems “Potential theory has developed out of the vector analysis created by Gauss, Green, and Kelvin for the mathematical theories of gravitational attraction, of electrostatics and of the hydrodynamics of perfect fluids (i.e., incompressible and inviscid fluids). The first stage of abstraction was the study of harmonic functions, i.e., potential functions in space free from masses, charges, sources, or sinks. This led to the inspired intuition of Dirichlet and the early attempts to justify his ‘principle’.” George Temple 11.1 Introduction Boundary-value problems associated with either ordinary or partial differential equations arise most frequently in mathematics, mathematical physics and engineering science. The linear superposition principle is one of the most elegant and effective methods to represent solutions of boundaryvalue problems in terms of an auxiliary function known as Green’s function. Such a function was first introduced by George Green as early as 1828. Subsequently, the method of Green’s functions became a very useful analytical method in mathematics and in many of the applied sciences. In previous chapters, it has been shown that the eigenfunction method can effectively be used to express the solutions of differential equations as infinite series. On the other hand, solutions of differential equations can be obtained as an integral superposition in terms of Green’s functions. So the method of Green’s functions offers several advantages over eigenfunction expansions. First, an integral representation of solutions provides a direct way of describing the general analytical structure of a solution that may be obscured by an infinite series representation. Second, from an analytical point of view, the evaluation of a solution from an integral representation may prove simpler than finding the sum of an infinite series, particularly near 408 11 Green’s Functions and Boundary-Value Problems rapidly-varying features of a function, where the convergence of an eigenfunction expansion may be slow. Third, in view of the Gibbs phenomenon discussed in Chapter 6, the integral representation seems to impose less stringent requirements on the functions that describe the values that the solution must assume on a given boundary than the expansion based on eigenfunctions. Many physical problems are described by second-order nonhomogeneous differential equations with homogeneous boundary conditions or by secondorder homogeneous equations with nonhomogeneous boundary conditions. Such problems can be solved by the method of Green’s functions. We consider a nonhomogeneous partial differential equation of the form Lxu (x) = f (x), (11.1.1) where x = (x, y, z) is a vector in three (or higher) dimensions, Lx is a linear partial differential operator in three or more independent variables with constant coefficients, and u (x) and f (x) are functions of three or more independent variables. The Green’s function G (x, ξ) of this problem satisfies the equation LxG (x, ξ) = δ (x − ξ) (11.1.2) and represents the effect at the point x of the Dirac delta function source at the point ξ = (ξ, η, ζ). Multiplying (11.1.2) by f (ξ) and integrating over the volume V of the ξ space, so that dV = dξ dη dζ, we obtain  V LxG (x, ξ) f (ξ) dξ =  V δ (x − ξ) f (ξ) dξ = f (x). (11.1.3) Interchanging the order of the operator Lx and integral sign in (11.1.3) gives Lx  V G (x, ξ) f (ξ) dξ = f (x). (11.1.4) A simple comparison of (11.1.4) with (11.1.1) leads to the solution of (11.1.1) in the form u (x) =  V G (x, ξ) f (ξ) dξ. (11.1.5) Clearly, (11.1.5) is valid for any finite number of components of x. Accordingly, the Green’s function method can be applied, in general, to any linear, constant coefficient, nonhomogeneous partial differential equation in any number of independent variables. Another way to approach the problem is by looking for the inverse operator L −1 x . If it is possible to find L −1 x , then the solution of (11.1.1) can 11.2 The Dirac Delta Function 409 be obtained as u (x) = L −1 x (f (x)). It turns out that in many important cases it is possible, and the inverse operator can be expressed as an integral operator of the form u (x) = L −1 x (f (ξ)) =  V G (x, ξ) f (ξ) dξ. (11.1.6) The kernel G (x, ξ) is called the Green’s function which is, in fact, the characteristic of the operator Lx for any finite number of independent variables. In our study of partial differential equations with the aid of Green’s functions, special attention will be given to those three partial differential equations which occur most frequently in mathematics, mathematical physics and engineering science; the wave equation utt − c 2∇2u = f (x), (11.1.7) the heat or diffusion equation ut − κ∇2u = f (x), (11.1.8) and the potential or the Laplace equation ∇2u = f (x), (11.1.9) where the Laplacian ∇2 in an n-dimensional Euclidean space is given by ∇2 ≡ ∂ 2 ∂x2 1 + ∂ 2 ∂x2 2 + ... + ∂ 2 ∂x2 n , (11.1.10) and x = (x1, x2,...,xn). Clearly, the solutions of the wave and heat equations are functions of (n + 1) coordinates consisting of n space dimensions, x = (x1, x2,...,xn) and one time dimension t, whereas the solutions of the Laplace equation are functions of n space dimensions. This chapter deals with the basic idea and properties of Green’s functions and how to construct such functions for finding solutions of partial differential equations. Some examples of applications are provided in this chapter and in the next chapter. 11.2 The Dirac Delta Function The application of Green’s functions to boundary-value problems in ordinary differential equations was described earlier in Chapter 8. The Green’s function method is applied here to boundary-value problems in partial differential equations. The method provides solutions in integral form and is applicable to a wide class of problems in applied mathematics and mathematical physics. 410 11 Green’s Functions and Boundary-Value Problems Before developing the method of Green’s functions, we will first define the Dirac delta function δ (x − ξ,y − η) in two dimensions by (a) δ (x − ξ,y − η)=0, x = ξ, y = η, (11.2.1) (b)  Rε δ (x − ξ,y − η) dx dy = 1, Rε : (x − ξ) 2 + (y − η) 2 < ε2 , (11.2.2) (c)  R F (x, y) δ (x − ξ,y − η) dx dy = F (ξ,η), (11.2.3) for arbitrary continuous function F in the region Rε. The delta function is not a function in the ordinary sense. For an elegant treatment of the delta function as a generalized function, see L. Schwartz, Th´eorie des Distributions (1950, 1951). It is a symbolic function, and is often viewed as the limit of a distribution. If δ (x − ξ) and δ (y − η) are one-dimensional delta functions, we have  R F (x, y) δ (x − ξ) δ (y − η) dx dy = F (ξ,η). (11.2.4) Since (11.2.3) and (11.2.4) hold for an arbitrary continuous function F, we conclude that δ (x − ξ,y − η) = δ (x − ξ) δ (y − η). (11.2.5) Thus, we may state that the two-dimensional delta function is the product of two one-dimensional delta functions. Higher dimensional delta functions can be defined in a similar manner. δ (x1, x2,...,xn) = δ (x1) δ (x2)...δ (xn). (11.2.6) The expression for the δ-function become much more complicated when we introduce curvilinear coordinates. However, for simplicity, we transform the two-dimensional delta function from Cartesian coordinates x, y to curvilinear coordinates α, β by means of the transformation x = u (α, β) and y = v (α, β), (11.2.7) where u and v are single-valued continuous and differentiable functions of their arguments. We assume that under this transformation α = α1 and β = β1 correspond to x = ξ and y = η respectively. Changing the coordinates according to (11.2.7), we reduce equation (11.2.4) to  F (u, v) δ (u − ξ) δ (v − η)|J| dα dβ = F (ξ,η), (11.2.8) where J is the Jacobian of the transformation defined by 11.2 The Dirac Delta Function 411 J = ∂ (u, v) ∂ (α, β) =       uα uβ vα vβ       = 0. (11.2.9) Consequently, we can write δ (u − ξ) δ (v − η)|J| = δ (α − α1) δ (β − β1). (11.2.10) In particular, the transformation from rectangular Cartesian coordinates (x, y) to polar coordinates (r, θ) is defined by x = r cos θ, y = r sin θ, (11.2.11) so that the Jacobian J is J =       xr yr xθ yθ       = xryθ − yrrθ = r. (11.2.12) In this case, J vanishes at the origin and the transformation is singular at r = 0 for any θ. Hence, θ can be ignored and δ (x) δ (y) = δ (r) |J1| = 1 2π δ (r) r , (11.2.13) where J1 =  π 0 J dθ = 2πr. Similarly, the transformation from three-dimensional rectangular Cartesian coordinates (x, y, z) to spherical polar coordinates (r, θ, φ) is given by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, (11.2.14) where 0 ≤ r < ∞, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π. The Jacobian of the transformation is J = r 2 sin θ. This Jacobian vanishes for all points on the z-axis, that is, for θ = 0, and hence, the coordinate φ may be ignored. Also, J vanishes at the origin (r = 0) in which case both θ and φ may be ignored. Consequently, δ (x) δ (y) δ (z) = δ (r) |J2| = δ (r) 4πr2 , (11.2.15) where J2 =  π 0  2π 0 J dθ dφ =  π 0  2π 0 r 2 sin θ dφ = 4πr2 . 412 11 Green’s Functions and Boundary-Value Problems 11.3 Properties of Green’s Functions The solution of the Dirichlet problem in a domain D with boundary B ∇2u = h (x, y) in D u = f (x, y) on B (11.3.1) is given in Section 11.5 and has the form u (x, y) =  D G (x, y; ξ,η) h (ξ,η) dξ dη +  B f ∂G ∂n ds, (11.3.2) where G is the Green’s function and n denotes the outward normal to the boundary B of the region D. It is rather obvious then that the solution u (x, y) can be determined as soon as the Green’s function G is ascertained, so the problem in this technique is really to find the Green’s function. First, we shall define the Green’s function for the Dirichlet problem involving the Laplace operator. Then, the Green’s function for the Dirichlet problem involving the Helmholtz operator may be defined in a completely analogous manner. The Green’s function for the Dirichlet problem involving the Laplace operator is the function which satisfies (a) ∇2G = δ (x − ξ,y − η) in D, (11.3.3) G = 0 on B. (11.3.4) (b) G is symmetric, that is, G (x, y; ξ,η) = G (ξ,η; x, y), (11.3.5) (c) G is continuous in x, y, ξ, η, but (∂G/∂n) has a discontinuity at the point (ξ,η) which is specified by the equation limε→0  Cε ∂G ∂n ds = 1, (11.3.6) where n is the outward normal to the circle Cε : (x − ξ) 2 + (y − η) 2 = ε 2 . The Green’s function G may be interpreted as the response of the system at a field point (x, y) due to a δ function input at the source point (ξ,η). G is continuous everywhere in D, and its first and second derivatives are continuous in D except at (ξ,η). Thus, property (a) essentially states that ∇2G = 0 everywhere except at the source point (ξ,η). We will now prove property (b). Theorem 11.3.1. The Green’s function is symmetric. 11.3 Properties of Green’s Functions 413 Proof. Applying the Green’s second formula  D  φ∇2ψ − ψ∇2φ dS =  B  φ ∂ψ ∂n − ψ ∂φ ∂n ds, (11.3.7) to the functions φ = G (x, y; ξ,η) and ψ = G (x, y; ξ ∗ , η∗ ), we obtain  D G (x, y; ξ,η) ∇2G (x, y; ξ ∗ , η∗ ) − G (x, y; ξ ∗ , η∗ ) ∇2G (x, y; ξ,η) ! dx dy =  B G (x, y; ξ,η) ∂G ∂n (x, y; ξ ∗ , η∗ ) − G (x, y; ξ ∗ , η∗ ) ∂G ∂n (x, y; ξ,η) ds. Since G (x, y; ξ,η) and hence, G (x, y; ξ ∗ , η∗ ) must vanish on B, we have  D G (x, y; ξ,η) ∇2G (x, y; ξ ∗ , η∗ ) − G (x, y; ξ ∗ , η∗ ) ∇2G (x, y; ξ,η) ! dx dy = 0. But ∇2G (x, y; ξ,η) = δ (x − ξ,y − η), and ∇2G (x, y; ξ ∗ , η∗ ) = δ (x − ξ ∗ , y − η ∗ ). Since  D G (x, y; ξ,η) δ (x − ξ ∗ , y − η ∗ ) dx dy = G (ξ ∗ , η∗ ; ξ,η), and  D G (x, y; ξ ∗ , η∗ ) δ (x − ξ,y − η) dx dy = G (ξ,η; ξ ∗ , η∗ ), we obtain G (ξ,η; ξ ∗ , η∗ ) = G (ξ ∗ , η∗ ; ξ,η). Theorem 11.3.2. ∂G/∂n is discontinuous at (ξ,η); in particular limε→0  Cε ∂G ∂n ds = 1, Cε : (x − ξ) 2 + (y − η) 2 = ε 2 . Proof. Let Rε be the region bounded by Cε. Then, integrating both sides of equation (11.3.3), we obtain  Rε ∇2G dx dy =  R δ (x − ξ,y − η) dx dy = 1. 414 11 Green’s Functions and Boundary-Value Problems It therefore follows that limε→0  Rε ∇2G dx dy = 1. (11.3.8) Thus, by the Divergence theorem of calculus, limε→0  Cε ∂G ∂n ds = 1. 11.4 Method of Green’s Functions It is often convenient to seek G as the sum of a particular integral of the nonhomogeneous equation and the solution of the associated homogeneous equation. That is, G may assume the form G (ξ,η; x, y) = F (ξ,η; x, y) + g (ξ,η; x, y), (11.4.1) where F, known as the free-space Green’s function, satisfies ∇2F = δ (ξ − x, η − y) in D, (11.4.2) and g satisfies ∇2 g = 0 in D, (11.4.3) so that by superposition G = F + g satisfies equation (11.3.3). Also G = 0 on B requires that g = −F on B. (11.4.4) Note that F need not satisfy the boundary condition. Hereafter, (x, y) will denote the source point. Before we determine the solution of a particular problem, let us first find F for the Laplace and Helmholtz operators. (1) Laplace Operator In this case, F must satisfy the equation ∇2F = δ (ξ − x, η − y) in D. Then, for r = " (ξ − x) 2 + (η − y) 2 # 1 2 > 0, that is, for ξ = x, η = y, we have with (x, y) as the center ∇2F = 1 r ∂ ∂r  r ∂F ∂r  = 0, 11.4 Method of Green’s Functions 415 since F is independent of θ. Therefore, the solution is F = A + B log r. Applying condition (11.3.6), it follows directly from equation (11.3.8) with ∇2 g = 0, that limε→0  Cε ∂F ∂n ds = limε→0  2π 0 B r r dθ = 1. Thus, B = 1/2π and A is arbitrary. For simplicity, we choose A = 0. Then F takes the form F = 1 2π log r. (11.4.5) (2) Helmholtz Operator Here F is required to satisfy ∇2F + κ 2F = δ (x − ξ,y − η). Again for r > 0, we find 1 r ∂ ∂r  r ∂F ∂r  + κ 2F = 0, or, r 2Frr + rFr + κ 2 r 2F = 0. This is the Bessel equation of order zero, the solution of which is F (κr) = AJ0 (κr) + BY0 (κr). Since the behavior of J0 at r = 0 is not singular, we set A = 0. Thus, we have F (κr) = BY0 (κr). But, for very small r, Y0 (κr) ∼ 2 π log r. Applying condition (11.3.6), we obtain 1 = limε→0  Cε ∂F ∂n ds = limε→0  Cε B ∂Y0 ∂r ds = B · 2 πr · 2πr and hence, B = 1/4. Thus, F (κr) becomes 416 11 Green’s Functions and Boundary-Value Problems F (κr) = 1 4 Y0 (κr). (11.4.6) We may point out that, since  ∇2 + κ 2 approaches ∇2 as κ → 0, it should (and does) follow that 1 4 Y0 (κr) → 1 2 log r as κ → 0 + . 11.5 Dirichlet’s Problem for the Laplace Operator We are now in a position to determine the solution of the Dirichlet problem ∇2u = h in D, (11.5.1) u = f on B, by the method of Green’s function. By putting φ (ξ,η) = G (ξ,η; x, y) and ψ (ξ,η) = u (ξ,η) in equation (11.3.7), we obtain  D G (ξ,η; x, y) ∇2u − u (ξ,η) ∇2G ! dξ dη =  B G (ξ,η; x, y) ∂u ∂n − u (ξ,η) ∂G ∂n ds. But ∇2u = h (ξ,η) in D, and ∇2G = δ (ξ − x, η − y) in D. Thus, we have  D [G (ξ,η; x, y) h (ξ,η) − u (ξ,η) δ (ξ − x, η − y)] dξ dη =  B G (ξ,η; x, y) ∂u ∂n − u (ξ,η) ∂G ∂n ds. (11.5.2) Since G = 0 and u = f on B, and since G is symmetric, it follows that u (x, y) =  D G (x, y; ξ,η) h (ξ,η) dξ dη +  B f ∂G ∂n ds (11.5.3) 11.5 Dirichlet’s Problem for the Laplace Operator 417 which is the solution given by (11.3.2). As a specific example, consider the Dirichlet problem for a unit circle. Then ∇2 g = gξξ + gηη = 0 in D, (11.5.4) g = −F on B. But we already have from equation (11.4.5) that F = (1/2π) log r. If we introduce the polar coordinates (see Figure 11.5.1) ρ, θ, σ, β by means of the equations x = ρ cos θ, ξ = σ cos β, (11.5.5) y = ρ sin θ, η = σ sin β, then the solution of equation (11.5.4) is [see Section 9.4] g (σ, β) = a0 2 + ∞ n=1 σ n (an cos nβ + bn sin nβ), where g = − 1 4π log 1 + ρ 2 − 2ρ cos (β − θ) ! on B. Figure 11.5.1 Image point. 418 11 Green’s Functions and Boundary-Value Problems By using the relation log 1 + ρ 2 − 2ρ cos (β − θ) ! = −2 ∞ n=1 ρ n cos n (β − θ) n , and equating the coefficients of sin nβ and cos nβ to determine an and bn, we find an = ρ n 2πn cos nθ, bn = ρ n 2πn sin nθ. It therefore follows that g (ρ, θ; σ, β) = 1 2π ∞ n=1 (σρ) n n cos n (β − θ) = − 1 4π log " 1+(σρ) 2 − 2 (σρ) cos (β − θ) # . Hence, the Green’s function for the problem is G (ρ, θ; σ, β) = 1 4π log σ 2 + ρ 2 − 2σρ cos (β − θ) ! − 1 4π log " 1+(σρ) 2 − 2σρ cos (β − θ) # , (11.5.6) from which we find ∂G ∂n     on B =  ∂G ∂σ  σ=1 = 1 2π 1 − ρ 2 [1 + ρ 2 − 2ρ cos (β − θ)]. If h = 0, then solution (11.5.3) reduces to the Poisson integral formula similar to (9.4.10) and assumes the form u (ρ, θ) = 1 2π  2π 0 1 − ρ 2 1 + ρ 2 − 2ρ cos (β − θ) f (β) dβ. 11.6 Dirichlet’s Problem for the Helmholtz Operator We will now determine the Green’s function solution of the Dirichlet problem involving the Helmholtz operator, namely, ∇2u + κ 2u = h in D, (11.6.1) u = f on B, where D is a circular domain of unit radius with boundary B. Then, the Green’s function must satisfy 11.6 Dirichlet’s Problem for the Helmholtz Operator 419 ∇2G + κ 2G = δ (ξ − x, η − y) in D, (11.6.2) G = 0 on B. Again, we seek the solution in the form G (ξ,η; x, y) = F (ξ,η; x, y) + g (ξ,η; x, y). (11.6.3) From equation (11.4.6), we have F = 1 4 Y0 (κr), (11.6.4) where r = " (ξ − x) 2 + (η − y) 2 # 1 2 . The function g must satisfy ∇2 g + κ 2 g = 0 in D, (11.6.5) g = − 1 4 Y0 (κr) on B. This solution can be determined easily by the method of separation of variables. Thus, the solution in the polar coordinates defined by equation (11.5.5) may be written in the form g (ρ, θ, σ, β) = ∞ n=0 Jn (κσ) [an cos nβ + bn sin nβ] , (11.6.6) where a0 = − 1 8πJ0 (κ)  π −π Y0 " κ  1 + ρ 2 − 2ρ cos (β − θ) # dβ, an bn = = − 1 4πJn(κ) * π −π Y0 " κ  1 + ρ 2 − 2ρ cos (β − θ) # cos nβ dβ − 1 4πJn(κ) * π −π Y0 " κ  1 + ρ 2 − 2ρ cos (β − θ) # sin nβ dβ ⎫ ⎪⎪⎬ ⎪⎪⎭ n = 1, 2,.... To find the solution of the Dirichlet problem, we multiply both sides of the first equation of equation (11.6.1) by G and integrate. Thus, we have  D  ∇2u + κ 2u G (ξ,η; x, y) dξ dη =  D h (ξ,η) G (ξ,η; x, y) dξ dη. We then apply Green’s theorem on the left side of the preceding equation and obtain  D h (ξ,η) G (ξ,η; x, y) dξ dη −  D u  ∇2G + κ 2G dξ dη =  B (G un − u Gn) ds. 420 11 Green’s Functions and Boundary-Value Problems But ∇2G + κ 2G = δ (ξ − x, η − y) in D and G = 0 on B. We, therefore, have u (x, y) =  D h (ξ,η) G (ξ,η; x, y) dξ dη +  B f (ξ,η) Gnds, (11.6.7) where G is given by equation (11.6.3) with equations (11.6.4) and (11.6.6). 11.7 Method of Images We shall describe another method of obtaining Green’s functions. This method, called the method of images, is based essentially on the construction of Green’s function for a finite domain from that of an infinite domain. The disadvantage of this method is that it can be applied only to problems with simple boundary geometry. As an illustration, we consider the same Dirichlet problem solved in Section 11.5. Let P (ξ,η)be a point in the unit circle D, and let Q (x, y) be the source point also in D. The distance between P and Q is r. Let Q′ be the image which lies outside of D on the ray from the origin opposite to the source point Q (as shown in Figure 11.7.1) such that OQ/σ = σ/OQ′ , where σ is the radius of the circle passing through P centered at the origin. Figure 11.7.1 Image point. 11.7 Method of Images 421 Since the two triangles OPQ and OPQ′ are similar by virtue of the hypothesis (OQ) (OQ′ ) = σ 2 and by possessing a common angle at O, we have r ′ r = σ ρ , (11.7.1) where r ′ = P Q′ and ρ = OQ. If σ = 1, equation (11.7.1) becomes 4 r r ′ 5 1 ρ = 1. Then, we clearly see that the quantity 1 2π log  r r ′ 1 ρ  = 1 2π log r − 1 2π log r ′ + 1 2π log 1 ρ (11.7.2) which vanishes on the boundary σ = 1, is harmonic in D except at Q, and satisfies equation (11.3.3). (Note the log r ′ is harmonic everywhere except at Q′ , which is outside the domain D.) This suggests that we should choose the Green’s function G = 1 2π log r − 1 2π log r ′ + 1 2π log 1 ρ . (11.7.3) Noting that Q′ is at (1/ρ, θ), the function G in polar coordinates takes the form G (ρ, θ, σ, β) = 1 4π log σ 2 + ρ 2 − 2σρ cos (β − θ) ! − 1 4π log 1 σ 2 + ρ 2 − 2 ρ σ cos (β − θ) + 1 2π log 1 σ (11.7.4) which is the same as G given by (11.5.6). It is quite interesting to observe the physical interpretation of the Green’s function (11.7.3) and (11.7.4). The first term represents the potential due to a unit line charge at the source point, whereas the second term represents the potential due to a negative unit charge at the image point. The third term represents a uniform potential. The sum of these potentials makes up the total potential field. Example 11.7.1. To illustrate an obvious and simple case, consider the semiinfinite plane η > 0. The problem is to solve ∇2u = h in η > 0, u = f on η = 0. 422 11 Green’s Functions and Boundary-Value Problems The image point should be obvious by inspection. Thus, if we construct G = 1 4π log " (ξ − x) 2 + (η − y) 2 # − 1 4π " (ξ − x) 2 + (η + y) 2 # , (11.7.5) the condition that G = 0 on η = 0 is clearly satisfied. It is also evident that G is harmonic in η > 0 except at the source point, and that G satisfies equation (11.3.3). With Gn|B = [−Gη] η=0, the solution (11.5.3) is given by u (x, y) = y π  ∞ −∞ f (ξ) dξ (ξ − x) 2 + y 2 + 1 4π  ∞ 0  ∞ −∞ log 1 (ξ − x) 2 + (η − y) 2 (ξ − x) 2 + (η + y) 2 3 h (ξ,η) dξ dη. (11.7.6) Example 11.7.2. Another example that illustrates the method of images well is the Robin’s problem on the quarter infinite plane, that is, ∇2u = h (ξ,η) in ξ > 0, η> 0, u = f (η) on ξ = 0, (11.7.7) un = g (ξ) on η = 0. This illustrated in Figure 11.7.2. Figure 11.7.2 Images in the Robin problem. 11.8 Method of Eigenfunctions 423 Let (−x, y), (−x, −y), and (x, −y) be the three image points of the source point (x, y). Then, by inspection, we can immediately construct Green’s function G = 1 4π log " (ξ − x) 2 + (η − y) 2 #"(ξ − x) 2 + (η + y) 2 # " (ξ + x) 2 + (η − y) 2 #"(ξ + x) 2 + (η + y) 2 # . (11.7.8) This function satisfies ∇2G = 0 except at the source point, and G = 0 on ξ = 0 and Gη = 0 on η = 0. The solution from equation (11.3.3) is thus given by u (x, y) =  D G h dξ dη +  B (G un − u Gn) ds, =  ∞ 0  ∞ 0 G h dξ dη +  ∞ 0 g (ξ) G (ξ, 0; x, y) dξ, +  ∞ 0 f (η) Gξ (0, η; x, y) dξ. 11.8 Method of Eigenfunctions In this section, we will apply the method of eigenfunctions, described in Chapter 10, to obtain the Green’s function. We consider the boundary-value problem ∇2u = h in D, (11.8.1) u = f on B. For this problem, G must satisfy ∇2G = δ (ξ − x, η − y) in D, (11.8.2) G = 0 on B, and hence, the associated eigenvalue problem is ∇2φ + λφ = 0 in D, (11.8.3) φ = 0 on B. Let φmn be the eigenfunctions and λmn be the corresponding eigenvalues. We then expand G and δ in terms of the eigenfunctions φmn. Consequently, we write 424 11 Green’s Functions and Boundary-Value Problems G (ξ,η; x, y) =  m  n amn (x, y) φmn (ξ,η), (11.8.4) δ (ξ − x, η − y) =  m  n bmn (x, y) φmn (ξ,η), (11.8.5) where bmn = 1 φmn 2  D δ (ξ − x, η − y) φmn (ξ,η) dξ dη = φmn (x, y) φmn 2 (11.8.6) in which φmn 2 =  D φ 2 mndξ dη. Now substituting equations (11.8.4) and (11.8.5) into equation (11.8.2) and using the relation from equation (11.8.3) that ∇2φmn + λmnφmn = 0, we obtain −  m  n λmnamn (x, y) φmn (ξ,η) =  m  n φmn (x, y) φmn (ξ,η) φmn 2 . Hence, amn (x, y) = − φmn (x, y) λmn φmn 2 , (11.8.7) and the Green’s function is therefore given by G (ξ,η; x, y) = −  m  n φmn (x, y) φmn (ξ,η) λmn φmn 2 . (11.8.8) Example 11.8.1. As a particular example, consider the Dirichlet problem in a rectangular domain ∇2u = h in D {0 < x < a, 0 <y<b} ,="" u="0" on="" b.="" the="" eigenfunctions="" can="" be="" obtained="" explicitly="" by="" method="" of="" separation="" variables.="" we="" assume="" a="" nontrivial="" solution="" in="" form="" (ξ,η)="X" (ξ)="" y="" (η).="" substituting="" this="" following="" system="" 11.9="" higher-dimensional="" problems="" 425="" ∇2u="" +="" λu="0" d,="" b,="" yields,="" with="" α="" 2="" as="" constant,="" x′′="" 2x="0," ′′="" ="" λ="" −="" homogeneous="" boundary="" conditions="" x="" (0)="X" (a)="0" and="" (b)="0," functions="" are="" found="" to="" xm="" sin="" ="" mπξ="" ="" yn="" (η)="Bn" 4nπη="" b="" 5="" .="" then="" have="" λmn="π" m2="" n="" thus,="" obtain="" φmn="" knowing="" φmn,="" compute="" φmn="" 0="" ="" sin2="" dξ="" dη="" ab="" 4="" from="" equation="" (11.8.8)="" green’s="" function="" g="" (ξ,η;="" x,="" y)="−" 4ab="" π="" ∞="" m="1" mπx="" nπy="" 4="" nπη="" (m2b="" n2a="" 2)="" easily="" extended="" for="" applications="" three="" more="" dimensions.="" since="" most="" encountered="" physical="" sciences="" dimensions,="" will="" illustrate="" some="" examples="" suitable="" practical="" application.="" first="" extend="" our="" definition="" dirichlet="" problem="" involving="" laplace="" operator="" is="" that="" satisfies="" 426="" 11="" boundary-value="" ∇2g="δ" (x="" ξ,y="" η,="" z="" ζ)="" r,="" (11.9.1)="" s.="" (11.9.2)="" (x,="" y,="" z;="" ξ,="" (ξ,="" ζ;="" z).="" (11.9.3)="" (c)="" limε→0="" ="" sε="" ∂g="" ∂n="" ds="1," (11.9.4)="" where="" outward="" unit="" normal="" surface="" :="" ξ)="" (y="" η)="" (z="" proceeding="" two-dimensional="" case,="" (11.9.5)="" s,="" z)="" r="" h="" dr="" s="" f="" gn="" ds.="" (11.9.6)="" again="" let="" z),="" ∇2f="δ" ∇2="" example="" 11.9.1.="" consider="" spherical="" domain="" radius="" a.="" must="" except="" at="" source="" point.="" τ="" 2ρ2="" a2="" 2τ="" ρ="" cos="" γ="" #="" 1="" (11.9.10)="" angle="" between="" ′="" now="" differentiating="" g,="" ∂τ="" =="" 4πa="" (a="" 2aρ="" γ)="" 428="" (ρ,="" θ,="" φ)="a" 4π="" 2π="" (α,="" ψ)="" dα="" dψ="" 3="" (11.9.11)="" θ="" (ψ="" φ).="" integral="" called="" three-dimensional="" poisson="" formula.="" exterior="" radially="" inward="" towards="" origin,="" simply="" replacing="" (11.9.11).="" 11.9.2.="" an="" helmholtz="" threedimensional="" radiation="" κ="" 2u="0," (11.9.12)="" limr→∞="" (ur="" iκu)="0," i="√" −1;="" limit="" condition="" condition,="" field="" point="" distance.="" satisfy="" 2g="δ" (ξ="" η="" ζ="" (11.9.13)="" dependent="" only="" write="" grr="" gr=""> 0. Note that the source point is taken as the origin. If we write the above equation in the form (G r) rr + κ 2 (G r) = 0 for r > 0 (11.9.14) then the solution can easily be seen to be G r = Aeiκr + Be−iκr , or, equivalently, G = A e iκr r + B e −iκr r . (11.9.15) In order for G to satisfy the radiation condition limr→∞ r (Gr + iκG)=0, the constant A = 0, and G then takes the form G = B e −iκr r . 11.9 Higher-Dimensional Problems 429 To determine B, we have limε→0  Sε ∂G ∂n dS = − limε→0  Sε B e −iκr r  1 r + iκ dS = 1 from which we obtain B = −1/4π, and consequently, G = − e −iκr 4πr . (11.9.16) Note that this reduces to (1/4πr) when κ = 0. Example 11.9.3. Show that the solution of the Poisson equation −∇2u = f (x, y, z), (11.9.17) is u (x, y, z) =  G (r) f (ξ, η, ζ) dξ dη dζ, (11.9.18) where the Green’s function G (r) is G (r) = 1 4πr = 1 4π ( (x − ξ) 2 + (y − η) 2 + (z − ζ) 2 )− 1 2 . (11.9.19) The Green’s function G satisfies the equation −∇2G = δ (x − ξ) δ (y − η) δ (z − ζ). (11.9.20) It is noted that everywhere except at (x, y, z)=(ξ, η, ζ), equation (11.9.20) is a homogeneous equation that can be solved by the method of separation of variables. However, at the point (ξ, η, ζ) this equation is no longer homogeneous. Usually, this point (ξ, η, ζ) represents a source point or a source point singularity in electrostatics or fluid mechanics. In order to solve (11.9.17), it is necessary to take into account the source point at (ξ, η, ζ). Without loss of generality, it is convenient to transform the frame of reference so that the source point is at the origin. This can be done by the transformation x1 = x − ξ, y1 = y − η, and z1 = z − ζ. Consequently, equation (11.9.20) becomes ∇2G = −δ (x1) δ (y1) δ (z1), (11.9.21) where ∇2 is the Laplacian in terms of x1, y1, and z1. Introducing the spherical polar coordinates x1 = r sin θ cos φ, y1 = r sin θ sin φ, z1 = r cos θ, equation (11.9.21) reduces to the form 430 11 Green’s Functions and Boundary-Value Problems ∇2G = − δ (r) 4πr2 , (11.9.22) where ∇2G ≡ 1 r 2 ∂ ∂r  r 2 ∂G ∂r  + 1 r 2 sin θ ∂ ∂θ  sin θ ∂G ∂θ  + 1 r 2 sin2 θ ∂ 2G ∂φ2 .(11.9.23) Since the right hand side of (11.9.22) is a function of r alone, and hence, G must be a function of r alone, we write (11.9.22) with (11.9.23) as 1 r 2 ∂ ∂r  r 2 ∂G ∂r  = − δ (r) 4πr2 . (11.9.24) We assume that G tends to zero as r → ∞. The solution of the corresponding homogeneous equation of (11.9.24) is G (r) = a r + b, (11.9.25) where a and b are constants of integration. Since G → 0 as r → ∞, b = 0 and we set a = 1 4π . Consequently, the solution for G is G (r) = 1 4πr . (11.9.26) This solution can be interpreted as the potential produced by a point charge at the point (ξ, η, ζ). Finally, the solution of (11.9.17) is then given by u (x, y, z) = −  ∞ −∞ G (r) f (ξ, η, ζ) dξ dη dζ = 1 4π  ∞ −∞ " (x − ξ) 2 + (y − η) 2 + (z − ζ) 2 #− 1 2 ×f (ξ, η, ζ) dξ dη dζ. (11.9.27) Physically, this solution of the Poisson equation represents the potential u (x, y, z) produced by a charge distribution of volume density f (x, y, z). 11.10 Neumann Problem We have noted in the chapter on boundary-value problems that the Neumann problem requires more attention than Dirichlet’s problem because an additional condition is necessary for the existence of a solution of the Neumann problem. We now consider the Neumann problem 11.10 Neumann Problem 431 ∇2u + κ 2u = h in R, ∂u ∂n = 0 on S. By the divergence theorem of calculus, we have  R ∇2u dR =  S ∂u ∂n dS. Thus, if we integrate the Helmholtz equation and use the preceding result, we obtain κ 2  R u dR =  R h dR. In the case of Poisson’s equation where κ = 0, this relation is satisfied only when  R h dR = 0. If we consider a heat conduction problem, this condition may be interpreted as the requirement that the net heat generation be zero. This is physically reasonable since the boundary is insulated in such a way that the net flux across it is zero. If we define Green’s function G, in this case, by ∇2G + κ 2G = δ (ξ − x, η − y, ζ − z) in R, ∂G ∂n = 0 on S. Then we must have κ 2  R G dR = 1 which cannot be satisfied for κ = 0. But, we know from a physical point of view that a solution exists if  R h dR = 0. Hence, we will modify the definition of Green’s function so that ∂G ∂n = C on S, where C is a constant. Integrating ∇2G = δ over R, we obtain C  S dS = 1. 432 11 Green’s Functions and Boundary-Value Problems It is not difficult to show that G remains symmetric if  S G dS = 0. Thus, under this condition, if we take C to be reciprocal of the surface area, the solution of the Neumann problem for Poisson’s equation is u (x, y, z) = C ∗ +  R G (x, y, z; ξ, η, ζ) h (ξ, η, ζ) dξ dη dζ, where C ∗ is a constant. We should remark here that the method of Green’s functions provides the solution in integral form. This is made possible by replacing a problem involving nonhomogeneous boundary conditions with a problem of finding Green’s function G with homogeneous boundary conditions. Regardless of method employed, the Green’s function of a problem with nonhomogeneous equation and homogeneous boundary conditions is the same as the Green’s function of a problem with homogeneous equation and nonhomogeneous boundary conditions, since one problem can be transferred to the other without difficulty. To illustrate, we consider the problem Lu = f in R, u = 0 on ∂R, where ∂R denotes the boundary of R. If we let v = w − u, where w satisfies Lw = f in R, then the problem becomes Lv = 0 in R, v = w on ∂R. Conversely, if we consider the problem Lu = 0 in R, u = g on ∂R, we can easily transform this problem into Lv = Lw ≡ w ∗ in R, v = 0 on ∂R, by putting v = w − u and finding w that satisfies w = g on ∂R. In fact, if we have Lu = f in R, u = g on ∂R, we can transform this problem into either one of the above problems. 11.11 Exercises 433 11.11 Exercises 1. If L denotes the partial differential operator Lu = Auxx + Buxy + Cuyy + Dux + Euy + F u, and if M denotes the adjoint operator Mv = (Av)xx + (Bv)xy + (Cv)yy − (Dv)x − (Ev)y + Fv, show that  R (vLu − uMv) dx dy =  ∂R [U cos (n, x) + V cos (n, y)] ds, where U = Avux − u (Av)x − u (Bv)y + Duv, V = Bvux + Cvuy − u (Cv)y + Euv, and ∂R is the boundary of a region R. 2. Prove that the Green’s function for a problem, if it exists, is unique. 3. Determine the Green’s function for the exterior Dirichlet problem for a unit circle ∇2u = 0 in r > 1, u = f on r = 1. 4. Prove that for x = x (ξ,η) and y = y (ξ,η) δ (x − x0) δ (y − y0) = 1 |J| δ (ξ − ξ0) δ (η − η0), where J is the Jacobian and (x0, y0) corresponds to (ξ0, η0). Hence, show that for polar coordinates δ (x − x0) δ (y − y0) = 1 r δ (r − r0) δ (θ − θ0). 5. Determine, for an infinite wedge, the Green’s function that satisfies ∇2G + κ 2G = 1 r δ (r − r0, θ − θ0), G = 0, θ = 0, and θ = α. 6. Determine, for the Poisson’s equation, the Green’s function which vanishes on the boundary of a semicircular domain of radius R. 434 11 Green’s Functions and Boundary-Value Problems 7. Find the solution of the Dirichlet problem ∇2u = 0, 0 < x < a, 0 < y < b, u (0, y) = u (a, y) = u (x, b)=0, u (x, 0) = f (x). 8. Determine the solution of Dirichlet’s problem ∇2u = f (r, θ) in D, u = 0, on ∂D, where ∂D is the boundary of a circle D of radius R. 9. Determine the Green’s function for the semi-infinite region ζ > 0 for ∇2G + κ 2G = δ (ξ − x, η − y, ζ − z), G = 0, on ζ = 0. 10. Determine the Green’s function for the semi-infinite region ζ > 0 for ∇2G + κ 2G = δ (ξ − x, η − y, ζ − z), ∂G ∂n = 0, on ζ = 0. 11. Find the Green’s function in the quarter plane ξ > 0, η > 0 which satisfies ∇2G = δ (ξ − x, η − y), G = 0, on ξ = 0 and η = 0. 12. Find the Green’s function in the quarter plane ξ > 0, η > 0 which satisfies ∇2G = δ (ξ − x, η − y), Gξ (0, η)=0, G (ξ, 0) = 0. 13. Find the Green’s function in the half plane 0 <x< ∞,="" −∞="" <y<="" ∞="" for="" the="" problem="" ∇2u="f" in="" r,="" u="0," on="" x="0." 14.="" determine="" green’s="" function="" that="" satisfies="" ∇2g="δ" (x="" −="" ξ,y="" η)="" d="" :="" 0="" <="" a,="" g="0," ∂d="" y="0," is="" bounded="" at="" infinity.="" 11.11="" exercises="" 435="" 15.="" find="" r="" δ="" (r="" ρ,="" θ="" β),="" <θ<="" π="" 3="" ,="" <r<="" 1,="" and="" ∂g="" ∂n="0," 16.="" solve="" boundary-value="" 1="" ∂r="" ="" ∂u="" ="" +="" ∂="" 2u="" ∂z2="" κ="" ≥="" 0,="" z=""> 0, ∂u ∂z = ⎧ ⎨ ⎩ 0, r>a, z C, r < a, z = = 0, 0, C = constant. 17. Obtain the solution of the Laplace equation ∇2u = 0, 0 <r< ∞,="" 0="" <θ<="" 2π,="" u="" (r,="" 0+)="u" 2π−)="0." 18.="" determine="" the="" green’s="" function="" for="" equation="" ∇2u="" −="" κ="" 2u="0," vanishing="" on="" all="" sides="" of="" rectangle="" ≤="" x="" a,="" y="" b.="" 19.="" helmholtz="" +="" <="" −∞="" <y<="" and="" 20.="" solve="" exterior="" dirichlet="" problem="" in="" r=""> 1, u (1, θ, φ) = f (θ, φ). 21. By the method of images, determine the potential due to a point charge q near a conducting sphere of radius R with potential V . 22. By the method of images, show that the potential due to a conducting sphere of radius R in a uniform electric field E0 is given by U = −E0  r − R2 r 2  cos θ, where r, θ are polar coordinates with origin at the center of the sphere. 436 11 Green’s Functions and Boundary-Value Problems 23. Determine the potential in a cylinder of radius R and length l. The potential on the ends is zero, while the potential on the cylindrical surface is prescribed to be f (θ, z). 24. Consider the fundamental solution of the Fokker–Planck equation defined by ∂ ∂t − ∂ ∂x  ∂ ∂x + x  G (x, x′ ;t, t′ ) = δ (x − x ′ ) δ (t − t ′ ). Using the transformation of variables employed in Example 7.8.4, show that the above equation becomes ∂ ∂τ − ∂ ∂ξ2 G (ξ, ξ′ ; τ, τ ′ ) = δ (ξ − ξ ′ ) δ (τ − τ ′ ). Show that (a) the fundamental solution of the Fokker–Planck equation (see Reif (1965)) is G (x, x′ ;t, t′ ) = [2π {1 − exp [−2 (t − t ′ )]}] − 1 2 × exp 1 − 1 2 [x − x ′ exp {− (t − t ′ )}] 2 1 − exp {−2 (t − t ′)} 3 , (b) limt→∞ G (x, x′ ;t, t′ ) = 1 √ 2π exp  − 1 2 x 2  , (c) limt→∞ u (x, t) = 1 √ 2π exp  − 1 2 x 2   ∞ −∞ f (x ′ ) dx′ , where u (x, 0) = f (x). Give an interpretation of this asymptotic solution u (x, t) as t → ∞. 25. (a) Use the transformation u = ve−t to show that the telegraph equation utt − c 2uxx + 2ut = 0, can be reduced to the form vtt − c 2 vxx + v = 0. (b) Show that the fundamental solution of the transformed telegraph equation is given by G (x − x ′ , t − t ′ ) = 1 2 I0 % (t − t ′) 2 − (x − x ′) 2 ×H [(t − t ′ ) − (x − x ′ )] H [(t − t ′ )+(x − x ′ )] . 11.11 Exercises 437 (c) If the initial data for the telegraph equation are u (x, 0) = f (x), ut (x, 0) = g (x), show that the solution of the telegraph equation is given by u (x, t) =  1 2  x+t x−t g (ξ) I0 % t 2 − (x − ξ) 2 dξ + 1 2 ∂ ∂t  x+t x−t f (ξ) I0 % t 2 − (x − ξ) 2 dξ0 e −t , which is, by evaluating the second term, = e −t  1 2 [f (x − t) + f (x + t)] + 1 2  x+t x−t g (ξ) I0 % t 2 − (x − ξ) 2 dξ + t 2  x+t x−t f (ξ) " t 2 − (x − ξ) 2 #− 1 2 I1 % t 2 − (x − ξ) 2 dξ0 . 12 Integral Transform Methods with Applications “The theory of Fourier series and integrals has always had major difficulties and necessitated a large mathematical apparatus in dealing with questions of convergence. It engendered the development of methods of summation, although these did not lead to a completely satisfactory solution of the problem.... For the Fourier transform, the introduction of distribution (hence the space S) is inevitable either in an explicit or hidden form.... As a result one may obtain all that is desired from the point of view of the continuity and inversion of the Fourier transform.” L. Schwartz “In every mathematical investigation, the question will arise whether we can apply our mathematical results to the real world.” V. I. Arnold 12.1 Introduction The linear superposition principle is one of the most effective and elegant methods to represent solutions of partial differential equations in terms of eigenfunctions or Green’s functions. More precisely, the eigenfunction expansion method expresses the solution as an infinite series, whereas the integral solution can be obtained by integral superposition or by using Green’s functions with initial and boundary conditions. The latter offers several advantages over eigenfunction expansion. First, an integral representation provides a direct way of describing the general analytical structure of a solution that may be obscured by an infinite series representation. Second, from a practical point of view, the evaluation of a solution from an integral representation may prove simpler than finding the sum of an infinite series, 440 12 Integral Transform Methods with Applications particularly near rapidly-varying features of a function, where the convergence of an eigenfunction expansion is expected to be slow. Third, in view of the Gibbs phenomenon discussed in Chapter 6, the integral representation seems to be less stringent requirements on the functions that describe the initial conditions or the values of a solution are required to assume on a given boundary than expansions based on eigenfunctions. Integral transform methods are found to be very useful for finding solutions of initial and/or boundary-value problems governed by partial differential equations for the following reason. The differential equations can readily be replaced by algebraic equations that are inverted by the inverse transform so that the solution of the differential equations can then be obtained in terms of the original variables. The aim of this chapter is to provide an introduction to the use of integral transform methods for students of applied mathematics, physics, and engineering. Since our major interest is the application of integral transforms, no attempt will be made to discuss the basic results and theorems relating to transforms in their general forms. The present treatment is restricted to classes of functions which usually occur in physical and engineering applications. 12.2 Fourier Transforms We first give a formal definition of the Fourier transform by using the complex Fourier integral formula (6.13.10). Definition 12.2.1. If u (x, t) is a continuous, piecewise smooth, and absolutely integrable function, then the Fourier transform of u (x, t) with respect to x ∈ R is denoted by U (k, t) and is defined by F {u (x, t)} = U (k, t) = 1 √ 2π  ∞ −∞ e −ikx u (x, t) dx, (12.2.1) where k is called the Fourier transform variable and exp (−ikx) is called the kernel of the transform. Then, for all x ∈ R, the inverse Fourier transform of U (k, t) is defined by F −1 {U (k, t)} = u (x, t) = 1 √ 2π  ∞ −∞ e ikx U (k, t) dk. (12.2.2) We may note that the factor (1/2π) in the Fourier integral formula (6.13.9) has been split and placed in front of the integrals (12.2.1) and (12.2.2). Often the factor (1/2π) can be placed in only one of the relations (12.2.1) and (12.2.2). It is not uncommon to adopt the kernel exp (ikx) in (12.2.1) instead of exp (−ikx), and as a consequence, exp (−ikx) would be replaced by exp (ikx) in (12.2.2). 12.2 Fourier Transforms 441 Example 12.2.1. Show that (a) F & exp  −ax2 ' = 1 √ 2a exp  − k 2 4a  , a > 0, (12.2.3) (b) F {exp (−a |x|)} = 2 2 π a (a 2 + k 2) , a > 0, (12.2.4) (c) F & χ[−a,a] (x) ' = 2 2 π  sin ak k  , (12.2.5) where χ[−a,a] (x) = H (a − |x|) = ⎧ ⎨ ⎩ 1, |x| < a 0, |x| > a ⎫ ⎬ ⎭ . (12.2.6) Proof. We have, by definition (12.2.1), F & exp  −ax2 ' = 1 √ 2π  ∞ −∞ e −ikx−ax2 dx = 1 √ 2π  ∞ −∞ exp 1 −a  x + ik 2a 2 − k 2 4a 3 dx = 1 √ 2π exp  − k 2 4a   ∞ −∞ e −ay2 dy = 1 √ 2a exp  − k 2 4a  , in which the change of variable y =  x + ik 2a is used. The above result is correct, and the change of variable can be justified by methods of complex analysis because (ik/2a) is complex. If a = 1 2 , then F  exp  − 1 2 x 2 0 = exp  − 1 2 k 2  . (12.2.7) This shows that F {f (x)} = f (k). Such a function is said to be selfreciprocal under the Fourier transformation. The graphs of f (x) = e −ax2 and F (k) = F {f (x)} are shown in Figure 12.2.1 for a = 1. To prove (b), we write F {exp (−a |x|)} = 1 √ 2π  ∞ −∞ exp (−a |x| − ikx) dx = 1 √ 2π  0 −∞ exp {(a − ik) x} dx +  ∞ 0 exp {− (a + ik) x} dx = 1 √ 2π 1 a − ik + 1 a + ik = 2 2 π a a 2 + k 2 . 442 12 Integral Transform Methods with Applications Figure 12.2.1 Graphs of f (x) = exp(−ax2 ) and F (k). It is noted that f (x) = exp (−a |x|) decreases rapidly at infinity, and it is not differentiable at x = 0. The graphs of f (x) and its Fourier transform F (k) are shown in Figure 12.2.2. To prove (c), we have Fa (k) = F & χ[−a,a] (x) ' = 1 √ 2π  ∞ −∞ e −ikxχ[−a,a] (x) dx = 1 √ 2π  a −a e −ikxdx = 2 2 π  sin ak a  . The graphs of χ[−a,a] (x) and Fa (k) are shown in Figure 12.2.3 with a = 1. Analogous to the Fourier cosine and sine series, there are Fourier cosine and sine integral transforms for odd and even functions respectively. Definition 12.2.2. Let f (x) be defined for 0 ≤ x < ∞, and extended as an even function in (−∞,∞) satisfying the conditions of Fourier InteFigure 12.2.2 Graphs of f (x) = exp (−a |x|) and F (k). 12.2 Fourier Transforms 443 Figure 12.2.3 Graphs of χ[−a,a] (x) and Fa (k). gral formula (6.13.9). Then, at the points of continuity, the Fourier cosine transform of f (x) and its inverse transform are defined by Fc {f (x)} = Fc (k) = 2 2 π  ∞ 0 cos kx f (x) dx, (12.2.8) F −1 c {Fc (k)} = f (x) = 2 2 π  ∞ 0 cos kx Fc (k) dk, (12.2.9) where Fc is the Fourier cosine transformation and F −1 c is its inverse transformation respectively. Definition 12.2.3. Similarly, the Fourier sine integral formula (6.13.3) leads to the Fourier sine transform and its inverse defined by Fs {f (x)} = Fs (k) = 2 2 π  ∞ 0 sin kx f (x) dx, (12.2.10) F −1 s {Fs (k)} = f (x) = 2 2 π  ∞ 0 sin kx Fs (k) dk, (12.2.11) where Fs is called the Fourier sine transformation and F −1 s is its inverse. Example 12.2.2. Show that (a) Fc & e −ax' = 2 2 π a (a 2 + k 2) , a > 0, (12.2.12) (b) Fs & e −ax' = 2 2 π k (a 2 + k 2) , a > 0, (12.2.13) (c) F −1 s  1 k e −sk0 = 2 2 π tan−1 4x s 5 . (12.2.14) 444 12 Integral Transform Methods with Applications We have, by definition, Fc & e −ax' = 2 2 π  ∞ 0 e −ax cos kx dk, = 1 2 2 2 π  ∞ 0 " e −(a−ik)x + e −(a+ik)x # dx, = 1 2 2 2 π 1 a − ik + 1 a + ik = 2 2 π a (a 2 + k 2) . The proof of (b) is similar and is left to the reader as an exercise. To prove (c), we use the standard definite integral 2 π 2 F −1 s & e −sk' =  ∞ 0 e −sk sin kx dk = x s 2 + x 2 . Integrating both sides with respect to s from s to ∞ gives  ∞ 0 e −sk k sin kx dk =  ∞ s x ds x 2 + s 2 = " tan−1 s x # = (π/2) − tan−1 4 s x 5 . Consequently, F −1 s  1 k e −sk0 = 2 2 π  ∞ 0 1 k e −sk sin kx dk = 2 2 π tan−1 4x s 5 . 12.3 Properties of Fourier Transforms Theorem 12.3.1. (Linearity). The Fourier transformation F is linear. Proof. We have F [f (x)] = 1 √ 2π  ∞ −∞ e −ikxf (x) dx. Then, for any constants a and b, F [af (x) + bg (x)] = 1 √ 2π  ∞ −∞ [af (x) + bg (x)] e −ikxdx, = a √ 2π  ∞ −∞ f (x) e −ikxdx + b √ 2π  ∞ −∞ g (x) e −ikxdx, = a F [f (x)] + b F [g (x)] . Theorem 12.3.2. (Shifting). Let F [f (x)] be a Fourier transform of f (x). Then F [f (x − c)] = e −ixcF [f (x)] , where c is a real constant. 12.3 Properties of Fourier Transforms 445 Proof. From the definition, we have, for c > 0, F [f (x − c)] = 1 √ 2π  ∞ −∞ f (x − c) e −ikxdx, = 1 √ 2π  ∞ −∞ f (ξ) e −ik(ξ+c) dξ, where ξ = x − c = e −ikcF [f (x)] . Theorem 12.3.3. (Scaling). If F is the Fourier transform of f, then F [f (cx)] = (1/ |c|) F (k/c), where c is a real nonzero constant. Proof. For c = 0, F [f (cx)] = 1 √ 2π  ∞ −∞ f (cx) e −ikxdx. If we let ξ = cx, then F [f (cx)] = 1 |c| 1 √ 2π  ∞ −∞ f (ξ) e −i(k/c)ξ dξ = (1/ |c|) F (k/c). Theorem 12.3.4. (Differentiation). Let f be continuous and piecewise smooth in (−∞,∞). Let f (x) approach zero as |x|→∞. If f and f ′ are absolutely integrable, then F [f ′ (x)] = ikF [f (x)] = ikF (k). Proof. F [f ′ (x)] = 1 √ 2π  ∞ −∞ f ′ (x) e −ikxdx, = 1 √ 2π f (x) e −ikx  ∞ −∞ + ik √ 2π  ∞ −∞ f (x) e −ikxdx , = ikF [f (x)] = ikF (k). This result can be easily extended. If f and its first (n − 1) derivatives are continuous, and if its nth derivative is piecewise continuous, then F " f (n) (x) # = (ik) n F [f (x)] = (ik) n F (k), n = 0, 1, 2,... (12.3.1) provided f and its derivatives are absolutely integrable. In addition, we assume that f and its first (n − 1) derivatives tend to zero as |x| tends to infinity. 446 12 Integral Transform Methods with Applications If u (x, t) → 0 as |x|→∞, then F  ∂u ∂x0 = 1 √ 2π  ∞ −∞ e −ikx  ∂u ∂x dx, which is, integrating by parts, = 1 √ 2π e −ikxu (x, t) !∞ −∞ + ik √ 2π  ∞ −∞ e −ikxu (x, t) dx, = ik F {u (x, t)} = ik U (k, t). (12.3.2) Similarly, if u (x, t) is continuously n times differentiable, and ∂mm ∂xm → 0 as |x|→∞ for m = 1, 2, 3,..., (n − 1) then F  ∂ nu ∂xn 0 = (ik) n F {u (x, t)} = (ik) n U (k, t). (12.3.3) It also follows from the definition (12.2.1) that F  ∂u ∂t 0 = dU dt , F  ∂ 2u ∂t2 0 = d 2U dt2 , ..., F  ∂ nu ∂tn 0 = d nU dtn . (12.3.4) The definition of the Fourier transform (12.2.1) shows that a sufficient condition for u (x, t) to have a Fourier transform is that u (x, t) is absolutely integrable in −∞ <x< ∞.="" this="" existence="" condition="" is="" too="" strong="" for="" many="" practical="" applications.="" simple="" functions,="" such="" as="" a="" constant="" function,="" sin="" ωx,="" and="" x="" nh="" (x),="" do="" not="" have="" fourier="" transforms="" even="" though="" they="" occur="" frequently="" in="" the="" above="" definition="" of="" transform="" has="" been="" extended="" more="" general="" class="" functions="" to="" include="" other="" functions.="" we="" simply="" state="" fact="" that="" there="" sense,="" useful="" applications,="" which="" stated="" others="" transforms.="" following="" are="" examples="" their="" (see="" lighthill,="" 1964):="" f="" {h="" (a="" −="" |x|)}="2" 2="" π="" ="" ak="" k="" ="" ,="" (12.3.5)="" where="" h="" (x)="" heaviside="" unit="" step="" {δ="" (x="" a)}="1" √="" 2π="" exp="" (−iak),="" (12.3.6)="" δ="" a)="" dirac="" delta="" 1="" iπk="" +="" (k)="" (−iak).="" (12.3.7)="" 12.3="" properties="" 447="" example="" 12.3.1.="" find="" solution="" dirichlet="" problem="" half-plane="" y=""> 0 uxx + uyy = 0, −∞ <x< ∞,="" y=""> 0, u (x, 0) = f (x), −∞ <x< ∞,="" u="" and="" ux="" vanish="" as="" |x|→∞,="" is="" bounded="" y="" →="" ∞.="" let="" (k,="" y)="" be="" the="" fourier="" transform="" of="" (x,="" with="" respect="" to="" x.="" then="" √="" 2π="" ="" ∞="" −∞="" e="" −ikxdx.="" application="" x="" gives="" uyy="" −="" k="" 2u="0," (12.3.8)="" 0)="F" (k)="" 0="" (12.3.9)="" solution="" this="" transformed="" system="" −|k|y="" .="" inverse="" in="" form="" 1="" f="" (ξ)="" −ikξdξ="" ikxdk,="1" dξ="" −k[i(ξ−x)]−|k|y="" dk.="" it="" follows="" from="" proof="" example="" 12.2.1="" (b)="" that="" dk="2y" (ξ="" x)="" 2="" +="" hence,="" dirichlet="" problem="" half-plane=""> 0 is u (x, y) = y π  ∞ −∞ f (ξ) (ξ − x) 2 + y 2 dξ. From this solution, we can readily deduce a solution of the Neumann problem in the half-plane y > 0. Example 12.3.2. Find the solution of Neumann’s problem in the half-plane y > 0 uxx + uyy = 0, −∞ <x< ∞,="" y=""> 0, uy (x, 0) = g (x), −∞ <x< ∞,="" u="" is="" bounded="" as="" y="" →="" and="" ux="" vanish="" |x|→∞.="" 448="" 12="" integral="" transform="" methods="" with="" applications="" let="" v="" (x,="" y)="uy" y).="" then="" η)="" dη="" the="" neumann="" problem="" becomes="" ∂="" 2v="" ∂x2="" +="" ∂y2="∂" 2uy="" ∂y="" (uxx="" uyy)="0." 0)="uy" (x).="" this="" dirichlet="" for="" y),="" its="" solution="" given="" by="" π="" ="" ∞="" −∞="" g="" (ξ)="" dξ="" (ξ="" −="" x)="" 2="" .="" thus,="" we="" have="" η="" dη,="1" 2π="" 2η="" ,="1" log="" "="" (x="" ξ)="" #="" dξ,="" where="" an="" arbitrary="" constant="" can="" be="" added="" to="" solution.="" in="" other="" words,="" of="" any="" neumann’s="" uniquely="" determined="" up="" constant.="" 12.4="" convolution="" theorem="" fourier="" function="" (f="" ∗="" g)="" (x)="1" √="" f="" (12.4.1)="" called="" functions="" over="" interval="" (−∞,∞).="" 12.4.1.="" (convolution="" theorem).="" if="" (k)="" are="" transforms="" respectively,="" product="" (k).="" that="" is,="" {f="" (x)}="F" (12.4.2)="" or,="" equivalently,="" 449="" −1="" (k)}="f" (12.4.3)="" more="" explicitly,="" 1="" e="" ikxdk="(f" dξ.="" (12.4.4)="" proof.="" definition,="" [(f="" (x)]="1" −ikxdx="" −ikξdξ="" −ik(x−ξ)="" dx.="" change="" variable="" ξ,="" (η)="" −ikηdη="F" satisfies="" following="" properties:="" 1.="" (commutative).="" 2.="" (g="" h)="(f" h="" (associative).="" 3.="" (ag="" bh)="a" b="" h),="" (distributive),="" a="" constants.="" 12.4.2.="" (parseval’s="" formula).="" |f="" (x)|="" dx="" (k)|="" dk.="" (12.4.5)="" formula="" gives="" ikξdk="" which="" putting="" ξ="0," (−x)="" (12.4.6)="" (x),="" −ikxdx,="1" e−ikxdx="F" (k),="" 450="" bar="" denotes="" complex="" conjugate.="" result="" dk,="" terms="" notation="" norm,="" f="F" physical="" systems,="" quantity="" |f|="" measure="" energy,="" represents="" power="" spectrum="" example="" 12.4.3.="" obtain="" initial-value="" heat="" conduction="" infinite="" rod="" ut="κ" uxx,="" <x<="" t=""> 0, (12.4.7) u (x, 0) = f (x), −∞ <x< ∞,="" (12.4.8)="" u="" (x,="" t)="" →="" 0,="" as="" |x|→∞,="" where="" represents="" the="" temperature="" distribution="" and="" is="" bounded,="" κ="" a="" constant="" of="" diffusivity.="" fourier="" transform="" with="" respect="" to="" x="" defined="" by="" (k,="" √="" 2π="" ="" ∞="" −∞="" e="" −ikxu="" dx.="" in="" view="" this="" transformation,="" equations="" (12.4.7)–(12.4.8)="" become="" ut="" +="" k2="" (12.4.9)="" 0)="F" (k).="" (12.4.10)="" solution="" transformed="" system="" (k)="" −k="" 2κ="" t="" .="" inverse="" transformation="" gives="" f="" 2κte="" ikxdk="" which="" is,="" convolution="" theorem="" 12.4.1,="" 12.4="" 451="1" (ξ)="" g="" (x="" −="" ξ)="" dξ,="" (x)="" 2κt="" has="" form="" −1="" (="" −κk2="" )="1" 2κt+ikxdk="1" −x="" 2="" 4κt="" consequently,="" final="" 4πκt="" exp="" 1="" 3="" (12.4.11)="" ξ,="" (12.4.12)="" ,="" (12.4.13)="" called="" green’s="" function="" (or="" fundamental="" solution)="" diffusion="" equation.="" means="" that="" at="" any="" point="" time="" represented="" definite="" integral="" made="" up="" contribution="" due="" initial="" source="" t).="" response="" along="" rod="" an="" unit="" impulse="" heat="" physical="" meaning="" decomposed="" into="" spectrum="" impulses="" magnitude="" each="" resulting="" thus,="" integrated="" find="" (12.4.11).="" so-called="" principle="" superposition.="" using="" change="" variable="" ξ="" κt="ζ," dζ="dξ" we="" obtain="" π="" 4="" 2√="" ζ5="" −ζ="" dζ.="" (12.4.14)="" or="" poisson="" representation="" distribution.="" convergent="" for=""> 0, and integrals obtained from it by differentiation under the integral sign with respect to x and t are uniformly convergent in the neighborhood of 452 12 Integral Transform Methods with Applications the point (x, t). Hence, u (x, t) and its derivatives of all orders exist for t > 0. In the limit t → 0+, solution (12.4.12) becomes formally u (x, 0) = f (x) =  ∞ −∞ f (ξ) limt→0+ G (x − ξ, t) dξ. This limit represents the Dirac delta function δ (x − ξ) = limt→0+ 1 √ 4πκt e −(x−ξ) 2/4κt . (12.4.15) Consider a special case where f (x) = ⎧ ⎨ ⎩ 0, x< 0 a, x > 0 ⎫ ⎬ ⎭ = a H (x). Then, the solution (12.4.11) gives u (x, t) = a 2 √ πκt  ∞ 0 exp 1 − (x − ξ) 2 4κt 3 dξ. If we introduce a change of variable η = ξ − x 2 √ κt then the above solution becomes u (x, t) = a √ π  ∞ −x/2 √ κt e −η 2 dη = a √ π 1 0 −x/2 √ κt e −η 2 dη +  ∞ 0 e −η 2 dη3 = a √ π 1 x/2 √ κt 0 e −η 2 dη + √ π 2 3 = a 2 1 + erf  x 2 √ κt , where erf (x) is called the error function and is defined by erf (x) = 2 √ π  x 0 e −η 2 dη. (12.4.16) This is a widely used and tabulated function. 12.5 The Fourier Transforms of Step and Impulse Functions 453 Figure 12.5.1 The Heaviside unit step function. 12.5 The Fourier Transforms of Step and Impulse Functions In this section, we shall determine the Fourier transforms of the step function and the impulse function, functions which occur frequently in applied mathematics and mathematical physics. The Heaviside unit step function is defined by H (x − a) = ⎧ ⎨ ⎩ 0, x</x<></x<></x<></x<></x<></x<></r<></x<></y<b}></x<></x<></a,></r</a,></r<></r<r1,> 0, and ε is a small positive constant, as shown in Figure 12.5.2. This type of function appears in practical applications; for instance, a force of large magnitude may act over a very short period of time. The Fourier transform of the impulse function is 12.5 The Fourier Transforms of Step and Impulse Functions 455 F [p (x)] = 1 √ 2π  ∞ −∞ p (x) e −ikxdx = 1 √ 2π  a+ε a−ε h e−ikxdx = h √ 2π e −iak ik  e ikε − e −ikε = 2hε √ 2π e −iak  sin kε kε  . Now if we choose the value of h to be (1/2ε), then the impulse defined by I (ε) =  ∞ −∞ p (x) dx becomes I (ε) =  a+ε a−ε 1 2ε dx = 1 which is a constant independent of ε. In the limit as ε → 0, this particular function pε (x) with h = (1/2ε) satisfies limε→0 pε (x)=0, x = a, limε→0 I (ε)=1. Thus, we arrive at the result δ (x − a)=0, x = a,  ∞ −∞ δ (x − a) dx = 1. (12.5.4) This is the Dirac delta function which was defined earlier in Section 8.11. We now define the Fourier transform of δ (x) as the limit of the transform of pε (x). We then consider F [δ (x − a)] = limε→0 F [pε (x)] = limε→0 e −iak √ 2π  sin kε kε  = e −iak √ 2π (12.5.5) in which we note that, by L’Hospital’s rule, limε→0 (sin kε/kε) = 1. When a = 0, we obtain F [δ (x)] = 4 1/ √ 2π 5 . (12.5.6) 456 12 Integral Transform Methods with Applications Example 12.5.1. Slowing-down of Neutrons (see Sneddon (1951), p. 215). Consider the following physical problem ut = uxx + δ (x) δ (t), (12.5.7) u (x, 0) = δ (x), (12.5.8) lim |x|→∞ u (x, t)=0. (12.5.9) This is the problem of an infinite medium which slows neutrons, in which a source of neutrons is located. Here u (x, t) represents the number of neutrons per unit volume per unit time and δ (x) δ (t) represents the source function. Let U (k, t) be the Fourier transform of u (x, t). Then the Fourier transformation of equation (12.5.7) yields dU dt + k 2U = 1 √ 2π δ (t). The solution of this, after applying the condition U (k, 0) =  1/ √ 2π , is U (k, t) = 1 √ 2π e −k 2 t . Hence, the inverse Fourier transform gives the solution of the problem u (x, t) = 1 √ 2π  ∞ −∞ e −k 2 t+ikxdk, = 1 √ 4πt e −x 2/4t . 12.6 Fourier Sine and Cosine Transforms For semi-infinite regions, the Fourier sine and cosine transforms determined in Section 12.2 are particularly appropriate in solving boundary-value problems. Before we illustrate their applications, we must first prove the differentiation theorem. Theorem 12.6.1. Let f (x) and its first derivative vanish as x → ∞. If Fc (k) is the Fourier cosine transform, then Fc [f ′′ (x)] = −k 2Fc (k) − 2 2 π f ′ (0). (12.6.1) Proof. Fc [f ′′ (x)] = 2 2 π  ∞ 0 f ′′ (x) cos kx dx 12.6 Fourier Sine and Cosine Transforms 457 = 2 2 π [f ′ (x) cos kx] ∞ 0 + 2 2 π k  ∞ 0 f ′ (x) sin kx dx = − 2 2 π f ′ (0) + 2 2 π k [f (x) sin kx] ∞ 0 − 2 2 π k 2  ∞ 0 f (x) cos kx dx = − 2 2 π f ′ (0) − k 2Fc (k). In a similar manner, the Fourier cosine transforms of higher-order derivatives of f (x) can be obtained. Theorem 12.6.2. Let f (x) and its first derivative vanish as x → ∞. If Fs (k) is the Fourier sine transform, then Fs [f ′′ (x)] = 2 2 π k f (0) − k 2Fs (k). (12.6.2) The proof is left to the reader. Example 12.6.2. Find the temperature distribution in a semi-infinite rod for the following cases with zero initial temperature distribution: (a) The heat supplied at the end x = 0 at the rate g (t); (b) The end x = 0 is kept at a constant temperature T0. The problem here is to solve the heat conduction equation ut = κ uxx, x > 0, t> 0, u (x, 0) = 0, x > 0. (a) ux (0, t) = g (t) and (b) u (0, t) = T0, t ≥ 0. Here we assume that u (x, t) and ux (x, t) vanish as x → ∞. For case (a), let U (k, t) be the Fourier cosine transform of u (x, t). Then the transformation of the heat conduction equation yields Ut + κ k2 U = − 2 2 π g (t) κ. The solution of this equation with U (k, 0) = 0 is u (x, t) = 2 2 π  ∞ 0 U (k, t) cos kx dk = − 2κ π  t 0 g (τ ) dτ  ∞ 0 e −κk2 (t−τ) cos kx dk. 458 12 Integral Transform Methods with Applications The inner integral is given by (see Problem 6, Exercises 12.18)  ∞ 0 e −k 2κ(t−τ) cos kx dk = 1 2 2 π κ (t − τ ) exp − x 2 4κ (t − τ ) . The solution, therefore, is u (x, t) = − 2 κ π  t 0 g (τ ) √ t − τ e −x 2/4κ(t−τ) dτ. (12.6.3) For case (b), we apply the Fourier sine transform U (k, t) of u (x, t) to obtain the transformed equation Ut + κ k2U = 2 2 π k T0 κ. The solution of this equation with zero initial condition is U (k, t) = T0 2 2 π 4 1 − e −κtk2 5 k . Then the inverse Fourier sine transformation gives u (x, t) = 2T0 π  ∞ 0 sin kx k 4 1 − e −κtk2 5 dk. Making use of the integral  ∞ 0 e −a 2x 2  sin kx k  dk = π 2 erf 4 x 2a 5 , the solution is found to be u (x, t) = 2T0 π π 2 − π 2 erf  x 2 √ κt = T0 erfc  x 2 √ κt , (12.6.4) where erfc (x)=1 − erf (x) is the complementary error function defined by erfc (x) = 2 √ π  ∞ x e −α 2 dα. 12.7 Asymptotic Approximation of Integrals by Stationary Phase Method Although definite integrals represent exact solutions for many physical problems, the physical meaning of the solutions is often difficult to determine. In many cases the exact evaluation of the integrals is a formidable task. It is then necessary to resort to asymptotic methods. 12.7 Asymptotic Approximation of Integrals by Stationary Phase Method 459 We consider the typical integral solution u (x, t) =  b a F (k) e itθ(k) dk, (12.7.1) where F (k) is called the spectral function determined by the initial or boundary data in a<k<b, and="" θ="" (k),="" known="" as="" the="" phase="" function,="" is="" given="" by="" (k)="" ≡="" k="" x="" t="" −="" ω=""> 0. (12.7.2) We examine the asymptotic behavior of (12.7.1) for both large x and large t; one of the interesting limits is t → ∞ with (x/t) held fixed. Integral (12.7.1) can be evaluated by the Kelvin stationary phase method for large t. As t → ∞, the integrand of (12.7.1) oscillates very rapidly; consequently, the contributions to u (x, t) from adjacent parts of the integrand cancel one another except in the neighborhood of the points, if any, at which the phase function θ (k) is stationary, that is, θ ′ (k) = 0. Thus, the main contribution to the integral for large t comes from the neighborhood of the point k = k1 which determined by the solution of θ ′ (k1) = x t − ω ′ (k1)=0, a < k1 < b. (12.7.3) The point k = k1 known as the point of stationary phase, or simply, stationary point. We expand both F (k) and θ (k) in Taylor series about k = k1 so that u (x, t) =  b a F (k1)+(k − k1) F ′ (k1) + 1 2 (k − k1) 2 F ′′ (k1) + ... × exp  it θ (k1) + 1 2 (k − k1) 2 θ ′′ (k1) + 1 6 (k − k1) 3 θ ′′′ (k1) + ...0 dk (12.7.4) provided that θ ′′ (k1) = 0. Introducing the change of variable k − k1 = εα, where ε (t) =  2 t|θ ′′ (k1)| 01 2 , (12.7.5) we find that the significant contribution to integral (12.7.4) is u (x, t) ∼ ε  (b−k1)/ε −(k1−a)/ε F (k1) + εαF′ (k1) + 1 2 ε 2α 2F ′′ (k1) + ... × exp  i t θ (k1) + α 2 sgn θ ′′ (k1) + 1 3 ε  θ ′′′ (k1) |θ ′′ (k1)|  α 3 + ...0dα, (12.7.6) 460 12 Integral Transform Methods with Applications where sgn x denotes the signum function defined by sgn x = 1, x > 0 and sgn x = −1, x < 0. We then proceed to the limit as ε → 0 (t → ∞) and use the standard integral  ∞ −∞ exp  ±iα2 dα = √ π exp  ± iπ 4  (12.7.7) to obtain the asymptotic approximation as t → ∞, u (x, t) ∼ F (k1) 2π t|θ ′′ (k1)| 1 2 exp ( i " t θ (k1) + π 4 sgn θ ′′ (k1) #) + O  ε 2 , (12.7.8) where O  ε 2 means that a function tends to zero like ε 2 (t) as t → ∞. If there is more than one stationary point, each one contributes a term similar to (12.7.8) and we obtain, for n stationary points k = kr, r = 1, 2,...n; u(x, t) ∼ n r=1 F(kr)  2π t|θ ′′ (kr)| 01 2 exp ( i " t θ (kr) + π 4 sgn θ ′′(kr) #), t → ∞. (12.7.9) If θ ′′ (k1) = 0, but θ ′′′ (k1) = 0, then asymptotic approximation (12.7.8) fails. This important special case can be handled in a similar fashion. The asymptotic approximation of (12.7.1) is then given by u (x, t) = F (k1) exp {itθ (k1)}  ∞ −∞ exp i 6 t θ′′′ (k1) (k − k1) 3 dk ∼ Γ  4 3  6 t|θ ′′′ (k1)| 1 3 F (k1) exp itθ (k1) + πi 6 + O 4 t − 2 3 5 as t → ∞. (12.7.10) For an elaborate treatment of the stationary phase method, see Copson (1965). 12.8 Laplace Transforms Because of their simplicity, Laplace transforms are frequently used to solve a wide class of partial differential equations. Like other transforms, Laplace transforms are used to determine particular solutions. In solving partial differential equations, the general solutions are difficult, if not impossible, to obtain. The transform technique sometimes offers a useful tool for finding particular solutions. The Laplace transform is closely related to the complex Fourier transform, so the Fourier integral formula (6.13.10) can be used to define 12.8 Laplace Transforms 461 the Laplace transform and its inverse. We replace f (x) in (6.13.10) by H (x) e −cxf (x) for x > 0 to obtain f (x) H (x) e −cx = 1 2π  ∞ −∞ e ikxdk  ∞ 0 f (t) e −t(c+ik) dt or f (x) H (x) = 1 2π  ∞ −∞ e x(c+ik) dk  ∞ 0 f (t) e −t(c+ik) dt. Substituting s = c + ik so that ds = idk, we obtain, for x > 0, f (x) H (x) = 1 2πi  c+i∞ c−i∞ e xsds  ∞ 0 f (t) e −stdt. (12.8.1) Thus, we give the following definition of the Laplace transform: If f (t) is defined for all values of t > 0, then the Laplace transform of f (t) is denoted by ¯f (s) or L {f (t)} and is defined by the integral ¯f (s) = L {f (t)} =  ∞ 0 e −stf (t) dt, (12.8.2) where s is a positive real number or a complex number with a positive real part so that the integral is convergent. Hence, (12.8.1) gives f (x) = L −1 & ¯f (s) ' = 1 2πi  c+i∞ c−i∞ e xs ¯f (s) ds, c > 0, (12.8.3) for x > 0 and zero for x < 0. This complex integral is used to define the inverse Laplace transform which is denoted by L −1 & ¯f (s) ' = f (t). It can be verified easily that both L and L −1 are linear integral operators. We now find the Laplace transforms of some elementary functions. 1. Let f (t) = c, c is a constant. L[c] =  ∞ 0 e −stc dt = − ce−st s ∞ 0 = c s . 2. Let f (t) = e at , a is a constant. L e at! =  ∞ 0 e −ste atdt = − e −(s−a)t (s − a) ∞ 0 = 1 s − a , s ≥ a. 3. Let f (t) = t 2 . Then 462 12 Integral Transform Methods with Applications L t 2 ! =  ∞ 0 e −st t 2 dt. Integration by parts yields L & t 2 ' = − t 2 e −st s ∞ 0 +  ∞ 0 e −st s 2 t dt. Since t 2 e −st → 0 as t → ∞, we have, integrating by parts again, L t 2 ! = 2 s − e −st s t + 2 s  ∞ 0 e −st s dt = 2 s 3 . 4. Let f (t) = sin ωt. ¯f (s) = L[sin ωt] =  ∞ 0 e −st sin ωt dt = − e −st s sin ωt∞ 0 +  ∞ 0 e −st s ω cos ωt dt = ω s − e −st s cos ωt − ω s  ∞ 0 e −st s ω sin ωt dt ¯f (s) = ω s 2 − ω 2 s 2 ¯f (s). Thus, solving for ¯f (s), we obtain L[sin ωt] = ω/  s 2 + ω 2 . A function f (t) is said to be of exponential order as t → ∞ if there exist real constants M and a such that |f (t)| ≤ Meat for 0 ≤ t < ∞. Theorem 12.8.1. Let f be piecewise continuous in the interval [0, T] for every positive T, and let f be of exponential order, that is, f (t) = O (e at) as t → ∞ for some a > 0. Then, the Laplace transform of f (t) exists for Re s>a. Proof. Since f is piecewise continuous and of exponential order, we have |L(f (t))| =      ∞ 0 e −stf (t) dt     ≤  ∞ 0 e −st |f (t)| dt ≤  ∞ 0 e −stMeatdt = M  ∞ 0 e −(s−a)t dt = M/ (s − a), Re s > a. Thus,  ∞ 0 e −stf (t) dt exists for Re s>a. 12.9 Properties of Laplace Transforms 463 12.9 Properties of Laplace Transforms Theorem 12.9.1. (Linearity) If L[f (t)]and L[g (t)] are Laplace transforms of f (t) and g (t) respectively, then L[af (t) + bg (t)] = aL[f (t)] + bL[g (t)] where a and b are constants. Proof. L[af (t) + bg (t)] =  ∞ 0 [af (t) + bg (t)] e −stdt = a  ∞ 0 f (t) e −stdt + b  ∞ 0 g (t) e −stdt = aL[f (t)] + bL[g (t)] . This shows that L is a linear operator. Theorem 12.9.2. (Shifting) If ¯f (s) is the Laplace transform of f (t), then the Laplace transform of e atf (t) is ¯f (s − a). Proof. By definition, we have L e atf (t) ! =  ∞ 0 e −ste atf (t) dt =  ∞ 0 e −(s−a)t f (t) dt = ¯f (s − a). Example 12.9.1. (a) If L t 2 ! = 2/s3 , then L[t s e t ]=2/ (s − 1)3 . (b) If L[sin ωt] = ω/  s 2 + ω 2 , then L[e at sin ωt] = ω/ " (s − 1)2 + ω 2 # . (c) If L {cos ωt} = s s 2+ω2 , then L {e at cos ωt} = s−a (s−a) 2+ω2 . (d) If L {t n} = n! sn+1 , then L {e att n} = n! (s−a) n+1 . Theorem 12.9.3. (Scaling) If the Laplace transform of f (t) is ¯f (s), then the Laplace transform of f (ct) with c > 0 is (1/c) ¯f (s/c). Proof. By definition, we have L[f (ct)] =  ∞ 0 e −stf (ct) dt =  ∞ 0 1 c e −(sξ/c) f (ξ) dξ (substituting ξ = ct) = (1/c) ¯f (s/c). 464 12 Integral Transform Methods with Applications Example 12.9.2. (a) If s s 2+1 = L[cost], then 1 ω s/ω (s/ω) 2 + 1 = s s 2 + ω2 = L[cos ωt] . (b) If 1 s−1 = L[e t ], then 1 a 1  s a − 1 = L e at! , or L e at! = 1 s − a . Theorem 12.9.4. (Differentiation) Let f be continuous and f ′ piecewise continuous, in 0 ≤ t ≤ T for all T > 0. Let f also be of exponential order as t → ∞. Then, the Laplace transform of f ′ (t) exists and is given by L[f ′ (t)] = sL[f (t)] − f (0) = s ¯f (s) − f (0). Proof. Consider the definite integral  T 0 e −stf ′ (t) dt = e −stf (t) !T 0 +  T 0 s e−stf (t) dt = e −sT f (T) − f (0) + s  T 0 e −stf (t) dt. Since |f (t)| ≤ Meat for large t, with a > 0 and M > 0,  e −sT f (T)   ≤ Me−(s−a)T . In the limit as T → ∞, e −sT f (T) → 0 whenever s>a. Hence, L[f ′ (t)] = sL[f (t)] − f (0) = s ¯f (s) − f (0). If f ′ and f ′′ satisfy the same conditions imposed on f and f ′ respectively, then, the Laplace transform of f ′′ (t) can be obtained immediately by applying the preceding theorem; that is L[f ′′ (t)] = sL[f ′ (t)] − f ′ (0) = s {sL[f (t)] − f (0)} − f ′ (0) = s 2L[f (t)] − sf (0) − f ′ (0) = s 2 ¯f (s) − sf (0) − f ′ (0). Clearly, the Laplace transform of f (n) (t) can be obtained in a similar manner by successive application of Theorem 12.9.4. The result may be written as L " f (n) (t) # = s nL[f (t)] − s n−1 f (0) − ... − s f(n−2) (0) − f (n−1) (0). 12.9 Properties of Laplace Transforms 465 Theorem 12.9.5. (Integration) If ¯f (s) is the Laplace transform of f (t), then L  t 0 f (τ ) dτ = ¯f (s) /s. Proof. L  t 0 f (τ ) dτ =  ∞ 0  t 0 f (τ ) dτ e −stdt = − e −st s  t 0 f (τ ) dτ∞ 0 + 1 s  ∞ 0 f (t) e −stdt = ¯f (s) /s since  t 0 f (τ ) dτ is of exponential order. In solving problems by the Laplace transform method, the difficulty arises in finding inverse transforms. Although the inversion formula exists, its evaluation requires a knowledge of functions of complex variables. However, for some problems of mathematical physics, we need not use this inversion formula. We can avoid its use by expanding a given transform by the method of partial fractions in terms of simple fractions in the transform variables. With these simple functions, we refer to the table of Laplace transforms given in the end of the book and obtain the inverse transforms. Here, we should note that we use the assumption that there is essentially a one-to-one correspondence between functions and their Laplace transforms. This may be stated as follows: Theorem 12.9.6. (Lerch) Let f and g be piecewise continuous functions of exponential order. If there exists a constant s0, such that L[f] = L[g] for all s>s0, then f (t) = g (t) for all t > 0 except possibly at the points of discontinuity. For a proof, the reader is referred to Kreider et al. (1966). In order to find a solution of linear partial differential equations, the following formulas and results are useful. If L {u (x, t)} = u (x, s), then L  ∂u ∂t 0 = s u (x, s) − u (x, 0), L  ∂ 2u ∂t2 0 = s 2 u (x, s) − s u (x, 0) − ut (x, 0), and so on. Similarly, it is easy to show that 466 12 Integral Transform Methods with Applications L  ∂u ∂x0 = du dx, L  ∂ 2u ∂x2 0 = d 2u dx2 ,...,L  ∂ nu ∂xn 0 = d nu dxn . The following results are useful for applications: L  erfc  a 2 √ t 0 = 1 s exp  −a √ s , a ≥ 0, (12.9.1) L ( exp (at) erf 4√ at5) = √ a √ s (s − a) , a > 0. (12.9.2) Example 12.9.3. Consider the motion of a semi-infinite string with an external force f (t) acting on it. One end is kept fixed while the other end is allowed to move freely in the vertical direction. If the string is initially at rest, the motion of the string is governed by utt = c 2uxx + f (t), 0 <x< ∞,="" t=""> 0, u (x, 0) = 0, ut (x, 0) = 0, u (0, t)=0, ux (x, t) → 0, as x → ∞. Let u (x, s) be the Laplace transform of u (x, t). Transforming the equation of motion and using the initial conditions, we obtain uxx −  s 2 /c2 u = −f (s) /c2 . The solution of this ordinary differential equation is u (x, s) = Aesx/c + Be−sx/c + f (s) /s2 ! . The transformed boundary conditions are given by u (0, s)=0, and limx→∞ ux (x, s)=0. In view of the second condition, we have A = 0. Now applying the first condition, we obtain u (0, s) = B + f (s) /s2 ! = 0. Hence u (x, s) = f (s) /s2 ! " 1 − e −sx/c# . (a) When f (t) = f0, a constant, then u (x, s) = f0  1 s 3 − 1 s 3 e −sx/c . The inverse Laplace transform gives the solution 12.10 Convolution Theorem of the Laplace Transform 467 u (x, t) = f0 2 t 2 − 4 t − x c 52 when t ≥ x/c, = (f0/2)t 2 when t ≤ x/c. (b) When f (t) = cos ωt, where ω is a constant, then ¯f (s) =  ∞ 0 e −st cos ωt dt = s/  ω 2 + s 2 . Thus, we have u (x, s) = 1 s (ω2 + s 2) 4 1 − e −sx/c5 . (12.9.3) By the method of partial fractions, we write 1 s (s 2 + ω2) = 1 ω2 1 s − 1 s 2 + ω2 . Hence L −1 1 s (s 2 + ω2) = 1 ω2 (1 − cos ωt) = 2 ω2 sin2  ωt 2  . If we denote ψ (t) = sin2  ωt 2  , then the Laplace inverse of equation (12.9.3) may be written in the form u (x, t) = 2 ω2 " ψ (t) − ψ 4 t − x c 5# when t ≥ x/c, = 2 ω2 ψ (t) when t ≤ x/c. 12.10 Convolution Theorem of the Laplace Transform The function (f ∗ g) (t) =  t 0 f (t − ξ) g (ξ) dξ (12.10.1) is called the convolution of the functions f and g. Theorem 12.10.1. (Convolution) If ¯f (s) and g¯ (s) are the Laplace transforms of f (t) and g (t) respectively, then the Laplace transform of the convolution (f ∗ g) (t) is the product ¯f (s) ¯g (s). 468 12 Integral Transform Methods with Applications Figure 12.10.1 Region of integration. Proof. By definition, we have L[(f ∗ g) (t)] =  ∞ 0 e −st  t 0 f (t − ξ) g (ξ) dξ dt =  ∞ 0  t 0 e −stf (t − ξ) g (ξ) dξ dt. The region of integration is shown in Figure 12.10.1. By reversing the order of integration, we have L[(f ∗ g) (t)] =  ∞ 0  ∞ ξ e −stf (t − ξ) g (ξ) dt dξ =  ∞ 0 g (ξ)  t ξ e −stf (t − ξ) dt dξ. If we introduce the new variable η = (t − ξ) in the inner integral, we obtain L[(f ∗ g) (t)] =  ∞ 0 g (ξ)  ∞ 0 e −s(ξ+η) f (η) dη dξ =  ∞ 0 g (ξ) e −sξdξ  ∞ 0 e −sηf (η) dη = ¯f (s) ¯g (s). (12.10.2) The convolution satisfies the following properties: 12.10 Convolution Theorem of the Laplace Transform 469 1. f ∗ g = g ∗ f (commutative). 2. f ∗ (g ∗ h)=(f ∗ g) ∗ h (associative). 3. f ∗ (αg + βh) = α (f ∗ g) + β (f ∗ h), (distributive), where α and β are constants. Example 12.10.1. Find the temperature distribution in a semi-infinite radiating rod. The temperature is kept constant at x = 0, while the other end is kept at zero temperature. If the initial temperature distribution is zero, the problem is governed by ut = kuxx − hu, 0 <x< ∞,="" t=""> 0, h = constant, u (x, 0) = 0, u (0, t) = u0, t> 0, u0 = constant, u (x, t) → 0, as x → ∞. Let u (x, s) be the Laplace transform of u (x, t). Then the transformation with respect to t yields uxx −  s + h k  u = 0, u (0, s) = u0/s, limx→∞ u (x, s)=0. The solution of this equation is u (x, s) = A ex √ (s+h)/k + B e−x √ (s+h)/k . The boundary condition at infinity requires that A = 0. Applying the other boundary condition gives u (0, s) = B = u0/s. Hence, the solution takes the form u (x, s)=(u0/s) exp " −x  (s + h) /k # . We find (by using the Table of Laplace Transforms) that L −1 " u0 s # = u0, and L −1 " exp ( −x  (s + h) /k )# = x exp −ht −  x 2/4kt ! 2 √ πkt3 . Thus, the inverse Laplace transform of u (x, s) is u (x, t) = L −1 " u0 s exp ( −x  (s + h) /k )# . 470 12 Integral Transform Methods with Applications By the Integration Theorem 12.9.5, we have u (x, t) =  t 0 u0 x exp −hτ −  x 2/4kτ ! 2 √ πk τ 3 2 dτ. Substituting the new variable η = 4 x/2 √ kτ5 yields u (x, t) = 2 u0 √ π  ∞ x/2 √ kt exp −η 2 +  hx2 /4kη2 ! dη. For the case h = 0, the solution u (x, t) becomes u (x, t) = 2 u0 √ π  ∞ x/2 √ kt e −η 2 dη = 2 u0 √ π  ∞ 0 e −η 2 dη − 2 u0 √ π  x/2 √ kt 0 e −η 2 dη = u0 1 − erf  x 2 √ kt = u0 erfc  x 2 √ kt . 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions We have defined the Heaviside unit step function. Now, we will find its Laplace transform. L[H (t − a)] =  ∞ 0 e −stH (t − a) dt =  ∞ a e −stdt =  1 s  e −as, s> 0. (12.11.1) Theorem 12.11.1. (Second Shifting) If ¯f (s) and g¯ (s) are the Laplace transforms of f (t) and g (t) respectively, then (a) L[H (t − a) f (t − a)] = e −as ¯f (s) = e −asL {f (t)} . (b) L {H (t − a) g (t)} = e −asL {g (t + a)} . Proof. (a) By definition L[H (t − a) f (t − a)] =  ∞ 0 e −stH (t − a) f (t − a) dt =  ∞ a e −stf (t − a) dt. Introducing the new variable ξ = t − a, we obtain 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions 471 L[H (t − a) f (t − a)] =  ∞ 0 e −(ξ+a)s f (ξ) dξ = e −as  ∞ 0 e −ξsf (ξ) dξ = e −as ¯f (s). To prove (b), we write L {H (t − a) g (t)} =  ∞ a e −stg (t) dt (t − a = τ ) =  ∞ 0 e −s(a+τ) g (a + τ ) dτ = e −saL {g (t + a)} . Example 12.11.1. (a) Given that f (t) = ⎧ ⎨ ⎩ 0, t < 2 t − 2, t ≥ 2 ⎫ ⎬ ⎭ = (t − 2) H (t − 2), find the Laplace transform of f (t). We have L[f (t)] = L[H (t − 2) (t − 2)] = e −2sL[t] =  1 s 2  e −2s . (b) Find the inverse Laplace transform of f (s) = 1 + e −2s s 2 . L −1 f (s) ! = L −1  1 s 2 + e −2s s 2  = L −1 1 s 2 + L −1 e −2s s 2 = t + H (t − 2) (t − 2) = ⎧ ⎨ ⎩ t, 0 ≤ t < 2, 2 (t − 1), t ≥ 2. The Laplace transform of the impulse function p (t) is given by L[p (t)] =  ∞ 0 e −stp (t) dt =  a+ε a−ε h e−stdt = h − e −st s a+ε a−ε = h e−as s  e εs − e −εs = 2 h e−as s sinh (εs). (12.11.2) 472 12 Integral Transform Methods with Applications If we choose the value of h to be (1/2ε), then the impulse is given by I (ε) =  ∞ −∞ p (t) dt =  a+ε a−ε 1 2ε dt = 1. Thus, in the limit as ε → 0, this particular impulse function satisfies limε→0 pε (t)=0, t = a, limε→0 I (ε)=1. From this result, we obtain the Dirac delta function which satisfies δ (t − a)=0, t = a,  ∞ −∞ δ (t − a) dt = 1. (12.11.3) Thus, we may define the Laplace transform of δ (t) as the limit of the transform of pε (t). L[δ (t − a)] = limε→0 L[pε (t)] , = limε→0 e −as sinh (εs) εs (12.11.4) = e −as . If a = 0, we have L[δ (t)] = 1. (12.11.5) One very useful result that can be derived is the integral of the product of the delta function and any continuous function f (t).  ∞ −∞ δ (t − a) f (t) dt = limε→0  ∞ −∞ pε (t) f (t) dt, = limε→0  a+ε a−ε f (t) 2ε dt, = limε→0 1 2ε · 2ε f (t ∗ ), a − ε<t∗ <="" a="" +="" ε="f" (a).="" (12.11.6)="" suppose="" that="" f="" (t)="" is="" periodic="" function="" with="" period="" t.="" let="" be="" piecewise="" continuous="" on="" [0,="" t].="" then,="" the="" laplace="" transform="" of="" l[f="" (t)]="" ∞="" 0="" e="" −stf="" dt,="∞" n="0" ="" (n+1)t="" nt="" dt.="" 12.11="" transforms="" heaviside="" and="" dirac="" delta="" functions="" 473="" if="" we="" introduce="" new="" variable="" ξ="t" −="" nt,="" then="" −nt="" s="" t="" −sξf="" (ξ)="" dξ,="∞" 1="" (s),="" where="" (s)="*" dξ="" over="" first="" period.="" since="" series="" geometric="" series,="" obtain="" for="" (1="" e−t="" s)="" .="" (12.11.7)="" example="" 12.11.2.="" find="" square="" wave="" 2c="" given="" by="" ⎨="" ⎩="" h,="" <t<c="" −h,="" c="" (t="" 2c)="f" (t),="" as="" shown="" in="" figure="" 12.11.1.="" 12.11.1="" function.="" 474="" 12="" integral="" methods="" applications="" −sξh="" −sξ="" (−h)="" ="" −cs="" 2="" thus,="" is,="" (12.11.7),="" e−2cs="h" −cs)="" e−2cs)="h" e−cs)="h" tanh="" 4cs="" 5="" 12.11.3.="" uniform="" bar="" length="" l="" fixed="" at="" one="" end.="" force="" f0,=""> 0 0, t< 0 be suddenly applied at the end x = l. If the bar is initially at rest, find the longitudinal displacement for t > 0. The motion of the bar is governed by the differential system utt = a 2uxx, 0 < x < l, t > 0, a = constant, u (x, 0) = 0, ut (x, 0) = 0, u (0, t)=0, ux (l, t)=(f0/E), where E is a constant and t > 0. Let u (x, s) be the Laplace transform of u (x, t). Then, u (x, s) satisfies the system uxx − s 2 a 2 u = 0, u (0, s)=0, ux (l, s)=(f0/Es). The solution of this differential equation is u (x, s) = Aexs/a + Be−xs/a . Applying the boundary conditions, we have A + B = 0, 4 s a e ls/a5 A + 4 − s a e −ls/a5 B = f0/Es. Solving for A and B, we obtain 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions 475 A = −B = af0 Es2  e ls/a + e−ls/a . Hence, the transform of the displacement function is given by u (x, s) = af0  e xs/a − e −xs/a Es2  e ls/a + e−ls/a . Before finding the inverse transform of u (x, s), multiply the numerator and denominator by  e −ls/a − e −3ls/a . Thus, we have u (x, s) =  af0 Es2  " e −(l−x)s/a − e −(l+x)s/a − e −(3l−x)s/a + e −(3l+x)s/a# × 1  1 − e−4ls/a . Since the denominator has the term  1 − e −4ls/a , the inverse transform u (x, t) is periodic with period (4l/a). Hence, the final solution may be written in the form u (x, t) = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ 0, 0 <t< l−x="" a="" ,="" af0="" e="" ="" t="" −="" <t<="" l+x="" t="" !="" 3l−x="" 3l+x="" +="" 0,="" 4l="" which="" may="" be="" simplified="" to="" obtain="" u="" (x,="" t)="⎧" ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨="" ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩="" 0="" (l="" x)="" a,="" <="" 2x="" (3l="" −t="" a.="" this="" result="" can="" clearly="" seen="" in="" figure="" 12.11.2.="" 476="" 12="" integral="" transform="" methods="" with="" applications="" example="" 12.11.4.="" consider="" semi-infinite="" string="" fixed="" at="" the="" end="" x="0." is="" initially="" rest.="" let="" there="" an="" external="" force="" f="" δ="" 4="" v="" 5="" acting="" on="" string.="" concentrated="" f0="" point="" motion="" of="" governed="" by="" initial="" boundary-value="" problem="" utt="c" 2uxx="" 0)="0," ut="" (0,="" bounded="" as="" →="" ∞.="" s)="" laplace="" t).="" transforming="" wave="" equation="" and="" using="" conditions,="" we="" uxx="" s="" 2="" c="" exp="" (−xs="" v).="" solution="" aesx="" be−sx="" c5="" ⎧="" ⎪⎨="" ⎪⎩="" f0v="" −sx="" (c="" 2−v="" 2)s="" for="" f0xe−sx="" 2cs="" 12.11.2="" graph="" 12.11="" transforms="" heaviside="" dirac="" delta="" functions="" 477="" condition="" that="" must="" infinity="" requires="" application="" yields="" b="⎧" ⎨="" ⎩="" −f0v="" hence,="" given="" (e="" −xs="" v−e="" c)="" f0xe−xs="" inverse="" therefore="" 2)="" f0x="" 2c="" 12.11.5.="" (the="" stokes="" rayleigh="" fluid="" dynamics).="" solve="" concerned="" unsteady="" boundary="" layer="" flows="" induced="" viscous="" infinite="" horizontal="" disk="" z="0" due="" oscillations="" its="" own="" plane="" frequency="" ω.="" velocity="" (z,="" uzz,=""> 0, t> 0, with the boundary and initial conditions u (z, t) = U0e iωt, z = 0, t> 0, u (z, t) → 0, as z → ∞, t> 0, u (z, 0) = 0, at t ≤ 0 for all z > 0, where u (z, t) is the velocity of fluid of kinematic viscosity ν and U0 is a constant. The Laplace transform solution of the equation with the transformed boundary conditions is u (z, s) = U0 (s − iω) exp  −z 2 s ν  . 478 12 Integral Transform Methods with Applications Using a standard table of inverse Laplace transforms, we obtain the solution u (z, t) = U0 2 e iωt " exp (−λz) erfc 4 ζ − √ iωt5 + exp (λz) erfc 4 ζ + √ iωt5# , where ζ =  z/2 √ νt is called the similarity variable of the viscous boundary layer theory, and λ = (iω/ν) 1 2 . This result describes the unsteady boundary layer flow. In view of the asymptotic formula for the complementary error function erfc 4 ζ + √ iωt5 ∼ (2, 0) as t → ∞, the above solution for u (z, t) has the asymptotic representation u (z, t) ∼ U0 exp (iωt − λz) = U0 exp iωt − 4 ω 2ν 5 1 2 (1 + i) z . (12.11.8) This is called the Stokes steady-state solution. This represents the propagation of shear waves which spread out from the oscillating disk with velocity ω/k = √ 2νω 4 k = (ω/2ν) 1 2 5 and exponentially decaying amplitude. The boundary layer associated with the solution has thickness of the order (ν/ω) 1 2 in which the shear oscillations imposed by the disk decay exponentially with distance z from the disk. This boundary layer is called the Stokes layer. In other words, the thickness of the Stokes layer is equal to the depth of penetration of vorticity which is essentially confined to the immediate vicinity of the disk for high frequency ω. The Stokes problem with ω = 0 becomes the Rayleigh problem. In other words, the motion is generated in the fluid from rest by moving the disk impulsively in its own plane with constant velocity U0. In this case, the Laplace transform solution is u (z, s) = U0 s exp  −z 2 s ν  , so that the inversion gives the Rayleigh solution u (z, t) = U0 erfc  z 2 √ νt . (12.11.9) This describes the growth of a boundary layer adjacent to the disk. The associated boundary layer is called the Rayleigh layer of thickness of the order δ ∼ √ νt which grows with increasing time t. The rate of growth is of the order dδ/dt ∼  ν/t, which diminishes with increasing time. 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions 479 The vorticity of the unsteady flow is given by ∂u ∂z = U0 √ πνt exp  −ζ 2 (12.11.10) which decays exponentially to zero as z ≫ δ. Note that the vorticity is everywhere zero at t = 0 except at z = 0. This implies that it is generated at the disk and diffuses outward within the Rayleigh layer. The total viscous diffusion time is Td ∼ δ 2/ν. Another physical quantity related to the Stokes and Rayleigh problems is the skin friction on the disk defined by τ0 = µ  ∂u ∂z  z=0 , (12.11.11) where µ = νρ is the dynamic viscosity and ρ is the density of the fluid. The skin friction can readily be calculated from the flow field given by (12.11.8) or (12.11.9). Example 12.11.6. (The Nonhomogeneous Cauchy Problem for the Wave Equation). We consider the nonhomogeneous Cauchy problem utt − c 2uxx = q (x, t), x ∈ R, t > 0, (12.11.12) u (x, 0) = f (x), ut (x, 0) = g (x) for all x ∈ R, (12.11.13) where q (x, t) is a given function representing a source term. We use the joint Laplace and Fourier transform of u (x, t) U (k, s) = L[F {u (x, t)}] = 1 √ 2π  ∞ −∞ e −ikxdx  ∞ 0 e −stu (x, t) dt. (12.11.14) Application of the joint transform leads to the solution of the transformed Cauchy problem in the form U (k, s) = s F (k) + G (k) + Q (k, s) (s 2 + c 2k 2) . (12.11.15) The inverse Laplace transform of (12.11.15) gives U (k, t) = F (k) cos (ckt) + 1 ckG (k) sin (ckt) + 1 ckL −1  ck s 2 + c 2k 2 · Q (k, s) 0 = F (k) cos (ckt) + G (k) ck sin (ckt) + 1 ck  t 0 sin ck (t − τ ) Q (k, τ ) dτ. (12.11.16) The inverse Fourier transform leads to the exact integral solution 480 12 Integral Transform Methods with Applications u (x, t) = 1 2 √ 2π  ∞ −∞  e ickt + e −ickt e ikxF (k) dk + 1 2 √ 2π  ∞ −∞  e ickt − e −ickt e ikx · G (k) ick dk + 1 √ 2π · 1 2c  t 0 dτ  ∞ −∞ Q (k, τ ) ik " e ick(t−τ) − e −ick(t−τ) # e ikxdk = 1 2 [f (x + ct) + f (x − ct)] + 1 2c  x+ct x−ct g (ξ) dξ + 1 2c  t 0 dτ 1 √ 2π  ∞ −∞ Q (k, τ ) dk  x+c(t−τ) x−c(t−τ) e ikξdξ = 1 2 [f (x − ct) + f (x + ct)] + 1 2c  x+ct x−ct g (ξ) dξ + 1 2c  t 0 dτ  x+c(t−τ) x−c(t−τ) q (ξ, τ ) dξ. (12.11.17) In the case of the homogeneous Cauchy problem, q (x, t) ≡ 0, the solution of (12.11.17) reduces to the famous d’Alembert solution (5.3.8). Example 12.11.7. (The Heat Conduction Equation in a Semi-Infinite Medium and Fractional Derivatives). Solve the one-dimensional diffusion equation ut = κ uxx, x > 0, t> 0, (12.11.18) with the initial and boundary conditions u (x, 0) = 0, x > 0, (12.11.19) u (0, t) = f (t), t> 0, (12.11.20) u (x, t) → 0, as x → ∞, t> 0. (12.11.21) Application of the Laplace transform with respect to t to (12.11.18) gives d 2u dx2 − s κ u = 0. (12.11.22) The general solution of this equation is u (x, s) = A exp  −x 2 s κ  + B exp  x 2 s κ  , where A and B are integrating constants. For bounded solutions, B ≡ 0, and using u (0, s) = f (s), we obtain the solution u (x, s) = f (s) exp  −x 2 s κ  . (12.11.23) 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions 481 The Laplace inversion theorem gives the solution u (x, t) = x 2 √ πκ  t 0 f (t − τ ) τ − 3 2 exp  − x 2 4κτ  dτ, (12.11.24) which, by setting λ = x 2 √ κτ , or dλ = − x 4 √ κ τ − 3 2 dτ , = 2 √ π  ∞ x 2 √κt f  t − x 2 4κλ2  e −λ 2 dλ. (12.11.25) This is the formal solution of the heat conduction problem. In particular, if f (t) = T0 = constant, solution (12.11.25) becomes u (x, t) = 2T0 √ π  ∞ x 2 √κt e −λ 2 dλ = T0 erfc  x 2 √ κt . (12.11.26) Clearly, the temperature distribution tends asymptotically to the constant value T0, as t → ∞. Alternatively, solution (12.11.23) can be written as u (x, s) = f (s) s u0 (x, s), (12.11.27) where s u0 (x, s) = exp  −x 2 s κ  . (12.11.28) Consequently, the inversion of (12.11.27) gives a new representation u (x, t) =  t 0 f (t − τ )  ∂u0 ∂τ  dτ. (12.11.29) This is called the Duhamel formula for the diffusion equation. We consider another physical problem: determining the temperature distribution of a semi-infinite solid when the rate of flow of heat is prescribed at the end x = 0. Thus, the problem is to solve diffusion equation (12.11.18) subject to conditions (12.11.19), (12.11.21), and −k  ∂u ∂x = g (t) at x = 0, t> 0, (12.11.30) where k is a constant called thermal conductivity. Application of the Laplace transform gives the solution of the transformed problem u (x, s) = 1 k 2 κ s g (s) exp  −x 2 s κ  . (12.11.31) 482 12 Integral Transform Methods with Applications The inverse Laplace transform yields the solution u (x, t) = 1 k 2 κ π  t 0 g (t − τ ) τ − 1 2 exp  − x 2 4κτ  dτ, (12.11.32) which is, by the change of variable λ = x 2 √ κτ , = x k √ π  ∞ x 2 √κt g  t − x 2 4κλ2  λ −2 e −λ 2 dλ. (12.11.33) In particular, if g (t) = T0 = constant, this solution becomes u (x, t) = T0 x k √ π  ∞ √x 4κt λ −2 e −λ 2 dλ. Integrating this result by parts gives u (x, t) = T0 k 1 2 2 κt π exp  − x 2 4κt − x erfc  x 2 √ κt3 . (12.11.34) Alternatively, the heat conduction problem (12.11.18)–(12.11.21) can be solved by using fractional derivatives (Debnath 1995). We recall (12.11.23) and rewrite it as ∂u ∂x = − 2 s κ u. (12.11.35) This can be expressed in terms of a fractional derivative of order 1 2 as ∂u ∂x = − 1 √ κ L −1 &√ s u (x, s) ' = − 1 √ κ 0D 1 2 t u (x, t). (12.11.36) Thus, the heat flux is expressed in terms of the fractional derivative. In particular, when u (0, t) = constant = T0, then the heat flux at the surface is given by −k  ∂u ∂x x=0 = k √ κ D 1 2 t T0 = kT0 √ πκt . (12.11.37) Example 12.11.8. (Diffusion Equation in a Finite Medium). Solve the diffusion equation ut = κ uxx, 0 < x < a, t > 0, (12.11.38) with the initial and boundary conditions u (x, 0) = 0, 0 < x < a, (12.11.39) u (0, t) = U, t > 0, (12.11.40) ux (a, t)=0, t> 0, (12.11.41) 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions 483 where U is a constant. We introduce the Laplace transform of u (x, t) with respect to t to obtain d 2u dx2 − s κ u = 0, 0 < x < a, (12.11.42) u (0, s) = U s ,  du dx x=a = 0. (12.11.43) The general solution of (12.11.42) is u (x, s) = A cosh  x 2 s κ  + B sinh  x 2 s κ  , (12.11.44) where A and B are constants of integration. Using (12.11.43), we obtain the values of A and B, so that the solution (12.11.44) becomes u (x, s) = U s · cosh (a − x) s κ ! cosh  a s κ . (12.11.45) The inverse Laplace transform gives the solution u (x, t) = UL −1 / cosh (a − x) s κ s cosh  a s κ 0 . (12.11.46) The inversion can be carried out by the Cauchy residue theorem to obtain the solution u (x, t) = U 1 1 + 4 π ∞ n=1 (−1)n (2n − 1) cos  (2n − 1) (a − x) π 2a 0 × exp  − (2n − 1)2 4 π 2a 52 κt0 . (12.11.47) By expanding the cosine term, this becomes u (x, t) = U 1 1 − 4 π ∞ n=1 1 (2n − 1) sin 2n − 1 2a  πx0 × exp  − (2n − 1)2 4 π 2a 52 κt0 . (12.11.48) This result can be obtained by solving the problem by the method of separation of variables. Example 12.11.9. (Diffusion in a Finite Medium). Solve the one-dimensional diffusion equation in a finite medium 0 <z<a, where="" the="" concentration="" function="" c="" (z,="" t)="" satisfies="" equation="" 484="" 12="" integral="" transform="" methods="" with="" applications="" ct="κ" czz,="" 0="" <="" z="" a,="" t=""> 0, (12.11.49) and the initial and boundary data C (z, 0) = 0 for 0 < z < a, (12.11.50) C (z, t) = C0 for z = a, t > 0, (12.11.51) ∂C ∂z = 0 for z = 0, t> 0, (12.11.52) where C0 is a constant. Application of the Laplace transform of C (z, t) with respect to t gives d 2C dz2 − 4 s κ 5 C = 0, 0 < z < a, C (a, s) = C0 s ,  dC dz  z=0 = 0. The solution of this differential equation system is C (z, s) = C0 cosh  z s κ s cosh  a s κ , (12.11.53) which, by writing α = s κ , = C0 s (e αz + e −αz) (e αa + e−αa) = C0 s [exp {−α (a − z)} + exp {−α (a + z)}] ∞ n=0 (−1)n exp (−2nαa) = C0 s 1∞ n=0 (−1)n exp [−α {(2n + 1) a − z}] + ∞ n=0 (−1)n exp [−α {(2n + 1) a + z}] 3 .(12.11.54) Using the result (12.9.1), we obtain the final solution C (z, t) = C0 /∞ n=0 (−1)n erfc  (2n + 1) a − z 2 √ κt 0 + erfc  (2n + 1) a + z 2 √ κt 00 . (12.11.55) This solution represents an infinite series of complementary error functions. The successive terms of this series are, in fact, the concentrations at depth a − z, a + z, 3a − z, 3a + z, ... in the medium. The series converges rapidly for all except large values of  κt a2 . 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions 485 Example 12.11.10. (The Wave Equation for the Transverse Vibration of a Semi-Infinite String). Find the displacement of a semi-infinite string, which is initially at rest in its equilibrium position. At time t = 0, the end x = 0 is constrained to move so that the displacement is u (0, t) = A f (t) for t ≥ 0, where A is a constant. The problem is to solve the one-dimensional wave equation utt = c 2uxx, 0 ≤ x < ∞, t> 0, (12.11.56) with the boundary and initial conditions u (x, t) = A f (t) at x = 0, t ≥ 0, (12.11.57) u (x, t) → 0 as x → ∞, t ≥ 0, (12.11.58) u (x, t) = 0= ∂u ∂t at t = 0 for 0 <x< ∞.="" (12.11.59)="" application="" of="" the="" laplace="" transform="" u="" (x,="" t)="" with="" respect="" to="" t="" gives="" d="" 2u="" dx2="" −="" s="" 2="" c="" for="" 0="" ≤="" x="" <="" ∞,="" s)="A" f="" (s)="" at="" →="" as="" solution="" this="" differential="" equation="" system="" is="" exp="" 4="" xs="" 5="" .="" (12.11.60)="" inversion="" h="" (12.11.61)="" in="" other="" words,="" ⎨="" ⎩="" a="" ="" ,=""> x c 0, t < x c . (12.11.62) This solution represents a wave propagating at a velocity c with the characteristic x = ct. Example 12.11.11. (The Cauchy–Poisson Wave Problem in Fluid Dynamics). We consider the two-dimensional Cauchy–Poisson problem (Debnath 1994) for an inviscid liquid of infinite depth with a horizontal free surface. We assume that the liquid has constant density ρ and negligible surface tension. Waves are generated on the free surface of liquid initially at rest for time t < 0 by the prescribed free surface displacement at t = 0. 486 12 Integral Transform Methods with Applications In terms of the velocity potential φ (x, z, t) and the free surface elevation η (x, t), the linearized surface wave motion in Cartesian coordinates (x, y, z) is governed by the following equation and free surface and boundary conditions: ∇2φ = φxx + φzz = 0, −∞ < z ≤ 0, x ∈ R, t < 0, (12.11.63) φz − ηt = 0 φt + gη = 00 on z = 0, t> 0, (12.11.64) φz → 0 as z → −∞. (12.11.65) The initial conditions are φ (x, 0, 0) = 0 and η (x, 0) = η0 (x), (12.11.66) where η0 (x) is a given initial elevation with compact support. We introduce the Laplace transform with respect to t and the Fourier transform with respect to x defined by " ˜ φ (k, z, s), η˜ (k, s) # = 1 √ 2π  ∞ −∞ e −ikxdx  ∞ 0 e −st [φ, η] dt. (12.11.67) Application of the joint transform method to the above system gives ˜ φzz − k 2 ˜ φ = 0, −∞ < z ≤ 0, (12.11.68) ˜ φ = s η˜ − η˜0 (k) s ˜ φ + gη˜ = 0 ⎫ ⎪⎬ ⎪⎭ on z = 0, (12.11.69) ˜ φz → 0 as z → −∞, (12.11.70) where η˜0 (k) = F {η0 (x)} . The bounded solution of equation (12.11.68) is ˜ φ (k, s) = A exp (|k| z), (12.11.71) where A = A (s) is an arbitrary function of s. Substituting (12.11.71) into (12.11.69) and eliminating η˜ from the resulting equations gives A. Hence, the solutions for ˜ φ and η˜ are " ˜ φ, η˜ # = − g η˜0 exp (|k| z) s 2 + ω2 , s η˜0 s 2 + ω2 , (12.11.72) and the associated the dispersion relation is 12.11 Laplace Transforms of the Heaviside and Dirac Delta Functions 487 ω 2 = g |k| . (12.11.73) The inverse Laplace and Fourier transforms give the solutions φ (x, z, t) = − g √ 2π  ∞ −∞ sin ωt ω exp (ikx + |k| z) ˜η0 (k) dk, (12.11.74) η (x, t) = 1 √ 2π  ∞ −∞ η˜0 (k) cos ωt eikxdk, = 1 √ 2π  ∞ 0 η˜0 (k) " e i(kx−ωt) + e i(kx+ωt) # dk, (12.11.75) in which ˜η0 (−k)=˜η0 (k) is assumed. Physically, the first and second integrals of (12.11.75) represent waves traveling in the positive and negative directions of x, respectively, with phase velocity ω k . These integrals describe superposition of all such waves over the wavenumber spectrum 0 <k< ∞.="" for="" the="" classical="" cauchy–poisson="" wave="" problem,="" η0="" (x)="aδ" (x),="" where="" δ="" is="" dirac="" delta="" function,="" so="" that="" ˜η0="" (k)="" a="" √="" 2π="" .="" thus,="" solution="" (12.11.75)="" becomes="" η="" (x,="" t)="a" ="" ∞="" 0="" "="" e="" i(kx−ωt)="" +="" i(kx+ωt)="" #="" dk.="" (12.11.76)="" integrals="" (12.11.74)="" and="" represent="" exact="" velocity="" potential="" φ="" free="" surface="" elevation="" all="" x="" t=""> 0. However, they do not lend any physical interpretations. In general, the exact evaluation of these integrals is a formidable task. So it is necessary to resort to asymptotic methods. It would be sufficient for the determination of the principal features of the wave motions to investigate (12.11.75) or (12.11.76) asymptotically for large time t and large distance x with (x, t) held fixed. The asymptotic solution for this kind of problem is available in many standard books; (for example, see Debnath 1994, p. 85). We use the stationary phase approximation of a typical wave integral (12.7.1), for t → ∞, given by (12.7.8) η(x, t) =  b a F(k) exp[itθ(k)]dk (12.11.77) ∼ f(k1) 2π t|θ ′′(k1)| 1 2 exp " i ( tθ(k1) + π 4 sgn θ ′′(k1) )#, (12.11.78) where θ (k) = kx t − ω (k), x > 0, and k = k1 is a stationary point that satisfies the equation θ ′ (k1) = x t − ω ′ (k1)=0, a < k1 < b. (12.11.79) 488 12 Integral Transform Methods with Applications Application of (12.11.78) to (12.11.75) shows that only the first integral in (12.11.75) has a stationary point for x > 0. Hence, the stationary phase approximation (12.11.78) gives the asymptotic solution, as t → ∞, x > 0, η(x, t) ∼ 1 t|ω′′(k1)| 1 2 η˜0(k1) exp i{k1x − tω(k1)} + iπ 4 sgn {−ω ′′(k1)} , (12.11.80) where k1 =  gt2/4x 2 is the root of the equation ω ′ (k) = x t . On the other hand, when x < 0, only the second integral of (12.11.75) has a stationary point k1 =  gt2/4x 2 , and hence, the same result (12.11.78) can be used to obtain the asymptotic solution for t → ∞ and x < 0 as η(x, t) ∼ 1 t|ω′′(k1)| 1 2 η˜0(k1) exp i {tω(k1) − k1 |x|} + iπ 4 sgn ω ′′(k1) . (12.11.81) In particular, for the classical Cauchy–Poisson solution (12.11.76), the asymptotic representation for η (x, t) follows from (12.11.81) in the form η (x, t) ∼ at 2 √ 2π √g x 3/2 cos  gt2 4x  , gt 2 ≫ 4x (12.11.82) and gives a similar result for η (x, t), when x < 0 and t → ∞. 12.12 Hankel Transforms We introduce the definition of the Hankel transform from the two-dimensional Fourier transform and its inverse given by F {f (x, y)} = F (k,l) = 1 2π  ∞ −∞  ∞ −∞ exp {−i(κ · r)} f (x, y) dx dy, (12.12.1) F −1 {F (k,l)} = f (x, y) = 1 2π  ∞ −∞  ∞ −∞ exp {i(κ · r)} F (k,l) dk dl, (12.12.2) where r = (x, y) and κ = (k,l). Introducing polar coordinates (x, y) = r (cos θ,sin θ) and (k,l) = κ (cos φ,sin φ), we find κ · r = κr cos (θ − φ) and then F (κ, φ) = 1 2π  ∞ 0 rdr  2π 0 exp [−iκr cos (θ − φ)] f (r, θ) dθ. (12.12.3) 12.12 Hankel Transforms 489 We next assume f (r, θ) = exp (inθ) f (r), which is not a very severe restriction, and make a change of variable θ−φ = α− π 2 to reduce (12.12.3) to the form F (κ, φ) = 1 2π  ∞ 0 rf (r) dr ×  2π+φ0 φ0 exp " in 4 φ − π 2 5 + i(nα − κr sin α) # dα, (12.12.4) where φ0 = π 2 − φ. We use the integral representation of the Bessel function of order n Jn (κr) = 1 2π  2π+φ0 φ0 exp [i(nα − κr sin α)] dα (12.12.5) so that integral (12.12.4) becomes F (κ, φ) = exp " in 4 φ − π 2 5#  ∞ 0 rJn (κr) f (r) dr (12.12.6) = exp " in 4 φ − π 2 5# ˜fn (κ), (12.12.7) where ˜fn (κ) is called the Hankel transform of f (r) and is defined formally by Hn {f (r)} = ˜fn (κ) =  ∞ 0 rJn (κr) f (r) dr. (12.12.8) Similarly, in terms of the polar variables with the assumption f (x, y) = f (r, θ) = e inθf (r) and with result (12.12.7), the inverse Fourier transform (12.12.2) becomes e inθf (r) = 1 2π  ∞ 0 κ dκ  2π 0 exp [iκr cos (θ − φ)] F (κ, φ) dφ = 1 2π  ∞ 0 κ ˜fn (κ) dκ  2π 0 exp " in 4 φ − π 2 5 + iκr cos (θ − φ) # dφ, which is, by the change of variables θ −φ = −  α + π 2 and θ0 = −  θ + π 2 , = 1 2π  ∞ 0 κ ˜fn (κ) dκ  2π+θ0 θ0 exp [in (θ + α) − iκr sin α] dα = e inθ  ∞ 0 κJn (κr) ˜fn (κ) dκ, by (12.12.5). (12.12.9) Thus, the inverse Hankel transform is defined by H−1 n " ˜fn (κ) # = f (r) =  ∞ 0 κ Jn (κr) ˜fn (κ) dκ. (12.12.10) 490 12 Integral Transform Methods with Applications Instead of ˜fn (κ), we often simply write ˜f (κ) for the Hankel transform specifying the order. Integrals (12.12.8) and (12.12.10) exist for certain large classes of functions, which usually occur in physical applications. Alternatively, the famous Hankel integral formula (Watson, 1966, p 453) f (r) =  ∞ 0 κJn (κr) dκ  ∞ 0 p Jn (κp) f (p) dp, (12.12.11) can be used to define the Hankel transform (12.12.8) and its inverse (12.12.10). In particular, the Hankel transforms of zero order (n = 0) and of order one (n = 1) are often useful for the solution of problems involving Laplace’s equation in an axisymmetric cylindrical geometry. Example 12.12.1. Obtain the zero-order Hankel transforms of (a) r −1 exp (−ar), (b) δ(r) r , (c) H (a − r), where H (r) is the Heaviside unit step function. (a) f50 (κ) = H0 &1 r exp (−ar) ' =  ∞ 0 exp (−ar) J0 (κr) dr = √ 1 κ2+a2 . (b) f50 (κ) = H0 ( δ(r) r ) =  ∞ 0 δ (r) J0 (κr) dr = 1. (c) f50 (κ) = H0 {H (a − r)} =  a 0 rJ0 (κr) dr = 1 κ2  aκ 0 pJ0 (p) dp = 1 κ2 [pJ1 (p)]aκ 0 = a κ J1 (aκ). Example 12.12.2. Find the first-order Hankel transform of the following functions: (a) f (r) = e −ar , (b) f (r) = 1 r e −ar . (a) ˜f (κ) = H1 {e −ar} =  ∞ 0 re−arJ1 (κr) dr = κ (a2+κ2) 3 2 . (b) ˜f (κ) = H1 ( e −ar r ) =  ∞ 0 e −arJ1 (κr) dr = 1 κ " 1 − a  κ 2 + a 2 − 1 2 # . Example 12.12.3. Find the nth-order Hankel transforms of (a) f (r) = r nH (a − r), (b) f (r) = r n exp  −ar2 . 12.13 Properties of Hankel Transforms and Applications 491 (a) ˜f (κ) = Hn [r nH (a − r)] =  a 0 r n+1Jn (κr) dr = a n+1 κ Jn+1 (aκ). (b) ˜f (κ) = Hn r n exp  −ar2 ! =  ∞ 0 r n+1Jn (κr) exp  −ar2 dr = κ n (2a) n+1 exp 4 − κ 2 4a 5 . 12.13 Properties of Hankel Transforms and Applications We state the following properties of the Hankel transforms: (i) The Hankel transform operator, Hn is a linear integral operator, that is, Hn {af (r) + bg (r)} = aHn {f (r)} + bHn {g (r)} for any constants a and b. (ii) The Hankel transform satisfies the Parseval relation  ∞ 0 rf (r) g (r) dr =  ∞ 0 k ˜f (k) ˜g (k) dk (12.13.1) where ˜f (k) and ˜g (k) are Hankel transforms of f (r) and g (r) respectively. To prove (12.13.1), we proceed formally to obtain  ∞ 0 k ˜f (k) ˜g (k) dk =  ∞ 0 k ˜f (k) dk  ∞ 0 rJn (kr) g (r) dr =  ∞ 0 rg (r) dr  ∞ 0 kJn (kr) ˜f (k) dk =  ∞ 0 rf (r) g (r) dr. (iii) Hn {f ′ (r)} = k 2n " (n − 1) ˜fn+1 (k) − (n + 1) ˜fn−1 (k) # provided rf (r) vanishes as r → 0 and as r → ∞. (iv) Hn  1 r d dr  r df dr  − n 2 r 2 f (r) 0 = −k 2 ˜fn (k) (12.13.2) provided both 4 r df dr5 and rf (r) vanish as r → 0 and as r → ∞. 492 12 Integral Transform Methods with Applications We have, by definition, Hn  1 r d dr  r df dr  − n 2 r 2 f (r) 0 =  ∞ 0 d dr  r df dr  Jn (kr) dr −  ∞ 0 n 2 r 2 rf (r) Jn (kr) dr = r df dr Jn (kr) ∞ 0 −  ∞ 0 kJ′ n (kr) r df dr dr −  ∞ 0 n 2 r 2 [rf (r)] Jn (kr) dr, by partial integration = − [f (r) krJ′ n (kr)]∞ 0 +  ∞ 0 d dr [k rJ′ n (kr)] f (r) dr −  ∞ 0 n 2 r 2 rf (r) Jn (kr) dr, by partial integration which is, by the given assumption and Bessel’s differential equation (8.6.1), = −  ∞ 0  k 2 − n 2 r 2  rf (r) Jn (kr) dr −  ∞ 0 n 2 r 2 [rf (r)] Jn (kr) dr = −k 2  ∞ 0 rf (r) Jn (kr) dr = −k 2Hn {f (r)} = −k 2 ˜fn (k). (v) (Scaling). If Hn {f (r)} = ˜fn (κ), then Hn {f (ar)} = 1 a 2 ˜fn 4κ a 5 , a > 0. (12.13.3) Proof. We have, by definition, Hn {f (ar)} =  ∞ 0 rJn (κr) f (ar) dr = 1 a 2  ∞ 0 s Jn 4κ a s 5 f (s) ds = 1 a 2 ˜fn 4κ a 5 . These results are used very widely in solving partial differential equations in the axisymmetric cylindrical configurations. We illustrate this point by considering the following examples of applications. Example 12.13.1. Obtain the solution of the free vibration of a large circular membrane governed by the initial-value problem ∂ 2u ∂r2 + 1 r ∂u ∂r = 1 c 2 ∂ 2u ∂t2 , 0 <r< ∞,="" t=""> 0, (12.13.4) u (r, 0) = f (r), ut (r, 0) = g (r), 0 ≤ r < ∞, (12.13.5) 12.13 Properties of Hankel Transforms and Applications 493 where c 2 = (T /ρ) = constant, T is the tension in the membrane, and ρ is the surface density of the membrane. Application of the Hankel transform of order zero u˜ (k, t) =  ∞ 0 r u (r, t) J0 (kr) dr to the vibration problem gives d 2u˜ dt2 + k 2 c 2u˜ = 0 u˜ (k, 0) = ˜f (k), u˜t (k, 0) = ˜g (k). The general solution of this transformed system is u˜ (k, t) = ˜f (k) cos (ckt) + g˜ (k) ck sin (ckt). The inverse Hankel transformation gives u (r, t) =  ∞ 0 k ˜f (k) cos (ckt) J0 (kr) dk + 1 c  ∞ 0 g˜ (k) sin (ckt) J0 (kr) dr. (12.13.6) This is the desired solution. In particular, we consider the following initial conditions u (r, 0) = f (r) = A  1 + r 2 a2 1 2 , ut (r, 0) = g (r)=0 so that ˜g (k) = 0 and ˜f (k) = Aa  ∞ 0 rJ0 (kr) dr (a 2 + r 2) 1 2 = Aa k e −ak by means of Example 12.12.1(a). Thus, solution (12.13.6) becomes u (r, t) = Aa  ∞ 0 e −akJ0 (kr) cos (ckt) dk = Aa Re  ∞ 0 e −k(a+ict)J0 (kr) dk = Aa Re ( r 2 + (a + ict) 2 )− 1 2 . (12.13.7) 494 12 Integral Transform Methods with Applications Example 12.13.2. Obtain the steady-state solution of the axisymmetric acoustic radiation problem governed by the wave equation in cylindrical polar coordinates (r, θ, z): c 2∇2u = utt, 0 <r< ∞,="" z=""> 0, t > 0 (12.13.8) uz = f (r, t) on z = 0, (12.13.9) where f (r, t) is a given function and c is a constant. We also assume that the solution is bounded and behaves as outgoing spherical waves. This is referred to as the Sommerfeld radiation condition. We seek a solution of the acoustic radiation potential u = e iωtφ (r, z) so that φ satisfies the Helmholtz equation φrr + 1 r φr + φzz + ω 2 c 2 φ = 0, 0 <r< ∞,="" z=""> 0 (12.13.10) with the boundary condition representing the normal velocity prescribed on the z = 0 plane φz = f (r) on z = 0, (12.13.11) where f (r) is a known function of r. We solve the problem by means of the zero-order Hankel transformation φ˜ (k, z) =  ∞ 0 rJ0 (kr) φ (r, z) dr so that the given differential system becomes φ˜ zz = κ 2φ, z > ˜ 0, φ˜ z = ˜f (k) on z = 0 where κ = k 2 −  ω 2/c2 ! 1 2 . The solution of this system is φ˜ (k, z) = −κ −1 ˜f (k) e −κz , (12.13.12) where κ is real and positive for k > ω/c, and purely imaginary for k < ω/c. The inverse transformation yields the solution φ (r, z) = −  ∞ 0 κ −1 ˜f (k) kJ0 (kr) e −κzdk. (12.13.13) Since the exact evaluation of this integral is difficult, we choose a simple form of f (r) as f (r) = A H (a − r), (12.13.14) 12.14 Mellin Transforms and their Operational Properties 495 where A is a constant and H (x) is the Heaviside unit step function so that ˜f (k) =  a 0 kJ0 (ak) dk = a k J1 (ak). Then the solution for this special case is given by φ (r, z) = −Aa  ∞ 0 κ −1J1 (ak) J0 (kr) e −κzdk. (12.13.15) For an asymptotic evaluation of this integral, we express it in terms of the spherical polar coordinates (R, θ, φ), (x = R sin θ cos φ, y = R sin θ sin φ, z = R cos θ), combined with the asymptotic result J0 (kr) ∼  2 πkr1 2 cos 4 kr − π 4 5 as r → ∞ so that the acoustic potential u = e iωtφ is u ∼ − Aa√ 2 e iωt √ πR sin θ  ∞ 0 J1 (ka) cos 4 kR sin θ − π 4 5 e −kzdk, where z = R cos θ. This integral can be evaluated asymptotically for R → ∞ by using the stationary phase approximation formula (12.7.8) to obtain u ∼ − Aac ωR sin θ J1 (k1a) e i(ωt−ωR/c) , (12.13.16) where k1 = ω/c sin θ is the stationary point. This solution represents the outgoing spherical waves with constant velocity c and decaying amplitude as R → ∞. 12.14 Mellin Transforms and their Operational Properties If f (t) is not necessarily zero for t < 0, it is possible to define the two-sided (or bilateral) Laplace transform f (p) =  ∞ −∞ e −ptf (t) dt. (12.14.1) Then replacing f (x) with e −cxf (x) in Fourier integral formula (6.13.9), we obtain e −cxf (x) = 1 2π  ∞ −∞ e −ikxdk  ∞ −∞ f (t) e −t(c−ik) dt, 496 12 Integral Transform Methods with Applications or f (x) = 1 2π  ∞ −∞ e x(c−ik) dk  ∞ −∞ f (t) e −t(c−ik) dt. Making a change of variable p = c − ik and using definition (12.14.1), we obtain the formal inverse transform after replacing x by t as f (t) = 1 2πi  c+i∞ c−i∞ e pt f (p) dp, c > 0. (12.14.2) If we put e −t = x into (12.14.1) with f (− log x) = g (x) and f (p) ≡ G (p), then (12.14.1)–(12.14.2) become G (p) = M {g (x)} =  ∞ 0 x p−1 g (x) dx, (12.14.3) g (x) = M−1 {G (p)} = 1 2πi  c+i∞ c−i∞ x −pG (p) dp. (12.14.4) The function G (p) is called the Mellin transform of g (x) defined by (12.14.3). The inverse Mellin transformation is given by (12.14.4). We state the following operational properties of the Mellin transforms: (i) Both M and M−1 are linear integral operators, (ii) M[f (ax)] = a −pF (p), (iii) M[x af (x)] = F (p + a), (iv) M[f ′ (x)] = − (p − 1) F (p − 1), provided that f (x) x p−1 !∞ 0 = 0, M[f ′′ (x)] = (p − 1) (p − 2) F (p − 2), ··· ··· ··· ··· ··· ··· , M f (n) (x) ! = (−1)nΓ(p) Γ(p−n) F (p − n), provided limx→0 x p−r−1f (r) (x) = 0, r = 0, 1, 2, ..., (n − 1), (v) M {xf′ (x)} = −pM {f (x)} = −pF (p), provided that [x pf (x)]∞ 0 = 0, M & x 2f ′′ (x) ' = (−1)2  p + p 2 F (p), ··· ··· ··· ··· ··· ··· , M & x nf (n) (x) ' = (−1)n Γ(p+n) Γ(p) F (p). (vi) M ( x d dx n f (x) ) = (−1)n p nF (p), n = 1, 2, .... 12.14 Mellin Transforms and their Operational Properties 497 (vii) Convolution Property M  ∞ 0 f (xξ) g (ξ) dξ = F (p) G (1 − p), M  ∞ 0 f 4 x ξ 5 g (ξ) dξ ξ = F (p) G (p). (viii) If F (p) = M(f (x)) and G (p) = M(g (x)), then, the following convolution result holds: M[f (x) g (x)] = 1 2πi  c+i∞ c−i∞ F (s) G (p − s) ds. In particular, when p = 1, we obtain the Parseval formula  ∞ 0 f (x) g (x) dx = 1 2πi c+i∞ c−i∞ F (s) G (1 − s) ds. The reader is referred to Debnath (1995) for other properties of the Mellin transform. Example 12.14.1. Show that the Mellin transform of (1 + x) −1 is π cosec πp, 0 < Re p < 1. We consider the standard definite integral  1 0 (1 − t) m−1 t p−1 dt = Γ (m) Γ (p) Γ (m + p) , Re p > 0, Re m > 0, and then change the variable t = x 1+x to obtain  ∞ 0 x p−1dx (1 + x) m+p = Γ (m) Γ (p) Γ (m + p) . If we replace m + p by α, this gives M " (1 + x) −α # = Γ (p) Γ (α − p) Γ (α) . Setting α = 1 and using the result Γ (p) Γ (1 − p) = π cosec πp, 0 < Re p < 1, we obtain M " (1 + x) −1 # = π cosec πp, 0 < Re p < 1. 498 12 Integral Transform Methods with Applications Example 12.14.2. Obtain the solution of the boundary-value problem x 2uxx + xux + uyy = 0, 0 ≤ x < ∞, 0 <y< 1,="" u="" (x,="" 0)="0," and="" 1)="⎧" ⎨="" ⎩="" a,="" 0="" ≤="" x="" 1="" 0,=""> 1 , where A is constant. We apply the Mellin transform U (p, y) =  ∞ 0 x p−1u (x, y) dx to reduce the system to the form Uyy + p 2U = 0, 0 <y< 1,="" u="" (p,="" 0)="0," and="" 1)="A" ="" 1="" 0="" x="" p−1="" dx="A" p="" .="" the="" solution="" of="" this="" differential="" system="" is="" y)="A" sin="" (py)="" ,="" <="" re="" 1.="" inverse="" mellin="" transform="" gives="" (x,="" 2πi="" c+i∞="" c−i∞="" −p="" dp,="" where="" analytic="" in="" a="" vertical="" strip="" p<π="" hence,="" <c<π.="" integrand="" has="" simple="" poles="" at="" r="1," 2,="" 3,="" ...="" which="" lie="" inside="" semi-circular="" contour="" right="" half-plane.="" application="" theory="" residues="" for=""> 1 u (x, y) = A π ∞ r=1 (−1)r x −rπ r sin (rπy). Example 12.14.3. Find the Mellin transform of the Weyl fractional integral ω (x, α) = Wα [f (ξ)] = 1 Γ (α)  ∞ x f (ξ) (ξ − x) α−1 dξ. We rewrite the Weyl integral by setting k (x) = x α f (x), g (x) = 1 Γ (α) (1 − x) α−1 H (1 − x), so that 12.15 Finite Fourier Transforms and Applications 499 ω (x, α) =  ∞ 0 k (ξ) g  x ξ  dξ ξ . The Mellin transform of this result is obtained by the convolution property (vii): Ω (p, a) = K (p) G (p), where K (p) = M[k (x)] = M[x αf (x)] = F (p + α) and G (p) = 1 Γ (α)  1 0 (1 − x) α−1 x p−1 dx = Γ (p) Γ (p + α) . Thus, Ω (p, a) = M[Wαf (ξ)] = Γ (p) Γ (p + α) F (p + α). (12.14.5) If α is complex with a positive real part such that n − 1 < Re α 0, u (x, 0) = 0, ut (x, 0) = 0, 0 < x < π, (12.15.7) u (0, t)=0, u (π, t)=0, t > 0. Applying the finite Fourier sine transform to the equation of motion with respect to x gives Fs utt − c 2uxx − f (x, t) ! = 0. Due to its linearity (see Problem 52, 12.18 Exercises), this can be written in the form Fs [utt] − c 2Fs [uxx] = Fs [f (x, t)] . (12.15.8) Let U (n, t) be the finite Fourier sine transform of u (x, t). Then we have Fs [utt] = 2 π  π 0 utt sin nx dx = d 2 dt2 2 π  π 0 u (x, t) sin nx dx = d 2Us (n, t) dt2 . We also have, from Theorem 12.15.1, Fs [uxx] = 2n π [u (0, t) − (−1)n u (π, t)] − n 2Us (n, t). Because of the boundary conditions u (0, t) = u (π, t)=0, Fs [uxx] becomes Fs [uxx] = −n 2Us (n, t). If we denote the finite Fourier sine transform of f (x, t) by Fs (n, t) = 2 π  π 0 f (x, t) sin nx dx, then, equation (12.15.8) takes the form d 2Us dt2 + n 2 c 2Us = Fs (n, t). This is a second-order ordinary differential equation, the solution of which is given by 502 12 Integral Transform Methods with Applications Us (n, t) = A cos nct + B sin nct + 1 nc  t 0 Fs (n, τ ) sin nc (t − τ ) dτ. Applying the initial conditions Fs [u (x, 0)] = 2 π  π 0 u (x, 0) sin nx dx = Us (n, 0) = 0, and Fs [ut (x, 0)] = d dtUs (n, 0) = 0, we have Us (n, t) = 1 nc  t 0 Fs (n, τ ) sin nc (t − τ ) dτ. Thus, the inverse transform of Us (n, t) is u (x, t) = ∞ n=1 Us (n, t) sin nx = ∞ n=1 1 nc  t 0 Fs (n, τ ) sin nc (t − τ ) dτ sin nx. In the case when f (x, t) = h which is a constant, then Fs [h] = 2 π  π 0 h sin nx dx = 2h nπ [1 − (−1)n ] . Now, we evaluate Us (n, t) = 1 nc  t 0 2h nπ [1 − (−1)n ] sin nc (t − τ ) dτ = 2h n3πc2 [1 − (−1)n ] (1 − cos nct). Hence, the solution is given by u (x, t) = 2h πc2 ∞ n=1 [1 − (−1)n ] n3 (1 − cos nct) sin nx. Example 12.15.2. Find the temperature distribution in a rod of length π. The heat is generated in the rod at the rate g (x, t) per unit time. The ends are insulated. The initial temperature distribution is given by f (x). The problem is to find the temperature function u (x, t) of the system ut = uxx + g (x, t), 0 < x < π, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ π, ux (0, t)=0, ux (π, t)=0, t ≥ 0. 12.15 Finite Fourier Transforms and Applications 503 Let Us (n, t) be the finite Fourier cosine transform of u (x, t). As before, transformation of the heat equation with respect to x, using the boundary conditions, yields dUs dt = −n 2Us + Gs (n, t), where Gs (n, t) = 2 π  π 0 g (x, t) cos nx dx. Rewriting this equation, we obtain d dt 4 e n 2 t Us 5 = Gs e n 2 t . Thus, the solution is Us (n, t) =  t 0 e −n 2 (t−τ)Gs (n, τ ) dτ + A e−n 2 t . Transformation of the initial condition gives Us (n, 0) = 2 π  π 0 u (x, 0) cos nx dx = 2 π  π 0 f (x) cos nx dx. Hence, Us (n, t) takes the form Us (n, t) =  t 0 e −n 2 (t−τ)Gs (n, τ ) dτ + Us (n, 0) e −n 2 t . The solution u (x, t), therefore, is given by u (x, t) = Us (0, 0) 2 + ∞ n=1 Us (n, t) cos nx. Example 12.15.3. A rod with diffusion constant κ contains a fuel which produces neutrons by fission. The ends of the rod are perfectly reflecting. If the initial neutron distribution is f (x), find the neutron distribution u (x, t) at any subsequent time t. The problem is governed by ut = κuxx + bu, u (x, 0) = f (x), 0 < x < l, t > 0, ux (0, t) = ux (l, t)=0. If U (n, t) is the finite Fourier cosine transform of u (x, t), then by transforming the equation and using the boundary conditions, we obtain 504 12 Integral Transform Methods with Applications Ut +  κn2 − b U = 0. The solution of this equation is U (n, t) = C e−(κn2−b)t where C is a constant. Then applying the initial condition, we obtain U (n, t) = U (n, 0) e −(κn2−b)t , where U (n, 0) = 2 l  l 0 f (x) cos nx dx. Thus, the solution takes the form u (x, t) = U (0, 0) 2 + ∞ n=1 U (n, t) cos nx. If for instance f (x) = x in 0 <x<π, then,="" u="" (0,="" 0)="π," and="" (n,="" n2π="" [(−1)n="" −="" 1]="" ,="" n="1," 2,="" 3,...="" the="" solution="" is="" given="" by="" (x,="" t)="π" 2="" +="" ∞="" exp="" &="" ="" κn2="" b="" t="" '="" cos="" nx.="" 12.16="" finite="" hankel="" transforms="" applications="" fourier–bessel="" series="" representation="" of="" a="" function="" f="" (r)="" defined="" in="" 0="" ≤="" r="" can="" be="" stated="" following="" theorem:="" theorem="" 12.16.1.="" if="" fn="" (ki)="Hn" {f="" (r)}="" rf="" jn="" (rki)="" dr,="" (12.16.1)="" then="" {fn="" (ki)}="2" i="1" j="" n+1="" (aki)="" (12.16.2)="" where="" ki="" (0="" <="" k1="" k2="" <...)="" are="" roots="" equation="" 505="" called="" nth-order="" transform="" (r),="" inverse="" (12.16.2).="" particular,="" when="" order="" zero="" its="" integral="" respectively="" j0="" (12.16.3)="" −1="" 1="" .="" (12.16.4)="" example="" find="" n.="" we="" have="" result="" for="" bessel="" ="" n+1jn="" (kir)="" dr="a" jn+1="" (aki),="" so="" that="" hn="" {r="" }="a" (aki).="" h0="" {1}="a" j1="" or,="" equivalently,="" ="" 0="2" kij1="" 12.16.2.="" &a="" '.="" definition="" rj0="" k="" 3="" 2a="" root="" (ax)="0." hence,="" state="" operational="" properties="" transform:="" 506="" 12="" methods="" with="" (i)="" df="" dr0="ki" 2n="" [(n="" 1)="" hn+1="" (n="" hn−1="" (r)}]="" (ii)="" h1="" (iii)="" d="" {rf′="" (a)="" ′="" (12.16.5)="" (iv)="" ′′="" (12.16.6)="" 12.16.3.="" axisymmetric="" heat="" conduction="" ut="κ" ="" urr="" ur="" ="" a,=""> 0, with the boundary and initial conditions u (r, t) = f (t) on r = a, t ≥ 0, u (r, 0) = 0, 0 ≤ r ≤ a, where u (r, t) represents the temperature distribution. We apply the finite Hankel transform defined by U (k, t) = H0 {u (r, t)} =  a 0 rJ0 (kir) u (r, t) dr so that the given equation with the boundary condition becomes dU dt + κk2 i U = κakiJ1 (aki) f (t). The solution of this equation with the transformed initial condition is U (k, t) = aκkiJ1 (aki)  t 0 f (τ ) e −κk2 i (t−τ) dτ. The inverse transformation gives the solution as u (r, t) =  2κ a ∞ i=1 kiJ0 (rki) J1 (aki)  t 0 f (τ ) e −κk2 i (t−τ) dτ. (12.16.7) In particular, if f (t) = T0 = constant, then this solution becomes u (r, t) =  2T0 a ∞ i=1 J0 (rki) kiJ1 (aki) 4 1 − e −κk2 i t 5 . (12.16.8) 12.16 Finite Hankel Transforms and Applications 507 In view of Example 12.16.1, result (12.16.8) becomes u (r, t) = T0 1 1 − 2 a ∞ i=1 J0 (rki) kiJ1 (aki) e −κk2 i t 3 . (12.16.9) This solution consists of the steady-state term, and the transient term which tends to zero as t → ∞. Consequently, the steady-state is attained in the limit t → ∞. Example 12.16.4. (Unsteady Viscous Flow in a Rotating Cylinder). The axisymmetric unsteady motion of a viscous fluid in an infinitely long circular cylinder of radius a is governed by vt = ν  vrr + 1 r vr − v r 2  , 0 ≤ r ≤ a, t > 0, where v = v (r, t) is the tangential fluid velocity and ν is the kinematic viscosity of the fluid. The cylinder is at rest until at t = 0+ it is caused to rotate, so that the boundary and initial conditions are v (r, t) = aΩf (t) H (t) on r = a, v (r, t)=0, at t = 0 for r < a, where f (t) is a physically realistic function of t. We solve the problem by using the joint Laplace and finite Hankel transforms of order one defined by V (km, s) =  a 0 rJ1 (rkm) dr  ∞ 0 e −st v (r, t) dt, where V (km, s) is the Laplace transform of V (km, t), and km are the roots of equation J1 (akm) = 0. Application of the transform yields 4 s ν 5 V (km, s) = −akmV (a, s) J ′ 1 (akm) − k 2 mV (km, s), V (a, s) = a Ω f (s), where f (s) is the Laplace transform of f (t). The solution of this system is V (km, s) = − a 2νkmΩ f (s) J ′ 1 (akm) (s + νk2 m) . The joint inverse transformation gives 508 12 Integral Transform Methods with Applications v (r, t) = 2 a 2 ∞ m=1 J1 (rkm) [J ′ 1 (akm)]2 1 2πi  c+i∞ c−i∞ e st V (km, s) ds = −2νΩ ∞ m=1 kmJ1 (rkm) J ′ 1 (akm) 1 2πi  c+i∞ c−i∞ e st f (s) (s + νk2 m) ds = −2νΩ ∞ m=1 kmJ1 (rkm) J ′ 1 (akm)  t 0 f (τ ) exp −νk2 m (t − τ ) ! dτ, by the Convolution Theorem of the Laplace transform. In particular, when f (t) = cos ωt, the velocity field becomes v (r, t) = −2νΩ ∞ m=1 kmJ1 (rkm) J ′ 1 (akm)  t 0 cos ωτ exp −νk2 m (t − τ ) ! dτ = 2νΩ ∞ m=1 kmJ1 (rkm) J ′ 1 (akm) × 1 νk2 m exp  −νtk2 m −  ω sin ωt + νk2 m cos ωt (ω2 + ν 2k 4 m) 3 = vst (s, t) + vtr (r, t), (12.16.10) where the steady-state flow field vst and the transient flow field vtr are given by vst (r, t) = −2νΩ ∞ m=1 kmJ1 (rkm)  ω sin ωt + νk2 m cos ωt J ′ 1 (akm) (ω2 + ν 2k 4 m) , (12.16.11) vtr (r, t)=2ν 2Ω ∞ m=1 J1 (rkm) k 3 me −νtk2 m J ′ 1 (akm) (ω2 + ν 2k 4 m) . (12.16.12) Thus, the solution consists of the steady-state and transient components. In the limit t → ∞, the latter decays to zero, and the ultimate steady-state is attained and is given by (12.16.11), which has the form vst (r, t) = −2νΩ ∞ m=1 kmJ1 (rkm) cos (ωt − α) J ′ 1 (akm) (ω2 + ν 2k 4 m) 1 2 , (12.16.13) where tan α =  ω/νk2 m . Thus, we see that the steady solution suffers from a phase change of α+π. The amplitude of the motion remains bounded for all values of ω. The frictional couple exerted on the fluid by unit length of the cylinder of radius r = a is given by C =  2π 0 [Prθ] r=a a 2 dθ = 2πa2 [Prθ] r=a , 12.16 Finite Hankel Transforms and Applications 509 where Prθ = µ r (d/dr) (v/r) with µ = νρ calculated from (12.16.10). Thus, C = 4πµΩ 1 −a ∞ m=1 νkmJ1 (akm) νk2 m exp  −νtk2 m −  νk2 m cos ωt + ω sin ωt ! (ω2 + ν 2k 4 m) J ′ 1 (akm) + a 2 ∞ m=1 νk2 m νk2 m exp  −νtk2 m −  νk2 m cos ωt + ω sin ωt ! (ω2 + ν 2k 4 m) J ′ 1 (akm) 3 . (12.16.14) A particular case corresponding to ω = 0 is of special interest. The solution assumes the form v (r, t) = −2Ω ∞ m=1 J1 (rkm) 4 1 − e −νtk2 m 5 km J ′ 1 (akm) = vst + vtr,(12.16.15) where vst and vtr represent the steady-state and the transient flow fields respectively given by vst (r, t) = −2Ω ∞ m=1 J1 (rkm) kmJ ′ 1 (akm) , (12.16.16) vtr (r, t)=2Ω ∞ m=1 J1 (rkm) kmJ ′ 1 (akm) e −νtk2 m. (12.16.17) It follows from (12.16.16) that vst (r, t)=2Ω ∞ m=1 J1 (rkm) kmJ2 (akm) = 2Ω a 2 ∞ m=1 a 2 km J2 (akm) · J1 (rkm) J 2 2 (akm) = ΩH −1 1  a 2 km J2 (akm) 0 = Ωr, by Example 12.16.1. Thus, the steady-state solution has the closed form vst (r, t) = rΩ. (12.16.18) This represents the rigid body rotation of the fluid inside the cylinder. Thus, the final form of (12.16.15) is given by v (r, t) = rΩ − 2Ω ∞ m=1 J1 (rkm) e −νtk2 m kmJ2 (akm) . (12.16.19) In the limit t → ∞, the transients die out and the ultimate steady-state is attained as the rigid body rotation about the axis of the cylinder. 510 12 Integral Transform Methods with Applications 12.17 Solution of Fractional Partial Differential Equations (a) Fractional Diffusion Equation The fractional diffusion equation is given by ∂ αu ∂tα = κ ∂ 2u ∂x2 , x ∈ R, t > 0, (12.17.1) with the boundary and initial conditions u (x, t) → 0 as |x|→∞, (12.17.2) 0D α−1 t u (x, t) ! t=0 = f (x) for x ∈ R, (12.17.3) where κ is a diffusivity constant and 0 < α ≤ 1. Application of the Fourier transform to (12.17.1) with respect to x and using the boundary conditions (12.17.2) and (12.17.3) yields 0Dα t u˜ (k, t) = −κ k2 u, ˜ (12.17.4) 0D α−1 t u˜ (k, t) ! t=0 = ˜f (k), (12.17.5) where ˜u (k, t) is the Fourier transform of u (x, t) defined by (12.2.1). The Laplace transform solution of (12.17.4) and (12.17.5) yields u˜¯ (k, s) = ˜f (k) (s α + κ k2) . (12.17.6) The inverse Laplace transform of (12.17.6) gives u˜ (k, t) = ˜f (k)t α−1Eα,α  −κ k2 t α , (12.17.7) where Eα,β is the Mittag-Leffler function defined by Eα,β (z) = ∞ m=0 z m Γ (αm + β) , α > 0, β> 0. (12.17.8) Finally, the inverse Fourier transform leads to the solution u (x, t) =  ∞ −∞ G (x − ξ, t) f (ξ) dξ, (12.17.9) where G (x, t) = 1 π  ∞ −∞ t α−1Eα,α  −κ k2 t α cos kx dk. (12.17.10) 12.17 Solution of Fractional Partial Differential Equations 511 This integral can be evaluated by using the Laplace transform of G (x, t) as G (x, s) = 1 π  ∞ −∞ cos kx dk s α + κk2 = 1 √ 4κ s −α/2 exp  − |x| √ κ s α/2  , (12.17.11) where L " t mα+β−1E (m) α,β (+atα ) # = m! s α−β (s α + a) m+1 , (12.17.12) and E (m) α,β (z) = d m dzm Eα,β (z). (12.17.13) The inverse Laplace transform of (12.17.11) gives the explicit solution G (x, t) = 1 √ 4κ t α 2 −1W 4 −ξ, − α 2 , α 2 5 , (12.17.14) where ξ = |x| √ κ tα/2 , and W (z, α, β) is the Wright function (see Erd´elyi 1953, formula 18.1 (27)) defined by W (z, α, β) = ∞ n=0 z n n! Γ (αn + β) . (12.17.15) It is important to note that when α = 1, the initial-value problem (12.17.1)–(12.17.3) reduces to the classical diffusion problem and solution (12.17.9) reduces to the classical solution because G (x, t) = 1 √ 4κt W  − x √ κt , − 1 2 , 1 2  = 1 √ 4πκt exp  − x 2 4κt . (12.17.16) The fractional diffusion equation (12.17.1) has also been solved by other authors including Schneider and Wyss (1989), Mainardi (1994, 1995), Debnath (2003) and Nigmatullin (1986) with a physical realistic initial condition u (x, 0) = f (x), x ∈ R. (12.17.17) The solutions obtained by these authors are in total agreement with (12.17.9). It is noted that the order α of the derivative with respect to time t in equation (12.17.1) can be of arbitrary real order including α = 2 so that equation (12.17.1) may be called the fractional diffusion-wave equation. For α = 2, it becomes the classical wave equation. The equation (12.17.1) with 1 < α ≤ 2 will be solved next in some detail. 512 12 Integral Transform Methods with Applications (b) Fractional Nonhomogeneous Wave Equation The fractional nonhomogeneous wave equation is given by ∂ αu ∂tα − c 2 ∂ 2u ∂x2 = q (x, t), x ∈ R, t > 0 (12.17.18) with the initial condition u (x, 0) = f (x), ut (x, 0) = g (x), x ∈ R, (12.17.19) where c is a constant and 1 < α ≤ 2. Application of the joint Laplace transform with respect to t and the Fourier transform with respect to x gives the transform solution u˜¯ (k, s) = ˜f (k) s α−1 s α + c 2k 2 + g˜ (k) s α−2 s α + c 2k 2 + q˜¯(k, s) s α + c 2k 2 , (12.17.20) where k is the Fourier transform variable and s is the Laplace transform variable. The inverse Laplace transform produces the following result: u˜ (k, t) = ˜f (k)L −1  s α−1 s α + c 2k 2 0 + ˜g (k)L −1  s α−2 s α + c 2k 2 0 +L −1  q˜¯(k, s) s α + c 2k 2 0 , (12.17.21) which, by (12.17.12), = ˜f (k) Eα,1  −c 2 k 2 t α + ˜g (k)tEα,2  −c 2 k 2 t α +  t 0 q˜(k, t − τ ) τ α−1Eα,α  −c 2 k 2 τ α dτ. (12.17.22) Finally, the inverse Fourier transform gives the formal solution u (x, t) = 1 √ 2π  ∞ −∞ ˜f (k) Eα,1  −c 2 k 2 t α e ikxdk + 1 √ 2π  ∞ −∞ t g˜ (k) Eα,2  −c 2 k 2 τ α e ikxdk + 1 √ 2π  t 0 τ α−1 dτ  ∞ −∞ q˜(k, t − τ ) Eα,α  −c 2 k 2 τ α e ikxdk. (12.17.23) In particular, when α = 2, the fractional wave equation (12.17.18) reduces to the classical nonhomogeneous wave equation. In this particular case, we use 12.17 Solution of Fractional Partial Differential Equations 513 E2,1  −c 2 k 2 t 2 = cosh (ickt) = cos (ckt), (12.17.24) tE2,2  −c 2 k 2 t 2 = t · sinh (ickt) ickt = 1 ck sin ckt. (12.17.25) Consequently, solution (12.17.23) reduces to solution (12.11.17) for α = 2 as u (x, t) = 1 √ 2π  ∞ −∞ ˜f (k) cos (ckt) e ikxdk + 1 √ 2π  ∞ −∞ g˜ (k) sin (ckt) ck e ikxdk + 1 √ 2π c  t 0 dτ  ∞ −∞ q˜(k, τ ) sin ck (t − τ ) k e ikxdk (12.17.26) = 1 2 [f (x − ct) + f (x + ct)] + 1 2c  x+ct x−ct g (ξ) dξ + 1 2c  t 0 dτ  x+c(t−τ) x−c(t−τ) q (ξ, τ ) dξ. (12.17.27) We now derive the solution of the nonhomogeneous fractional diffusion equation (12.17.18) with c 2 = κ and g (x) = 0. In this case, the joint transform solution (12.17.20) becomes u˜¯ (k, s) = ˜f (k) s α−1 (s α + κk2) + q˜¯(k, s) (s α + κk2) (12.17.28) which is inverted by using (12.17.12) to obtain ˜u (k, t) in the form u˜ (k, t) = ˜f (k) Eα,1  −κ k2 t α +  t 0 (t − τ ) α−1 Eα,α & −κ k2 (t − τ ) α ' q˜(k, τ ) dτ. (12.17.29) Finally, the inverse Fourier transform gives the exact solution u (x, t) = 1 √ 2π  ∞ −∞ ˜f (k) Eα,1  −κ k2 t α e ikxdk + 1 √ 2π  t 0 dτ  ∞ −∞ (t − τ ) α−1 Eα,α & −κ k2 (t − τ ) α ' q˜(k, τ ) e ikxdk. (12.17.30) Application of the Convolution Theorem of the Fourier transform gives the final solution in the form u (x, t) =  ∞ −∞ G1 (x − ξ, t) f (ξ) dξ +  t 0 (t − τ ) α−1 dτ  ∞ −∞ G2 (x − ξ, t − τ ) q (ξ, τ ) dξ, (12.17.31) 514 12 Integral Transform Methods with Applications where G1 (x, t) = 1 2π  ∞ −∞ e ikxEα,1  −κ k2 t α dk, (12.17.32) and G2 (x, t) = 1 2π  ∞ −∞ e ikxEα,α  −κ k2 t α dk. (12.17.33) In particular, when α = 1, the classical solution of the nonhomogeneous diffusion equation (12.17.18) is obtained in the form u (x, t) =  ∞ −∞ G1 (x − ξ, t) f (ξ) dξ +  t 0 dτ  ∞ −∞ G2 (x − ξ, t − τ ) q (ξ, τ ) dξ, (12.17.34) where G1 (x, t) = G2 (x, t) = 1 √ 4πκt exp  − x 2 4κt . (12.17.35) In the case of classical homogeneous diffusion equation (12.17.18), solutions (12.17.30) and (12.17.34) are in perfect agreement with those of Mainardi (1996), who obtained the solution by using the Laplace transform method together with complicated evaluation of the Laplace inversion integral and the auxiliary function M (z,α). He obtained the solution in terms of M  z, α 2 and discussed the nature of the solution for different values of α. He made some comparisons between ordinary diffusion (α = 1) and fractional diffusion  α = 1 2 and α = 2 3 . For cases α = 4 3 and α = 3 2 , the solution exhibits a striking difference from ordinary diffusion with a transition from the Gaussian function centered at z = 0 (ordinary diffusion) to the Dirac delta function centered at z = 1 (wave propagation). This indicates a possibility of an intermediate process between diffusion and wave propagation. A special difference is observed between the solutions of the fractional diffusion equation (0 < α ≤ 1) and the fractional wave equation (1 < α ≤ 2). In addition, the solution exhibits a slow process for the case with 0 < α ≤ 1 and an intermediate process for 1 < α ≤ 2. (c) Fractional-Order Diffusion Equation in Semi-Infinite Medium We consider the fractional-order diffusion equation in a semi-infinite medium x > 0, when the boundary is kept at a temperature u0f (t) and the initial temperature is zero in the whole medium. Thus, the initial boundaryvalue problem is governed by the equation ∂ αu ∂tα = κ ∂ 2u ∂x2 , 0 <x< ∞,="" t=""> 0, (12.17.36) 12.17 Solution of Fractional Partial Differential Equations 515 with u (x, 0) = 0, x > 0, (12.17.37) u (0, t) = u0f (t), t> 0, and u (x, t) → 0 as x → ∞. (12.17.38) Application of the Laplace transform with respect to t gives d 2u dx2 −  s α κ  u (x, s)=0, x > 0, (12.17.39) u (0, s) = u0f (s), u (x, s) → 0 as x → ∞.(12.17.40) Evidently, the solution of this transformed boundary-value problem is u (x, s) = u0 f (s) exp (−ax), (12.17.41) where a = (s α/κ) 1 2 . Thus, the solution (12.17.41) is given by u (x, t) = u0  t 0 f (t − τ ) g (x, τ ) dτ = u0f (t) ∗ g (x, t),(12.17.42) where g (x, t) = L −1 {exp (−ax)} . When α = 1 and f (t) = 1, solution (12.17.41) becomes u (x, s) = u0 s exp  −x 2 s κ  , (12.17.43) which yields the classical solution in terms of the complementary error function (see Debnath 1995) u (x, t) = u0 erfc  x 2 √ κt . (12.17.44) In the classical case (α = 1), the more general solution is given by u (x, t) = u0  t 0 f (t − τ ) g (x, τ ) dτ = u0f (t) ∗ g (x, t),(12.17.45) where g (x, t) = L −1  exp  −x 2 s κ 0 = x 2 √ πκt3 exp  − x 2 4κt . (12.17.46) (d) The Fractional Stokes and Rayleigh Problems in Fluid Dynamics The classical Stokes problem (see Debnath 1995) deals with the unsteady boundary layer flows induced in a semi-infinite viscous fluid bounded 516 12 Integral Transform Methods with Applications by an infinite horizontal disk at z = 0 due to nontorsional oscillations of the disk in its own plane with a given frequency ω. When ω = 0, the Stokes problem reduces to the classical Rayleigh problem where the unsteady boundary layer flow is generated in the fluid from rest by moving the disk impulsively in its own plane with constant velocity U. We consider the unsteady fractional boundary layer equation for the fluid velocity u (z, t) that satisfies the equation ∂ αu ∂tα = ν ∂ 2u ∂z2 , 0 <z< ∞,="" t=""> 0, (12.17.47) with the given boundary and initial conditions u (0, t) = Uf (t), u (z, t) → 0 as z → ∞, t> 0, (12.17.48) u (z, 0) = 0 for all z > 0, (12.17.49) where ν is the kinematic viscosity, U is a constant velocity, and f (t) is an arbitrary function of time t. Application of the Laplace transform with respect to t gives s α u (z, s) = ν d 2u dz2 , 0 <z< ∞,="" (12.17.50)="" u="" (0,="" s)="U" f="" (s),="" (z,="" →="" 0="" as="" z="" ∞.="" (12.17.51)="" use="" of="" the="" fourier="" sine="" transform="" (see="" debnath="" 1995)="" with="" respect="" to="" yields="" us="" (k,="" 2="" π="" ν="" u5="" k="" (s)="" (s="" α="" +="" νk2)="" .="" (12.17.52)="" inverse="" leads="" solution="" u="" ="" ∞="" sin="" kz="" dk,="" (12.17.53)="" and="" laplace="" gives="" for="" velocity="" t)="" dk="" t="" (t="" −="" τ="" )="" α−1eα,α="" ="" −νk2="" dτ.="" (12.17.54)="" when="" (t)="exp" (iωt),="" fractional="" stokes="" problem="" is="" 2νu="" ="" e="" iωt="" −iωτ="" (12.17.55)="" 12.17="" partial="" differential="" equations="" 517="" reduces="" classical="" 4="" 1="" −νtk2="" 5="" (iω="" dk.="" (12.17.56)="" rayleigh="" problem,="" follows="" from="" in="" form="" (12.17.57)="" this="" e1,1="" −ντk2="" dτ="" exp="" 2u="" which="" (by="" (2.10.10)="" 1995),="" π="" erf="" ="" √="" νt="Uerfc" (12.17.58)="" above="" analysis="" full="" agreement="" solutions="" problems="" 1995).="" (e)="" unsteady="" couette="" flow="" we="" consider="" viscous="" fluid="" between="" plate="" at="" rest="" motion="" parallel="" itself="" a="" variable="" x-direction.="" satisfies="" equation="" ∂="" αu="" ∂tα="P" ∂z2="" ,="" ≤="" h,=""> 0, (12.17.59) with the boundary and initial conditions u (0, t) = 0 and u (h, t) = U (t), t > 0, (12.17.60) u (z, t) = 0 at t ≤ 0 for 0 ≤ z ≤ h, (12.17.61) where − 1 ρ px = P (t) and ν is the kinematic viscosity of the fluid. We apply the joint Laplace transform with respect to t and the finite Fourier sine transform with respect to z defined by u¯˜s (n, s) =  ∞ 0 e −stdt  h 0 u (z, t) sin 4nπz h 5 dz (12.17.62) 518 12 Integral Transform Methods with Applications to the system (12.17.59)–(12.17.61) so that the transform solution is u¯˜s (n, s) = P (s) 1 a [1 − (−1)n ] (s α + νa2) + νa (−1)n+1 U (s) (s α + νa2) , (12.17.63) where a =  nπ h and n is the finite Fourier sine transform variable. Thus, the inverse Laplace transform yields u˜s (n, t) = 1 a [1 − (−1)n ]  t 0 P (t − τ ) τ α−1Eα,α  −νa2 τ α dτ +νa (−1)n+1  t 0 U (t − τ ) τ α−1Eα,α  −νa2 τ α dτ. (12.17.64) Finally, the inverse finite Fourier sine transform leads to the solution u (z, t) = 2 h ∞ n=1 u˜s (n, t) sin 4nπz h 5 . (12.17.65) In particular, when α = 1, P (t) = constant, and U (t) = constant, then solution (12.17.65) reduces to the solution of the generalized Couette flow (see p. 277 Debnath 1995). (f) Fractional Axisymmetric Wave-Diffusion Equation The fractional axisymmetric wave-diffusion equation in an infinite domain ∂ αu ∂tα = a  ∂ 2u ∂r2 + 1 r ∂u ∂r  , 0 <r< ∞,="" t=""> 0, (12.17.66) is called the diffusion or wave equation accordingly as a = κ or a = c 2 . For the fractional diffusion equation, we prescribe the initial condition u (r, 0) = f (r), 0 <r< ∞.="" (12.17.67)="" application="" of="" the="" joint="" laplace="" transform="" with="" respect="" to="" t="" and="" hankel="" zero="" order="" (see="" section="" 12.12)="" r="" (12.17.66)="" gives="" solution="" u¯˜="" (k,="" s)="s" α−1="" ˜f="" (k)="" (s="" α="" +="" κk2)="" ,="" (12.17.68)="" where="" k,="" s="" are="" variables="" respectively.="" inverse="" leads="" u="" (r,="" t)="" ∞="" 0="" kj0="" (kr)="" eα,1="" ="" −κk2="" dk,="" (12.17.69)="" 12.17="" fractional="" partial="" differential="" equations="" 519="" j0="" is="" bessel="" function="" first="" kind="" zero-order="" f="" (r).="" when="" reduces="" classical="" that="" was="" obtained="" by="" debnath="" p="" 66,="" 2005).="" on="" other="" hand,="" we="" can="" solve="" wave="" equation="" a="c" 2="" initial="" conditions="" 0)="f" (r),="" ut="" (r)="" for="" <r<="" ∞,="" (12.17.70)="" provided="" transforms="" g="" exist.="" c="" 2k="" 2)="" α−2="" g˜="" .="" (12.17.71)="" transformation="" r)="" −c="" k="" dk="" ="" ˜g="" (k)teα,2="" dk.="" (12.17.72)="" (12.13.6).="" in="" finite="" domain="" ≤="" a,="" diffusion="" has="" boundary="" data="" (t)=""> 0, (12.17.73) u (r, 0) = 0 for all r in (0, a). (12.17.74) Application of the joint Laplace and finite Hankel transform of zero order (see pp. 317, 318, Debnath 1995) yields the solution u (r, t) = 2 a 2 ∞ i=1 u˜ (ki , t) J0 (rki) J 2 1 (aki) , (12.17.75) where u˜ (ki , t)=(aκ ki) J1 (aki)  t 0 f (t − τ ) τ α−1Eα,α  −κ k2 i τ α dτ. (12.17.76) When α = 1, (12.17.75) reduces to (12.16.7). Similarly, the fractional wave equation (12.17.66) with a = c 2 in a finite domain 0 ≤ r ≤ a with the boundary and initial conditions u (r, t) = 0 on r = a, t > 0, (12.17.77) u (r, 0) = f (r) and ut (r, 0) = g (r) for 0 < r < a, (12.17.78) 520 12 Integral Transform Methods with Applications can be solved by means of the joint Laplace and finite Hankel transforms. The solution of this problem is u (r, t) = 2 a 2 ∞ i=1 u˜ (ki , t) J0 (rki) J 2 1 (aki) , (12.17.79) where u˜ (ki , t) = ˜f (ki) Eα,1  −c 2 k 2 i t α + ˜g (ki)tEα,2  −c 2 k 2 i t α . (12.17.80) When α = 2, solution (12.17.79) reduces to the solution (11.4.26) obtained by Debnath (1995). (g) The Fractional Schr¨odinger Equation in Quantum Mechanics The one-dimensional fractional Schr¨odinger equation for a free particle of mass m is i ∂ αψ ∂tα = −  2 2m ∂ 2ψ ∂x2 , −∞ <x< ∞,="" t=""> 0, (12.17.81) ψ (x, 0) = ψ0 (x), −∞ <x< ∞,="" (12.17.82)="" ψ="" (x,="" t)="" →="" 0="" as="" |x|→∞,="" (12.17.83)="" where="" is="" the="" wave="" function,="" h="2π" =="" 6.625="" ×="" 10−27erg="" sec="4.14" 10−21mev="" planck="" constant,="" and="" ψ0="" (x)="" an="" arbitrary="" function.="" application="" of="" joint="" laplace="" fourier="" transform="" to="" (12.17.81)–="" gives="" solution="" in="" space="" form="" Ψ="" (k,="" s)="s" α−1Ψ0="" (k)="" s="" α="" +="" ak2="" ,="" a="i" 2m="" (12.17.84)="" k,="" represent="" transforms="" variables.="" use="" inverse="" yields="" √="" 2π="" ="" ∞="" −∞="" e="" ikxψ˜="" eα,1="" ="" −ak2="" t="" dk.="" (12.17.85)="F" −1="" (="" ψ˜="" )="" (12.17.86)="" which="" is,="" by="" theorem="" 12.4.1,="" convolution="" g="" (x="" −="" ξ,="" (ξ)="" dξ,="" (12.17.87)="" f="" &="" '="1" ikxeα,1="" (12.17.88)="" 12.18="" exercises="" 521="" when="" becomes="" (12.17.89)="" green’s="" function="" given="" ikxe1,1="" dk="1" exp="" ikx="" atk2="" 4πat="" ="" x="" 2="" 4at="" .="" (12.17.90)="" this="" perfect="" agreement="" with="" classical="" obtained="" debnath="" (1995).="" 1.="" find="" (a)="" −ax2="" (b)="" (−a="" |x|),="" constant.="" 2.="" gate="" fa="" ⎨="" ⎩="" 1,="" |x|="" <="" a,="" positive="" 0,="" ≥="" a.="" 3.="" −a<x<a="" otherwise,="" (c)="" 1="" ≤="" (d)="" (x2+a2)="" 4.="" 522="" 12="" integral="" methods="" applications="" 5.="" show="" that="" i="" −a="" 2x="" dx="√" π="" 2a,=""> 0, by noting that I 2 =  ∞ 0  ∞ 0 e −a 2 (x 2+y 2 )dx dy =  π/2 0  ∞ 0 e −a 2 r 2 r dr dθ. 6. Show that  ∞ 0 e −a 2x 2 cos bx dx = √ π/2a e −b 2/4a 2 , a > 0. 7. Prove that (a) f (x) = √ 1 2π  ∞ −∞ e ikxF (k) dk = F −1 {F (k)}, (b) F [f (ax − b)] = 1 |a| e ikb/aF (k/a). 8. Prove the following properties of the Fourier convolution: (a) f (x) ∗ g (x) = g (x) ∗ f (x), (b) f ∗ (g ∗ h)=(f ∗ g) ∗ h, (c) f ∗ (ag + bh) = a (f ∗ g) + b (f ∗ h), where a and b are constants, (d) f ∗ 0=0 ∗ f = 0, (e) f ∗ 1 = f, (f) f ∗ √ 2π δ = f = √ 2π δ ∗ f, (g) F {f (x) g (x)} = (F ∗ G) (k) = √ 1 2π  ∞ −∞ F (k − ξ) G (ξ) dξ, 9. Prove the following properties of the Fourier convolution: (a) d dx {f (x) ∗ g (x)} = f ′ (x) ∗ g (x) = f (x) ∗ g ′ (x), (b) d 2 dx2 [(f ∗ g) (x)] = (f ′ ∗ g ′ ) (x)=(f ′′ ∗ g) (x), (c) (f ∗ g) (m+n) (x) = f (m) ∗ g (n) ! (x), (d)  ∞ −∞ (f ∗ g) (x) dx =  ∞ −∞ f (u) du ∞ −∞ g (v) dv. (e) If g (x) = 1 2a H (a − x), then (f ∗ g) (x) is the average value of f (x) in [x − a, x + a]. 12.18 Exercises 523 (f) If Gt (x) = √ 1 4πkt exp 4 − x 2 4κt5 , then (Gt ∗ Gs) (x) = Gt+s (x). 10. Prove the following results: (a) √ 1 2π  ∞ −∞ e −k 2 t−ikx dk = √ 1 2t e −x 2/4t , (b)  ∞ −∞ F (k) g (k) e ikx dk =  ∞ −∞ f (y) G (y − x) dy, (c)  ∞ −∞ F (k) g (k) dk =  ∞ −∞ f (y) G (y) dy, (d) sin x ∗ e −a|x| = % 2 π a sin x (1+a2) , (e) e ax ∗ χ[0,∞) (x) = 1 a e ax √ 2π , a> 0 , (f) √ 1 2a exp 4 − x 2 4a 5 ∗ √ 1 2b exp 4 − x 2 4b 5 = √ 1 2(a+b) exp 4 − x 2 4(a+b) 5 . 11. Determine the solution of the initial-value problem utt = c 2uxx, −∞ <x< ∞,="" t=""> 0, u (x, 0) = f (x), ut (x, 0) = g (x), −∞ <x< ∞.="" 12.="" solve="" ut="uxx," x=""> 0, t > 0, u (x, 0) = f (x), u (0, t)=0. 13. Solve utt = c 2uxxxx = 0, −∞ <x< ∞,="" t=""> 0, u (x, 0) = f (x), ut (x, 0) = 0, −∞ <x< ∞.="" 14.="" solve="" utt="" +="" c="" 2uxxxx="0," x=""> 0, t > 0, u (x, 0) = 0, ut (x, 0) = 0, x > 0, u (0, t) = g (t), uxx (0, t)=0, t > 0. 15. Solve φxx + φyy = 0, −a < x < a, 0 <y< ∞,="" φy="" (x,="" 0)="⎧" ⎨="" ⎩="" δ0,="" 0="" <="" |x|="" a,="" 0,=""> a. φ (x, y) → 0 uniformly in x as y → ∞. 524 12 Integral Transform Methods with Applications 16. Solve ut = uxx + t u, −∞ <x< ∞,="" t=""> 0, u (x, 0) = f (x), u (x, t) is bounded, −∞ <x< ∞.="" 17.="" solve="" ut="" −="" uxx="" +="" hu="δ" (x)="" δ="" (t),="" −∞="" <x<="" ∞,="" t=""> 0, u (x, 0) = 0, u (x, t) → 0 uniformly in t as |x|→∞. 18. Solve ut − uxx + h (t) ux = δ (x) δ (t), 0 <x< ∞,="" t=""> 0, u (x, 0) = 0, ux (0, t)=0, u (x, t) → 0 uniformly in t as x → ∞. 19. Solve uxx + uyy = 0, 0 <x< ∞,="" 0="" <y<="" u="" (x,="" 0)="f" (x),="" ≤="" x="" <="" ux="" (0,="" y)="g" (y),="" y="" →="" uniformly="" in="" as="" ∞="" and="" ∞.="" 20.="" solve="" uxx="" +="" uyy="0," −∞="" <x<="" a,="" a)="0," |x|→∞.="" 21.="" ut="uxx,"> 0, t> 0, u (x, 0) = 0, x > 0, u (0, t) = f (t), t> 0, u (x, t) is bounded for all x and t. 22. Solve uxx + uyy = 0, x > 0, 0 <y< 1,="" u="" (x,="" 0)="f" (x),="" 1)="0," x=""> 0, u (0, y)=0, u (x, y) → 0 uniformly in y as x → ∞. 12.18 Exercises 525 23. Find the Laplace transform of each of the following functions: (a) t n, (b) cos ωt, (c) sinh kt, (d) cosh kt, (e) teat, (f) e at sin ωt, (g) e at cos ωt, (h) tsinh kt, (i) t cosh kt, (j) % 1 t , (k) √ t, (l) sin at t . 24. Find the inverse transform of each of the following functions: (a) s (s 2+a2)(s 2+b 2) , (b) 1 (s 2+a2)(s 2+b 2) , (c) 1 (s−a)(s−b) , (d) 1 s(s+a) 2 , (e) 1 s(s+a) , (f) s 2−a 2 (s 2+a2) 2 . 25. The velocity potential φ (x, z, t) and the free-surface evaluation η (x, t) for surface waves in water of infinite depth satisfy the Laplace equation φxx + φzz = 0, −∞ <x< ∞,="" −∞="" <="" z="" ≤="" 0,="" t=""> 0, with the free-surface, boundary, and initial conditions φz = ηt on z = 0, t > 0, φt + gη = 0 on z = 0, t > 0, φz → 0 as z → −∞, φ (x, 0, 0) = 0 and η (x, 0) = f (x), −∞ <x< ∞,="" where="" g="" is="" the="" constant="" acceleration="" due="" to="" gravity.="" show="" that="" φ="" (x,="" z,="" t)="−" √g="" √="" 2π="" ="" ∞="" −∞="" k="" −="" 1="" 2="" f="" (k)="" e="" |k|z−ikx="" sin="" 4="" |k|t="" 5="" dk,="" η="" −ikx="" cos="" represents="" fourier="" transform="" variable.="" find="" asymptotic="" solution="" for="" as="" t="" →="" ∞.="" 26.="" use="" method="" of="" one-dimensional="" schr¨odinger="" equation="" a="" free="" particle="" mass="" m,="" iψt="−" 2m="" ψxx,="" <x<=""> 0, ψ (x, 0) = f (x), −∞ <x< ∞,="" 526="" 12="" integral="" transform="" methods="" with="" applications="" where="" ψ="" and="" ψx="" tend="" to="" zero="" as="" |x|→∞,="" h="2π" is="" the="" planck="" constant,="" given="" by="" (x,="" t)="1" √="" 2π="" ="" ∞="" −∞="" f="" (ξ)="" g="" (x="" −="" ξ)="" dξ,="" 2="" γt="" exp="" "="" x="" 4iγt="" #="" green’s="" function="" γ="" 2m="" .="" 27.="" prove="" following="" properties="" of="" laplace="" convolution:="" (a)="" ∗="" f,="" (b)="" (g="" h)="(f" g)="" h,="" (c)="" (αg="" +="" βh)="α" (f="" β="" h),="" α="" are="" constants,="" (d)="" 0="0" (e)="" d="" dt="" [(f="" (t)]="f" ′="" (t)="" (0)="" (t),="" (f)="" dt2="" ′′="" (g)="" n="" dtn="" (n)="" n−1="" k="0" (k)="" (n−k−1)="" (t).="" 28.="" obtain="" solution="" problem="" utt="c" 2uxx,="" <x<="" t=""> 0, u (x, 0) = f (x), ut (x, 0) = 0, u (0, t)=0, u (x, t) → 0 uniformly in t as x → ∞. 29. Solve utt = c 2uxx, 0 < x < l, t > 0, u (x, 0) = 0, ut (x, 0) = 0, u (0, t) = f (t), u (l, t)=0, t ≥ 0. 30. Solve ut = κuxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = f0, 0 <x< ∞,="" u="" (0,="" t)="f1," (x,="" →="" f0="" uniformly="" in="" t="" as="" x=""> 0. 31. Solve ut = κuxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = x, x > 0, u (0, t)=0, u (x, t) → x uniformly in t as x → ∞, t> 0. 12.18 Exercises 527 32. Solve ut = κuxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = 0, 0 <x< ∞,="" u="" (0,="" t)="t" 2="" ,="" (x,="" →="" 0="" uniformly="" in="" t="" as="" x="" ≥="" 0.="" 33.="" solve="" ut="κuxx" −="" hu,="" <x<=""> 0, h = constant, u (x, 0) = f0, x > 0, u (0, t)=0, ux (0, t) → 0 uniformly in t as x → ∞, t> 0. 34. Solve ut = κuxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = 0, 0 <x< ∞,="" u="" (0,="" t)="f0," (x,="" →="" 0="" uniformly="" in="" t="" as="" x=""> 0. 35. Solve utt = c 2uxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = 0, ut (x, 0) = f0, 0 <x< ∞,="" u="" (0,="" t)="0," ux="" (x,="" →="" 0="" uniformly="" in="" t="" as="" x=""> 0. 36. Solve utt = c 2uxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = f (x), ut (x, 0) = 0, 0 <x< ∞,="" u="" (0,="" t)="0," ux="" (x,="" →="" 0="" uniformly="" in="" t="" as="" x=""> 0. 37. A semi-infinite lossless transmission line has no initial current or potential. A time dependent EMF, V0 (t) H (t) is applied at the end x = 0. Find the potential V (x, t). Then determine the potential for cases: (i) V0 (t) = V0 = constant, and (ii) V0 (t) = V0 cos ωt. 38. Solve the Blasius problem of an unsteady boundary layer flow in a semiinfinite body of viscous fluid enclosed by an infinite horizontal disk at z = 0. The governing equation, boundary, and initial conditions are ∂u ∂t = ν ∂ 2u ∂z2 , z> 0, t > 0, u (z, t) = U t on z = 0, t > 0, u (z, t) → 0 as z → ∞, t > 0, u (z, t) = 0 at t ≤ 0, z > 0. Explain the implication of the solution. 528 12 Integral Transform Methods with Applications 39. The stress-strain relation and equation of motion for a viscoelastic rod in the absence of external force are ∂e ∂t = 1 E ∂σ ∂t + σ η , ∂σ ∂x = ρ ∂ 2u ∂t2 , where e is the strain, η is the coefficient of viscosity, and the displacement u (x, t) is related to the strain by e = ∂u/∂x. Prove that the stress σ (x, t) satisfies the modified wave equation ∂ 2σ ∂x2 − ρ η ∂σ ∂t = 1 c 2 ∂ 2σ ∂t2 , c2 = E/ρ. Show that the stress distribution in a semi-infinite viscoelastic rod subject to the boundary and initial conditions, u˙ (0, t) = U H (t), σ (x, t) → 0 as x → ∞, t > 0, σ (x, 0) = 0, u˙ (x, 0) = 0, is given by σ (x, t) = −U ρc exp  − Et 2η  I0 1 Et 2η  t 2 − x 2 c 2 1 2 3 H 4 t − x c 5 . 40. An elastic string is stretched between x = 0 and x = l and is initially at rest in the equilibrium position. Show that the Laplace transform solution for the displacement field subject to the boundary conditions y (0, t) = f (t) and y (l, t) = 0, t > 0 is y (x, s) = f (s) sinh &s c (l − x) ' sinh sl c . 41. The end x = 0 of a semi-infinite submarine cable is maintained at a potential V0H (t). If the cable has no initial current and potential, determine the potential V (x, t) at point x and at time t. 42. Obtain the solution of the Stokes–Ekman problem (see Debnath, 1995) of an unsteady boundary layer flow in a semi-infinite body of viscous fluid bounded by an infinite horizontal disk at z = 0, when both the fluid and the disk rotate with a uniform angular velocity Ω about the zaxis. The governing boundary layer equation, the boundary conditions, and the initial conditions are ∂q ∂t + 2Ωiq = ν ∂ 2 q ∂z2 , z > 0, t > 0, q (z, t) = aeiωt + be−iωt on z = 0, t > 0, q (z, t) → 0 as z → ∞, t > 0, q (z, t) = 0 at t ≤ 0, for all z > 0, 12.18 Exercises 529 where q = u + iv, is the complex velocity field, ω is the frequency of oscillations of the disk, and a and b are complex constants. Hence, deduce the steady-state solution, and determine the structure of the associated boundary layers. 43. Show that, when ω = 0 in the Stokes–Ekman problem 42, the steady flow field is given by q (z, t) ∼ (a + b) exp / −  2iΩ ν 1 2 z 0 . Hence determine the thickness of the Ekman layer. 44. For problem 14 (e) (iii) in 3.9 Exercises, show that the potential V (x, t) and the current I (x, t) satisfy the partial differential equation  ∂ 2 ∂t2 + 2k ∂ ∂t + k 2  (V, I) = c 2 ∂ 2 ∂x2 (V, I). Find the solution for V (x, t) with the boundary and initial data V (x, t) = V0 (t) at x = 0, t > 0, V (x, t) → 0 as x → ∞, t > 0, V (x, 0) = Vt (x, 0) = 0 for 0 ≤ x < ∞. 45. Use the Laplace transform to solve the Abel integral equation g (t) =  t 0 f ′ (τ ) (t − τ ) −α dτ, 0 <α< 1. 46. Solve Abel’s problem of tautochronous motion described in problem 17 of 14.11 Exercises. 47. The velocity potential φ (r, z, t) and the free-surface elevation η (r, t) for axisymmetric surface waves in water of infinite depth satisfy the equation φrr + 1 r φr + φzz = 0, 0 ≤ r < ∞, −∞ < z ≤ 0, t > 0, with the free-surface, boundary, and initial conditions φz = ηz on z = 0, t > 0, φt + gη = 0, on z = 0, t > 0, φz → 0, z → −∞, φ (r, 0, 0) = 0, and η (r, 0) = f (r), 0 ≤ r < ∞, where g is the acceleration due to gravity and f (r) represents the initial elevation. Show that 530 12 Integral Transform Methods with Applications φ (r, z, t) = − √ g  ∞ 0 √ k f5(k) J0 (kr) e kz sin 4 gk t5 dk, η (r, t) =  ∞ 0 k f5(k) J0 (kr) cos 4 gk t5 dk, where f5(k) is the zero-order Hankel transform of f (r). Derive the asymptotic solution η (r, t) ∼ gt2 2 3 2 r 3 f5  gt2 4r 2  cos  gt2 4r  as t → ∞. 48. Write the solution for the Cauchy–Poisson problem where the initial elevation is concentrated in the neighborhood of the origin, that is, f (r)=(a/2πr) δ (r), where a is the total volume of the fluid displaced. 49. The steady temperature distribution u (r, z) in a semi-infinite solid with z ≥ 0 is governed by the system urr + 1 r ur + uzz = −Aq (r), 0 <r< ∞,="" z=""> 0, u (r, 0) = 0, where A is a constant and q (r) represents the steady heat source. Show that the solution is given by u (r, z) = A  ∞ 0 q5(k) J0 (kr) k −1  1 − e −kz dk, where q5(k) is the zero-order Hankel transform of q (r). 50. Find the solution for the small deflection u (r) of an elastic membrane subjected to a concentrated loading distribution which is governed by urr + 1 r ur − κ 2u = 1 2π δ (r) r , 0 ≤ r < ∞, where u and its derivatives vanish as r → ∞. 51. Obtain the solution for the potential v (r, z) due to a flat electrified disk of radius unity with the center of the disk at the origin and the axis along the z-axis. The function v (r, z) satisfies the Laplace equation vrr + 1 r vr − vzz = 0, 0 <r< ∞,="" z=""> 0, with the boundary conditions v (r, 0) = v0, 0 ≤ r < 1, vz (r, 0) = 0, r > 1. 12.18 Exercises 531 52. Prove that the Fourier sine and cosine transforms are linear. 53. If Fs (n) is the Fourier sine transform of f (x) on 0 ≤ x ≤ l, show that Fs [f ′′ (x)] = 2nπ l 2 [f (0) − (−1)n f (l)] − 4nπ l 52 Fs (n). 54. If Fc (n) is the Fourier cosine transform of f (x) on 0 ≤ x ≤ l, show that Fc [f ′′ (x)] = 2 l [(−1)n f ′ (l) − f ′ (0)] − 4nπ l 52 Fc (n). When l = π, show that Fc [f ′′ (x)] = 2 π [(−1)n f ′ (π) − f ′ (0)] − n 2Fc (n). 55. By the transform method, solve ut = uxx + g (x, t), 0 < x < π, t > 0, u (x, 0) = f (x), 0 ≤ x ≤ π, u (0, t)=0, u (π, t) → 0 t > 0. 56. By the transform method, solve ut = uxx + g (x, t), 0 < x < π, t > 0, u (x, 0) = 0, 0 < x < π, u (0, t)=0, ux (π, t) + hu (π, t)=0, t > 0. 57. By the transform method, solve ut = uxx + g (x, t), 0 < x < π, t > 0, u (x, 0) = 0, 0 < x < π, u (0, t)=0, ux (π, t)=0, t > 0. 58. By the transform method, solve ut = uxx − hu, 0 < x < π, t > 0, u (x, 0) = sin x, 0 ≤ x ≤ π, u (0, t)=0, u (π, t)=0, t > 0. 59. By the transform method, solve utt = uxx + h, 0 < x < π, t > 0, h = constant, u (x, 0) = 0, ut (x, 0) = 0, 0 < x < π, ux (0, t)=0, ux (π, t)=0, t> 0. 532 12 Integral Transform Methods with Applications 60. By the transform method, solve utt = uxx + g (x), 0 < x < π, t > 0, u (x, 0) = 0, ut (x, 0) = 0, 0 < x < π, u (0, t)=0, u (π, t)=0, t > 0. 61. By the transform method, solve utt + c 2uxxxx = 0, 0 < x < π, t > 0, u (x, 0) = 0, ut (x, 0) = 0, 0 < x < π, u (0, t)=0, u (π, t)=0, t > 0. uxx (0, t)=0, uxx (π, t) = sin t, t ≥ 0. 62. Find the temperature distribution u (r, t) in a long cylinder of radius a when the initial temperature is constant, u0, and radiation occurs at the surface into a medium with zero temperature. Here u (r, t) satisfies the initial boundary-problem ut = κ  urr + 1 r ur  , 0 ≤ r < a, t > 0, ur + α u = 0 at r = a, t > 0, u (r, 0) = u0 for 0 ≤ r < a, where κ and α are constants. 63. Apply the finite Fourier sine transform to solve the longitudinal displacement field in a uniform bar of length l and cross section A subjected to an external force F A applied at the end x = l. The governing equation and boundary and initial conditions are c 2uxx = utt,  c 2 = E ρ  , 0 < x < l, t > 0, u (0, t)=0 E u (l, t) = F, t > 0, u (x, 0) = ut (x, 0) = 0, 0 < x < l, where E is the constant Young’s modulus, ρ is the density, and F is constant. 64. Use the finite Fourier cosine transform to solve the heat conduction problem ut = κuxx, 0 < x < l, t > 0, ux (x, t) = 0 at x = 0 and x = l, t > 0, u (x, 0) = u0 for 0 < x < l, where u0 and κ are constant. 12.18 Exercises 533 65. Use the Mellin transform to find the solution of the integral equation  ∞ −∞ f (x) k (xt) dx = g (t), t > 0. 66. Use the Mellin transform to show the following results: (a) ∞ n=1 f (n) = 1 2πi c+i∞ c−i∞ ζ (p) F (p) dp, (b) ∞ n=1 f (nx) = M−1 [ζ (p) F (p)], where ζ (s) is the Riemann zeta function defined by (6.7.13). 67. Show that the solution of the boundary-value problem urr + 1 r ur + uzz = 0, r ≥ 0, z> 0, u (r, 0) = u0 for 0 ≤ r ≤ a, u (r, z) → 0 as z → ∞, is u (r, z) = au0  ∞ 0 J1 (ak) J0 (kr) e −kzdk. 68. Show that the asymptotic representation of the Bessel function Jn (kr) for large kr is Jn (kr) = 1 π  π 0 cos (nθ − kr sin θ) dθ ∼  2 πkr1 2 cos 4 kr − nπ 2 − π 4 5 . 69. (a) Use the Laplace transform to solve the heat conduction problem ut = κuxx, 0 <x< ∞,="" t=""> 0, u (x, 0) = 0, x> 0, u (0, t) = f (t), u (x, t) → 0 as x → ∞, t> 0. (b) Derive Duhamel’s formula u (x, t) =  t 0 f (t − τ )  ∂u0 ∂τ  dτ, where  ∂u0 ∂t  = x √ 4πκ t −3/2 exp  − x 2 4κt . 13 Nonlinear Partial Differential Equations with Applications “True Laws of Nature cannot be linear.” Albert Einstein “... the progress of physics will to a large extent depend on the progress of nonlinear mathematics, of methods to solve nonlinear equations ... and therefore we can learn by comparing different nonlinear problems.” Werner Heisenberg 13.1 Introduction The three-dimensional linear wave equation utt = c 2 ∇2u, (13.1.1) arises in the areas of elasticity, fluid dynamics, acoustics, magnetohydrodynamics, and electromagnetism. The general solution of the one-dimensional equation (13.1.1) is u (x, t) = φ (x − ct) + ψ (x + ct), (13.1.2) where φ and ψ are determined by the initial or boundary conditions. Physically, φ and ψ represent waves moving with constant speed c and without change of shape, along the positive and the negative directions of x respectively. The solutions φ and ψ correspond to the two factors when the onedimensional equation (13.1.1) is written in the form  ∂ ∂t + c ∂ ∂x ∂ ∂t − c ∂ ∂x u = 0. (13.1.3) 536 13 Nonlinear Partial Differential Equations with Applications Obviously, the simplest linear wave equation is ut + c ux = 0, (13.1.4) and its solution u = φ (x − ct) represents a wave moving with a constant velocity c in the positive x-direction without change of shape. 13.2 One-Dimensional Wave Equation and Method of Characteristics The simplest first-order nonlinear wave equation is given by ut + c (u) ux = 0, −∞ <x< ∞,="" t=""> 0, (13.2.1) where c (u) is a given function of u. We solve this nonlinear equation subject to the initial condition u (x, 0) = f (x), −∞ <x< ∞.="" (13.2.2)="" before="" we="" discuss="" the="" method="" of="" solution,="" following="" comments="" are="" in="" order.="" first,="" unlike="" linear="" differential="" equations,="" principle="" superposition="" cannot="" be="" applied="" to="" find="" general="" solution="" nonlinear="" partial="" equations.="" second,="" effect="" nonlinearity="" can="" change="" entire="" nature="" solution.="" third,="" a="" study="" above="" initial-value="" problem="" reveals="" most="" important="" ideas="" for="" hyperbolic="" waves.="" finally,="" large="" number="" physical="" and="" engineering="" problems="" governed="" by="" system="" or="" an="" extension="" it.="" although="" (13.2.1)–(13.2.2)="" looks="" simple,="" it="" poses="" nontrivial="" mathematics,="" leads="" surprisingly="" new="" phenomena.="" solve="" characteristics.="" order="" construct="" continuous="" solutions,="" consider="" total="" du="" given="" ∂t="" dt="" +="" ∂u="" ∂xdx,="" (13.2.3)="" so="" that="" points="" (x,="" t)="" assumed="" lie="" on="" curve="" Γ.="" then,="" dx="" represents="" slope="" Γ="" at="" any="" point="" p="" thus,="" equation="" becomes="" ="" ="" ux.="" (13.2.4)="" follows="" from="" this="" result="" (13.2.1)="" regarded="" as="" ordinary="" (13.2.5)="" 13.2="" one-dimensional="" wave="" characteristics="" 537="" along="" member="" family="" curves="" which="" (u).="" (13.2.6)="" these="" called="" characteristic="" main="" (13.2.1).="" has="" been="" reduced="" pair="" simultaneous="" equations="" (13.2.6).="" clearly,="" both="" speed="" depend="" u.="" implies="" u="constant" each="" Γ,="" c="" (u)="" remains="" constant="" therefore,="" shows="" form="" straight="" lines="" t)-plane="" with="" indicates="" depends="" finding="" lines.="" also,="" line="" corresponds="" value="" if="" initial="" is="" denoted="" ξ="" one="" intersects="" t="0" x="ξ," then="" (ξ,="" 0)="f" (ξ)="" whole="" shown="" figure="" 13.2.1.="" have="" Γ:="" (u),="" (0)="ξ," (13.2.7)="" (ξ).="" (13.2.8)="" constitute="" coupled="" solved="" independently="" because="" function="" however,="" readily="" obtain="" 13.2.1="" curve.="" 538="" 13="" applications="" hence,="" (ξ),="" (13.2.9)="" where="" f="" (f="" (ξ)).="" (13.2.10)="" integrating="" gives="" ξ.="" (13.2.11)="" whose="" not="" constant,="" but="" combining="" results,="" parametric="" 0="" ,="" (13.2.12)="" next="" verify="" final="" analytic="" expression="" differentiating="" respect="" t,="" ux="f" ′="" ξx,="" ut="f" ξt,="" 1="{1" tf′="" (ξ)}="" 0="F" {1="" ξt.="" elimination="" ξx="" ξt="" .="" (13.2.13)="" since="" (ξ)),="" satisfied="" provided="" 1+tf′="" also="" satisfies="" condition="" unique.="" suppose="" v="" two="" solutions.="" tf="" t).="" proved="" following:="" 539="" theorem="" −∞="" <x<="" ∞,=""> 0, u (x, t) = f (x), at t = 0, −∞ <x< ∞,="" has="" a="" unique="" solution="" provided="" 1+tf′="" (ξ)="0," f="" and="" c="" are="" 1="" (r)="" functions="" where="" (f="" (ξ)).="" the="" is="" given="" in="" parametric="" form:="" u="" (x,="" t)="f" (ξ),="" x="ξ" +="" tf="" (ξ).="" remark:="" when="" (u)="constant" ==""> 0, equation (13.2.1) becomes the linear wave equation (13.1.4). The characteristic curves are x = ct + ξ and the solution u is given by u (x, t) = f (ξ) = f (x − ct). Physical Significance of (13.2.12). We assume c (u) > 0. The graph of u at t = 0 is the graph of f. In view of the fact u (x, t) = u (ξ + tF (ξ), t) = f (ξ) the point (ξ,f (ξ)) moves parallel to the x-axis in the positive direction through a distance tF (ξ) = ct, and the distance moved (x = ξ + ct) depends on ξ. This is a typical nonlinear phenomenon. In the linear case, the curve moves parallel to the x-axis with constant velocity c, and the solution represents waves travelling without change of shape. Thus, there is a striking difference between the linear and the nonlinear solution. Theorem 13.2.1 asserts that the solution of the nonlinear initial-value problem exists provided 1 + tF′ (ξ) = 0, x = ξ + tF′ (ξ). (13.2.14) However, the former condition is always satisfied for sufficiently small time t. By a solution of the problem, we mean a differentiable function u (x, t). It follows from results (13.2.13) that both ux and ut tend to infinity as 1 + tF′ (ξ) → 0. This means that the solution develops a singularity (discontinuity) when 1 + tF′ (ξ) = 0. We consider a point (x, t)=(ξ, 0) so that this condition is satisfied on the characteristics through the point (ξ, 0) at a time t such that t = − 1 F′ (ξ) (13.2.15) 540 13 Nonlinear Partial Differential Equations with Applications which is positive provided F ′ (ξ) = c ′ (f) f ′ (ξ) < 0. If we assume c ′ (f) > 0, the above inequality implies that f ′ (ξ) < 0. Hence, the solution (13.2.12) ceases to exist for all time if the initial data is such that f ′ (ξ) < 0 for some value of ξ. Suppose t = τ is the time when the solution first develops a singularity (discontinuity) for some value of ξ. Then τ = − 1 min−∞<ξ<∞ {c ′ (f) f ′ (ξ)} > 0. We draw the graphs of the nonlinear solution u (x, t) = f (ξ) below for different values of t = 0, τ , 2τ , .... The shape of the initial curve for u (x, t) changes with increasing values of t, and the solution becomes multiplevalued for t ≥ τ . Therefore, the solution breaks down when F ′ (ξ) < 0 for some ξ, and such breaking is a typical nonlinear phenomenon. In linear theory, such breaking will never occur. More precisely, the development of a singularity in the solution for t ≥ τ can be seen by the following consideration. If f ′ (ξ) < 0, we can find two values of ξ = ξ1, ξ2 (ξ1 < ξ2) on the initial line such that the characteristics through them have different slopes 1/c (u1) and 1/c (u2) where u1 = f (ξ1) and u2 = f (ξ2) and c (u2) < c (u1). Thus, these two characteristics will intersect at a point in the (x, t)-plane for some t > 0. Since the characteristics carry constant values of u, the solution ceases to be single-valued at their point of intersection. Figure 13.2.2 shows that the wave profile progressively distorts itself, and at any instant of time there exists an interval on the x-axis, where u assumes three values for a given x. The end result is the development of a nonunique solution, and this leads to breaking. Therefore, when conditions (13.2.14) are violated the solution develops a discontinuity known as a shock. The analysis of shock involves extension of a solution to allow for discontinuities. Also, it is necessary to impose on the solution certain restrictions to be satisfied across its discontinuity. This point will be discussed further in a subsequent section. 13.3 Linear Dispersive Waves We consider a single linear partial differential equation with constant coefficients in the form P  ∂ ∂t, ∂ ∂x, ∂ ∂y , ∂ ∂z  u (x, t)=0, (13.3.1) where P is a polynomial in partial derivatives and x = (x, y, z). We seek an elementary plane wave solution of (13.3.1) in the form u (x, t) = a ei(κ·x−ωt) , (13.3.2) where a is the amplitude, κ = (k,l, m) is the wavenumber vector, ω is the frequency and a, κ, ω are constants. When this plane wave solution is 13.3 Linear Dispersive Waves 541 Figure 13.2.2 The solution u (x, t) for different times t = 0, τ and 2τ ; the characteristics are shown by the dotted lines; two of them from x = ξ1 and x = ξ2 intersect at t>τ . substituted in the equation, ∂/∂t, ∂/∂x, ∂/∂y, and ∂/∂z produce factors −iω, ik, il, and im respectively, and the solution exists provided ω and κ are related by an equation P (−iω, ik, il, im)=0. (13.3.3) This equation is known as the dispersion relation. Evidently, we have a direct correspondence between equation (13.3.1) and the dispersion relation (13.3.3) through the correspondence ∂ ∂t ↔ −iω,  ∂ ∂x, ∂ ∂y , ∂ ∂z  ↔ i(k,l, m). (13.3.4) Equation (13.3.1) and the corresponding dispersion relation (13.3.3) indicate that the former can be derived from the latter and vice-versa by using (13.3.4). The dispersion relation characterizes the plane wave motion. In many problems, the dispersion relation can be written in the explicit form ω = W (k,l, m). (13.3.5) The phase and the group velocities of the waves are defined by Cp (κ) = ω κ κ, 6 (13.3.6) Cg (κ) = ∇κω, (13.3.7) 542 13 Nonlinear Partial Differential Equations with Applications where κ6 is the unit vector in the direction of wave vector κ. In the one-dimensional case, (13.3.5)–(13.3.7) become ω = W (k), Cp = ω k , Cg = dω dk . (13.3.8) The one-dimensional waves given by (13.3.2) are called dispersive if the group velocity Cg ≡ ω ′ (k) is not constant, that is, ω ′′ (k) = 0. Physically, as time progresses, the different waves disperse in the medium with the result that a single hump breaks into wavetrains. Example 13.3.1. (i) Linearized one-dimensional wave equation utt − c 2uxx = 0, ω = + ck. (13.3.9) (ii) Linearized Korteweg and de Vries (KdV) equation for long water waves ut + αux + βuxxx = 0, ω = αk − βk3 . (13.3.10) (iii) Klein–Gordon equation utt − c 2uxx + α 2u = 0, ω = +  c 2 k 2 + α 2 1 2 . (13.3.11) (iv) Schr¨odinger equation in quantum mechanics and de Broglie waves i ψt −  V −  2 2m ∇2  ψ = 0,  ω =  2κ 2 2m + V, (13.3.12) where V is a constant potential energy, and h = 2π is the Planck constant. The group velocity of de Broglie wave is (κ/m), and through the correspondence principle, ω is to be interpreted as the total energy,   2κ 2/2m as the kinetic energy, and κ as the particle momentum. Hence, the group velocity is the classical particle velocity. (v) Equation for vibration of a beam utt + α 2uxxxx = 0, ω = + αk2 . (13.3.13) (vi) The dispersion relation for water waves in an ocean of depth h ω 2 = gk tanh kh, (13.3.14) where g is the acceleration due to gravity. 13.3 Linear Dispersive Waves 543 (vii) The Boussinesq equation utt − α 2∇2u − β 2∇2utt = 0, ω = + ακ  1 + β 2κ 2 . (13.3.15) This equation arises in elasticity for longitudinal waves in bars, long water waves, and plasma waves. (viii) Electromagnetic waves in dielectrics  utt + ω 2 0u utt − c 2 0uxx − ω 2 putt = 0,  ω 2 − ω 2 0 ω 2 − c 2 0k 2 − ω 2 pω 2 = 0, (13.3.16) where ω0 is the natural frequency of the oscillator, c0 is the speed of light, and ωp is the plasma frequency. In view of the superposition principle, the general solution can be obtained from (13.3.2) with the dispersion solution (13.3.3). For the one-dimensional case, the general solution has the Fourier integral representation u (x, t) =  ∞ −∞ F (k) e i[kx−tW(k)]dk, (13.3.17) where F (k) is chosen to satisfy the initial or boundary data provided the data are physically realistic enough to have Fourier transforms. In many cases, as cited in Example 13.3.1, there are two modes ω = + W (k) so that the solution (13.3.17) has the form u (x, t) =  ∞ −∞ F1 (k) e i[kx−tW(k)]dk +  ∞ −∞ F2 (k) e i[kx−tW(k)]dk, (13.3.18) with the initial data at t = 0 u (x, t) = φ (x), ut (x, t) = ψ (x). (13.3.19) The initial conditions give φ (x) =  ∞ −∞ [F1 (k) + F2 (k)] e ikxdk, ψ (x) = −i  ∞ −∞ [F1 (k) + F2 (k)] W (k) e ikxdk. Applying the Fourier inverse transformations, we have F1 (k) + F2 (k) = Φ(k) = 1 √ 2π  ∞ −∞ φ (x) e −ikxdx, −iW (k) [F1 (k) − F2 (k)] = Ψ (k) = 1 √ 2π  ∞ −∞ ψ (x) e −ikxdx, 544 13 Nonlinear Partial Differential Equations with Applications so that [F1 (k) + F2 (k)] = 1 2 Φ(k) + i Ψ (k) W (k) . (13.3.20) The asymptotic behavior of u (x, t) for large t with fixed (x/t) can be obtained by the Kelvin stationary phase approximation. For real φ (x), ψ (x), Φ(−k) = Φ ∗ (k) and Ψ (−k) = Ψ ∗ (k), where the asterisk denotes a complex conjugate. It follows from (13.3.20) that, for W (k) even [F1 (−k), F2 (−k)] = [F ∗ 2 (k), F∗ 1 (k)] , (13.3.21) and for W (k) odd, [F1 (−k), F2 (−k)] = [F ∗ 1 (k), F∗ 2 (k)] . (13.3.22) In particular, when φ (x) = δ (x) and ψ (x) ≡ 0, then F1 (k) = F2 (k) = 1/ √ 8π, and the solution (13.3.18) reduces to the form u (x, t) = 2 2 π  ∞ 0 cos kx cos {tW (k)} dk. (13.3.23) In order to obtain the asymptotic approximation by the Kelvin stationary phase method (see Section 12.7) for t → ∞, we consider both cases when W (k) is even (W′ (k) is odd) and when W (k) is odd (W′ (k) is even) and make an extra reasonable assumption that W′ (k) is monotonic and positive for k > 0. It turns out that the asymptotic solution for t → ∞ is u (x, t) ∼ 2 Re / F1 (k)  2π t|W′′ (k)| 01 2 exp " i ( θ (x, t) − π 4 sgn W′′ (k) )#0 + O  1 t  , = Re " a (x, t) e iθ(x,t) # , (13.3.24) where k (x, t) is the positive root of the equation W′ (k) = x t , ω = W (k), x t > 0, (13.3.25ab) θ (x, t) = x k (x, t) − t ω (x, t), (13.3.26) and a (x, t)=2F1 (k)  2π t|W′′ (k)| 01 2 exp  − iπ 4 sgn W′′ (k) 0 . (13.3.27) It is important to point out that solution (13.3.24) has a form similar to that of the elementary plane wave solution, but k, ω, and a are no 13.4 Nonlinear Dispersive Waves and Whitham’s Equations 545 longer constants; they are functions of space variable x and time t. The solution still represents an oscillatory wavetrain with the phase function θ (x, t) describing the variations between local maxima and minima. Unlike the elementary plane wavetrain, the present asymptotic result (13.3.24) represents a nonuniform wavetrain in the sense that the amplitude, the distance, and the time between successive maxima are not constant. It also follows from (13.3.25a) that kt k = − W′ (k) kW′′ (k)t 1 t ∼ O  1 t  , (13.3.28) kx k = − 1 kW′′ (k) 1 t ∼ O  1 t  . (13.3.29) These results indicate the k (x, t) is a slowly varying function of x and t as t → ∞. Applying a similar argument to ω and a, we conclude that k (x, t), ω (x, t), and a (x, t) are slowly varying functions of x and t as t → ∞. Finally, all these results seem to provide an important clue for natural generalization of the concept of nonlinear and nonuniform wavetrains. 13.4 Nonlinear Dispersive Waves and Whitham’s Equations To describe a slowly varying nonlinear and nonuniform oscillatory wavetrain in a medium (see Whitham, 1974), we assume the existence of a solution in the form (13.3.24) so that u (x, t) = a (x, t) e iθ(x,t) + c.c., (13.4.1) where c.c. stands for the complex conjugate, a (x, t) is the complex amplitude given by (13.3.27), and the phase function θ (x, t) is θ (x, t) = x k (x, t) − t ω (x, t), (13.4.2) and k, ω, and a are slowly varying function of x and t. Due to slow variations of k and ω, it is reasonable to assume that these quantities still satisfy the dispersion relation ω = W (k). (13.4.3) Differentiating (13.4.2) with respect to x and t respectively, we obtain θx = k + {x − t W′ (k)} kx, (13.4.4) θt = −W (k) + {x − t W′ (k)} kt. (13.4.5) In the neighborhood of stationary points defined by (13.3.25a), these results become 546 13 Nonlinear Partial Differential Equations with Applications θx = k (x, t), θt = −ω (x, t). (13.4.6) These results can be used as a definition of local wavenumber and local frequency of the slowly varying nonlinear wavetrain. In view of (13.4.6), relation (13.4.3) gives a nonlinear partial differential equation for the phase θ in the form ∂θ ∂t + W  ∂θ ∂x = 0. (13.4.7) The solution of this equation determines the geometry of the wave pattern. However, it is convenient to eliminate θ from (13.4.6) to obtain ∂k ∂t + ∂ω ∂t = 0. (13.4.8) This is known as the Whitham equation for the conservation of waves, where k represents the density of waves and ω is the flux of waves. Using the dispersion relation (13.4.3), we obtain ∂k ∂t + Cg (k) ∂k ∂x = 0, (13.4.9) where Cg (k) = W′ (k) is the group velocity. This represents the simplest nonlinear wave (hyperbolic) equation for the propagation of k with the group velocity Cg (k). Since equation (13.4.9) is similar to (13.2.1), we can use the analysis of Section 13.2 to find the general solution of (13.4.9) with the initial condition k (x, 0) = f (x) at t = 0. In this case, the solution has the form k (x, t) = f (ξ), x = ξ + tF (ξ), (13.4.10) where F (ξ) = Cg (f (ξ)). This further confirms the propagation of k with the velocity Cg. Some physical interpretations of this kind of solution have already been discussed in Section 13.2. Equations (13.4.9) and (13.4.3) reveal that ω also satisfies the nonlinear wave (hyperbolic) equation ∂ω ∂t + W′ (k) ∂ω ∂x = 0. (13.4.11) It follows from equations (13.4.9) and (13.4.11) that both k and ω remain constant on the characteristic curves defined by dx dt = W′ (k) = Cg (k), (13.4.12) in the (x, t) plane. Since k and ω is constant on each curve, the characteristic curves are straight lines with slope Cg (k). The solution for k is given by (13.4.10). 13.4 Nonlinear Dispersive Waves and Whitham’s Equations 547 Finally, it follows from the above analysis that any constant value of the phase θ propagates according to θ (x, t) = constant, and hence, θt +  dx dt  θx = 0, (13.4.13) which gives, by (13.4.6), dx dt = − θt θx = ω k = Cp. (13.4.14) Thus, the phase of the waves propagates with the phase speed Cp. On the other hand, (13.4.9) ensures that the wavenumber k propagates with the group velocity Cg (k)=(dω/dk) = W′ (k). We next investigate how the wave energy propagates in the dispersive medium. We consider the following integral involving the square of the wave amplitude (energy) given by (13.3.24) between any two points x = x1 and x = x2 (0 < x1 < x2) Q (t) =  x2 x1 |a| 2 dx =  x2 x1 aa∗ dx, (13.4.15) = 8π  x2 x1 F1 (k) F ∗ 1 (k) t|W′′ (k)| dx, (13.4.16) which is, due to a change of variable x = t W′ (k), = 8π  k2 k1 F1 (k) F ∗ 1 (k) dk, (13.4.17) where kr = t W′ (kr), r = 1, 2. When kr is kept fixed as t varies, Q (t) remains constant so that 0 = dQ dt = d dt  x2 x1 |a| 2 dx, =  x2 x1 ∂ ∂t |a| 2 dx + |a| 2 2 W′ (k2) − |a| 2 1 W′ (k1). (13.4.18) In the limit x2 − x1 → 0, this result reduces to the partial differential equation ∂ ∂t |a| 2 + ∂ ∂x " W′ (k)|a| 2 # = 0. (13.4.19) This represents the equation for the conservation of wave energy, where |a| 2 and |a| 2 W′ (k) are the energy density and energy flux respectively. It also follows that the energy propagates with the group velocity W′ (k). It has been shown that the wavenumber k also propagates with the group velocity. Evidently, the group velocity plays a double role. 548 13 Nonlinear Partial Differential Equations with Applications The above analysis reveals another important fact; equations (13.4.3), (13.4.8), and (13.4.19) constitute a closed set of equations for the three quantities k, ω, and a. Indeed, these are the fundamental equations for nonlinear dispersive waves and are known as Whitham’s equations. 13.5 Nonlinear Instability For infinitesimal waves, the wave amplitude (ak ≪ 1) is very small, so that nonlinear effects can be neglected altogether. However, for finite amplitude waves the terms involving a 2 cannot be neglected, and the effects of nonlinearity become important. In the theory of water waves, Stokes first obtained the connection due to inherent nonlinearity between the waveprofile and the frequency of a steady periodic wave system. According to the Stokes theory, the remarkable fact is the dependence of ω on a which couples (13.4.8) to (13.4.19). This leads to a new nonlinear phenomenon. For finite amplitude waves, the frequency ω has the Stokes expansion ω = ω0 (k) + a 2ω2 (k) + ... = ω  k, a2 . (13.5.1) This can be regarded as the nonlinear dispersion relation which depends on both k and a 2 . In the linear case, the amplitude a → 0, (13.5.1) gives the linear dispersion relation (13.4.3), that is, ω = ω0 (k). In order to discuss nonlinear instability, we substitute (13.5.1) into (13.4.8) and retain (13.4.19) in the linear approximation to obtain the following coupled system: ∂k ∂t + ∂ ∂x + & ω0 (k) + ω2 (k) a 2 ' = 0, (13.5.2) ∂a2 ∂t + ∂ ∂x & ω ′ 0 (k) a 2 ' = 0, (13.5.3) where W (k) ≡ ω0 (k). These equations can be further approximated to obtain ∂k ∂t + ω ′ 0 ∂k ∂x + ω2 ∂a2 ∂x = O  a 2 , (13.5.4) ∂a2 ∂t + ω ′ 0 ∂a2 ∂x + ω ′′ 0 a 2 ∂k ∂x = 0. (13.5.5) In matrix form, these equations read ⎛ ⎝ ω ′ 0 ω2 ω ′′ 0 a 2 ω ′ 0 ⎞ ⎠ ⎛ ⎝ ∂k ∂x ∂a2 ∂x ⎞ ⎠ + ⎛ ⎝ 1 0 0 1 ⎞ ⎠ ⎛ ⎝ ∂k ∂t ∂a2 ∂t ⎞ ⎠ = 0. (13.5.6) Hence, the eigenvalues λ are the roots of the determinant equation 13.6 The Traffic Flow Model 549 |aij − λ bij | =       ω ′ 0 − λ ω2 ω ′′ 0 a 2 ω ′ 0 − λ       = 0, (13.5.7) where aij and bij are the coefficient matrices of (13.5.6). This equation gives the characteristic velocities λ = dx dt = ω ′ 0 + (ω2ω ′′ 0 ) 1 2 a + O  a 2 . (13.5.8) If ω2ω ′′ 0 > 0, the characteristics are real and the system is hyperbolic. The double characteristic velocity splits into two separate real velocities. This provides a new extension of the group velocity to nonlinear problems. If the disturbance is initially finite in extent, it would eventually split into two disturbances. In general, any initial disturbance or modulating source would introduce disturbances in both families of characteristics. In the hyperbolic case, compressive modulation will progressively distort and steepen so that the question of breaking will arise. These results are remarkably different from those found in linear theory, where there is only one characteristic velocity and any hump may distort, due to the dependence of ω ′ 0 (k) on k, but would never split. On the other hand, if ω2ω ′′ 0 < 0, the characteristics are complex and the system is elliptic. This leads to ill-posed problems. Any small perturbations in k and a will be given by the solutions of the form exp [iα (x − λt)] where λ is calculated from (13.5.8) for unperturbed values of k and a. In this elliptic case, λ is complex, and the perturbation will grow as t → ∞. Hence, the original wavetrain will become unstable. In the linear theory, the elliptic case does not arise at all. Example 13.5.1. For Stokes waves in deep water, the dispersion relation is ω = (gk) 1 2  1 + 1 2 k 2 a 2  , (13.5.9) so that ω0 (k)=(gk) 1 2 and ω2 (k) = 1 2 √g k 3 2 . In this case, ω ′′ 0 (k) = − 1 4 √g k− 3 2 so that ω ′′ 0ω2 = − g 8 k < 0. The conclusion is that Stokes waves in deep water are definitely unstable. This is one of the most remarkable results in the theory of nonlinear water waves discovered during the 1960’s. 13.6 The Traffic Flow Model We consider the flow of cars on a long highway under the assumptions that cars do not enter or leave the highway at any one of its points. We take the x-axis along the highway and assume that the traffic flows in the positive 550 13 Nonlinear Partial Differential Equations with Applications direction. Suppose ρ (x, t) is the density representing the number of cars per unit length at the point x of the highway at time t, and q (x, t) is the flow of cars per unit time. We assume a conservation law which states that the change in the total amount of a physical quantity contained in any region of space must be equal to the flux of that quantity across the boundary of that region. In this case, the time rate of change of the total number of cars in any segment x1 ≤ x ≤ x2 of the highway is given by d dt  x2 x1 ρ (x, t) dx =  x2 x1 ∂ρ ∂t dx. (13.6.1) This rate of change must be equal to the net flux across x1 and x2 given by q (x1, t) − q (x2, t) (13.6.2) which measures the flow of cars entering the segment at x1 minus the flow of cars leaving the segment at x2. Thus, we have the conservation equation d dt  x2 x1 ρ (x, t) dx = q (x1, t) − q (x2, t), (13.6.3) or  x2 x1 ∂ρ ∂t dx = −  x2 x1 ∂q ∂x dx, or  x2 x1  ∂ρ ∂t + ∂q ∂x dx = 0. (13.6.4) Since the integrand in (13.6.4) is continuous, and (13.6.4) holds for every segment [x1, x2], it follows that the integrand must vanish so that we have the partial differential equation ∂ρ ∂t + ∂q ∂x = 0. (13.6.5) We now introduce an additional assumption which is supported by both theoretical and experimental findings. According to this assumption, the flow rate q depends on x and t only through the density, that is, q = Q (ρ) for some function Q. This assumption seems to be reasonable in the sense that the density of cars surrounding a given car indeed controls the speed of that car. The functional relation between q and ρ depends on many factors, including speed limits, weather conditions, and road characteristics. Several specific relations are suggested by Haight (1963). We consider here a particular relation q = ρ v where v is the average local velocity of cars. We assume that v is a function of ρ to a first approximation. In view of this relation, (13.6.5) reduces to the nonlinear hyperbolic equation 13.6 The Traffic Flow Model 551 ∂ρ ∂t + c (ρ) ∂ρ ∂x = 0, (13.6.6) where c (ρ) = q ′ (ρ) = v + ρ v′ (ρ). (13.6.7) In general, the local velocity v (ρ) is a decreasing function of ρ so that v (ρ) has a finite maximum value vmax at ρ = 0 and decreases to zero at ρ = ρmax = ρm. For the value of ρ = ρm, the cars are bumper to bumper. Since q = ρ v, q (ρ) = 0 when ρ = 0 and ρ = ρm. This means that q is an increasing function of ρ until it attains a maximum value qmax = qM for some ρ = ρM and then decreases to zero at ρ = ρm. Both q (ρ) and v (ρ) are shown in Figure 13.6.1. Equation (13.6.6) is similar to (13.2.1) with the wave propagation velocity c (ρ) = v (ρ) + ρ v′ (ρ). Since v ′ (ρ) < 0, c (ρ) < v (ρ), that is, the propagation velocity is less than the car velocity. In other words, waves propagate backwards through the stream of cars, and drivers are warned of disturbances ahead. It follows from Figure 13.6.1a that q (ρ) is an increasing function in [0, ρM], a decreasing function in [ρM, ρm], and attains a maximum at ρM. Hence, c (ρ) = q ′ (ρ) is positive in [0, ρM], zero at ρM and negative in [ρM, ρm]. All these mean that waves propagate forward relative to the highway in [0, ρM], are stationary at ρM, and then travel backwards in [ρM, ρm]. We use Section 13.1 to solve the initial-value problem for the nonlinear equation (13.6.6) with the initial condition ρ (x, 0) = f (x). The solution is ρ (x, t) = f (ξ), x = ξ + tF (ξ), (13.6.8) where F (ξ) = c (f (ξ)). Figure 13.6.1 Graphs of q (ρ) and v (ρ). 552 13 Nonlinear Partial Differential Equations with Applications Since c ′ (ρ) = q ′′ (ρ) < 0, q (ρ) is convex, and c (ρ) is a decreasing function of ρ. This means that breaking occurs at the left due to formation of shock at the back. Waves propagate slower than the cars, so drivers enter such a local density increase from behind; they must decelerate rapidly through the shock but speed up slowly as they get out from the crowded area. These conclusions are in accord with observational results. Actual observational data of traffic flow indicate that a typical result on a single lane highway is ρm ≈ 225 cars per mile, ρM ≈ 80 cars per mile, and qM ≈ 1590 cars per hour. Thus, the maximum flow rate qM occurs at a low velocity v = qM/ρM ≈ 20 miles per hour. 13.7 Flood Waves in Rivers We consider flood waves in a long rectangular river of constant breadth. We take the x-axis along the river which flows in the positive x-direction and assume that the disturbance is approximately the same across the breadth. In this problem, the depth h (x, t) of the river plays the role of density in the traffic flow model discussed in Section 13.6. Let q (x, t) be the flow per unit breadth and per unit time. According to the Conservation Law, the rate of change of the mass of the fluid in any section x1 ≤ x ≤ x2 must be balanced by the net flux across x2 and x1 so that the conservation equation becomes d dt  x2 x1 h (x, t) dx + q (x2, t) − q (x1, t)=0. (13.7.1) An argument similar to the previous section gives ∂h ∂t + ∂q ∂x = 0. (13.7.2) Although the fluid flow is extremely complicated, we assume a simple function relation q = Q (h) as a first approximation to express the increase in flow as the water level arises. Thus, equation (13.7.2) becomes ht + c (h) hx = 0, (13.7.3) where c (h) = Q′ (h) and Q (h) is determined from the balance between the gravitational force and the frictional force of the river bed. This equation is similar to (13.2.1) and the method of solution has already been obtained in Section 13.2. Here we discuss the velocity of wave propagation for some particular values of Q (h). One such result is given by the Chezy result as Q (h) = hv, (13.7.4) 13.8 Riemann’s Simple Waves of Finite Amplitude 553 where v = α √ h is the velocity of fluid flow and α is a constant, so that the propagation velocity of flood waves is given by c (h) = Q ′ (h) = 3 2 α √ h = 3 2 v. (13.7.5) Thus, the flood waves propagate one and a half times faster than the stream velocity. For a general case where v = αhn, Q (h) = hv = αhn+1 , (13.7.6) so the propagation velocity of flood waves is c (h) = Q ′ (h)=(n + 1) v. (13.7.7) This result also indicates that flood waves propagate faster than the fluid. 13.8 Riemann’s Simple Waves of Finite Amplitude We consider a one-dimensional unsteady isentropic flow of gas of density ρ and pressure p with the direction of motion along the x-axis. Suppose u (x, t) is the x-component of the velocity at time t and A is an areaelement of the (y, z)-plane. The volume of the rectangular cylinder of height dx standing on the element A is then A dx and its mass A ρt dx dt is determined by the mass entering it, which is equal to −A (∂/∂x) (ρu) dx dt. Its acceleration is (ut + uux) and the force impelling it in the positive xdirection is −pxA dx = −c 2ρxA dx, where p = f (ρ) and c 2 = f ′ (ρ). These results lead to two coupled nonlinear partial differential equations ∂ρ ∂t + ∂ ∂x (ρu)=0, (13.8.1) (ut + uux) + c 2 ρ ρx = 0. (13.8.2) In matrix form, this system is A ∂U ∂x + I ∂U ∂t = 0, (13.8.3) where U, A and I are matrices given by U = ⎛ ⎝ ρ u ⎞ ⎠ , A = ⎛ ⎝ u ρ c 2/ρ u ⎞ ⎠ and I = ⎛ ⎝ 1 0 0 1 ⎞ ⎠ . (13.8.4) The concept of characteristic curves introduced briefly in Section 13.2 requires generalization if it is to be applied to quasi-linear systems of firstorder partial differential equations (13.8.1)–(13.8.2). 554 13 Nonlinear Partial Differential Equations with Applications It is of interest to determine how a solution evolves with time t. Hence, we leave the time variable unchanged and replace the space variable x by some arbitrary curvilinear coordinate ξ so that the semi-curvilinear coordinate transformation from (x, t) to (ξ, t′ ) can be introduced by ξ = ξ (x, t), t′ = t. (13.8.5) If the Jacobian of this transformation is nonzero, we can transform (13.8.3) by the following correspondence rule: ∂ ∂t ≡ ∂ξ ∂t ∂ ∂ξ + ∂t′ ∂t · ∂ ∂t′ = ∂ξ ∂t ∂ ∂ξ + ∂ ∂t′ , ∂ ∂x ≡ ∂ξ ∂x ∂ ∂ξ + ∂t′ ∂x ∂ ∂t′ = ∂ξ ∂x ∂ ∂ξ . This rule transforms (13.8.3) into the form I ∂U ∂t′ +  ∂ξ ∂t I + ∂ξ ∂xA  ∂U ∂ξ = 0. (13.8.6) This equation can be used to determine ∂U/∂ξ provided that the determinant of its coefficient matrix is non-zero. Obviously, this condition depends on the nature of the curvilinear coordinate curves ξ (x, t) = constant, which has been kept arbitrary. We assume now that the determinant vanishes for the particular choice ξ = η so that     ∂η ∂t I + ∂η ∂x A     = 0. (13.8.7) In view of this, ∂U/∂η will become indeterminate on the family of curves η = constant, and hence, ∂U/∂η may be discontinuous across the curves η = constant. This implies that each element of ∂U/∂η will be discontinuous across any of the curves η = constant. It is then necessary to find out how these discontinuities in the elements of ∂U/∂η are related across the curve η = constant. We next consider the solutions U which are everywhere continuous with discontinuous derivatives ∂U/∂η across the particular curve η = constant = η0. Since U is continuous, elements of the matrix A are not discontinuous across η = η0 so that A can be determined in the neighborhood of a point P on η = η0. And since ∂U/∂t′ is continuous everywhere, it is continuous across the curve η = η0 at P. In view of all of the above facts, it follows that differential equation (13.8.6) across the curve ξ = η = η0 at P becomes  ∂η ∂t I + ∂η ∂x A  P ∂U ∂η P = 0, (13.8.8) where [f]P = f (P+)−f (P−) denotes the discontinuous jump in the quantity f across the curve η = η0, and f (P−) and f (P+) represent the values 13.8 Riemann’s Simple Waves of Finite Amplitude 555 to the immediate left and immediate right of the curve at P. Since P is any arbitrary point on the curve, ∂/∂η denotes the differentiation normal to the curves η = constant so that equation (13.8.8) can be regarded as the compatibility condition satisfied by ∂U/∂η on either side of and normal to these curves in the (x, t)-plane. Obviously, equation (13.8.8) is a homogeneous system of equations for the two jump quantities [∂U/∂η]. Therefore, for the existence of a nontrivial solution, the coefficient determinant must vanish, that is,     ∂η ∂t I + ∂η ∂xA     = 0. (13.8.9) However, along the curves η = constant, we have 0 = dη = ηt +  dx dt  ηx, (13.8.10) so that these curves have the constant slope, λ dx dt = − ηt ηx = λ. (13.8.11) Consequently, equations (13.8.9) and (13.8.8) can be expressed in terms of λ in the form |A − λI| = 0, (13.8.12) (A − λI) ∂U ∂η = 0, (13.8.13) where λ represents the eigenvalues of the matrix A, and [∂U/∂η] is proportional to the corresponding right eigenvector of A. Since A is a 2 × 2 matrix, it must have two eigenvalues. If these are real and distinct, integration of (13.8.11) leads to two distinct families of real curves Γ1 and Γ2 in the (x, t)-plane: Γr : dx dt = λr, r = 1, 2. (13.8.14) The families of curves Γr are called the characteristic curves of the system (13.8.3). Any one of these families of curves Γr may be chosen for the curvilinear coordinate curves η = constant. The eigenvalues λr have the dimensions of velocity, and the λr associated with each family will then be the velocity of propagation of the matrix column vector [∂U/∂η] along the curves Γr belonging to that family. In this particular case, the eigenvalues λ of the matrix A are determined by (13.8.12), that is,       u − λ ρ c 2/ρ u − λ       = 0, (13.8.15) 556 13 Nonlinear Partial Differential Equations with Applications so that λ = λr = u+c, r = 1, 2. (13.8.16) Consequently, the families of the characteristic curves Γr (r = 1, 2) defined by (13.8.14) become Γ1 : dx dt = u + c, and Γ2 : dx dt = u − c. (13.8.17) In physical terms, these results indicate that disturbances propagate with the sum of the velocities of the fluid and sound along the family of curves Γ1. In the second family Γ2, they propagate with the difference of the fluid velocity u and the sound velocity c. The right eigenvectors µr ≡ ⎛ ⎜⎝ µ (1) r µ (2) r ⎞ ⎟⎠ are solutions of the equations (A − λrI) µr = 0, r = 1, 2, (13.8.18) or, ⎛ ⎝ u − λr ρ c 2/ρ u − λr ⎞ ⎠ ⎛ ⎜⎝ µ (1) r µ (2) r ⎞ ⎟⎠ = 0, r = 1, 2. (13.8.19) This result combined with (13.8.13) gives ⎛ ⎝ [ρη] [uη] ⎞ ⎠ = ⎛ ⎜⎝ µ (1) r µ (2) r ⎞ ⎟⎠ = α ⎛ ⎝ 1 + c/ρ ⎞ ⎠ , r = 1, 2, (13.8.20) where α is a constant. In other words, across a wavefront in the Γ1 family of characteristic curves, [∂ρ/∂η] 1 = [∂u/∂η] c/ρ , (13.8.21) and across a wavefront in the Γ2 family of characteristic curves, [∂ρ/∂η] 1 = [∂u/∂η] −c/ρ , (13.8.22) where c and ρ have values appropriate to the wavefront. The above method of characteristics can be applied to a more general system 13.8 Riemann’s Simple Waves of Finite Amplitude 557 ∂U ∂t + A ∂U ∂x = 0, (13.8.23) where U is an n × 1 matrix with elements u1, u2, ..., un and A is an n × n matrix with elements aij . An argument similar to that given above leads to n eigenvalues of (13.8.13). If these eigenvalues are real and distinct, integration of equations (13.8.14) with r = 1, 2, ..., n gives n distinct families of real curves Γr in the (x, t)-plane so that Γr : dx dt = λr, r = 1, 2, . . . n. (13.8.24) When the eigenvalues λr of A are all real and distinct, there are n distinct linearly independent right eigenvectors µr of A satisfying the equation Aµr = λrµr, where µr is an n × 1 matrix with elements µ (1) r , µ (2) r , ..., µ (n) r . Then across a wavefront belonging to the Γr family of characteristics, it turns out that [∂u1/∂η] µ (1) r = [∂u2/∂η] µ (2) r = ... = [∂un/∂η] µ (n) r , (13.8.25) where the elements of µr are known on the wavefront. In order to introduce the Riemann invariants, we form the linear combination of the eigenvectors (+c/ρ, 1) with equations (13.8.1)–(13.8.2) to obtain + c ρ (ρt + ρux + uρx) +  ut + uux + c 2 ρ ρx  = 0. (13.8.26) We use ∂u/∂ρ = + c/ρ from (13.8.21)–(13.8.22) and rewrite (13.8.26) as + c ρ [ρt + (u + c) ρx]+[ut + (u + c) ux]=0. (13.8.27) In view of (13.8.17), equation (13.8.27) becomes du + c ρ dρ = 0 on Γr, r = 1, 2, (13.8.28) or, d [F (ρ) + u] = 0 on Γr, (13.8.29) where F (ρ) =  ρ ρ0 c (ρ) ρ dρ. (13.8.30) Integration of (13.8.29) gives 558 13 Nonlinear Partial Differential Equations with Applications F (ρ) + u = 2r on Γ1 and F (ρ) − u = 2s on Γ2, (13.8.31) where 2r and 2s are constants of integration on Γ1 and Γ2, respectively. The quantities r and s are called the Riemann invariants. As stated above, r is an arbitrary constant on characteristics Γ1, and hence, in general, r will vary on each Γ2. Similarly, s is constant on each Γ2 but will vary on Γ1. It is natural to introduce r and s as new curvilinear coordinates. Since r is constant on Γ1, s can be treated as the parameter on Γ1. Similarly, r can be regarded as the parameter on Γ2. Then, dx = (u + c) dt on Γr implies that dx ds = (u + c) dt ds on Γ1, (13.8.32) dx dr = (u − c) dt dr on Γ2. (13.8.33) In fact, r is a constant on Γ1, and s is a constant on Γ2. Therefore, the derivatives in the two equations are really partial derivations with respect to s and r so that we can rewrite them as ∂x ∂s = (u + c) ∂t ∂s , (13.8.34) ∂x ∂r = (u − c) ∂t ∂r . (13.8.35) These two first-order PDE’s can, in general, be solved for x = x (r, s), t = t(r, s), and then, by inversion, r and s as functions x and t can be obtained. Once this is done, we use (13.8.31) to determine u (x, t) and ρ (x, t) in terms of r and s as u (x, t) = r − s, F (ρ) = r + s. (13.8.36) When one of the Riemann invariants r and s is constant throughout the flow, the corresponding solution is tremendously simplified. The solutions are known as simple wave motions representing simple waves in one direction only. The generating mechanisms of simple waves with their propagation laws can be illustrated by the piston problem in gas dynamics. Example 13.8.1. Determine the Riemann invariants for a polytropic gas characterized by the law p = kργ , where k and γ are constants. In this case c 2 = dp dρ = kγργ−1 , F (ρ) =  ρ 0 c (ρ) ρ = 2c γ − 1 . Hence, the Riemann invariants are given by  2c γ − 1  c + u = (2r, 2s) on Γr. (13.8.37) 13.8 Riemann’s Simple Waves of Finite Amplitude 559 It is also possible to express the dependent variables u and c in terms of the Riemann invariants. It turns out that u = r − s, c = γ − 1 2 (r + s). (13.8.38) Example 13.8.2. (The Piston Problem in a Polytropic Gas). The problem is to determine how a simple wave is produced by the prescribed motion of a piston in the closed end of a semi-infinite tube filled with gas. This is a one-dimensional unsteady problem in gas dynamics. We assume that the gas is initially at rest with a uniform state u = 0, ρ = ρ0, and c = c0. The piston starts from rest at the origin and is allowed to withdraw from the tube with a variable velocity for a time t1, after which the velocity of withdrawal remains constant. The piston path is shown by a dotted curve in Figure 13.8.1. In the (x, t)-plane, the path of the piston is given by x = X (t) with X (0) = 0. The fluid velocity u is equal to the piston velocity X˙ (t) on the piston x = X (t), which will be used as the boundary condition for the piston. The initial state of the gas is given by u = u0, ρ = ρ0, and c = c0 at t = 0, in x ≥ 0. The characteristic line Γ0 that bounds it and passes through the origin is determined by the equation dx dt = (u + c) t=0 = c0 so that the equation of the characteristic line Γ0 is x = c0t. Figure 13.8.1 Simple waves generated by the motion of a piston. 560 13 Nonlinear Partial Differential Equations with Applications In view of the uniform initial state, all the Γ2 characteristics start on the x-axis so that the Riemann invariants s in (12.8.37b) must be constant and of the form 2c γ − 1 − u = 2c0 γ − 1 , (13.8.39) or, u = 2 (c − c0) γ − 1 , c = c0 + (γ − 1) 2 u. (13.8.40ab) The characteristics Γ1 meeting the piston are given by 2c γ − 1 + u = 2r on each Γ1 and Γ1 : dx dt = u + c, (13.8.41) which is, since (13.8.40ab) holds everywhere, u = constant on Γ1 and Γ1 : dx dt = c0 + 1 2 (γ + 1) u. (13.8.42) Since the flow is continuous with no shocks, u = 0 and c = c0 ahead of and on Γ0, which separates those Γ1 meeting the x-axis from those meeting the piston. The family of lines Γ1 through the origin has the equation (dx/dt) = ξ, where ξ is a parameter with ξ = c0 on Γ0. The Γ1 characteristics are also defined by (dx/dt) = u+c so that ξ = u+c. Hence, elimination of c from (13.8.40b) gives u =  2 γ + 1 (ξ − c0). (13.8.43) Substituting this value of u in (13.8.40b), we obtain c =  γ − 1 γ + 1 ξ + 2c0 γ + 1 . (13.8.44) It follows from c 2 = γkργ−1 and (13.8.40b) with the initial data, ρ = ρ0, c = c0 that ρ = ρ0 1 + γ − 1 2c0 u 2/(γ−1) . (13.8.45) With ξ = (x/t), results (13.8.43) through (13.8.45) give the complete solution of the piston problem in terms of x and t. Finally, the equation of the characteristic line Γ1 is found by integrating the second equation of (13.8.42) and using the boundary condition on the piston. When a line Γ1 intersects the piston path at time t = τ , then u = X˙ (τ ) along it, and the equation becomes 13.9 Discontinuous Solutions and Shock Waves 561 x = X (τ ) +  c0 + γ + 1 2 X˙ (τ ) 0 (t − τ ). (13.8.46) It is noted that the family Γ1 represents straight lines with slope dx/dt increasing with velocity u. Consequently, the characteristics are likely to overlap on the piston, that is, X˙ (τ ) > 0 for any τ . If u increases, so do c, ρ, and p so that instability develops. It shows that shocks will be formed in the compressive part of the disturbance. 13.9 Discontinuous Solutions and Shock Waves The development of a nonunique solution of a nonlinear hyperbolic equation has already been discussed in connection with several different problems. In real physical situations, this nonuniqueness usually manifests itself in the formation of discontinuous solutions which propagate in the medium. Such discontinuous solutions across some surface are called shock waves. These waves are found to occur widely in high speed flows in gas dynamics. In order to investigate the nature of discontinuous solutions, we reconsider the nonlinear conservation equation (13.6.5) that is, ∂ρ ∂t + ∂q ∂x = 0. (13.9.1) This equation has been solved under two basic assumptions: (i) There exists a functional relation between q and ρ, that is, q = Q (ρ); (ii) ρ and q are continuously differentiable. In some physical situations, the solution of (13.9.1) leads to breaking phenomenon. When breaking occurs, questions arise about the validity of these assumptions. To examine the formation of discontinuities, we consider the following: (a) we assume the relation q = Q (ρ) but allow jump discontinuity for ρ and q; (b) in addition to the fact that ρ and q are continuously differentiable, we assume that q is a function of ρ and ρx. One of the simplest forms is q = Q (ρ) − νρx, ν > 0. (13.9.2) In case (a), we assume the conservation equation (13.6.1) still holds and has the form d dt  x2 x1 ρ (x, t) dx + q (x2, t) − q (x1, t)=0. (13.9.3) We now assume that there is a discontinuity at x = s (t) where s is a continuously differentiable function of t, and x1 and x2 are chosen so that x2 > s (t) > x1, and U (t)= ˙s (t). Equation (13.9.3) can be written as d dt 1 s − x1 ρ dx +  x2 s+ ρ dx3 + q (x2, t) − q (x1, t)=0, 562 13 Nonlinear Partial Differential Equations with Applications which implies that  s − x1 ρt dx + ˙sρ  s −, t +  x2 s+ ρt dx − sρ˙  s +, t + q (x2, t) − q (x1, t)=0, (13.9.4) where ρ (s −, t), ρ (s +, t) are the values of ρ (x, t) as x → s from below and above respectively. Since ρt is bounded in each of the intervals separately, the integrals tend to zero as x1 → s − and x2 → s +. Thus, in the limit, q  s +, t − q  s −, t = U & ρ  s +, t − ρ  s −, t ' . (13.9.5) In the conventional notation of shock dynamics, this can be written as q2 − q1 = U (ρ2 − ρ1), (13.9.6) or −U [ρ]+[q]=0, (13.9.7) where subscripts 1 and 2 are used to denote the values behind and ahead of the shock respectively, and [ ] denotes the discontinuous jump in the quantity involved. Equation (13.9.7) is called the shock condition. Thus, the basic problem can be written as ∂ρ ∂t + ∂q ∂x = 0 at points of continuity, (13.9.8) −U [ρ]+[q] = 0 at points of discontinuity. (13.9.9) Therefore, we have a nice correspondence ∂ρ ∂t ↔ −U [ ] , ∂ ∂x ↔ [ ] , (13.9.10) between the differential equation and the shock condition. It is now possible to find discontinuous solutions of (13.9.3). In any continuous part of the solution, equation (13.9.1) is still satisfied and the assumption q = Q (ρ) remains valid. But we have q1 = Q (ρ1) and q2 = Q (ρ2) on the two sides of any shock, and the shock condition (13.9.6) has the form U (ρ2 − ρ1) = Q (ρ2) − Q (ρ1). (13.9.11) Example 13.9.1. The simplest example in which breaking occurs is ρt + c (ρ) ρx = 0, with discontinuous initial data at t = 0 13.10 Structure of Shock Waves and Burgers’ Equation 563 ρ = ⎧ ⎨ ⎩ ρ2, x< 0 ρ1, x> 0 , (13.9.12) and F (x) = ⎧ ⎨ ⎩ c2 = c2 (ρ), x< 0 c1 = c1 (ρ), x> 0 , (13.9.13) where ρ1 > ρ2 and c2 > c1. In this case, breaking will occur immediately and this can be seen from Figure 13.9.1ab with c ′ (ρ) > 0. The multivalued region begins at the origin ξ = 0 and is bounded by the characteristics x = c1t and x = c2t with c1 < c2. This corresponds to a centered compression wave with overlapping characteristics in the (x, t)-plane. On the other hand, if the initial condition is expansive with c2 < c1, there is a continuous solution obtained from (13.2.12) in which all values of F (x) in [c2, c1] are taken on characteristics through the origin ξ = 0. This corresponds to a centered fan of characteristics x = ct, c2 ≤ c ≤ c1 in the (x, t)-plane so that the solution has the explicit form c = (x/t), c2 < (x/t) < c1. The density distribution and the expansion wave are shown in Figure 13.9.2ab. In this case, the complete solution is given by c = ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ c2, x ≤ c2t x t , c2t<x 0. (13.10.1) Note that near breaking where ρx is large, (13.10.1) gives a better approximation. With (13.10.1), the basic equation (13.9.1) becomes ρt + c (ρ) ρx = νρxx, (13.10.2) 564 13 Nonlinear Partial Differential Equations with Applications Figure 13.9.1ab Density distribution and centered compression wave with overlapping characteristics. 13.10 Structure of Shock Waves and Burgers’ Equation 565 Figure 13.9.2ab Density distribution and centered expansion wave. 566 13 Nonlinear Partial Differential Equations with Applications where c (ρ) = Q′ (ρ), the second and the third terms represent the effects on nonlinearity and diffusion. We first solve (13.10.2) for two simple cases: (i) c (ρ) = constant = c, and (ii) c (ρ) ≡ 0. In the first case, equation (13.10.1) becomes linear and we seek a plane wave solution ρ (x, t) = a exp {i(kx − ωt)} . (13.10.3) Substituting this solution into the linear equation (13.10.2), we have the dispersion relation ω = ck − iνk2 , (13.10.4) where Im ω = −νk2 < 0, since ν > 0. Thus, the wave profile has the form ρ (x, t) = a e−νk2 t exp [ik (x − ct)] (13.10.5) which represents a diffusive wave (Im ω < 0) with wavenumbers k and phase velocity c whose amplitude decays exponentially with time t. The decay time is given by t0 =  νk2 −1 which becomes smaller and smaller as k increases with fixed ν. Thus, the waves of smaller wavelengths decay faster than waves of longer wavelengths. On the other hand, for a fixed wavenumber k, t0 decreases as ν increases so that waves of a given wavelength attenuate faster in a medium with larger ν. The quantity ν may be regarded as a measure of diffusion. Finally, after a sufficiently long time (t ≫ t0) only disturbances of long wavelength will survive, while all short wavelength disturbances will decay rapidly. In the second case, (13.10.2) reduces to the linear diffusion equation ρt = νρxx. (13.10.6) This equation with the initial data at t = 0 ρ = ⎧ ⎨ ⎩ ρ1, x< 0 ρ1, x> 0, ρ1 > ρ2, (13.10.7) can readily be solved, and the solution for t > 0 is ρ (x, t) = ρ1 2 √ πνt  0 −∞ e −(x−ξ) 2/4νtdξ + ρ2 2 √ πνt  ∞ 0 e −(x−ξ) 2/4νtdξ. (13.10.8) After some manipulation involving changes of variables of integration (x − ξ) /2 √ νt = η, the solution is simplified to the form 13.10 Structure of Shock Waves and Burgers’ Equation 567 u (x, t) = 1 2 (ρ1 + ρ2)+(ρ2 − ρ1) 1 √ π  x/2 √ νt 0 e −η 2 dη, (13.10.9) = 1 2 (ρ1 + ρ2) + 1 2 (ρ2 − ρ1) erf  x 2 √ νt . (13.10.10) This shows that the effect of the term νρxx is to smooth out the initial distribution (νt) − 1 2 . The solution tends to values ρ1, ρ2 as x → +∞. The absence of the term νρxx in (13.10.2) leads to nonlinear steepening and breaking. Indeed, equation (13.10.2) combines the two opposite effects of breaking and diffusion. The sign of ν is important; indeed, solutions are stable or unstable according as ν > 0 or ν < 0. In order to investigate solutions that balance between steepening and diffusion, we seek solutions of (13.10.2) in the form ρ = ρ (X), X = x − U t, (13.10.11) where U is a constant to be determined. It follows from (13.10.2) that [c (ρ) − U] ρX = νρXX. (13.10.12) Integrating this equation gives Q (ρ) − U ρ + A = νρX, (13.10.13) where A is a constant of integration. Integrating (13.10.13) with respect to X gives an implicit relation for ρ (X) in the form X ν =  dρ Q (ρ) − U ρ + A . (13.10.14) We would like to have a solution which tends to ρ1, ρ2 as X → +∞. If such a solution exists with ρX → 0 as |X|→∞, the quantities U and A must satisfy Q (ρ1) − U ρ1 + A = Q (ρ2) − U ρ2 + A = 0, (13.10.15) which implies that U = Q (ρ1) − Q (ρ2) ρ1 − ρ2 . (13.10.16) This is exactly the same as the shock velocity obtained before. Result (13.10.15) shows that ρ1, ρ2 are zeros of Q (ρ) − U ρ + A. In the limit ρ → ρ1 or ρ2, the integral (13.10.14) diverges and X → + ∞. If c ′ (ρ) > 0, then Q (ρ)−U ρ+A ≤ 0 in ρ2 ≤ ρ ≤ ρ1 and then ρX ≤ 0 because 568 13 Nonlinear Partial Differential Equations with Applications Figure 13.10.1 Shock structure and shock thickness. of (13.10.11). Thus, ρ decreases monotonically from ρ1 at X = −∞ to ρ2 at X = ∞ as shown in Figure 13.10.1. Physically, a continuous waveform carrying an increase in ρ will progressively distort itself and eventually break forward and require a shock with ρ1 < ρ2 provided c ′ (ρ) > 0. It will break backward and require a shock with ρ1 > ρ2 and c ′ (ρ) < 0. Example 13.10.1. Obtain the solution of (13.10.2) with the initial data (13.10.7) and Q (ρ) = αρ2 + βρ + γ, α > 0. We write Q (ρ) − U ρ + A = −α (ρ1 − ρ) (ρ − ρ2), where U = β + α (ρ1 + ρ2) and A = αρ1ρ2 − γ. Integral (13.10.14) becomes X ν = − 1 α  dρ (ρ − ρ2) (ρ1 − ρ) = 1 α (ρ1 − ρ2) log  ρ1 − ρ ρ − ρ2  , which gives the solution ρ (X) = ρ2 + (ρ1 − ρ2) exp αX ν (ρ2 − ρ1) ! 1 + exp αX ν (ρ2 − ρ1) ! . (13.10.17) 13.10 Structure of Shock Waves and Burgers’ Equation 569 The exponential factor in the solution indicates the existence of a transition layer of thickness δ of the order of ν/ [α (ρ1 − ρ2)]. This can also be referred to as the shock thickness. The thickness δ increases as ρ1 → ρ2 for a fixed ν. It tends to zero as ν → 0 for a fixed ρ1 and ρ2. In this case, the shock velocity (13.10.16) becomes U = α (ρ1 − ρ2) + β = 1 2 (c1 + c2), (13.10.18) where c (ρ) = Q′ (ρ), c1 = c (ρ1), and c2 = c (ρ2). We multiply (13.10.2) by c ′ (ρ) and simplify to obtain ct + ccx = νcxx − ν c′′ (ρ) ρ 2 x . (13.10.19) Since Q (ρ) is a quadratic expression in ρ, then c (ρ) = Q′ (ρ) becomes linear in ρ and c ′′ (ρ) = 0. Thus, (13.10.19) leads to Burgers’ equation replacing c with u ut + uux = ν uxx. (13.10.20) This equation incorporates the combined opposite effects of nonlinearity and diffusion. It is the simplest nonlinear model equation for diffusive waves in fluid dynamics. Using the Cole–Hopf transformation u = −2ν φx φ . (13.10.21) Burgers’ equation can be solved exactly, and the opposite effects of nonlinearity and diffusion can be investigated in some detail. We introduce the transformation in two steps. First, we write u = ψx so that (13.10.20) can readily be integrated to obtain ψt + 1 2 ψ 2 x = ν ψxx. (13.10.22) The next step is to introduce ψ = −2ν log φ and to transform this equation into the so called diffusion equation φt = ν φxx. (13.10.23) This equation was solved in earlier chapters. We simply quote the solution of the initial-value problem of (13.10.23) with the initial data φ (x, 0) = Φ(x), −∞ <x< ∞.="" (13.10.24)="" the="" solution="" for="" φ="" is="" (x,="" t)="1" 2="" √="" πνt="" ="" ∞="" −∞="" Φ(ζ)="" exp="" 1="" −="" (x="" ζ)="" 4νt="" 3="" dζ,="" (13.10.25)="" 570="" 13="" nonlinear="" partial="" differential="" equations="" with="" applications="" where="" can="" be="" written="" in="" terms="" of="" initial="" value="" u="" 0)="F" (x)="" by="" using="" (13.10.21).="" it="" turns="" out="" that,="" at="" t="0," =="" ="" 1="" 2ν="" x="" 0="" f="" (α)="" dα0="" .="" (13.10.26)="" then="" convenient="" to="" write="" down="" form="" ="" ="" (13.10.27)="" (ζ,="" x,="" ζ="" dα="" +="" 2t="" (13.10.28)="" consequently,="" φx="" 4ν="" dζ.="" (13.10.29)="" therefore,="" follows="" from="" (13.10.21)="" and="" has="" 4="" x−ζ="" 5="" dζ="" *="" (13.10.30)="" although="" this="" exact="" burgers’="" equation,="" physical="" interpretation="" hardly="" given="" unless="" a="" suitably="" simple="" specified.="" even="" then,="" finding="" an="" evaluation="" integrals="" formidable="" task.="" necessary="" resort="" asymptotic="" methods.="" before="" we="" deal="" analysis,="" following="" example="" may="" considered="" investigation="" shock="" formation.="" 13.10.2.="" find="" equation="" significance="" case="" ⎨="" ⎩="" δ="" (x),="" x<="" 0,=""> 0. We first find f (ζ, x, t) = A  ζ 0+ δ (α) dα + (x − ζ) 2 t = ⎧ ⎪⎨ ⎪⎩ (x−ζ) 2 2t − A, ζ < 0 (x−ζ) 2 2t , ζ > 0. 13.10 Structure of Shock Waves and Burgers’ Equation 571 Thus,  ∞ −∞ x − ζ t exp  − f 2ν  dζ =  0 −∞  x − ζ t  exp 1 A 2ν − (x − ζ) 2 4νt 3 dζ +  ∞ 0  x − ζ t  exp 1 − (x − ζ) 2 4νt 3 dζ = 2ν 4 e A/2ν − 1 5 exp  − x 2 4νt , which is obtained by substitution, x − ζ 2 √ νt = α. Similarly,  ∞ −∞ exp  − f 2ν  dζ = 2√ νt √ π + 4 e A/2ν − 1 5 erfc  x 2 √ νt , where erfc (x) is the complementary error function defined by erfc (x) = 2 √ π  ∞ x e −η 2 dη. (13.10.31) Therefore, the solution for u (x, t) is u (x, t) = 2 ν t  e A/2ν − 1 exp 4 − x 2 4νt5 √ π +  eA/2ν − 1 ( √ π/2) erfc 4 x 2 √ νt5. (13.10.32) In the limit as ν → ∞, the effect of diffusion would be more significant than that of nonlinearity. Since erfc  x 2 √ νt → 0, eA/2ν ∼ 1 + A 2ν as ν → ∞, the solution (13.10.32) tends to the limiting value u (x, t) ∼ A 2 √ πνt exp  − x 2 4νt . (13.10.33) This represents the well-known source solution of the classical linear heat equation ut = ν uxx. On the other hand, when ν → 0 nonlinearity would dominate over diffusion. It is expected that solution (13.10.32) tends to that of Burgers’ equation as ν → 0. 572 13 Nonlinear Partial Differential Equations with Applications We next introduce the similarity variable η = x/√ 2At to rewrite (13.10.32) in the form u (x, t) = 4ν t 5 1 2  e A/2ν − 1 exp 4 − Aη2 2ν 5 √ π +  eA/2ν − 1 ( √ π/2) erfc 4% A 2ν η 5, (13.10.34) ∼ 4ν t 5 1 2 exp & A 2ν  1 − η 2 ' √ π + (√ π/2) exp  A 2ν erfc 4% A 2ν η 5 as ν → 0 for all η, (13.10.35) ∼ 0 as ν → 0, for η < 0 and η > 1. (13.10.36) Invoking the asymptotic result, erfc (x) ∼  2/ √ π e −x 2 2x as x → ∞, (13.10.37) the solution (13.10.34) for 0 <η< 1 has the form, u (x, t) ∼ 4ν t 5 1 2 2η  A 2ν 1 2 exp & A 2ν  1 − η 2 ' 2η  Aπ 2ν 1 2 + exp & A 2ν (1 − η 2) ' . =  2A t 1 2 η 1+2η  Aπ 2ν 1 2 exp & A 2ν (η 2 − 1)' ∼  2A t 1 2 as ν → 0. The final asymptotic solution as ν → 0 is u (x, t) ∼ ⎧ ⎨ ⎩ x t , 0 <x< (2at)="" 1="" 2="" 0,="" otherwise.="" (13.10.38)="" this="" result="" represents="" a="" shock="" at="" x="(2At)" with="" the="" velocity="" u="(A/2t)" .="" solution="" has="" jump="" from="" 0="" to="" t="(2A/t)" so="" that="" condition="" is="" fulfilled.="" asymptotic="" behavior="" of="" burgers’="" as="" ν="" →="" 0.="" we="" use="" kelvin="" stationary="" phase="" approximation="" method="" discussed="" in="" section="" 12.7="" examine="" (13.10.30).="" according="" method,="" significant="" contribution="" integrals="" involved="" (13.10.30)="" comes="" points="" for="" fixed="" and="" t,="" is,="" roots="" equation="" 13.11="" korteweg–de="" vries="" solitons="" 573="" ∂f="" ∂ζ="F" (ζ)="" −="" (x="" ζ)="" (13.10.39)="" suppose="" ζ="ξ" (x,="" t)="" root.="" (12.7.8),="" yield="" ="" ∞="" −∞="" ="" ξ="" ="" exp="" f="" 2ν="" dζ="" ∼="" ="" 4πν="" |f="" ′′="" (ξ)|="" 01="" (ξ)="" 0="" therefore,="" final="" ,="" (13.10.40)="" where="" satisfies="" (13.10.39).="" other="" words,="" can="" be="" rewritten="" form="" +="" tf="" (13.10.41)="" identical="" (13.2.12)="" which="" was="" obtained="" 13.2.="" case,="" point="" characteristic="" variable.="" although="" exact="" single-valued="" continuous="" function="" all="" time="" exhibits="" instability.="" it="" already="" been="" shown="" progressively="" distorts="" itself="" becomes="" multiple-valued="" after="" sufficiently="" long="" time.="" eventually,="" breaking="" will="" definitely="" occur.="" follows="" analysis="" nonlinear="" diffusion="" terms="" show="" opposite="" effects.="" former="" introduces="" steepening="" profile,="" whereas="" latter="" tends="" diffuse="" (spread)="" sharp="" discontinuities="" into="" smooth="" profile.="" view="" property,="" diffusive="" wave.="" context="" fluid="" flows,="" denotes="" kinematic="" viscosity="" measures="" viscous="" dissipation.="" finally,="" arises="" many="" physical="" problems,="" including="" one-dimensional="" turbulence="" (where="" had="" its="" origin),="" sound="" waves="" media,="" fluid-filled="" elastic="" pipes,="" magnetohydrodynamic="" media="" finite="" conductivity.="" celebrated="" dispersion="" relation="" (13.3.14)="" dispersive="" surface="" on="" water="" constant="" depth="" h0="" 574="" 13="" partial="" differential="" equations="" applications="" ω="(gk" tanh="" kh0)="" 3="" k="" 2h="" 1="" ≈="" c0k="" 6="" (13.11.1)="" c0="(gh0)" shallow="" wave="" speed.="" motions="" small="" exhibit="" such="" term="" contrast="" linearized="" theory="" value="" c0k.="" an="" free="" elevation="" η="" given="" by="" ηt="" c0ηx="" σηxxx="0," (13.11.2)="" σ="1" c0h="" fairly="" waves.="" called="" (kdv)="" moving="" positive="" direction="" only.="" group="" velocities="" are="" found="" they="" cp="ω" σk2="" (13.11.3)="" cg="dω" dk="c0" 3σk2="" (13.11.4)="" noted=""> Cg, and the dispersion comes from the term involving k 3 in the dispersion relation (13.11.1) and hence, from the term σηxxx. For sufficiently long waves (k → 0), Cp = Cg = c0, and hence, these waves are nondispersive. In 1895, Korteweg–de Vries derived the nonlinear equation for long water waves in a channel of depth h0 which has the remarkable form ηt + c0  1 + 3 2 η h0  ηx + σηxxx = 0. (13.11.5) This is the simplest nonlinear model equation for dispersive waves, and combines nonlinearity and dispersion. The KdV equation arises in many physical problems, which include water waves of long wavelengths, plasma waves, and magnetohydynamics waves. Like Burgers’ equation, the nonlinearity and dispersion have opposite effects on the KdV equation. The former introduces steepening of the wave profile while the latter counteracts waveform steepening. The most remarkable features is that the dispersive term in the KdV equation does allow the solitary and periodic waves which are not found in shallow water wave theory. In Burgers’ equation the nonlinear term leads to steepening which produces a shock wave; on the other hand, in the KdV equation the steepening process is balanced by dispersion to give a rise to a steady solitary wave. We now seek the traveling wave solution of the KdV equation (13.11.5) in the form 13.11 The Korteweg–de Vries Equation and Solitons 575 η (x, t) = h0f (X), X = x − U t, (13.11.6) for some function f and constant wave velocity U. We determine f and U by substitution of the form (13.11.6) into (13.11.5). This gives, with σ = 1 6 c0h 2 0 , 1 6 h 2 0 f ′′′ + 3 2 ff′ +  1 − U c0  f ′ = 0, (13.11.7) and then integration leads to 1 6 h 2 0 f ′′ + 3 4 f 2 +  1 − U c0  f + A = 0, where A is an integrating constant. We next multiply this equation by f ′ and integrate again to obtain 1 3 h 2 0 f ′2 + f 3 + 2  1 − U c0  f 2 + 4 Af + B = 0, (13.11.8) where B is a constant of integration. We now seek a solitary wave solution under the boundary conditions f, f ′ , f ′′ → 0 as |X|→∞. Therefore, A = B = 0 and (13.11.8) assumes the form 1 3 h 2 0 f ′2 + f 2 (f − α)=0, (13.11.9) where α = 2  U c0 − 1  . (13.11.10) Finally, we obtain X =  f 0 df f ′ =  h 2 0 3 1 2  f 0 df f  (α − f) , which is, by the substitution f = α sech2 θ, X − X0 =  4h 2 0 3α 1 2 θ, (13.11.11) for some integrating constant X0. Therefore, the solution for f (X) is f (X) = α sech2 1 3α 4h 2 0 1 2 (X − X0) 3 . (13.11.12) 576 13 Nonlinear Partial Differential Equations with Applications Figure 13.11.1 A soliton. The solution f (X) increases from f = 0 as X → −∞ so that it attains a maximum value f = fmax = α at X = 0, and then decreases symmetrically to f = 0 as X → ∞ as shown in Figure 13.11.1. These features also imply that X0 = 0, so that (13.11.12) becomes f (X) = α sech2 1 3α 4h 2 0 1 2 X 3 . (13.11.13) Therefore, the final solution is η (x, t) = η0 sech2 1 3η0 4h 3 0 1 2 (x − U t) 3 , (13.11.14) where η0 = (αh0). This is called the solitary wave solution of the KdV equation for any positive constant η0. However, it has come to be known as soliton since Zabusky and Kruskal coined the term in 1965. Since η > 0 for all X, the soliton is a wave of elevation which is symmetrical about X = 0. It propagates in the medium without change of shape with velocity U = c0 4 1 + α 2 5 = c0  1 + 1 2 η0 h0  , (13.11.15) which is directly proportional to the amplitude η0. The width,  3η0/4h 3 0 − 1 2 is inversely proportional to √η0. In other words, the solitary wave propagates to the right with a velocity U which is directly proportional to the amplitude, and has a width that is inversely proportional to the square root of the amplitude. Therefore, taller solitons travel faster and are narrower than the shorter (or slower) ones. They can overtake the shorter ones, and surprisingly, they emerge from the interaction without change of shape as shown in Figure 13.11.2. Indeed the discovery of soliton interactions confirms that solitons behave like elementary particles. 13.11 The Korteweg–de Vries Equation and Solitons 577 Figure 13.11.2 Interaction of two solitons (U1 > U2, t2 > t1,). General Waves of Permanent Form. We now consider the general case given by (13.11.8) which can be written  h 2 0 3  f ′2 = −f 3 + αf 2 − 4Af − B ≡ F (f). We seek real bounded solutions for f (X). Therefore, f ′2 ≥ 0 and varies monotonically until f ′ is zero. Hence, the zeros of the cubic F (f) are crucial. For bounded solutions, all the three zeros f1, f2, f3 must be real. Without loss of generality, we choose f1 = 0 and f2 = α. The third zero must be negative so we set f3 = α − β with 0 <α<β. Therefore, the equation for f (X) is 1 3 h 2 0  df dX 2 = f (α − f) (f − α + β), (13.11.16) or $ 3 h 2 0 dX = − df [f (α − f) (f − α + β)] 1 2 , (13.11.17) where U = c0  1 + 2α − β 2  . (13.11.18) 578 13 Nonlinear Partial Differential Equations with Applications We put α − f = p 2 in (13.11.17) to obtain  3 4h 2 0 1 2 dX = dp [(α − p 2) (β − p 2)] 1 2 . (13.11.19) We next substitute p = √ α q into (13.11.19) to transform it into the standard form  3β 4h 2 0 1 2 X =  q 0 dq [(1 − q 2) (1 − m2q 2)] 1 2 (13.11.20) where m = (α/β) 1 2 . The right hand side is an integral of the first kind, and hence, q can be expressed in terms of the Jacobian sn function (see Dutta and Debnath (1965)) q = sn 1 3β 4h 2 0 1 2 X, m3 , (13.11.21) where m is the modulus of the Jacobian elliptic function sn (z,m). Therefore, f (X) = α 1 1 − sn2 / 3β 4h 2 0 1 2 X 03 = α cn2 1 3β 4h 2 0 1 2 X 3 , (13.11.22) where cn (z,m) is also the Jacobian elliptic function of modulus m and cn2 (z)=1 − sn2 (z). From (13.11.20), the period P is given by P = 2  4h 2 0 3β 1 2  1 0 dq [(1 − q 2) (1 − m2q 2)] 1 2 (13.11.23) = 4h0 √ 3β K (m) ≡ λ, (13.11.24) where K (m) is the complete elliptic integral of the first kind defined by K (m) =  π/2 0  1 − m sin2 θ − 1 2 dθ (13.11.25) and λ denotes the wavelength of the cnoidal wave. It is important to note that cn (z,m) is periodic, and hence, η (X) represents a train of periodic waves in shallow water. Thus, these waves are called cnoidal waves with wavelength 13.11 The Korteweg–de Vries Equation and Solitons 579 Figure 13.11.3 A cnoidal wave. λ = 2  4h 3 3b 1/2 K (m). (13.11.26) The outcome of this analysis is that solution (13.11.22) represents a nonlinear wave whose shape and wavelength (or period) all depend on the amplitude of the wave. A typical cnoidal wave is shown in Figure 13.11.3. Sometimes, the cnoidal waves with slowly varying amplitude are observed in rivers. More often, wavetrains behind a weak bore (called an undular bore) can be regarded as cnoidal waves. Two limiting cases are of special interest: (i) m → 1 and (ii) m → 0. When m → 1 (α → β), it is easy to show that cn (z) → sech z. Hence, the cnoidal wave solution (13.11.22) tends to the solitary wave with the wavelength λ, given by (13.11.24) which approaches infinity because K (1) = ∞, K (0) = π/2. The solution identically reduces to (13.11.14) with (13.11.15). In the other limit m → 0 (α → 0), sn z → sin z and cn z → cos z so that solution (13.11.22) becomes f (X) = α cos2 1 3β 4h 2 0 1 2 X 3 , (13.11.27) where U = c0  1 − β 2  . (13.11.28) Using cos 2θ = 2 cos2 θ − 1, we can rewrite (13.11.27) in the form f (X) = α 2 1 + cos √ 3β h0  X . (13.11.29) We next introduce k = √ 3β/h0 (or β = 1 3 k 2h 2 0 ) to simplify (13.11.29) as f (X) = α 2 [1 + cos (kx − ωt)] , (13.11.30) where ω = U k = c0k  1 − 1 6 k 2h 2 0  . (13.11.31) This corresponds to the first two terms of the series of (gk tanh kh0) 1/2 . Thus, these results are in perfect agreement with the linearized theory. 580 13 Nonlinear Partial Differential Equations with Applications Remark: It is important to point out that the phase velocity (13.11.3) becomes negative for k 2 > (c0/σ) which indicates that waves propagate in the negative x direction. This contradicts the original assumption of forward travelling waves. Moreover, the group velocity given by (13.11.4) assumes large negative values for large k so that the fine-scale features of the solution are propagated in the negative x direction. The solution of (13.11.2) involves the Airy function which shows fiercely oscillatory character for large negative arguments. This leads to a lack of continuity and a tendency to emphasize short wave components which contradicts the KdV model representing fairly long waves. In order to eliminate these physically undesirable features of the KdV equation, Benjamin, Bona, and Mahony (1972) proposed a new nonlinear model equation in the form ηt + ηx + ηηx − ηxxt = 0. (13.11.32) This is known as the Benjamin, Bona and Mahony (BBM) equation. The advantage of this model over the KdV equation becomes apparent when we examine their linearized forms and the corresponding solutions. The linearized form (13.11.32) gives the dispersion relation ω = k 1 + k 2 , (13.11.33) which shows that both the phase velocity and the group velocity are bounded for all k, and both velocities tend to zero for large k. In other words, the model has the desirable feature of responding very insignifi- cantly to short wave components that may be introduced into the initial wave form. Thus, the BBM model seems to be a preferable long wave model of physical interest. However, whether the BBM equation is a better model than the KdV equation has not yet been established. Another important property of the KdV equation is that it satisfies the conservation law of the form Tt + Xx = 0, (13.11.34) where T is called the density and the X is called the flux. If T and X are integrable in −∞ <x< ∞,="" and="" x="" →="" 0="" as="" |x|→∞,="" then="" d="" dt="" ="" ∞="" −∞="" t="" dx="−" |x|="" therefore,="" so="" that="" the="" density="" is="" conserved.="" 13.12="" nonlinear="" schr¨odinger="" equation="" solitary="" waves="" 581="" canonical="" form="" of="" kdv="" ut="" −="" 6uux="" +="" uxxx="0," (13.11.35)="" can="" be="" written="" (u)="" ="" −3u="" 2="" uxx="" uxx.="" (13.11.36)="" if="" we="" assume="" u="" periodic="" or="" its="" derivatives="" decay="" very="" rapidly="" this="" leads="" to="" conservation="" mass,="" is,="" (13.11.37)="" often="" called="" time="" invariant="" function="" solutions="" equation.="" second="" law="" for="" (13.11.34)="" obtained="" by="" multiplying="" it="" ="" 1="" ="" −2u="" 3="" uux="" (13.11.38)="" gives="" (13.11.39)="" principle="" energy.="" well="" known="" has="" an="" infinite="" number="" polynomial="" laws.="" generally="" believed="" existence="" a="" soliton="" solution="" closely="" related="" first="" derive="" one-dimensional="" linear="" from="" fourier="" integral="" representation="" plane="" wave="" 582="" 13="" partial="" differential="" equations="" with="" applications="" φ="" (x,="" t)="" f="" (k)="" exp="" [i(kx="" ωt)]="" dk,="" (13.12.1)="" where="" spectrum="" determined="" given="" initial="" boundary="" conditions.="" slowly="" modulated="" propagates="" in="" dispersive="" medium.="" such="" wave,="" most="" energy="" confined="" neighborhood="" k="k0" dispersion="" relation="" ω="ω" expanded="" about="" point="" (k="" k0)="" ′="" ′′="" ...,="" (13.12.2)="" ω0="" ≡="" (k0),="" (k0).="" view="" (13.12.2),="" rewrite="" [i(k0x="" ω0t)]="" ,="" (13.12.3)="" amplitude="" ψ="" i(k="" i="" ="" 0="" dk.="" (13.12.4)="" evidently,="" represents="" varying="" (or="" modulated)="" part="" basic="" wave.="" simple="" computation="" ψt,="" ψx,="" ψxx="" ψt="−i" ψx="i(k" i(ψt="" ψx)="" (13.12.5)="" associated="" ω′="" 2ω="" .="" (13.12.6)="" choose="" frame="" reference="" moving="" group="" velocity,="" ∗="x" t,="" term="" involving="" dropped="" then,="" satisfies="" equation,="" dropping="" asterisks,="" 583="" (13.12.7)="" next="" both="" frequency="" general="" k,="" a2="" (13.12.8)="" expand="" taylor="" series="" |a|="" ∂ω="" ∂k="" ∂="" ∂k2="" 4="" 5="" (13.12.9)="" now="" replace="" i(∂="" ∂t)="" k0="" −i(∂="" ∂x),="" resulting="" operators="" act="" on="" a,="" obtain="" i(at="" ax)="" axx="" γ="" (13.12.10)="" ω′′="" constant.="" (nls)="" velocity="" ax="" will="" drop="" out="" (13.12.10),="" normalized="" nls="" at="" (13.12.11)="" corresponding="" (13.12.12)="" according="" stability="" criterion="" established="" section="" 13.5,="" modulation="" stable="" <="" unstable=""> 0. To study the solitary wave solution, it is convenient to use the NLS equation in the standard form i ψt + ψxx + γ |ψ| 2 ψ = 0, −∞ <x< ∞,="" t="" ≥="" 0.="" (13.12.13)="" 584="" 13="" nonlinear="" partial="" differential="" equations="" with="" applications="" we="" seek="" waves="" of="" permanent="" form="" by="" assuming="" the="" solution="" ψ="f" (x)="" e="" i(mx−nt)="" ,="" x="x" −="" u="" (13.12.14)="" for="" some="" functions="" f="" and="" constant="" wave="" speed="" to="" be="" determined,="" m,="" n="" are="" constants.="" substitution="" into="" gives="" ′′="" +="" i(2m="" u)="" ′="" ="" m2="" γ="" |f|="" 2="" (13.12.15)="" eliminate="" setting="" 2m="" then,="" write="" α="" so="" that="" can="" assumed="" real.="" thus,="" equation="" becomes="" αf="" 3="0." (13.12.16)="" multiplying="" this="" 2f="" integrating,="" find="" ′2="A" 4="" ≡="" (f),="" (13.12.17)="" where="" (f)="" α1="" α2f="" β1="" β2f="" (α1β2="" α2β1),="" a="α1β1," (α2β2),="" s="" β="" real="" distinct.="" evidently,="" 0="" df="" ="" (α1="" 2)="" (β1="" .="" (13.12.18)="" putting="" (α2="" α1)="" 1="" in="" integral,="" deduce="" following="" elliptic="" integral="" first="" kind="" (see="" dutta="" debnath="" (1965)):="" σx="" du="" (1="" κ="" 2u="" (13.12.19)="" σ="(α2β1)" (β1α2).="" final="" expressed="" terms="" jacobian="" sn="" function="" (σx,="" κ),="" or,="" α2="" 1="" κ).="" (13.12.20)="" particular,="" when=""> 0, and γ > 0, we obtain a solitary wave solution. In this case, equation (13.12.17) can be rewritten as 13.12 The Nonlinear Schr¨odinger Equation and Solitary Waves 585 √ α X =  f 0 df f  1 − γ 2α f 2 1 2 . (13.12.21) Substitution of (γ/2α) 1 2 f = sech θ in this integral gives the exact solution f (X) =  2α γ 1 2 sech √ α (x − U t) ! . (13.12.22) This represents a solitary wave which propagates without change of shape with constant velocity U. Unlike the solution of the KdV equation, the amplitude and the velocity of the wave are independent parameters. It is noted that the solitary wave exists only for the unstable case (γ > 0). This means that small modulations of the unstable wavetrain lead to a series of solitary waves. The nonlinear dispersion relation for deep water waves is ω =  gk  1 + 1 2 a 2 k 2  . (13.12.23) Therefore, ω ′ 0 = ω0 2k0 , ω′′ 0 = − ω0 4k 2 0 , and γ = − 1 2 ω0k 2 0 , (13.12.24) and the NLS equation for deep water waves is obtained from (13.12.10) in the form i  at + ω0 2k0 ax  −  ω0 8k 2 0  axx − 1 2 ω0 k 2 0 |a| 2 a = 0. (13.12.25) The normalized form of this equation in a frame of reference moving with the linear group velocity ω ′ 0 is i at −  ω0 8k 2 0  axx = 1 2 ω0 k 2 0 |a| 2 a. (13.12.26) Since γ ω′′ 0 =  ω 2 0/8 > 0, this equation confirms the instability of deep water waves. This is one of the most remarkable recent results in the theory of water waves. We next discuss the uniform solution and the solitary wave solution of (13.12.26). We look for solutions in the form a (x, t) = A (X) exp  i γ2 t , X = x − ω ′ 0 t (13.12.27) and substitute this into equation (13.12.26) to obtain AXX = −  8k 2 0 ω0 γ 2A + 1 2 ω0 k 2 0 A 3  . (13.12.28) 586 13 Nonlinear Partial Differential Equations with Applications We multiply this equation by 2AX and then integrate to find A 2 X = −  A 4 0 m′2 + 8 ω0 γ 2 k 2 0A 2 + 2k 4 0A 4  =  A 2 0 − A 2 A 2 − m′2A 2 0 , (13.12.29) where  A4 0m′2 is an integrating constant and 2k 4 0 = 1, m′2 = 1 − m2 , and A2 0 = 4γ 2/ω0k 2 0  m2 − 2 which relates A0, γ, and m. Finally, we rewrite equation (13.12.29) in the form A 2 0 dX = dA "41 − A2 A2 0 54 A2 A2 0 − m′2 5# 1 2 , (13.12.30) or, A0 (X − X0) =  ′ ds [(1 − s 2) (s 2 − m′2)] 1 2 , s = (A/A0). This can readily be expressed in terms of the Jacobian dn function (see Dutta and Debnath (1965)) A = A0 dn [A0 (X − X0), m] , (13.12.31) where m is the modulus of the dn function. In the limit m → 0, dn z → 1 and γ 2 → −1 2 ω0k 2 0A2 0 . Hence, the solution is a (x, t) = A (t) = A0 exp  − 1 2 i ω0 k 2 0 A 2 0 t  . (13.12.32) On the other hand, when m → 1, dn z → sech z and γ 2 → −1 4 ω0k 2 0A2 0 . Therefore, the solitary wave solution is a (x, t) = A0 sech [A0 (x − ω ′ 0 t − X0)] exp  − 1 4 ω0 k 2 0 A 2 0 t  .(13.12.33) We next use the NLS equation (13.12.26) to discuss the instability of deep water waves, which is known as the Benjamin and Feir instability. We consider a perturbation of (13.12.32) and write a (X, t) = A (t) [1 + B (X, t)] , (13.12.34) where A (t) is the uniform solution given by (13.12.32). Substituting equation (13.12.34) into (13.12.26) gives iAt (1 + B) + iA (t) Bt −  ω0 8k 2 0  A (t) BXX = 1 2 ω0k 2 0A 2 0 [(1 + B) + BB∗ (1 + B)+(B + B ∗ ) B + (B + B ∗ )] A, 13.12 The Nonlinear Schr¨odinger Equation and Solitary Waves 587 where B∗ is the complex conjugate of B. Neglecting squares of B, it follows that i Bt −  ω0 8k 2 0  BXX = 1 2 ω0k 2 0A 2 0 (B + B ∗ ). (13.12.35) We now seek a solution of the form B (X, t) = B1 e Ωt+iκX + B2 e Ωt−iκX, (13.12.36) where B1, B2 are complex constants, κ is a real wavenumber, and Ω is a growth rate (possibly complex) to be determined. Substitution of B into (13.12.35) leads to the pair of coupled equations  iΩ + ω0κ 2 8k 2 0  B1 − 1 2 ω0k 2 0A 2 0 (B1 + B ∗ 2 )=0, (13.12.37)  iΩ + ω0κ 2 8k 2 0  B2 − 1 2 ω0k 2 0A 2 0 (B ∗ 1 + B2)=0. (13.12.38) It is convenient to take the complex conjugate of (13.12.38) so that it assumes the form  −iΩ + ω0κ 2 8k 2 0  B ∗ 2 − 1 2 ω0k 2 0A 2 0 (B1 + B ∗ 2 )=0. (13.12.39) The pair of linear homogeneous equations (13.12.37) and (13.12.39) for B1 and B∗ 2 admits a nontrivial solution provided         iΩ + 4 ω0κ 2 8k 2 0 5 − 1 2 ω0k 2 0A2 0 − 1 2 ω0k 2 0A2 0 − 1 2 ω0k 2 0A2 0 iΩ + 4 ω0κ 2 8k 2 0 5 − 1 2 ω0k 2 0A2 0         = 0, or Ω 2 =  ω 2 0κ 2 8k 2 0 k 2 0A 2 0 − κ 2 8k 2 0  . (13.12.40) The growth rate Ω is purely imaginary or real and positive depending on whether  κ 2/k2 0 > 8k 2 0A2 0 or  κ 2/k2 0 < 8k 2 0A2 0 . The former case corresponds to a wave (an oscillatory solution) for B, and the latter case represents the Benjamin and Feir instability criterion with κ5 = (κ/k0) as the non-dimensional wavenumber so that κ5 2 < 8k 2 0A 2 0 . (13.12.41) The range of instability is given by 0 < κ <5 κ5c = 2√ 2 (k0A0). (13.12.42) 588 13 Nonlinear Partial Differential Equations with Applications Since Ω is a function of κ5, maximum instability occurs at κ5 = κ5max = 2k0A0 with a maximum growth rate given by (Re Ω)max = 1 2 ω0k 2 0A 2 0 . (13.12.43) To establish the connection with the Benjamin–Feir instability, we have to find the velocity potential for the fundamental wave mode multiplied by exp (kz). It turns out that the term proportional to B1 is the upper sideband, whereas that proportional to B2 is the lower sideband. The main conclusion of the preceding analysis is that Stokes water waves are definitely unstable. In 1967, Benjamin and Feir (see Whitham (1976) or Debnath (2005)) confirmed these remarkable results both theoretically and experimentally. Conservation Laws for the NLS Equation Zakharov and Shabat (1972) proved that equation (13.12.13) has an infinite number of polynomial conservation laws. Each has the form of an integral, with respect to x, of a polynomial expression in terms of the function ψ (x, t) and its derivatives with respect to x. These laws are somewhat similar to those already proved for the KdV equation. Therefore, the proofs of the conservation laws are based on similar assumptions used in the context of the KdV equation. We prove here three conservation laws for the nonlinear Schr¨odinger equation (13.12.13):  ∞ −∞ |ψ| 2 dx = constant = C1, (13.12.44)  ∞ −∞ i  ψ ψx − ψ ψx dx = constant = C2, (13.12.45)  ∞ −∞  |ψx| 2 − 1 2 γ |ψ| 4  dx = constant = C3, (13.12.46) where the bar denotes the complex conjugate. We multiply (13.12.13) by ψ and its complex conjugate by ψ and subtract the latter from the former to obtain i d dt  ψ ψ + d dx  ψx ψ − ψx ψ = 0. (13.12.47) Integration with respect to x in −∞ <x< ∞="" gives="" i="" d="" dt="" ="" −∞="" |ψ|="" 2="" dx="0." this="" proves="" result="" (13.12.44).="" 13.12="" the="" nonlinear="" schr¨odinger="" equation="" and="" solitary="" waves="" 589="" we="" multiply="" (13.12.13)="" by="" ψx="" its="" complex="" conjugate="" then,="" add="" them="" to="" obtain="" ="" ψt="" −="" +="" ψxx="" γ="" ψ="" (13.12.48)="" differentiate="" with="" respect="" x,="" former="" latter="" then="" together.="" leads="" ψxψxt="" ψxt="" ψxxx="" +γ="" "="" 4="" 5="" x="" #="0." (13.12.49)="" if="" subtract="" from="" simplify,="" have="" 4="" .="" integrating="" second="" result.="" resulting="" equations="" derive="" or,="" |ψx|="" (13.12.50)="" (13.12.46).="" above="" three="" conservation="" integrals="" a="" simple="" physical="" meaning.="" in="" fact,="" constants="" of="" motion="" c1,="" c2="" c3="" are="" related="" number="" particles,="" momentum,="" energy="" system="" governed="" equation.="" 590="" 13="" partial="" differential="" applications="" an="" analysis="" section="" reveals="" several="" remarkable="" features="" can="" also="" be="" used="" investigate="" instability="" phenomena="" many="" other="" systems.="" like="" various="" forms="" kdv="" equation,="" nls="" arises="" problems,="" including="" water="" ocean="" waves,="" plasma,="" propagation="" heat="" pulses="" solid,="" self-trapping="" optics,="" fluid="" filled="" viscoelastic="" tube,="" fluids="" plasmas="" (see="" debnath="" (2005)).="" 13.13="" lax="" pair="" zakharov="" shabat="" scheme="" his="" 1968="" seminal="" paper,="" developed="" elegant="" formalism="" for="" finding="" isospectral="" potentials="" as="" solutions="" evolution="" all="" integrals.="" work="" deals="" some="" new="" fundamental="" ideas,="" deeper="" results,="" their="" application="" model.="" subsequently="" paved="" way="" generalizations="" technique="" method="" solving="" equations.="" introducing="" heisenberg="" picture,="" inverse="" scattering="" based="" upon="" abstract="" formulation="" certain="" properties="" operators="" on="" hilbert="" space,="" which="" familiar="" context="" quantum="" mechanics.="" has="" feature="" associating="" linear="" that="" analogs="" formulate="" lax’s="" (1968),="" consider="" two="" l="" m.="" eigenvalue="" operator="" corresponds="" general="" form="" is="" lψ="λψ," (13.13.1)="" where="" eigenfunction="" λ="" corresponding="" eigenvalue.="" m="" describes="" change="" eigenvalues="" parameter="" t,="" usually="" represents="" time="" (13.13.2)="" differentiating="" t="" ltψ="" lψt="λtψ" λψt.="" (13.13.3)="" next="" eliminate="" using="" lmψ="λtψ" λmψ="λtψ" mλψ="λtψ" mlψ,="" (13.13.4)="" 591="" equivalently,="" ∂l="" ∂t="" (ml="" lm)="" ψ.="" (13.13.5)="" thus,="" constant="" nonzero="" eigenfunctions="" only="" (lm="" ml)="" [l,="" m]="" ψ,="" (13.13.6)="" called="" commutator="" m,="" derivative="" left-hand="" side="" interpreted="" alone.="" it="" picture="" problem,="" course,="" how="" determine="" these="" given="" there="" no="" systematic="" solution="" problem.="" negative="" integrable="" hierarchy,="" qiao="" (1995)="" strampp="" (2002)="" suggest="" approach="" generate="" equations;="" they="" devise="" strategies="" spectral="" initial-value="" problem="" u="" (x,="" t)="" satisfies="" ut="N" (u)="" (13.13.7)="" 0)="f" (x),="" (13.13.8)="" ∈="" y="" suitable="" function="" n="" :="" →="" independent="" but="" may="" involve="" or="" derivatives="" x.="" must="" assume="" expressed="" lt="" (13.13.9)="" space="" h="" depend="" scalar="" operator.="" self-adjoint="" so="" (lφ,="" ψ)="(φ," lψ)="" φ="" (·,="" ·)="" inner="" product.="" now="" h:="" (t)="" ≥="" 0,="" r.="" (13.13.10)="" making="" use="" (13.13.9),="" λtψ="(L" λ)="" (ψt="" mψ).="" (13.13.11)="" product="" yields="" (ψ,="" λt="((L" mψ),="" (13.13.12)="" 592="" which,="" since="" self-adjoint,="" mψ)="0." hence,="" confirming="" each="" constant.="" consequently,="" becomes="" l(ψt="" (13.13.13)="" shows="" λ.="" always="" possible="" redefine="" adding="" identity="" original="" remains="" unchanged.="" 0.="" (13.13.14)="" following.="" theorem="" 13.13.1.="" (13.13.15)="" holds,="" (13.13.14).="" not="" yet="" clear="" find="" satisfy="" preceding="" conditions.="" illustrate="" method,="" choose="" ≡="" ∂="" ∂x2="" u,="" (13.13.16)="" sturm–liouville="" l.="" l,="" theory="" unitary="" h,="" chosen="" antisymmetric,="" (mφ,="" (φ,="" h.="" so,="" combination="" odd="" natural="" choice="" follows="" nφ="" ∂xn="" ψdx="−" nψ="" mψ),(13.13.17)="" provided="" n,="" φ,="" tend="" zero,="" |x|→∞.="" moreover,="" require="" sufficient="" freedom="" any="" unknown="" functions="" make="" multiplicative="" operator,="" is,="" degree="" zero.="" simplest="" (∂="" ∂x),="" c="" automatically="" 593="" cux="0," (13.13.18)="" one-dimensional="" wave="" (13.13.19)="" associated="" motion.="" 3="" ∂x3="" ∂x="" ∂xa="" b,="" (13.13.20)="" constant,="" t),="" b="B" third="" term="" right-hand="" dropped,="" retain="" convenience.="" algebraic="" calculation="" uxxx="" axxx="" bxx="" uxa="" (3auxx="" 4axx="" 2bx)="" (3aux="" 4ax)="" would="" au="" (t).="" auux="" (13.13.21)="" standard="" defined="" reduces="" ="" ∂xu="" ="" (13.13.22)="" simplified="" (u="" uψ)x="" 3ψx="" +3(uψ)x="" bψ="2(u" 2λ)="" uxψ="" bψ.="" (13.13.23)="" close="" comments.="" first,="" solvable="" transform="" (ist),="" form.="" however,="" main="" difficulty="" completely="" determining="" whether="" produces="" and,="" indeed,="" proved="" infinite="" operators,="" one="" order="" ∂x,="" family="" flows="" under="" spectrum="" 594="" preserved.="" second,="" study="" choosing="" alternative="" third,="" restriction="" should="" limited="" class="" could="" removed.="" matrix="" operators.="" already="" been="" extended="" such="" fourth,="" (1972,="" 1974)="" published="" series="" notable="" papers="" field="" extending="" (nls)="" first="" time,="" generalized="" more="" than="" spatial="" variable.="" extension="" known="" (zs)="" scheme,="" essentially,="" recasts="" form,="" leading="" marchenko="" finally,="" briefly="" discuss="" zs="" nonself-adjoint="" n-soliton="" introduced="" ingenious="" (13.13.24)="" ml),="" (13.13.25)="" include="" coefficients,="" refers="" t.="" lφ="λφ." (13.13.26)="" differentiation="" dλ="" (iφt="" mφ).="" (13.13.27)="" initially="" changes="" manner="" iφt="Mφ," (13.13.28)="" (13.13.26).="" coupling="" coefficients="" nature="" determines="" potential="" (13.13.26),="" (13.13.28).="" although="" quite="" general,="" crucial="" step="" factor="" according="" (13.13.25).="" (1972)="" introduce="" ×="" matrices="" follows:="" ⎡="" ⎣="" 1="" α="" 0="" ⎤="" ⎦="" ∗="" ⎦,="" (13.13.29)="" ⎢="" |u|="" 1+α="" iu∗="" −iux="" −|u|="" 1−α="" ⎥="" (13.13.30)="" 13.14="" exercises="" 595="" iut="" uxx="" (13.13.31)="" complete="" solved="" initial="" condition="" 0).="" seems="" significant="" contribution="" come="" point="" large="" times="" (t="" ∞).="" physically,="" disturbance="" tends="" disintegrate="" into="" waves.="" mathematical="" asymptotic="" |x|→∞,="" expected="" end="" wavetrains="" modulations.="" 1.="" flow="" density="" relation="" q="vρ" (1="" ρ="" ρ1),="" traffic="" (x)="" let="" f="" ⎪⎪⎪⎪⎨="" ⎪⎪⎪⎪⎩="" ,="" ≤="" 5="" 12x,="" show="" (i)="" ρ0="" along="" characteristic="" lines="" ct="3(x" x0),="" x0="" (ii)="" x),="" 1,="" (iii)="" 12x0ρ0="" (x="" what="" happens="" at="" intersection="" x).="" draw="" versus="" 2.="" mountain="" height="" vulnerable="" erosion="" slope="" hx="" very="" large.="" ht="" functional="" (hx),="" ux="0," ′="" (u).="" 3.="" river="" carrying="" particles="" through="" solid="" bed.="" during="" sedimentation="" process,="" will="" 596="" deposited="" assuming="" v="" velocity="" ρb="" density,="" ρf="" carried="" material="" bed,="" law="" ∂ρ="" ∂q="" (a)="" ∂ρf="" (ρf="" )="" q′="" (b)="" chemical="" engineering="" between="" ∂ρb="" (α="" ρb)="" k2="" (β="" ρb,="" k1,="" represent="" reaction="" rates="" α,="" β="" values="" saturation="" levels="" bed="" respectively.="" speed="" (k1α="" k2β)="" densities="" small.="" 4.="" steady="" (ζ),="" ζ="x" burgers’="" boundary="" conditions="" +∞="" tanh="" ="" ∞)="" 4ν="" ct)="" 0="" 2a="" roots="" 2cf="" integration.="" 5.="" transformations="" −6γ="" reduce="" uux="" γuxxx="0" canonical="" 6uux="" hence="" otherwise,="" prove="" zero="" |x|→∞="" sech2="" (a="" 597="" 6.="" verify="" riccati="" transformation="" +vx="" transforms="" vt="" 6v="" vx="" vxxx="0." (without="" energy-level="" term)="" 7.="" apply="" characteristics="" solve="" ∂u="" ∂v="" data="" −x="" riemann="" invariants="" r="" (α)="2" cosh="" s="" (β)="2" sinh="" β,="" also,="" t).="" 8.="" isentropic="" flow,="" euler="" ρt="" (ρu)x="0," px="0," st="" usx="0," direction,="" p="" pressure,="" entropy.="" families="" characteristics.="" Γ0="" Γ+="" c,="" ∂p="" ∂ρ5="" following="" full="" set="" ds="" Γ0,="" dp="" ρc="" du="" Γ+.="" particular,="" when="" (s="constant" everywhere),="" (ρ)="" dρ="" c.="" 598="" 9.="" (13.8.32)–(13.8.36),="" second-order="" trs="" (r="" s)="" (tr="" ts)="0," 2c="" dc="" dρ="" polytropic="" gas,="" 4c="α" (c)="" γ−1="r" s,="" 9(b)="" euler–poisson–darboux="" tuu="" 2="" tcc="" 2α="" tc="" variables.="" 10.="" uxx,="" 11.="" uuxx="" 3u="" 2uxx="" xx="" uxut.="" 12.="" laws="" (v)="" vxx="" vvxx="" 599="" 13.="" (13.12.13),="" 14.="" (13.10.20)="" ν="" 15.="" uxxt="0" uxt="" uuxt="" 16.="" <x<="" ∞,=""> 0, ψ → 0, |x|→∞, ψ (x, 0) = ψ (x) with  ∞ −∞ |ψ| 2 dx = 1. has the conservation law 4 i|ψ| 2 5 t + (ψ ∗ψx − ψψ∗ x )x = 0 and the energy integral  ∞ −∞ |ψ| 2 dx = 1. 17. Seek a dispersive wave solution of the telegraph equation (see problem 14, 3.9 Exercises) utt − c 2uxx + ac2ut + bc2u = 0 in the form u (x, t) = A exp [i(kx − ωt)] . (a) Show that ω = − 1 2  iac2 + 1 2 4c 2k 2 +  4b − c 2 c 2 ! 1 2 . (b) If 4b = a 2 c 2 , show that the solution u (x, t) = A exp  − 1 2 ac2 t  exp [ik (x + ct)] represents nondispersive waves with attenuation. 14 Numerical and Approximation Methods “The strides that have been made recently, in the theory of nonlinear partial differential equations, are as great as in the linear theory. Unlike the linear case, no wholesale liquidation of broad classes of problems has taken place; rather, it is steady progress on old fronts and on some new ones, the complete solution of some special problems, and the discovery of some brand new phenomena. The old tools – variational methods, fixed point theorems, mapping degree, and other topological tools have been augmented by some new ones. Pre-eminent for discovering new phenomena is numerical experimentation; but it is likely that in the future numerical calculations will be parts of proofs.” Peter Lax “Almost everyone using computers has experienced instances where computational results have sparked new insights.” Norman J. Zabusky 14.1 Introduction The preceding chapters have been devoted to the analytical treatment of linear and nonlinear partial differential equations. Several analytical methods to find the exact analytical solution of these equations within simple domains have been discussed. The boundary and initial conditions in these problems were also relatively simple, and were expressible in simple mathematical form. In dealing with many equations arising from the modelling of physical problems, the determination of such exact solutions in a simple domain is a formidable task even when the boundary and/or initial data are simple. It is then necessary to resort to numerical or approximation methods in order to deal with the problems that cannot be solved analytically. In 602 14 Numerical and Approximation Methods view of the widespread accessibility of today’s high speed electronic computers, numerical and approximation methods are becoming increasingly important and useful in applications. In this chapter some of the major numerical and approximation approaches to the solution of partial differential equations are discussed in some detail. These include numerical methods based on finite difference approximations, variational methods, and the Rayleigh–Ritz, Galerkin, and Kantorovich methods of approximation. The chapter also contains a large section on analytical treatment of variational methods and the Euler– Lagrange equations and their applications. A short section on the finite element method is also included. 14.2 Finite Difference Approximations, Convergence, and Stability The Taylor series expansion of a function u (x, y) of two independent variables x and y is u (xi + h, yj ) = ui + 1,j = ui,j + h (ux) i,j + h 2 2! (uxx) i,j + h 3 3! (uxxx) i,j + ..., (14.2.1ab) u (xi , yj + k) = ui,j + 1 = ui,j + k (uy) i,j + k 2 2! (uyy) i,j + k 3 3! (uyyy) i,j + ..., (14.2.2ab) where ui,j = u (x, y), ui + 1,j = u (x + h, y), and ui,j + 1 = u (x, y + k). We choose a set of uniformly spaced rectangles with vertices at Pi,j with coordinates (ih, jk), where i, j, are positive or negative integers or zero, as shown in Figure 14.2.1. We denote u (ih, jk) by ui,j . Using the above Taylor series expansion, we write approximate expressions for ux at the vertex Pi,j in terms of ui,j , ui + 1,j : ux = 1 h [u (x + h, y) − u (x, y)] ∼ 1 h (ui+1,j − ui,j ) + O (h), (14.2.3) ux = 1 h [u (x, y) − u (x − h, y)] ∼ 1 h (ui,j − ui−1,j ) + O (h), (14.2.4) ux = 1 2h [u (x + h, y) − u (x − h, y)] ∼ 1 2h (ui+1,j − ui−1,j ) + O  h 2 . (14.2.5) These expressions are called the forward first difference, backward first difference, and central first difference of ux, respectively. The quantity O (h) or O  h 2 is known as the truncation error in this discretization process. 14.2 Finite Difference Approximations, Convergence, and Stability 603 Figure 14.2.1 Uniformly spaced rectangles. A similar approximate result for uxx at the vertex Pi,j is uxx = 1 h 2 [u (x + h, y) − 2 u (x, y) + u (x − h, y)] ∼ 1 h 2 [ui+1,j − 2 ui,j + ui−1,j ] + O  h 2 . (14.2.6) Similarly, the approximate formulas for uy and uyy at Pi,j are uy = 1 k [u (x, y + k) − u (x, y)] ∼ 1 k (ui,j+1 − ui,j ) + O (k), (14.2.7) uy = 1 k [u (x, y) − u (x, y − k)] ∼ 1 k (ui,j − ui,j−1) + O (k), (14.2.8) uy = 1 2k [u (x, y + k) − u (x, y − k)] ∼ 1 2k (ui,j+1 − ui,j−1) + O  k 2 , (14.2.9) uyy = 1 k 2 [u (x, y + k) − 2 u (x, y) + u (x, y − k)] ∼ 1 k 2 [ui,j+1 − 2ui,j + ui,j−1] + O  k 2 . (14.2.10) All these difference formulas are extremely useful in finding numerical solutions of first or second order partial differential equations. Suppose U (x, y) represents the exact solution of a partial differential equation L(U) = 0 with independent variables x and y, and ui,j is the exact solution of the corresponding finite difference equation F (ui,j ) = 0. Then, the finite difference scheme is said to be convergent if ui,j tends to U as h and k tend to zero. The difference, di,j ≡ (Ui,j − ui,j ) is called the cummulative truncation (or discretization) error. This error can generally be minimized by decreasing the grid sizes h and k. However, this error depends not only on h and k, but also on the 604 14 Numerical and Approximation Methods number of terms in the truncated series which is used to approximate each partial derivative. Another kind of error is introduced when a partial differential equation is approximated by a finite difference equation. If the exact finite difference solution ui,j is replaced by the exact solution Ui,j of the partial differential equation at the grid points Pi,j , then the value F (Ui,j ) is called the local truncation error at Pi,j . The finite difference scheme and the partial differential equation are said to be consistent if F (Ui,j ) tends to zero as h and k tend to zero. In general, finite difference equations cannot be solved exactly because the numerical computation is carried out only up to a finite number of decimal places. Consequently, another kind of error is introduced in the finite difference solution during the actual process of computation. This kind of error is called the round-off error, and it also depends upon the type of computer used. In practice, the actual computational solution is u ∗ i,j , but not ui,j , so that the difference ri,j =  ui,j − u ∗ i,j is the roundoff error at the grid point Pi,j . In fact, this error is introduced into the solution of the finite difference equation by round-off errors. In reality, the round-off error depends mainly on the actual computational process and the finite difference itself. In contrast to the cummulative truncation error, the round-off error cannot be made small by allowing h and k to tend to zero. Thus, the total error involved in the finite difference analysis at the point Pi,j is given by  Ui,j − u ∗ i,j = (Ui,j − ui,j ) +  ui,j − u ∗ i,j = di,j − ri,j . (14.2.11) Usually the discretization error di,j is bounded when ui,j is bounded because the value of Ui,j is fixed for a given partial differential equation with the prescribed boundary and initial data. This fact is used or assumed in order to introduce the concept of stability. The finite difference algorithm is said to be stable if the round-off errors are sufficiently small for all i as j → ∞, that is, the growth of ri,j can be controlled. It should be pointed out again that the round-off error depends not only on the actual computational process and the type of computer used, but also on the finite difference equation itself. Lax (1954) proved a remarkable theorem which establishes the relationship between consistency, stability, and convergence for the finite difference algorithm. Theorem 14.2.1. (Lax’s Equivalence Theorem). Given a properly posed linear initial-value problem and a finite difference approximation to it that satisfies the consistency criterion, stability is a necessary and sufficient condition for convergence. 14.3 Lax–Wendroff Explicit Method 605 Von Neumann’s Stability Method This method is essentially based upon a finite Fourier series. It expresses the initial errors on the line t = 0 in terms of a finite Fourier series and then examines the propagation of errors as t → ∞. It is convenient to denote the error function by er,s instead of ei,j so that er,s gives the initial values er,0 = e (rh) = er on the line t = 0 between x = 0 and x = l, where r = 0, 1, 2, ..., N and Nh = l. The finite Fourier series expansion of er is er =  N n=0 An exp (inπx/l) =  N n=0 An exp (iαnrh), (14.2.12) where αn = (nπ/l), x = rh, and An are the Fourier coefficients which are determined from the (N + 1) equations (14.2.12). Since we are concerned with the linear finite difference scheme, errors form an additive system so that the total error can be found by the superposition principle. Thus, it is sufficient to consider a single term exp (iαrh) in the Fourier series (14.2.12). Following the method of separation of variables commonly used for finding the analytical solution of a partial differential equation, we seek a separable solution of the finite difference equation for er,s in the form er,s = exp (iαrh + βsk) = exp (iαrh) p s (14.2.13) which reduces to exp (iαrh) at s = 0(t = sk = 0), where p = exp (βk), and β is a complex constant. This shows that the error is bounded as (t → ∞) provided that |p| ≤ 1 (14.2.14) is satisfied. This condition is found to be necessary and sufficient for the stability of the finite difference algorithm. 14.3 Lax–Wendroff Explicit Method To describe this method, we consider the first-order conservation equation ∂u ∂t + c ∂u ∂x = 0 (14.3.1) where u ≡ u (x, t) is some physical function of space variable x and time t. This equation occurs frequently in applied mathematics. Lax and Wendroff use the Taylor series expansion in t in the form ui,j+1 = ui,j + k (ut) i,j + k 2 2! (utt) i,j + k 3 3! (uttt) i,j + ..., (14.3.2) 606 14 Numerical and Approximation Methods where k ≡ δt. The partial derivatives in t in (14.3.2) can easily be eliminated by using ut = −c ux so that (14.3.2) becomes ui,j+1 = ui,j − c k (ux) i,j + c 2k 2 2 (uxx) i,j − .... (14.3.3) Replacing ux, uxx by the central difference formulas, (14.3.3) becomes ui,j+1 = ui,j −  ck 2h  (ui+1,j − ui−1,j ) + 1 2  ck h 2 (ui+1,j − 2 ui,j + ui−1,j ), or ui,j+1 =  1 − ε 2 ui,j + ε 2 (1 + ε) ui−1,j − ε 2 (1 + ε) ui+1,j + O  ε 3 ,(14.3.4) where ε = (ck/h). This is called the Lax–Wendroff second-order finite difference scheme; it has been widely used to solve first-order hyperbolic equations. Von Neumann criterion (14.2.14) can be applied to investigate the stability of the Lax–Wendroff scheme. It is noted that the error function er,s given by (14.2.13) satisfies the finite difference equation (14.3.4). We then substitute (14.2.13) into (14.3.4) and cancel common factors to obtain p =  1 − ε 2 + ε 2 (1 + ε) e −iαh − (1 − ε) e iαh! = 1 − 2 ε 2 sin2  αh 2  − 2iε sin  αh 2  cos  αh 2  , so that |p| 2 = 1 − 4ε 2  1 − ε 2 sin4  αh 2  . (14.3.5) According to the von Neumann criterion, the Lax–Wendroff scheme (14.3.4) is stable as t → ∞ if |p| ≤ 1, which gives 4ε 2  1 − ε 2 ≥ 0, that is, 0 < ε ≤ 1. The local truncation error of the Lax–Wendroff equation (14.3.4) at Pi,j is Ti,j = 1 k (ui,j+1 − ui−1,j ) which is, by (14.2.2a) and (14.2.1b) with ck = h (ε = 1), = (ut + cux) i,j + k 2  utt − c 2uxxx i,j + k 2 6  uttt + c 3uxxx i,j + O 4 (ck) 3 5 . (14.3.6) 14.3 Lax–Wendroff Explicit Method 607 The first two terms on the right side of (14.3.6) vanish by equation (14.3.1) so that the local truncation error becomes Ti,j = 1 6  k 2uttt + c h2uxxx i,j . (14.3.7) Another approximation to (14.3.1) with first-order accuracy is 1 k (ui,j+1 − ui,j ) + c h (ui,j − ui−1,j )=0. (14.3.8) A final explicit scheme for (14.3.1) is based on the central difference approximation. This scheme is called the leap frog algorithm. In this method, the finite difference approximation to (14.3.1) is 1 2k (ui,j+1 − ui,j−1) + c 2h (ui+1,j − ui−1,j )=0, or, ui,j+1 = ui,j−1 − ε (ui+1,j − ui−1,j ). (14.3.9) As shown in Figure 14.3.1, this equation shows that the value of u at Pi,j+1 is computed from the previously computed values at three grid points at two previous time steps. Figure 14.3.1 Grid system for the leap frog algorithm. 608 14 Numerical and Approximation Methods 14.4 Explicit Finite Difference Methods (A) Wave Equation and the Courant–Friedrichs–Lewy Convergence Criterion The method of characteristics provides the most convenient and accurate procedure for solving Cauchy problems involving hyperbolic equations. One of the main advantages of this method is that discontinuities of the initial data propagate into the solution domain along the characteristics. However, when the initial data are discontinuous, the finite difference algorithm for the hyperbolic systems is not very convenient. Problems concerning hyperbolic equations with continuous initial data can be solved successfully by finite difference methods with rectangular grid systems. A commonly cited problem is the propagation of a one-dimensional wave governed by the system utt = c 2 uxx, −∞ <x< ∞,="" t=""> 0, (14.4.1) u (x, 0) = f (x), ut (x, 0) = g (x) for all x ∈ R. (14.4.2) Using a rectangular grid system with h = δx, k = δt, ui,j = u (ih, jk), −∞ <x< ∞,="" and="" 0="" ≤="" j="" <="" the="" central="" difference="" approximation="" to="" equation="" (14.4.1)="" is="" 1="" k="" 2="" (ui,j+1="" −="" ui,j="" +="" ui,j−1)="c" h="" (ui+1,j="" ui−1,j="" ),="" or,="" ui,j+1="ε" )+2="" ="" ε="" ui,j−1,="" (14.4.3)="" where="" ≡="" (ck="" h),="" often="" called="" courant="" parameter.="" this="" explicit="" formula="" allows="" us="" determine="" approximate="" values="" at="" grid="" points="" on="" lines="" t="2k," 3k,="" 4k,="" ...,="" when="" have="" been="" obtained.="" of="" initial="" data="" line="" are="" ui,0="fi" ,="" 2k="" (ui,1="" ui,−1)="gi,0" (14.4.4)="" so="" that="" second="" result="" gives="" ui,−1="ui,1" gi,0.="" (14.4.5)="" in="" used,="" we="" obtain="" ui,1="1" (fi−1="" fi+1)="" fi="" (14.4.6)="" determines="" 14.4="" finite="" methods="" 609="" figure="" 14.4.1="" computational="" systems="" characteristics.="" value="" u="" pi,j+1="" obtained="" terms="" its="" previously="" calculated="" pi="" 1,j="" pi,j="" pi,j−1,="" which="" determined="" from="" computed="" 1)="" k,="" (j="" 2)="" 3)="" k.="" thus,="" computation="" suggests="" will="" represent="" a="" function="" within="" domain="" bounded="" by="" drawn="" back="" toward="" p="" whose="" gradients="" (+="" ε)="" as="" shown="" 14.4.1.="" thus="" triangular="" regions="" ab,="" pcd="" domains="" dependence="" solutions="" differential="" (14.4.1).="" analogy="" with="" real="" characteristic="" c="" d="" equation,="" straight="" b="" numerical="" it="" follows="" ∆p="" ab="" lies="" inside="" cd,="" means="" solution="" system="" would="" remain="" unchanged="" even="" along="" changed.="" courant,="" friedrichs="" lewy="" (cfl,="" 1928)="" proved="" converges="" tend="" zero="" provided="" partial="" equation.="" condition="" for="" convergence="" known="" cfl="" condition,="" ≥="" h,="" is,="" 1.="" if="" parameter="" reduces="" simple="" form="" 610="" 14="" ui,j−1.="" (14.4.7)="" 14.3.1,="" shows="" three="" two="" previous="" time="" steps.="" leap="" frog="" algorithm.="" (5.3.4)="" chapter="" 5,="" know="" cauchy="" problem="" wave="" has="" (x,="" t)="φ" (x="" ct)="" ψ="" ct),="" functions="" φ="" waves="" propagating="" without="" changing="" shape="" negative="" positive="" x="" directions="" constant="" speed="" c.="" slope="" (dt="" dx)="+" (1="" c)="" plane,="" trace="" progress="" waves,="" characteristics="" point="" (xi="" tj="" above="" )="" ).="" (14.4.8)="" 14.3.1="" xi="x1" (i="" takes="" (α="" ih="" jck)="" (β="" jck),="" (14.4.9)="" α="(x1" h)="" (t1="" k),="" β="(x1" k).="" since="" ck="h," becomes="" j)="" h).="" satisfies="" (14.4.7).="" method="" exact="" apply="" von="" neumann="" stability="" analysis="" investigate="" seek="" separable="" error="" er,s="" (iαrh)="" s="" (14.4.10)="" (βk).="" (14.4.3).="" substituting="" into="" cancelling="" common="" factors,="" quadratic="" +1="0," (14.4.11)="" 2ε="" sin2="" (αh="" 2),="" all="" α.="" complex="" roots="" p1="" p2="" ·="" one="" always="" modulus="" greater="" 611="" than="" unless="" |p1|="|p2|" =="" scheme="" unstable="" →="" ∞="" exceeds="" unity.="" other="" hand,="" −1="" 1,="" then="" stable="" leads="" useful="" cosec2="" ="" αh="" ="" .="" (14.4.12)="" limit="" space-grid="" size="" h.="" however,="" true="" example="" find="" utt="" uxx="0," <x<=""> 0, with the boundary conditions u (0, t) = u (1, t)=0, t ≥ 0, and the initial conditions u (x, 0) = sin πx, ut (x, 0) = 0, 0 ≤ x ≤ 1. Compare the numerical solution with the analytical solution u (x, t) = cos πtsin πx at several points. The explicit finite difference approximation to the wave equation with ε = (k/h) = 1 is found from (14.4.3) in the form ui,j+1 = ui−1,j + ui+1,j − ui,j−1, j ≥ 1. The problem is symmetric with respect to x = 1 2 , so we need to calculate the solution only for 0 ≤ x ≤ 1 2 . We take h = k = 1 10 = 0.1. The boundary conditions give u0,j = 0 for j = 0, 1, 2, 3, 4, 5. The initial condition ut (x, 0) = 0 yields ut (x, 0) = 1 2 (ui,1 − ui,−1)=0, or, ui,1 = ui,−1. The explicit formula with j = 0 gives ui,1 = 1 2 (ui−1,0 + ui+1,0), i = 1, 2, 3, 4, 5. Thus, 612 14 Numerical and Approximation Methods u1,1 = 1 2 (u0,0 + u2,0) = 1 2 u2,0 = 1 2 sin (0.2π)=0.2939. Similarly, u2,1 = 0.5590, u3,1 = 0.7695, u4,1 = 0.9045, u5,1 = 0.9511. We next use the basic explicit formula to compute u1,2 = u0,1 + u2,1 − u1,0 =0+0.5590 − 0.3090 = 0.2500, u2,2 = u1,1 + u3,1 − u2,0 = 0.2939 + 0.7695 − 0.5878 = 0.4756. Similarly, we compute other values for ui,j which are shown in Table 14.4.1. The analytical solutions at (x, t) = (0.1, 0.1) and (0.2, 0.3) are given by u (0.1, 0.1) = cos (0.1π) sin (0.1π) = (0.9511) (0.3090) = 0.2939, u (0.2, 0.2) = cos (0.2π) sin (0.2π) = (0.8090) (0.5878) = 0.4577, u (0.2, 0.3) = cos (0.3π) sin (0.2π) = (0.5878) (0.5878) = 0.3455. Comparison of the analytical solutions with the above finite difference solutions shows that the latter results are very accurate. (B) Parabolic Equations As a prototype diffusion problem, we consider ut = κ uxx, 0 <x< 1,="" t=""> 0, (14.4.13) u (0, t) = u (1, t)=0, for all t, (14.4.14) u (x, 0) = f (x), for all x in (0, 1), (14.4.15) where f (x) is a given function. Table 14.4.1. i 0 1 2 3 4 5 x 0.0 0.1 0.2 0.3 0.4 0.5 j t 1 0.1 0 0.2939 0.5590 0.7695 0.9045 0.9511 2 0.2 0 0.2500 0.4577 0.6545 0.7695 0.8090 3 0.3 0 0.1817 0.3455 0.4756 0.5590 0.5878 4 0.4 0 0.9045 0.7695 0.2500 0.2939 0.3090 5 0.5 0 0 0 0 0 0 14.4 Explicit Finite Difference Methods 613 The explicit finite difference approximation to (14.4.13) is 1 k (ui,j+1 − ui,j ) = κ h 2 (ui+1,j − 2 ui,j + ui−1,j ), (14.4.16) or, ui,j+1 = ε (ui+1,j + ui−1,j ) + (1 − 2 ε) ui,j , (14.4.17) where ε =  κk/h2 . This explicit finite difference formula gives approximate values of u on t = (j + 1) k in terms of values on t = jk with given ui,0 = fi . Thus, ui,j can be obtained for all j by successive use of (14.4.17). The problems of stability and convergence of the parabolic equation are similar to those of the wave equation. It can be shown that the solution of the finite difference equation converges to that of the differential equation system (14.4.13)–(14.4.15) as h and k tend to zero provided ε ≤ 1 2 . In particular, when ε = 1 2 , equation (14.4.17) takes a simple form ui,j+1 = 1 2 (ui+1,j + ui−1,j ). (14.4.18) This is called the Bender–Schmidt explicit formula which determines the solution at (xi , tj+1) as the mean of the values at the grid points (i + 1, j). However, more accurate results can be found from (14.4.17) for ε < 1 2 . To investigate the stability of the numerical scheme, we assume that the error function is er,s = exp (iαrh) p s , (14.4.19) where p = e βk. The error function and urs satisfy the same difference equation. Hence, we substitute (14.4.19) into (14.4.17) to obtain p = 1 − 4 ε sin2  αh 2  . (14.4.20) Clearly, p is always less than 1 because ε > 0. If p ≥ 0, the function given by (14.4.19) will decay steadily as s = j → ∞. If −1 <p< 0,="" then="" the="" solution="" will="" have="" a="" decaying="" amplitude="" as="" s="" →="" ∞.="" therefore,="" finite="" difference="" scheme="" be="" stable="" if="" p=""> −1, that is, if 0 < ε ≤ 1 2 cosec2  αh 2  . (14.4.21) This shows that the stability limit depends on h. However, in view of the inequality ε ≤ 1 2 ≤ 1 2 cosec2  αh 2  , we conclude that the stability condition is ε ≤ 1 2 . Finally, if p < −1, the solution oscillates with increasing amplitude as s → ∞, and hence, the scheme will be unstable for ε > 1 2 . 614 14 Numerical and Approximation Methods Example 14.4.2. Show that the Richardson explicit finite difference scheme for (14.4.13) is unconditionally unstable. The Richardson finite difference approximation to (14.4.13) is 1 2k (ui,j+1 − ui,j−1) = κ h 2 (ui+1,j − 2 ui,j + ui−1,j ). (14.4.22) To establish the instability of this equation, we use the Fourier method and assume that er,s = exp (iαrh) p s , p = e βk . This function satisfies the Richardson difference equation as does ur,s. Consequently, p − 1 p = −8 ε sin2  αh 2  , or p 2 + 8 p ε sin2  αh 2  − 1=0. This quadratic equation has two roots p1, p2 = −4 ε sin2  αh 2  +  1 + 16 ε 2 sin4 αh 2 1 2 , (14.4.23) or p1, p2 = + 1 − 4 ε sin2  αh 2 1 + 2 ε sin2 αh 2  + O  ε 4 . This gives |p1| ≤ 1 and |p2| > 1+4 ε sin2  αh 2  > 1 for all positive ε and, consequently, the Richardson scheme is always unstable. The unstable feature of the Richardson scheme can be eliminated by replacing ui,j with 1 2 (ui,j+1 + ui,j−1) in (14.4.22), which now becomes (1 + 2 ε) ui,j+1 = 2 ε (ui+1,j + ui−1,j ) + (1 − 2 ε) ui,j−1. (14.4.24) This is called the Du Fort–Frankel explicit algorithm, and it can be shown to be stable for all ε. 14.4 Explicit Finite Difference Methods 615 Example 14.4.3. Prove that the solution of the finite difference equation for the diffusion equation (14.4.13) in −∞ <x< ∞="" with="" the="" initial="" condition="" u="" (x,="" 0)="e" iαx="" converges="" to="" exact="" solution="" of="" (14.4.13)="" as="" h="" and="" k="" tend="" zero.="" we="" obtain="" by="" seeking="" a="" separable="" form="" t)="e" v="" (t),="" where="" (t)="" is="" function="" t="" alone="" which="" be="" determined.="" substituting="" this="" into="" gives="" dv="" dt="" +="" κ="" α2="" admits="" solutions="" e−κα2="" ,="" an="" integrating="" constant.="" (0)="1" hence,="" ="" −="" α="" 2κt="" .="" (14.4.25)="" now="" solve="" corresponding="" finite="" difference="" equation="" (14.4.17)="" replacing="" i,="" j="" r,="" s.="" seek="" ur,s="e" iαrh="" vs="" ur,0="e" v0,="" so="" that="" v0="1." (14.4.7)="" yields="" vs+1="" 1="" 4="" ε="" sin2="" αh="" 2="" ="" vs,="" can="" obtained="" simple="" inspection="" s="" (14.4.26)="" ="" 4ε="" (14.4.27)="" κk="" h2="" for="" small="" h,="" (αh="" 2)="" ∼="" α2h="" ≈="" exp="" −ε="" α2k.="" consequently,="" final="" becomes="" 616="" 14="" numerical="" approximation="" methods="" e="" iαrh−κα2ks="" →="" 0.="" (14.4.28)="" identical="" differential="" rh="x" sk="t." example="" shows="" reasonably="" good.="" 14.4.4.="" calculate="" boundaryvalue="" problem="" ut="uxx," 0="" <x<="" 1,=""> 0, with the boundary conditions u (0, t) = u (1, t)=0, t ≥ 0, and the initial condition u (x, 0) = x (1 − x), 0 ≤ x ≤ 1. Compare the numerical solution with the exact analytical solution at x = 0.04 and t = 0.02. The explicit finite difference approximation to the parabolic equation is ui,j+1 = ε ui−1,j + (1 − 2 ε) ui,j + ε ui+1,j , where ε =  k/h2 . This gives the unknown value ui,j+1 at the (i, j + 1) th grid point in terms of given values of u along the jth time row. We set h = 1 5 and k = 1 100 so that ε =  k/h2 = 1 4 and the above formula becomes ui,j+1 = 1 4 (ui−1,j + 2 ui,j + ui+1,j ). With the notation ui,0 = u (ih, 0), the initial condition gives u4,0 = 0.16, and u5,0 = 0. The boundary conditions yield u0,j = u (0, jk) = 0 and u5,j = u (5h, jk) = u (1, jk) = 0, for all j = 0, 1, 2, .... Using these initial and boundary data, we calculate ui,j as follows: u1,1 = 1 4 (u0,0 + 2u1,0 + u2,0)=0.14, u1,2 = 1 4 (u0,1 + 2u1,1 + u2,1) = 0.125, u2,1 = 1 4 (u1,0 + 2u2,0 + u3,0)=0.22, u2,2 = 1 4 (u1,1 + 2u2,1 + u3,1) = 0.200, u3,1 = 1 4 (u2,0 + 2u3,0 + u4,0)=0.22, u3,2 = 1 4 (u2,1 + 2u3,1 + u4,1) = 0.200, u4,1 = 1 4 (u3,0 + 2u4,0 + u5,0)=0.14, u4,2 = 1 4 (u3,1 + 2u4,1 + u5,1) = 0.125, 14.4 Explicit Finite Difference Methods 617 u1,3 = 1 4 (u0,2 + 2u1,2 + u2,2)=0.1125, u1,4 = 1 4 (u0,3 + 2u1,3 + u2,3) = 0.1016, u2,3 = 1 4 (u1,2 + 2u2,2 + u3,2)=0.1813, u2,4 = 1 4 (u1,3 + 2u2,3 + u3,3) = 0.1641, u3,3 = 1 4 (u2,2 + 2u3,2 + u4,2)=0.1813, u3,4 = 1 4 (u2,3 + 2u3,3 + u4,3) = 0.1641, u4,3 = 1 4 (u3,2 + 2u4,2 + u5,2)=0.1125, u4,4 = 1 4 (u3,3 + 2u4,3 + u5,3) = 0.1016. The method of separation of variables gives the analytical solution of the problem as u (x, t) = 8 π 3 ∞ n=0 1 (2n + 1)3 exp " − (2n + 1)2 π 2 t # sin (2n + 1) πx. This exact solution u (x, t) at x = 0.4 (i = 2) and t = 0.02 (j = 2) gives u ∼ 8 3 1 1 3 exp  −0.02 π 2 sin (0.4) π + 1 3 3 exp  −0.18 π 2 sin (1.2) π = 0.2000. The analytical solution is seen to be identical with the numerical value. Example 14.4.5. Obtain the numerical solution of the initial boundaryvalue problem ut = κ uxx, 0 ≤ x ≤ 1, t > 0, u (0, t)=1, u (1, t)=0, t ≥ 0, u (x, 0) = 0, 0 ≤ x ≤ 1. We use the explicit finite-difference formula (14.4.17) ui.j+1 = ε (ui+1,j + ui−1,j ) + (1 − 2 ε) ui,j , where ε =  κk/h2 . We set h = 0.25 = 1 4 and ε = 2 5 = 0.4 to compute ui.j for i, j = 0, 1, 2, 3, 4 as follows: 618 14 Numerical and Approximation Methods i 0 1 2 3 4 j 0 1.000 0.000 0.000 0.000 0.000 1 1.000 0.400 0.000 0.000 0.000 2 1.000 0.480 0.160 0.000 0.000 3 1.000 0.560 0.224 0.064 0.000 4 1.000 0.602 0.295 0.103 0.000 (C) Elliptic Equations As a prototype boundary-value problem, we consider the Dirichlet problem for the Laplace equation ∇2u ≡ uxx + uyy = 0, 0 ≤ x ≤ a, 0 ≤ y ≤ b, (14.4.29) where the value of u (x, y) is prescribed everywhere on the boundary of the rectangular domain. The rectangular grid system is the most common and convenient system for this problem. We choose the vertices of the rectangular domain as the nodal points and set h = a/m and k = b/n where m and n are positive integers so that the domain is divided into mn subrectangles. The finite difference approximation to the Laplace equation (14.4.29) is 1 h 2 (ui+1,j − 2ui,j + ui−1,j ) + 1 k 2 (ui,j+1 − 2ui,j + ui,j−1)=0,(14.4.30) or, 2  h 2 + k 2 ui,j = k 2 (ui+1,j + ui−1,j ) + h 2 (ui,j+1 + ui,j−1), (14.4.31) where 1 ≤ i ≤ m − 1 and 1 ≤ j ≤ n − 1. The prescribed conditions on the boundary of the rectangular domain determine the values u0,j , um,j , ui,0, and ui,n. For a square grid system (k = h), equation (14.4.30) becomes ui,j = 1 4 (ui+1,j + ui−1,j + ui,j+1 + ui,j−1). (14.4.32) This means that the value of u at an interior point is equal to the average of the value of u at four adjacent points. This is the well known mean value theorem for harmonic functions that satisfy the Laplace equation. As i and j vary, the present scheme reduces to a set of (m − 1) (n − 1) linear non-homogeneous algebraic equations for (m − 1) (n − 1) unknown values of u at interior grid points. It can be shown the solution of the finite 14.4 Explicit Finite Difference Methods 619 difference equation (14.4.31) converges to the exact solution of the problem as h, k → 0. The proof of the existence of a solution and its convergence to the exact solution as h and k tend to zero is essentially based on the Maximum Modulus Principle. It follows from the finite difference equation (14.4.30) or (14.4.31) that the value of |u| at any interior grid point does not exceed its value at any of the four adjoining nodal points. In other words, the value of u at Pi,j cannot exceed its values at the four adjoining points Pi + 1,j and Pi,j + 1. The successive application of this argument at all interior grid points leads to the conclusion that |u| at the interior grid points cannot be greater than the maximum value of |u| on the boundary. This may be recognized as the finite difference analogue of the Maximum Modulus Principle discussed in Section 9.2. Thus, the success of the numerical method is directly associated with the existence of the Maximum Modulus Principle. Clearly, the present numerical algorithm deals with a large number of algebraic equations. Even though numerical accuracy can be improved by making h and k sufficiently small, there is a major computational difficulty involved in the numerical solution of a large number of equations. It is possible to handle such a large number of algebraic equations by direct methods or by iterative methods, but it would be very difficult to obtain a numerical solution with sufficient accuracy. It is therefore necessary to develop some alternative methods of solution that can be conveniently and efficiently carried out on a computer. In order to eliminate some of the drawbacks stated above, one of the numerical schemes, the Liebmann’s iterative method, is useful. In this method values of u are first guessed for all interior grid points in addition to those given as the boundary points on the edges of the given domain. These values are denoted by u (0) i,j where the superscript 0 indicates the zeroth iteration. It is convenient to choose a square grid so that the simplified finite difference equation (14.4.32) can be used. The values of u are calculated for the next iteration by using (14.4.32) at every interior point based on the values of u at the present iteration. The sequence of computation starts from the interior grid point located at the lowest left corner, proceeds upward until reaching the top, and then goes to the bottom of the next vertical line on the right. This process is repeated until the new value of u at the last interior grid point at the upper right corner has been obtained. At the starting point, formula (14.4.32) gives u (1) 2,2 = 1 4 4 u (0) 3,2 + u (0) 1,2 + u (0) 2,3 + u (0) 2,1 5 , (14.4.33) where u (0) 1,2 and u (0) 2,1 are boundary values which remain constant during the iteration process. They may be replaced, respectively, with u (1) 1,2 , u (1) 2,1 in (14.4.33). The computation at the next step involves u (0) 2,2 . Since an improved value u (1) 2,2 is available at this time, it will be utilized instead. Hence, 620 14 Numerical and Approximation Methods u (1) 2,3 = 1 4 4 u (0) 3,3 + u (1) 1,3 + u (0) 2,4 + u (1) 2,2 5 , (14.4.34) where u (1) 1,3 is used to replace the constant boundary value u (0) 1,3 . We repeat this argument to obtain a general iteration formula for computation of u at step (n + 1) u (n+1) i,j = 1 4 4 u (n) i+1,j + u (n+1) i−1,j + u (n) i,j+1 + u (n+1) i,j−1 5 . (14.4.35) This result is valid for any interior point, whether it is next to some boundary point or not. If Pi,j is a true point, the second and fourth terms on the right side of (14.4.35) represent, respectively, the values of u at the grid points to the left of and below that point. These values have already been recomputed according to our scheme, and therefore, carry the superscript (n + 1). Result (14.4.35) is known as the Liebmann (n + 1) th iteration formula. It can be proved that u (n) i,j converges to ui,j as n → ∞. Another iteration scheme similar to (14.4.35) is given by u (n+1) i,j = 1 4 4 u (n) i+1,j + u (n) i−1,j + u (n) i,j+1 + u (n) i,j−1 5 . (14.4.36) This is called the Richardson iteration formula, and it is also useful. However, this scheme converges more slowly than that based on (14.4.35). One of the major difficulties of the above methods is the slow rate of convergence. An improved numerical method, the Successive Over-Relaxation (SOR) scheme gives a faster convergence than the Liebmann or Richardson method in solving the Laplace (or the Poisson) equation. For a rectangular domain of square grids, the successive iteration scheme is given by u (n+1) i,j = u (n) i,j + ω 4 4 u (n+1) i−1,j + u (n) i+1,j + u (n+1) i,j−1 + u (n) i,j+1 − 4 u (n) i,j 5 , (14.4.37) where ω is called the acceleration parameter (or relaxation factor ) to be determined. In general, ω lies in the range 1 ≤ ω < 2. The successive iterations converge fairly rapidly to the desired solution for 1 ≤ ω < 2. The most rapid rate of convergence is achieved for the optimum value of ω. Example 14.4.6. Obtain the standard five-point formula for the Poisson equation uxx + uyy = −f (x, y) in D ⊂ R 2 with the prescribed value of u (x, y) on the boundary ∂D. We assume that the domain D is covered by a system of squares with sides of length h parallel to the x and y axes. Using the central difference approximation to the Laplace operator, we obtain 1 h 2 (ui+1,j − 2 ui,j + ui−1,j ) + 1 h 2 (ui,j+1 − 2 ui,j + ui,j−1) = −fi,j , 14.4 Explicit Finite Difference Methods 621 or, ui,j = 1 4 (ui+1,j + ui−1,j + ui,j+1 + ui,j−1) + 1 4 h 2 fi,j where fi,j = f (ih, jh). This is known as the five-point formula. Example 14.4.7. Find the numerical solution of the torsion problem in a square beam governed by ∇2u = −2 in D = {(x, y):0 ≤ x ≤ 1, 0 ≤ y ≤ 1} with u (x, y) = 0 on ∂D. From the above five-point formula, we obtain ui,j = 1 4 (ui+1,j + ui−1,j + ui,j+1 + ui,j−1) − 1 2 h 2 where h is the side-length of the unit square net. We choose h = 1 2 , 1/2 2 , 1/2 3 , 1/2 4 to calculate the corresponding numerical values ui,j = 0.1250, 0.1401, 0.1456, 0.1469. Note that the known exact analytical solution is 0.1474. Example 14.4.8. Using the explicit finite difference method, find the solution of the Dirichlet problem uxx + uyy = 0, in 0 <x< 1,="" 0="" <y<="" u="" (x,="" 0)="x," 1)="0," on="" ≤="" x="" y)="0," for="" and="" y="" 1.="" we="" use="" four="" interior="" grid="" points="" (that="" is,="" i,="" j="1," 2,="" 3,="" 4)="" as="" shown="" in="" figure="" 14.4.2="" the="" y)-plane.="" apply="" explicit="" finite="" difference="" formula="" (14.4.32)="" to="" obtain="" algebraic="" equations="" −4u2,2="" +="" u3,2="" u1,2="" u2,3="" u2,1="0," −4u2,3="" u3,3="" u1,3="" u2,4="" u2,2="0," −4u3,2="" u4,2="" u3,1="0," −4u3,3="" u4,3="" u3,4="" given="" boundary="" conditions="" imply="" that="" =="" 0,="" 3="" so="" above="" system="" of="" becomes="" 1="" 2="" 622="" 14="" numerical="" approximation="" methods="" square="" system.="" matrix="" notation,="" this="" reads="" ⎡="" ⎢="" ⎣="" −41="" −40="" −4="" 011="" ⎤="" ⎥="" ⎦="" −="" .="" solutions="" are="" 72="" ,="" (d)="" simultaneous="" first-order="" recall="" wave="" equation="" (14.4.1)="" <x<="" t=""> 0. Introducing two auxiliary variables v and w by v = ut and w = c 2ux, the wave equation gives two simultaneous first-order equations vt = wx, wt = c 2 vx. (14.4.38) 14.4 Explicit Finite Difference Methods 623 The initial values of v and w are given at t = 0 for all x in 0 <x< 1.="" the="" boundary="" condition="" on="" v="" and="" w="" is="" also="" prescribed="" lines="" x="0" for="" t=""> 0. The explicit finite difference method can be used to determine v and w in the triangular domain of dependence bounded by the characteristics x − ct = 0 and x + ct = 1. The finite difference approximations to the differential equations (14.4.38) are 1 k (vi,j+1 − vi,j ) = 1 2h (wi+1,j − wi−1,j ), (14.4.39) 1 k (wi,j+1 − wi,j ) = c 2 2h (vi+1,j − vi−1,j ), (14.4.40) where the forward difference for vt or wt and the central difference for vx or wx are used. However, the central difference approximations to (14.4.38) can also be utilized to obtain 1 2k (vi,j+1 − vi,j−1) = 1 2h (wi+1,j − wi−1,j ), (14.4.41) 1 2k (wi,j+1 − wi,j−1) = c 2 2h (vi+1,j − vi−1,j ). (14.4.42) We examine the stability of the above two sets of finite difference formulas with c = 1. The von Neumann stability method is applied by replacing i and j by r and s respectively. The error function er,s given by (14.4.10) is substituted in (14.4.39)–(14.4.40) to obtain the stability relations A (p − 1) = εiB sin αh, (14.4.43) B (p − 1) = εiA sin αh, (14.4.44) where the initial perturbations in v and w along t = 0 are A exp (iαrh) and B exp (iαrh) respectively with two different constants A and B. Elimination of A and B from the above relations gives (p − 1)2 + ε 2 sin2 αh = 0 or p = 1+ iε sin αh, and |p| =  1 + ε 2 sin2 αh 1 2 ∼ 1 + 1 2 ε 2 sin2 αh =1+ O  ε 2 . (14.4.45) Since |p| > 1 + O (ε), the finite difference scheme for the finite time-step t = sk would be unstable as the grid sizes tend to zero. A similar stability analysis for (14.4.41)–(14.4.42) leads to the condition 624 14 Numerical and Approximation Methods  p − 1 p 2 + 4 ε 2 sin2 αh = 0. (14.4.46) This scheme is stable for ε ≤ 1. Another finite difference approximation to the coupled system (14.4.38) is 1 2h (vr+1,s − vr−1,s) = 1 k wr,s+1 − 1 2 (wr+1,s − wr−1,s) , (14.4.47) 1 2h (wr+1,s − wr−1,s) = 1 k vr,s+1 − 1 2 (vr+1,s − vr−1,s) . (14.4.48) A similar stability analysis can be carried out for these systems by substituting vr,s = Aps e iαrh and wr,s = Bps e iαrh into the equations. Elimination of A/B yields the stability equation p = cos αh + i ε sin αh, or |p| 2 = cos2 αh + 1 ε 2 sin2 αh ≤ 1. (14.4.49) Hence, the scheme is stable provided that ε ≥ 1, that is, k ≤ h. (14.4.50) 14.5 Implicit Finite Difference Methods From a computational point of view, the explicit finite difference algorithm is simple and convenient. However, as shown in Section 14.4(B), the major difficulty in the method for solving parabolic partial differential equations is the severe restriction on the time-step imposed by the stability condition ε ≤ 1 2 or k ≤ h 2/2κ. This difficulty is also present in the explicit finite difference method for the solution of hyperbolic equations. In order to overcome the above difficulty, we develop implicit finite difference schemes for solving partial differential equations. (A) Parabolic Equations One of the successful implicit finite difference schemes is the Crank and Nicolson Method (1947), which is based on six grid points. This method eliminates the major difficulty involved in the explicit scheme. When the Crank–Nicolson implicit scheme is applied to the parabolic equation (14.4.13), uxx is replaced by the mean value of the finite difference values 14.5 Implicit Finite Difference Methods 625 in the jth and the (j + 1) th row so that the finite difference approximation (14.4.13) becomes 1 k (ui,j+1 − ui,j ) = κ 2h 2 [(ui+1,j+1 − 2ui,j+1 + ui−1,j+1) + (ui+1,j − 2ui,j + ui−1,j )] , (14.5.1) or 2 (1 + ε) ui,j+1 − ε (ui−1,j+1 + ui+1,j+1) = 2 (1 − ε) ui,j + ε (ui−1,j + ui+1,j ), (14.5.2) where ε =  kκ/h2 is a parameter. The left side of (14.5.2) is a linear combination of three unknowns in the (j + 1) th row, and the right side involves three known values of u in the jth row of the grid system in the (x, t)-plane. Equation (14.5.2) is called the Crank–Nicolson implicit formula. This formula (or its suitable modification) is widely used for solving parabolic equations. If there are n internal grid points along each jth row, then, for j = 0 and i = 1, 2, 3, ..., n, the implicit formula (14.5.2) gives n simultaneous algebraic equations for n unknown values of u along the first jth row (j = 0) in terms of given boundary and initial data. Similarly, if j = 1 and i = 1, 2, 3, ..., n, equation (14.5.2) represents n unknown values of u along the second jth row (j = 1) and so on. This means that the method involves the solution of a system of simultaneous algebraic equations. In practice, the Crank–Nicolson scheme is convergent and unconditionally stable for all finite values of ε, and has the advantage of reducing the amount of numerical computation. This implicit scheme can be further generalized by introducing a numerical weight factor λ in the modified version of the explicit equation (14.4.16) which is written below by approximating uxx in (14.4.13) in the (j + 1) th row instead of the jth row. 1 k (ui,j+1 − ui,j ) = κ h 2 (ui+1,j+1 − 2ui,j+1 + ui−1,j+1), (14.5.3) or ui,j+1 − ui,j = ε (ui+1,j+1 − 2ui,j+1 + ui−1,j+1). (14.5.4) Introducing the numerical factor λ, this can be replaced by a more general difference equation in the form ui,j+1 − ui,j = ε λ δ2 x ui,j+1 + (1 − λ) δ 2 x ui,j ! , (14.5.5) where 0 ≤ λ ≤ 1 and δ 2 x is the difference operator defined by δ 2 xui,j = ui+1,j − 2ui,j + ui−1,j . (14.5.6) Another equivalent form of (14.5.5) is 626 14 Numerical and Approximation Methods (1 + 2ελ) ui,j+1 − ελ (ui+1,j+1 + ui−1,j+1) = {1 − 2ε (1 − λ)} ui,j + ε (1 − λ) (ui+1,j − ui−1,j ). (14.5.7) This is a fairly general implicit formula which reduces to (14.5.4) when λ = 1. When λ = 1 2 , (14.5.7) becomes the Crank–Nicolson formula (14.5.2). Finally, if λ = 0, this implicit difference equation reduces to the explicit equation (14.4.17). The Richardson explicit scheme was found to be unconditionally unstable in Section 14.4. This undesirable feature of the scheme can be eliminated by considering the corresponding implicit scheme. In terms of δ 2 x , the Richardson equation (14.4.22) can be expressed as ui,j+1 = 2 ε δ2 x ui,j + ui,j−1. (14.5.8) To obtain the implicit Richardson formula, we replace δ 2 x ui,j by 1 3 δ 2 x (ui,j+1 + ui,j + ui,j−1) in (14.5.8) and we obtain  1 − 2 ε 3 δ 2 x  ui,j+1 = 2ε 3 δ 2 xui,j +  1 + 2ε 3  ui,j−1. (14.5.9) This implicit scheme can be shown to be unconditionally stable. To prove this result, we apply the von Neumann stability method with the error function (14.5.9) to obtain the equation for p as (1 + a) p 2 + ap + (a − 1) = 0, (14.5.10) where a ≡  8ε 3  sin2  αh 2  . (14.5.11) The roots of the quadratic equation are p = −a +  4 − 3a 2 1 2 2 (1 + a) . (14.5.12) This gives |p| ≤ 1 for all values of a. Hence, the result is proved. Example 14.5.1. Obtain the numerical solution of the following parabolic system by using the Crank–Nicolson method ut = uxx, 0 <x< 1,="" t=""> 0, u (0, t) = u (1, t)=0, t ≥ 0, u (x, 0) = x (1 − x), 0 ≤ x ≤ 1. We recall the Crank–Nicolson equation (14.5.2) and then set h = 0.2 and k = 0.01 so that ε = 1 4 . The boundary and initial conditions give 14.5 Implicit Finite Difference Methods 627 u0,0 = u5,0 = u0,1 = u5,1 = 0 and ui,0 = u (ih, 0) = ih (1 − ih), i = 1, 2, 3, 4. Consequently, formula (14.5.2) leads to the following system of four equations: −u0,1 − u2,1 + 10u1,1 = u0,0 + u2,0 + 6u1,0 −u1,1 − u3,1 + 10u2,1 = u1,0 + u3,0 + 6u2,0 −u2,1 − u4,1 + 10u3,1 = u2,0 + u4,0 + 6u3,0 −u3,1 − u5,1 + 10u4,1 = u3,0 + u5,0 + 6u4,0. Using the boundary and initial conditions, the above system becomes −u2,1 + 10u1,1 = 1.20 −u1,1 + 10u2,1 − u3,1 = 1.84 −u2,1 − u4,1 + 10u3,1 = 1.84 −u3,1 + 10u4,1 = 1.20. These equations can be solved by direct elimination to obtain the solutions as u1,1 = 0.1418, u2,1 = 0.2202, u3,1 = 0.2202, u4,1 = 0.1420. (B) Hyperbolic Equations We consider an implicit finite difference scheme to solve the initial boundaryvalue problem consisting of the first-order hyperbolic equation ∂u ∂t + c ∂u ∂x = 0, (c > 0), (14.5.13) with the initial data u (x, 0) = U (x) and the boundary condition u (0, t) = V (t) where 0 ≤ x, t < ∞. The implicit finite difference approximation to (14.5.13) is 1 k (ui,j+1 − ui,j ) + c h (ui,j+1 − ui−1,j+1)=0, or ui,j = (1 + ε) ui,j+1 − ε ui−1,j+1, (14.5.14) where ε = (ck/h). The stability of the scheme can be examined by using the von Neumann method with the error function (14.4.10). It turns out that p = [1 − ε + ε exp (−iαh)]−1 , (14.5.15) from which it follows that |p| ≤ 1 for all h. Hence, the implicit scheme is unconditionally stable. 628 14 Numerical and Approximation Methods We next solve the wave equation utt = c 2uxx by an implicit finite difference scheme. In this case, utt is replaced by the central difference formula, and uxx by the mean value of the central difference values in the (j − 1) th and (j + 1) th rows. Consequently, the implicit difference approximation to the wave equation is ui,j+1 − 2 ui,j + ui,j−1 = ε 2 2 [(ui+1,j+1 − 2ui,j+1 + ui−1,j+1) + (ui+1,j−1 − 2 ui,j−1 + ui−1,j−1)] , (14.5.16) where ε = (ck/h). Expressing the solution for the (j + 1) th step in terms of the two preceding steps gives 2  1 + ε 2 ui,j+1 − ε 2 (ui−1,j+1 + ui+1,j+1) = 4ui,j + ε 2 (ui−1,j−1 + ui+1,j−1) − 2  1 + ε 2 ui,j−1. (14.5.17) The N grid points along the time step, j = 0, i = 1, 2, 3, ..., N, (14.5.17) along with the finite difference approximation to the boundary condition give N simultaneous equations for the N unknown values of u along the first time step. This constitutes a tridiagonal system of equations that can be solved by direct or iterative numerical methods. To investigate the stability of the implicit scheme, we apply the von Neumann stability method with the error function (14.4.10). This leads to the equation p + 1 p = 2  1+2 ε 2 sin2 αh 2 −1 , or p 2 − 2bp +1=0, (14.5.18) where b =  1+2ε 2 sin2 αh/2 −1 so that 0 < b ≤ 1. Hence, the stability condition is |p| ≤ 1 (14.5.19) which is always satisfied provided 0 < b ≤ 1, that is, ε < 1 for all positive h. This confirms the unconditional stability of the scheme. A more general implicit scheme can be introduced by replacing uxx in the wave equation (14.4.1) with uxx ∼ 1 h 2 λ  δ 2 x ui,j+1 + δ 2 x ui,j−1 + (1 + 2λ) δ 2 x ui,j ! , (14.5.20) 14.6 Variational Methods and the Euler–Lagrange Equations 629 where λ is a numerical weight (relaxation) factor and the central difference operator δ 2 x is given by (14.5.6). This general scheme allows us to approximate the wave equation with c = 1 by the form δ 2 t ui,j = ε 2 λ  δ 2 x ui,j+1 + δ 2 x ui,j−1 + (1 − 2λ) δ 2 x ui,j ! , (14.5.21) where ε = k/h. This equation reduces to (14.5.16) when λ = 1 2 , and to the explicit finite difference result when λ = 0. It follows from von Neumann stability analysis that the implicit scheme is unconditionally stable for λ ≥ 1 4 . Von Neumann introduced another fairly general finite difference algorithm for the wave equation (14.4.1) in the form δ 2 t ui,j = ε 2 δ 2 xui,j + ω h 2 δ 2 t δ 2 xui,j . (14.5.22) This equation with appropriate boundary conditions can be solved by the tridiagonal method. Von Neumann discussed the question of stability of this implicit scheme and proved that the scheme is conditionally stable if ω ≤ 1 4 and unconditionally stable if ω > 1 4 . 14.6 Variational Methods and the Euler–Lagrange Equations To describe the variational methods and Rayleigh–Ritz approximate method, it is convenient to introduce the concepts of the inner product (pre-Hilbert) and Hilbert spaces. An inner product space X consisting of elements u, v, w, ... over the complex number field C is a complex linear space with an inner product u, v : X × X → C such that (i) u, v = v, u, where the bar denotes the complex conjugate of v, u, (ii) αu + βv, w = α u, w + β v, w for any scalars α, β ∈ C, (iii) u, u ≥ 0; equality holds if and only if u = 0. By (i) u, u = u, u, and so u, u is real. We denote u, u 1 2 = u, which is called the norm of u. Thus, the norm is induced by the inner product. Thus, every inner product space is a normed linear space under the norm u =  u, u. Let X be an inner product space. A sequence {un} where un ∈ X for every n is called a Cauchy sequence in X if and only if for every given ε > 0 (no matter how small) we can find an N (ε) such that un − um < ε for all n, m > N (ε). The space X is called complete if every Cauchy sequence converges to a point in X. A complete normed linear space is called a Banach Space. A complete linear inner product space is called a Hilbert Space and is usually denoted by H. 630 14 Numerical and Approximation Methods Example 14.6.1. Let C n be the set of all n-tuples of complex numbers. Thus, C n is an n-dimensional Hilbert space with the inner product x, y = n k=1 xk yk. Obviously, the set of all n-tuples of real numbers Rn is an n-dimensional Hilbert space. Example 14.6.2. Let l2 be the set of all sequences with entries from C such that 2∞ k=1 |xk| 2 < ∞. This forms a Hilbert space with the inner product x, y = ∞ k=1 xk yk. Example 14.6.3. Let L2 ([a, b]) be the set of all square integrable functions in the Lebesgue sense in an interval [a, b]. L2 ([a, b]) is a Hilbert space with the inner product u, v =  b a u (x) v (x) dx. We next introduce the notion of an operator in a Hilbert space H. An operator A is a mapping from H to H (that is, A : H → H). It assigns to an element u in H a new element Au in H. An operator A is called linear if it satisfies the property A (αu + βv) = αAu + βAv for every α, β ∈ C. An operator is said to be bounded if there exists a constant k such that Au ≤ k u for all u ∈ H. We consider a bounded operator A on a Hilbert space H. For a fixed element v in H, the inner product Au, v in H can be regarded as a number I (u) which varies with u. Thus, Au, v = I (u) is a linear functional on H. If there exists an operator A∗ on a Hilbert space (A∗ : H → H) such that Au, v = u, A∗ v for all u, v ∈ H, then A∗ is called the adjoint of A. In general, A = A∗ . If A = A∗ , that is, Au, v = u, Av for all u, v in H, then A is called self-adjoint. It is important to note that any bounded operator T on a real Hilbert space (T : H → H) of the form T = A∗A is self-adjoint. This follows from the fact that T u, v = A ∗Au, v = Au, Av = u, A∗Au = u, T v. 14.6 Variational Methods and the Euler–Lagrange Equations 631 A self-adjoint operator A on a Hilbert space H is said to be positive if Au, u ≥ 0 for all u in H, where equality implies that u = 0 in H. Further, if there exists a positive constant k such that Au, u ≥ k u, u for all u in H, then A is called positive definite in H. The rest of this section is essentially concerned with linear operators in a real Hilbert space, which means that the associated scalar field involved is real. Some specific inner products which will be used in the subsequent sections include u, v =  b a u (x) v (x) dx, u, v =  D u (x, y) v (x, y) dx dy, where D ⊂ R2 . Example 14.6.4. Determine whether the differentiable operators (i) A = d/dx, (ii) A = d 2/dx2 , and (iii) A = ∇2 =  ∂ 2/∂x2 +  ∂ 2/∂y2 are selfadjoint for functions that are differentiable in a ≤ x ≤ b or in D ⊂ R2 and vanish on the boundary. (i) Au, v =  b a  du dx v dx =  b a u  − dv dx dx + [u, v] b a = u, A∗ v where A ∗ = − d dx = A. Hence, A is not self-adjoint. (ii) Au, v =  b a  d 2u dx2  v dx =  b a  du dx− dv dx dx + v du dxb a =  b a u  d 2v dx2  dx + v du dx − u dv dxb a =  b a u  d 2v dx2  dx = u, Av. Thus, A is self-adjoint. (iii) Au, v =  D  ∇2u v dx dy =  D [∇ · (∇u) v − ∇u · ∇v] dx dy =  ∂D (n6 · ∇u) v dS −  D ∇u · ∇v dx dy = −  D (∇u · ∇v) dx dy, where the divergence theorem is used with the unit outward normal vector n6. Noting the symmetry of the right hand side in u and v, it follows that Au, v = Av, u = u, Av. This means that the operator A = ∇2 is self-adjoint. 632 14 Numerical and Approximation Methods Example 14.6.5. Use the inner product u, v = *** D (u · v) dV and the operator A = grad to show that A∗ = −div provided the functions vanish on the boundary surface ∂D of D. We use the divergence theorem to obtain Aφ, v =  D (grad φ · v) dV =  D [div (φv) − φ div v] dV, =  D φ (−div v) dV +  ∂D (n6 · φv) dS, =  D φ (−div v) dV = φ, −div v = φ, A∗ v. In the theory of calculus of variations it is a common practice to use δu, δ 2u, etc. to denote the first and second variations of a function u. Thus, δ can be regarded as an operator that changes u into δu, ux into δ (ux), and uxx into δ (uxx) with the meaning, δu = εv, δ (ux) = εvx, δ (uxx) = εvxx, where ε is a small arbitrary real parameter. The operators δ, δ 2 are called the first and second variational operators respectively. Some simple properties of the operator δ are given by ∂ ∂x (δu) = ∂ ∂x (εv) = ε ∂v ∂x = δ  ∂u ∂x , (14.6.1) δ 1 b a u dx3 = ε  b a v dx =  b a ε v dx =  b a (δu) dx. (14.6.2) The variational operator can be interchanged with the differential and integral operators, and proves to be very useful in the calculation of the variation of a functional. The main task of the calculus of variations is concerned with the problem of minimizing or maximizing functionals involved in mathematical, physical and engineering problems. The variational principles have their origins in the simplest kind of variational problem, which was first considered by Euler in 1744 and Lagrange in 1760-61. The classical Euler–Lagrange variational problem is to determine the extremum value of the functional I (u) =  b a F (x, u, u′ ) dx, u′ = du dx, (14.6.3) with the boundary conditions u (a) = α and u (b) = β, (14.6.4) where u belongs to the class C 2 ([a, b]) of functions which have continuous derivatives up to second-order in a ≤ x ≤ b, and F has continuous secondorder derivatives with respect to all of its arguments. 14.6 Variational Methods and the Euler–Lagrange Equations 633 We assume that I (u) has an extremum at some u ∈ C (2) ([a, b]). Then we consider the set of all variations u + ǫv for fixed u where v is an arbitrary function belonging to C 2 ([a, b]) such that v (a) = v (b) = 0. We next consider the increment of the functional δI = I (u + εv) − I (u) =  b a [F (x, u + ε v, u′ + ε v′ ) − F (x, u, u′ )] dx. (14.6.5) From the Taylor series expansion F (x, u + ε v, u′ + ε v′ ) = F (x, u, u′ ) + ε  v ∂F ∂u + v ′ ∂F ∂u′  + ε 2 2!  v ∂F ∂u + v ′ ∂F ∂u′ 2 + ··· , it follows from (14.6.5) that I (u + εv) = I (u) + ε δI + ε 2 2! δ 2 I + ··· , (14.6.6) where the first and second variations of I are given by δI =  b a  v ∂F ∂u + v ′ ∂F ∂u′  dx, (14.6.7) δ 2 I =  b a  v ∂F ∂u + v ′ ∂F ∂u′ 2 dx. (14.6.8) The necessary condition for the functional I (u) to have an extremum (that is, I (u) is stationary at u) is that the first variation becomes zero at u so 0 = δI =  b a  v ∂F ∂u + v ′ ∂F ∂u′  dx (14.6.9) which is, by partial integration of the second integral, =  b a ∂F ∂u − d dx  ∂F ∂u′  v dx + v ∂F ∂u′ b a . Because v (a) = v (b) = 0, this means that  b a ∂F ∂u − d dx  ∂F ∂u′  v dx = 0. (14.6.10) Since v is arbitrary in a ≤ x ≤ b, it follows from (14.6.10) that ∂F ∂u − d dx  ∂F ∂u′  = 0. (14.6.11) This is the famous Euler–Lagrange equation. We therefore can state: 634 14 Numerical and Approximation Methods Theorem 14.6.1. A necessary condition for the functional I (u) to be stationary at u is that u is a solution of the Euler–Lagrange equation ∂F ∂u − d dx  ∂F ∂u′  = 0, a ≤ x ≤ b (14.6.12) with u (a) = α, u (b) = β. (14.6.13) This is called the Euler–Lagrange variational principle. Note that, in general, equation (14.6.12) is a nonlinear second-order ordinary differential equations, and, although such an equation is very difficult to solve, still it seems to be more accessible analytically than the functional (14.6.3) from which it is derived. The derivative d dx in (14.6.12) can be computed by recalling u = u (x) and u ′ = du dx , and equation (14.6.12) becomes ∂F ∂u − ∂ 2F ∂x∂u′ − ∂ 2F ∂u∂u′  du dx − ∂ 2F ∂u′2 d 2u ∂x2 = 0. It is left to the reader to verify that the functional with one independent variable and nth-order derivatives in the form I (u) =  b a F (x, u, ux, uxx,...,uxn ,...,) dx admits the Euler–Lagrange equation ∂F ∂u − d dx  ∂F ∂ux  + d 2 dx2  ∂F ∂uxx  − ...(−1)n d n dxn  ∂ 2F ∂uxn  = 0. After we have determined the function u which makes I (u) stationary, the question of the nature of the extremum arises, that is, its minimum, maximum, or saddle point properties. To answer this question, we look at the second variation defined in (14.6.8). If terms of O  ε 3 can be neglected in (14.6.6), or if they vanish for the case of quadratic F, it follows from (14.6.6) that a necessary condition for the functional I (u) to have a minimum I (u) ≥ I (u0) at u = u0 is that δ 2 I ≥ 0, for I (u) to have a maximum I (u) ≤ I (u0) at u = u0 is that δ 2 I ≤ 0 at u = u0 respectively for all admissible values of v. These results enable us to determine the upper or lower bounds for the stationary value I (u0) of the functional. Example 14.6.6. Find out the shortest distance between given points A and B in the (x, y)-plane. Suppose AP B is any curve in the plane through A and B, and s = arcAP. The problem is to determine the curve for which the functional 14.6 Variational Methods and the Euler–Lagrange Equations 635 I (y) =  B A ds, (14.6.14) is a minimum. Since ds/dx =  1 + y ′2 1 2 , functional (14.6.14) becomes I (y) =  x2 x1  1 + y ′2 1 2 dx. (14.6.15) In this case, F =  1 + y ′2 1 2 which depends on y ′ only, so ∂F/∂y = 0. Hence, the Euler–Lagrange equation (14.6.12) becomes d dx  ∂F ∂y′  = 0. This gives the differential equation y ′′ = 0. (14.6.16) This means that the curvature for all points on the curve AB is zero. Hence, the path AB is a straight line. It follows from the integration of (14.6.16) that y = mx + c is a two-parameter family of straight lines. Example 14.6.7. (Fermat principle in optics). In an optically homogeneous isotro-pic medium, light travels from one point A to another point B along the path for which the travel time is minimum. The velocity of light v is the same at all points of the medium; hence, the minimum time is equivalent to the minimum path length. For simplicity, consider a path joining the two points A and B in the (x, y)-plane. The time to travel an elementary arc length ds is ds/v. Thus, the variational problem is to find the path for which  B A ds v =  x2 x1  1 + y ′2 1 2 dx v =  x2 x1 F (y, y′ ) dx (14.6.17) is a minimum, where y ′ = dy/dx, and v = v (y). When F is a function of y and y ′ , the Euler–Lagrange equation (14.6.12) becomes d dx (F − y ′Fy′ )=0. (14.6.18) This follows from the result d dx (F − y ′Fy′ ) = d dxF (y, y′ ) − y ′′Fy′ − y ′ d dx (Fy′ ) = y ′Fy + y ′′Fy − y ′′Fy − y ′ d dx (Fy′ ) = y ′ Fy − d dx (Fy′ ) = 0, by (14.6.12). 636 14 Numerical and Approximation Methods Hence, F − y ′Fy′ = constant, (14.6.19) or  1 + y ′2 1 2 v − y ′2 v (1 + y ′2) 1 2 = constant, or v −1  1 + y ′2 − 1 2 = constant. (14.6.20) In order to give a simple physical interpretation, we rewrite (14.6.20) in terms of the angle φ made by the tangent to the minimum path with the vertical y-axis so that sin φ =  1 + y ′2 − 1 2 . Hence, 1 v sin φ = constant = K (14.6.21) for all points on the minimum curve. For a ray of light, (1/v) must be directly proportional to the refractive index n of the medium through which light is travelling. Equation (14.6.21) is called the Snell law of refraction of light. Often this law is stated as n sin φ = constant. (14.6.22) (A) Hamilton Principle The difference between the kinetic energy T and the potential energy V of a dynamical system is denoted by L = T − V . The quantity L is called the Lagrangian of the system. The Hamilton principle states that the first variation of the time integral of L is zero, that is, δ  t2 t1 L dt = δ  t2 t1 (T − V ) dt = 0. (14.6.23) This result is supposed to be valid for all dynamical systems whether they are conservative or nonconservative. For a conservative system the force field F = −∇V and T + V = C, where C is a constant, and so (14.6.23) gives the principle of least action δA = 0, A =  t2 t1 L dt, (14.6.24) where A is called the action integral or simply the action of the system. 14.6 Variational Methods and the Euler–Lagrange Equations 637 Example 14.6.8. Derive the Newton second law of motion from the Hamilton principle. Consider a particle of mass m at the position r = (x, y, z) which is moving under the action of a field of force F. The kinetic energy of the particle is T = 1 2mr˙ 2 , and the variation of work done is δW = F · δr and δV = −δW. Thus, the Hamilton principle for the system is 0 = δ  t2 t1 (T − V ) dt =  t2 t1 (δT − δV ) dt =  t2 t1 (mr˙ · δr˙ + F · δr) dt. Integrating this result by parts and noting that δr vanishes at t = t1 and t = t2, we obtain  t2 t1 (m¨r − F) · δr dt = 0. This is true for every virtual displacement δr, and hence, the integrand must vanish, that is, m¨r = F. (14.6.25) This is the celebrated Newton second law of motion. Example 14.6.9. Derive the equation for a simple harmonic oscillator in a non-resisting medium from the Hamilton principle. For a simple harmonic oscillator, T = 1 2mx˙ 2 and V = 1 2mω2x 2 . According to the Hamilton principle δ  t2 t1  1 2 mx˙ 2 − 1 2 mω2x 2  dt = δ  t2 t1 F (x, x˙) dt = 0. This leads to the Euler–Lagrange equation ∂F ∂x − d dt (mx˙)=0, or x¨ + ω 2x = 0. (14.6.26) This is the equation for the simple harmonic oscillator. Example 14.6.10. A straight uniform elastic beam of length l, line density ρ, cross-sectional moment of inertia I, and modulus of elasticity E is fixed at each end. The beam performs small transverse oscillations in the horizontal (x, y)-plane. Derive the equation of motion of the beam. The potential energy of the elastic beam is V = 1 2  l 0 M2 EI dx = 1 2  l 0 EIy′′2 dx, 638 14 Numerical and Approximation Methods where the bending moment M is proportional to the curvature so that M = EI y ′′ (1 + y ′2) 1 2 ∼ EIy′′ for small y ′ . The variational principle gives δ  t2 t1 (T − V ) dt = δ  t2 t1 F (y ′′ , y˙) dt = 0, where F (y ′′ , y˙) = 1 2  l 0  ρy˙ 2 − EIy′′2 dx. This principle leads to the Euler–Lagrange equation −  l 0 4 ρy¨ + EIy(iv) 5 dx = 0, or ρy¨ + EIy(iv) = 0. (14.6.27) This represents the partial differential equation of the transverse vibration of the beam. (B) The Generalized Coordinates, Lagrange Equation, and Hamilton Equation The Euler–Lagrange analysis of a dynamical system can be extended to more complex cases where the configuration of the system is described by generalized coordinates q1, q2, ..., qn. Without loss of generality, we consider a system of three variables where the familiar Cartesian coordinates x, y, z can be expressed in terms of the generalized coordinates q1, q2, q3 as x = x (q1, q2, q3), y = y (q1, q2, q3), z = z (q1, q2, q3). (14.6.28) For example, if (q1, q2, q3) represents the cylindrical polar coordinates (r, θ, z), the above result becomes x = r cos θ, y = r sin θ, z = z. Since the coordinates are functions of time t, we obtain the following result by differentiation x˙ = ∂x ∂q1 q˙1 + ∂x ∂q2 q˙2 + ∂x ∂q3 q˙3 (14.6.29) 14.6 Variational Methods and the Euler–Lagrange Equations 639 with similar expressions for ˙y and ˙z. If these results are substituted into T = 1 2m  x˙ 2 + ˙y 2 + ˙z 2 and V = V (x, y, z), then both T and V can be written in terms of the generalized coordinates qi and the generalized velocities ˙qi , as T = T (q1, q2, q3; ˙q1, q˙2, q˙3), V = V (q1, q2, q3), (14.6.30) so that the Lagrangian has the form L = T − V = L(qi , q˙i). (14.6.31) The Hamilton principle gives δ  t2 t1 L(qi , q˙i) dt = 0. (14.6.32) The simple variation of this integral with fixed end points, the interchange of the variation operations and time derivatives for the variation of the generalized velocities, and then integration by parts yield  t2 t1 1 3 i=1  ∂L ∂qi − d dt  ∂L ∂q˙i 0 δqi 3 dt = 0, (14.6.33) where the integrated components vanish because of the conditions δqi = 0 (i = 1, 2, 3) at t = t1 and t = t2. When the generalized coordinates are independent and the variations δqi are independent for all t in (t1, t2), the coefficients of the variations δqi vanish independently for arbitrary values of t1 and t2. This means that the integrand in (14.6.33) vanishes, that is, d dt  ∂L ∂q˙i  − ∂L ∂qi = 0, i = 1, 2, 3. (14.6.34) These are called the Lagrange equations of motion. If a particle of mass m at position r = (x1, x2, x3) moves under the action of a conservative force field Fi = −∂V /∂xi , the Lagrangian function is L = T − V = 1 2 m  x˙ 2 1 + ˙x 2 2 + ˙x 2 3 − V (x1, x2, x3). (14.6.35) Consequently, ∂L ∂x˙ i = mx˙ i , ∂L ∂xi = − ∂V ∂xi = Fi . (14.6.36) The former represents the momentum of the particle and the latter is the force acting on the particle. In view of (14.6.36), the Lagrange equation (14.6.34) gives the Newton second law of motion in the form d dt (mx˙ i) = Fi . (14.6.37) 640 14 Numerical and Approximation Methods Example 14.6.11. Apply the Lagrange equations of motion to derive the equations of motion of a particle under the action of a central force, −mF (r) where r is the distance of the particle of mass m from the center of force. It is convenient to use the polar coordinates r and θ. In terms of the generalized coordinates q1 = r and q2 = θ, we write x = r cos θ = q1 cos q2, y = r sin θ = q1 sin q2. The kinetic energy T is T = 1 2 m  x˙ 2 + ˙y 2 = 1 2 m 4 r˙ 2 + r 2 ˙θ 2 5 = 1 2 m  q˙ 2 1 + q 2 1 q˙ 2 2 . (14.6.38) Since F = ∇V , the potential is V (r) =  r F (r) dr =  q1 F (q1) dq1. (14.6.39) Then, the Lagrangian L is L = T − V = 1 2 m  q˙ 2 1 + q 2 1 q˙ 2 2 − 2  q1 F (q1) dq1 . (14.6.40) Thus, the Lagrange equations (14.6.34) with i = 1, 2, 3 give the equations of motion q¨1 − q1q˙ 2 2 + F (q1)=0, d dt  q 2 1 q˙2 = 0. (14.6.41) In term of the polar coordinates, these equations become r¨ − r ˙θ 2 = −F (r), d dt 4 r 2 ˙θ 5 = 0. (14.6.42ab) Equation (14.6.42b) gives immediately r 2 ˙θ = h, (14.6.43) where h is a constant. In this case, r ˙θ represents the transverse velocity component, and mr2 ˙θ = mh is the constant angular momentum of the particle about the center of force. Introducing r = 1/u, we obtain r˙ = dr dt = − 1 u 2 du dt = − 1 u 2 du dθ · dθ dt = −h du dθ , r¨ = d 2 r dt2 = −h d dt  du dθ  = −h d 2u dθ2 dθ dt = −h 2u 2 d 2u dθ2 . Substituting these into (14.6.42a) gives 14.6 Variational Methods and the Euler–Lagrange Equations 641 −h 2u 2 d 2u dθ2 − h 2u 3 = −F  1 u  , or d 2u dθ2 + u = 1 h 2u 2 F  1 u  . (14.6.44) This is the differential equation of the central orbit and can be solved by standard methods. In particular, if the law of force is the attractive inverse square, F (r) = µ/r2 so that the potential V (r) = −µ/r, the differential equation (14.6.44) becomes d 2u dθ2 + u = µ h 2 , (14.6.45) if the particle is projected initially from distance a with velocity V at an angle β that the direction of motion makes with the outward radius vector. Thus, the constant h in (14.6.43) is h = V a sin β. The angle φ between the tangent and radius vector of the orbit at any point is given by cot φ = 1 r dr dθ = u d dθ  1 u  = − 1 u du dθ . (14.6.46) At t = 0, the initial conditions are u = 1 a , du dθ = − 1 a cot β when θ = 0. (14.6.47) The general solution of equation (14.6.45) is u = µ h 2 [1 + e cos (θ + α)] , (14.6.48) where e and α are constants to be determined from the initial data. Finally, the solution can be written as l r =1+ e cos (θ + α), (14.6.49) where l = h 2 µ = (V a sin β) 2 /µ. (14.6.50) This represents a conic section of semi-latus rectum l and eccentricity e with its axis inclined at an angle α to the radius vector at the point of projection. The initial conditions (14.6.47) lead to 642 14 Numerical and Approximation Methods l a =1+ e cos α, − l a cot β = −e sin α, (14.6.51) which give tan α = l cot β l − a , e 2 =  l a − 1 2 + l 2 a 2 cot2 β = l 2 a 2 cosec2β − 2l a + 1, = 1 − 2aV 2 sin2 β µ + a 2V 4 sin2 β µ2 . (14.6.52) Thus, the conic is an ellipse, parabola, or hyperbola accordingly as e <=> 1 that is, V 2 < = > 2µ/a. To derive the Hamilton equations, we introduce the concept of generalized momentum, pi and generalized force, Fi as pi = ∂L ∂q˙i , Fi = ∂L ∂qi . (14.6.53ab) Consequently, the Lagrange equations (14.6.34) become ∂L ∂qi = d dt pi = ˙pi . (14.6.54) The Hamiltonian function H is defined by H = n i=1 pi q˙i − L. (14.6.55) In general, L = L(qi , q˙i , t) is a function of qi , ˙qi and t, where ˙qi enters through the kinetic energy as a quadratic term. Hence, equation (14.6.53a) will give pi as a linear function of ˙qi . This system of linear equations involving pi and ˙qi can be solved to determine ˙qi in terms of pi , and then, the q˙i can, in principle, be eliminated from (14.6.55). This means that H can always be expressed as a function of pi , qi and t so that H = H (pi , qi , t). Thus, dH =  ∂H ∂pi dpi +  ∂H ∂qi dqi + ∂H ∂t dt. (14.6.56) On the other hand, differentiating H in (14.6.55) with respect to t gives dH dt = pi d dt q˙i + q˙i d dt pi −  ∂L ∂qi d dt qi −  ∂L ∂q˙i d dt q˙i − ∂L ∂t , (14.6.57) or 14.6 Variational Methods and the Euler–Lagrange Equations 643 dH = pi dq˙i + q˙i dpi −  ∂L ∂qi dqi −  ∂L ∂q˙i dq˙i − ∂L ∂t dt, (14.6.58) which becomes, in view of (14.6.53a), dH = q˙i dpi −  ∂L ∂qi dqi − ∂L ∂t dt. (14.6.59) Evidently, two expressions of dH in (14.6.56) and (14.6.59) must be equal so that the coefficients of the corresponding differentials can be equated to obtain q˙i = ∂H ∂pi , − ∂L ∂qi = ∂H ∂qi , − ∂L ∂t = ∂H ∂t . (14.6.60abc) Using the Lagrange equation (14.6.54), the first two of the above equations become q˙i = ∂H ∂pi , p˙i = − ∂H ∂qi . (14.6.61ab) These are commonly known as the Hamilton canonical equations of motion. They play a fundamental role in advanced analytical dynamics. Finally, the Lagrange–Hamilton theory can be used to derive the law of conservation of energy. In general, the Lagrangian L is independent of time t and hence, (14.6.60c) implies that H = constant. Again, T involved in L = T − V is given by T = 1 2 n i=1 n j=1 aij q˙i q˙j , (14.6.62) where the coefficients aij are symmetric functions of the generalized coordinates qij , that is, aij = aji. On the other hand, V is, in general, independent of qi and hence, pi = ∂L ∂q˙i = ∂T ∂q˙i = n j=1 aij q˙j . (14.6.63) Thus, the Hamiltonian H becomes H = n i=1 piq˙i − L = n i=1 ⎛ ⎝ n j=1 aij q˙j ⎞ ⎠ q˙i − L = 2T − L = T + V. (14.6.64) Thus, H is equal to the total energy. It has already been observed that, if L does not contain t explicitly, H is a constant. This means that the sum of the potential and kinetic energies is constant. This is the law of the conservation of energy. 644 14 Numerical and Approximation Methods Example 14.6.12. Use the Hamiltonian equations to derive the equations of motion for the problem stated in Example 14.6.11. The Lagrangian L for this problem is given by (14.6.40) with q1 = r and q2 = θ. It follows from the definition (14.6.53a) of the generalized momentum that p1 = mq˙1 = mr, p ˙ 2 = mq2 1 q˙2 = mr2 ˙θ. (14.6.65) Expressing the results of the kinetic energy (14.6.38) and the potential energy (14.6.39) in terms of p1 and p2 the Hamiltonian H = T + V can be written as H = 1 2m  p 2 1 + p 2 2 q 2 1  + m  q1 F (q1) dq1. (14.6.66) Then, equations (14.6.65) and the Hamilton equation (14.6.61b) give p1 = mr, p ˙ 2 = mr2 ˙θ, (14.6.67) p˙1 = 1 m p 2 2 q 3 1 + mF (q1), p˙2 = 0. (14.6.68) Clearly, these equations are identical with the equations of motion (14.6.42ab). Example 14.6.13. Derive the equation of a simple pendulum by using (i) the Lagrange equations and (ii) the Hamilton equations. We consider the motion of simple pendulum of mass m attached at the end of a rigid massless string of length l that pivots about a fixed point. We suppose that the pendulum makes an angle θ with its vertical position. The force F acting on the mass m is F = −mg sin θ, so that the potential V is obtained from F = −∇V as V = mgl(1 − cos θ). The kinetic energy T = 1 2ml2 ˙θ 2 . Thus the Lagrangian L is L = T − V = 1 2 ml2 ˙θ 2 − mgl(1 − cos θ) = L 4 θ, ˙θ 5 . (14.6.69) The Lagrange equation is ∂L ∂θ − d dt  ∂L ∂ ˙θ  = 0, (14.6.70) or −mglsin θ − d dt 4 ml2 ˙θ 5 = 0, or ¨θ + ω 2 sin θ = 0, ω2 = g/l. (14.6.71ab) 14.6 Variational Methods and the Euler–Lagrange Equations 645 This is the equation of the simple pendulum. To derive the same equation from the Hamilton equations, we choose q1 = l( ˙q1 = 0) and q2 = θ as the generalized (polar) coordinates. The kinetic and potential energies are T = 1 2 ml2 q˙ 2 2 , V = mgl(1 − cos q2). (14.6.72ab) Thus, H = T + V and L = T − V are given by (H, L) = 1 2 ml2 q˙ 2 2 + mgl(1 − cos q2). (14.6.73ab) From the definition of the generalized momentum, we find that p2 = ∂L ∂q˙2 = ml2 q˙2 so that the Hamiltonian H in terms of p2 and q2 is H = 1 2 p 2 2 ml2 + mgl(1 − cos q2). Thus, the Hamilton equation (14.6.61ab) gives ¨θ + ω 2 sin θ = 0, ω2 = g l . (14.6.74) The variational methods can be further extended for functionals depending on functions or more independent variables in the form I [u (x, y)] =  D F (x, y, u, ux, uy) dx dy (14.6.75) where the values of the function u (x, y) are prescribed on the boundary ∂D of a finite domain D in the (x, y)-plane. We assume that F is differentiable and the surface u = u (x, y) giving an extremum is also continuously differentiable twice. The first variation δI of I is defined by δI [u, ε] = I (u + ε) − I (u) (14.6.76) which is, by Taylor’s expansion theorem =  D [εFu + εxFp + εyFq] dx dy (14.6.77) where ε ≡ ε (x, y) is small and p = ux and q = uy. According to the variational principle, δI = 0 for all admissible values of ε. The partial integration of (14.6.77) combined with ε = 0 on ∂D gives 646 14 Numerical and Approximation Methods 0 = δI =  D Fu − ∂ ∂xFp − ∂ ∂yFq ε (x, y) dx dy. (14.6.78) This is true for all arbitrary ε, and hence, the integrand must vanish, that is ∂ ∂xFp + ∂ ∂yFq − Fu = 0. (14.6.79) This is the Euler–Lagrange equation which is the second-order partial differential equation to be satisfied by the extremizing function u (x, y). Example 14.6.14. Derive the equation of motion for the free vibration of an elastic string of length l. The potential energy V of the string is V = 1 2 T ∗  l 0 u 2 xdx (14.6.80) where u = u (x, y) is the displacement of the string from its equilibrium position and T ∗ is the constant tension of the string. The kinetic energy T is T = 1 2  l 0 ρu2 t dx (14.6.81) where ρ is the constant line-density of the string. According to the Hamilton principle δI = δ  t2 t1 (T − V ) dt = δ  t2 t1  l 0 1 2  ρu2 t − T ∗u 2 x dx dt = 0 (14.6.82) which has the form δ  t2 t1  l 0 L(ut, ux)=0, (14.6.83) where L = 1 2  ρ u2 t − T ∗u 2 x . Then the Euler–Lagrange equation is given by ∂ ∂t (ρut) − ∂ ∂x (T ∗ux)=0, (14.6.84) or utt − c 2uxx = 0, c2 = T ∗ /ρ. (14.6.85) This is the wave equation of motion of the string. 14.7 The Rayleigh–Ritz Approximation Method 647 Example 14.6.15. Derive the Laplace equation from the functional I (u) =  D  u 2 x + u 2 y dx dy with a boundary condition u = f (x, y) on ∂D. The variational principle gives δI = δ  D  u 2 x + u 2 y dx dy = 0. This leads to the Euler–Lagrange equation uxx + uyy = 0 in D. Similarly, the functional I [u (x, y, z)] =  D  u 2 x + u 2 y + u 2 z dx dy dz will lead to the three-dimensional Laplace equation ∇2u = uxx + uyy + uzz = 0. 14.7 The Rayleigh–Ritz Approximation Method We consider the boundary-value problem governed by the differential equation Au = f in D (14.7.1) with the boundary condition B (u) = 0 on ∂D (14.7.2) where A is a self-adjoint differential operator in a Hilbert space H and f ∈ H. In general, the determination of the exact solution of the problem is often a difficult task. However, it can be shown that the solution of (14.7.1)– (14.7.2) is equivalent to finding the minimum of a functional I (u) associated with the differential system. In other words, the solution can be characterized as the function which minimizes (or maximizes) the functional I (u). A simple and efficient method for an approximate solution of the extremum problem was independently formulated by Lord Rayleigh and W. Ritz. We next prove a fundamental result which states that the solution of the equation (14.7.1) is equivalent to finding the minimum of the quadratic functional 648 14 Numerical and Approximation Methods I (u) ≡ A u, u − 2 f, u. (14.7.3) Suppose that u = u0 is the solution of (14.7.1) so that Au0 = f. Consequently, I (u) ≡ A u, u − 2 Au0, u = A (u − u0), u−Au0, u. Since the inner product is symmetrical and Au0, u = u0, Au = Au, u0, I (u) can be written as I (u) = A (u − u0), u−Au, u0 + Au0, u−Au0, u0, = A (u − u0), u − u0−Au0, u0, = A (u − u0), u − u0 + I (u0). (14.7.4) Since A is a positive operator, A (u − u0), u − u0 ≥ 0 where equality holds if and only if u − u0 = 0. It follows that I (u) ≥ I (u0), (14.7.5) where equality holds if and only if u = u0. We conclude from this inequality that I (u) assumes its minimum at the solution u = u0 of equation (14.7.1). Conversely, the function u = u0 that minimizes I (u) is a solution of equation (14.7.1). Clearly, I (u) ≥ I (u0), that is, in particular, I (u0 + αv) ≥ I (u0) for any real α and any function v. Explicitly, I (u0 + αv) = A (u0 + αv), u0 + αv − 2 f, u0 + αv, = Au0, u0 + 2α Au0, v + α 2 Av, v − 2 f, u0 − 2α f,v. This means that I (u0 + αv) is a quadratic expression in α. Since I (u) is minimum at u = u0, then δI (u0, v) = 0, that is, 0 = d dαI (u0 + αv) α=0 = 2 Au0, v − 2 f,v = 2 Au0 − f,v. This is true for any arbitrary but fixed v. Hence, Au0 − f = 0. This proves the assertion. In the Rayleigh–Ritz method an approximate solution of (14.7.1)– (14.7.2) is sought in the form un (x) = n i=1 aiφi (x), (14.7.6) where a1, a2, ..., an are n unknown coefficients to be determined so that I (un) is minimum, and φ1, φ2, ..., φn represent a linearly independent and 14.7 The Rayleigh–Ritz Approximation Method 649 complete set of arbitrarily chosen functions that satisfy (14.7.2). This set of functions is often called a trial set. We substitute (14.7.6) into (14.7.3) to obtain I (un) = ;n i=1 aiA (φi), n j=1 ajφj < − 2 ; f, n i=1 aiφi < . Then the necessary condition for I to obtain a minimum (or maximum) is that ∂I ∂aj (a1, a2,...,an)=0, j = 1, 2, . . . , n, (14.7.7) or ∂I ∂aj ⎡ ⎣ ;n i=1 aiA (φi), n j=1 ajφj < − 2 ; f, n i=1 aiφi <⎤ ⎦ = 0, or n i=1 A (φi), φj  ai + n i=1 A (φj ), φi aj − 2 f,φj  = 0, or 2 n i=1 A (φi), φj  ai = 2 f,φj . Therefore, n i=1 A (φi), φj  ai = f,φj , j = 1, 2, . . . , n. (14.7.8) This is a linear system of n equations for the n unknown coefficients aj . Once a1, a2, ..., an are determined, the approximate solution is given by (14.7.6). In particular, when A (φi), φj  = ⎧ ⎨ ⎩ 0, i = j 1, i = j, (14.7.9) equation (14.7.8) gives aj as aj = φj , f, (14.7.10) so that the Rayleigh–Ritz approximate series (14.7.6) becomes 650 14 Numerical and Approximation Methods un (x) = n i=1 φi , f φi (x). (14.7.11) This is similar to the Fourier series solution with known Fourier coefficients ai . In the limit n → ∞, a limit function can be obtained from (14.7.6) as u (x) = limn→∞ n i=1 ai φi (x) = ∞ i=1 ai φi (x), (14.7.12) provided that the series converges. Under certain assumptions imposed on the functional I (u) and the trial functions φ1, φ2, ..., φn, the limit function u (x) represents an exact solution of the problem. In any event, (14.7.6) or (14.7.11) gives a reasonable approximate solution. In the simplest case corresponding to n = 1, the Rayleigh–Ritz method gives a simple form of the functional I (u1) = I (a1φ1) = a 2 1 Aφ1, φ1 − 2a1 f,φ1, where a1 is readily determined from the necessary condition for extremum 0 = ∂ ∂a1 I (a1φ1)=2a1 Aφ1, φ1 − 2 f,φ1, or a1 = f,φ1 Aφ1, φ1 . (14.7.13) The corresponding minimum value of the functional is given by I (a1φ1) = − f,φ1 2 Aφ1, φ1 . (14.7.14) Thus, the essence of the Rayleigh–Ritz method is as follows. For a given boundary-value problem, an approximate series solution is sought so that the trial functions φi satisfy the boundary conditions. We solve the system of algebraic equations (14.7.7) to determine the coefficients ai . We now illustrate the method by several examples. Example 14.7.1. Find an approximate solution of the Dirichlet problem ∇2u ≡ uxx + uyy = 0 in D u = f on ∂D, where D ⊂ R2 , and f is a given function. This problem is equivalent to finding the minimum of the associated functional 14.7 The Rayleigh–Ritz Approximation Method 651 I (u) =  D  u 2 x + u 2 y dx dy. We seek an approximate series solution in the form u2 (x, y) = a1φ1 + a2φ2 with a1 = 1 so that u2 satisfies the given boundary conditions, that is, φ1 = f and φ2 = 0 on ∂D. Substituting u2 into the functional gives I (u2) =  D 1 ∂u2 ∂x 2 +  ∂u2 ∂y 2 3 dx dy =  D (∇φ1) 2 dx dy + 2a2  D (∇φ1 · ∇φ2) dx dy + a 2 2  D |∇φ2| 2 dx dy. The necessary condition for an extremum of I (u2) is ∂I ∂a2 = 0, or 2  D (∇φ1 · ∇φ2) dx dy + 2a2  D |∇φ2| 2 dx dy = 0. Therefore, a2 = − ** (∇φ1 · ∇φ2) dx dy ** |∇φ2| 2 dx dy . This a2 minimizes the functional and the approximate solution is obtained. However, this procedure can be generalized by seeking an approximate solution in the form un = n i=1 aiφi (a1 = 1) so that φ1 = f and φi = 0(i = 2, 3,...,n) on ∂D. The coefficients ai can be obtained by solving the system (14.7.7) with j = 2, 3, ..., n. Example 14.7.2. A uniform elastic beam of length l carrying a uniform load W per unit length is freely hinged at x = 0 and x = l. Find the approximate solution of the boundary-value problem EI (iν) y (x) = W, y = y ′′ = 0 at x = 0 and x = l, 652 14 Numerical and Approximation Methods where y = y (x) is the displacement function. This problem is equivalent to finding a function y (x) that minimizes the energy functional I (y) =  l 0  W y − EI 2 y ′′2  dx. We seek an approximate solution yn (x) = n r=1 ar sin 4rπx l 5 which satisfies the boundary conditions. Substitution of this solution into the energy functional gives I (yn) = n r=1 1 l 0 W ar sin 4rπx l 5 dx − EI 2  l 0 r 4π 4 l 4 a 2 r sin2 4rπx l 5 dx3 = 2Wl π n r=1 ar r − EIπ4 4l 3 n r=1 r 4 a 2 r . The necessary conditions for extremum are 0 = ∂I ∂ar = 2Wl rπ − EIπ4 4l 3 2arr 4 , r = 1, 2,...,n which give ar as ar = 4Wl4 π 5r 5EI , r = 1, 2, . . . , n. Thus, the approximate function y (x) is yn (x) = 4Wl4 π 5EI n r=1 1 r 5 sin 4rπx l 5 . The maximum deflection at x = l/2 is ymax = 4Wl4 π 5EI  1 − 1 3 5 + 1 5 5 − ... . In this case, the first term of the series solution gives a reasonably good approximate solution as y1 (x) ∼ 4Wl4 EIπ5 sin 4πx l 5 . 14.7 The Rayleigh–Ritz Approximation Method 653 Example 14.7.3. Apply the Rayleigh–Ritz method to investigate the free vibration of a fixed elastic wedge of constant thickness governed by the energy functional I (y) =  1 0  αx3 y ′′2 − ωxy2 dx, y (1) = y ′′ (1) = 0, where the free vibration is described by the function u (x, t) = e iωty (x), ω is the frequency. We seek an approximate solution in the form yn (x) = n r=1 aryr (x) = n r=1 ar (x − 1)2 x r−1 which satisfies the given boundary conditions. We take only the first two terms so that y2 (x) = a1y1 +a2y2 = (x − 1)2 (a1 + a2x). Substituting y2 into the functional we obtain I2 = I (y2) =  1 0 " αx3 (6a2x + 2a1 − 4a2) 2 − ωx (x − 1)4 (a1 + a2x) 2 # dx = α (a1 − 2a2) 2 + 24 5 (a1 − 2a2) a2 + 6a 2 2 − ω 5 a 2 1 6 + 2a1a2 21 + a 2 2 56 . The necessary conditions for an extremum are ∂I2 ∂a1 = 2a1 4 α − ω 30 5 + 2 5 a2 4 2α − ω 21 5 = 0, ∂I2 ∂a2 = 2a1 5 4 2α − ω 21 5 + 2a2 5 4 2α − ω 56 5 = 0. For nontrivial solutions, the determinant of this algebraic system must be zero, that is,       α − ω 30 1 5  2α − ω 21 2α − ω 21 2α − ω 56       = 0, or 5 4 α − ω 30 542α − ω 56 5 − 4 2α − ω 21 52 = 0. This represents the frequency equation of the vibration which has two roots ω1 and ω2. The smaller of these two frequencies gives an approximate value of the fundamental frequency of the vibration of the wedge. Example 14.7.4. An elastic beam of length l, density ρ, cross-sectional area A, and modulus of elasticity E has its end x = 0 fixed and the other end 654 14 Numerical and Approximation Methods connected to a rigid support through a linear elastic spring with spring constant k. Apply the Rayleigh–Ritz method to investigate the harmonic axial motion of the beam. The kinetic energy and the potential energy associated with the axial motion of the beam are T =  l 0 ρA 2 U 2 t dx, V =  l 0 EA 2 U 2 x dx + k 2 U 2 (l, t), where U (x, t) is the displacement function. Since the axial motion is simple harmonic, U (x, t) = u (x) e iωt, where ω is the frequency of vibration. Consequently, the expressions for T and V can be written in terms of u (x). We then apply the Hamilton variational principle δI (u) = δ 1 t2 t1  l 0 1 2  ρAω2 u 2 − EAu2 x dx − k 2 u 2 (l) 3 dt = 0. The Euler–Lagrange equation for the variational principle is d dx  EA du dx + ρAω2 u = 0, 0 < x < l, EA du dx + ku = 0, at x = l. In terms of nondimensional variables (x ∗ , u∗ ) = (1/l) (x, u) and parameters λ =  ω 2ρl2/E and α = (kl/EA), this system becomes, dropping the asterisks, uxx + λ u = 0, 0 <x< 1,="" ux="" +="" α="" u="0," at="" x="1." the="" associated="" functional="" for="" system="" is="" i="" (u)="1" 2="" ="" 1="" 0="" ="" λ="" u2="" −="" dx="" (1).="" according="" to="" rayleigh–ritz="" method,="" we="" seek="" approximate="" solution="" with="" in="" form="" (x)="a1x" a2x="" so="" that="" (u2)="" minimum.="" substitute="" into="" obtain="" i2="I" l="" "="" a1x="" (a1="" 2a2x)="" #="" a2)="" .="" 14.8="" galerkin="" approximation="" method="" 655="" necessary="" conditions="" extremum="" of="" are="" ∂a1="a1" ="" 3="" ="" a2="" 4="" ,="" ∂a2="a1" 5="" 7="" nontrivial="" solutions,="" determinant="" must="" be="" zero,="" is,="" ="" or="" 3λ="" 128λ="" 480="0." this="" quadratic="" equation="" gives="" two="" solutions:="" λ1="4.155," λ2="38.512." corresponding="" values="" frequency="" given="" by="" ω1="2.038" e="" ρ="" l2="" 1="" ω2="6.206" exact="" determined="" transcendental="" √="" tan="" first="" roots="" can="" obtained="" graphically="" as="" ω01="" ∼="" 2.0288="" ω02="" 4.9132="" an="" extension="" formulated="" ingenious="" which="" may="" applied="" a="" problem="" no="" simple="" variational="" principle="" exists.="" differential="" operator="" (14.7.1)="" need="" not="" linear="" equation.="" order="" solve="" boundary-value="" (14.7.1)–(14.7.2),="" construct="" un="" +n="" aiφi="" (x),="" (14.8.1)="" 656="" 14="" numerical="" and="" methods="" where="" φi="" known="" functions,="" u0="" introduced="" satisfy="" boundary="" conditions,="" coefficients="" ai="" determined.="" substituting="" non-zero="" residual="" rn="" (a1,="" a2,...,an,="" x,="" y)="A" (un)="A" (u0)="" aia="" (φi).="" (14.8.2)="" unknown="" solving="" following="" equations="" rn,="" φj="" ="0," j="1," 2,...,n.="" (14.8.3)="" since="" linear,="" written="" n="" a="" (φi),="" au0,="" ,="" (14.8.4)="" determines="" ′="" s.="" substitution="" s="" from="" required="" un.="" find="" interesting="" connection="" between="" fourier="" representation="" function="" u.="" (14.8.5)="" special="" restriction="" on="" satisfies="" condition="" aφi="" ⎨="" ⎩="" 0,="" (14.8.6)="" thus,="" application="" (14.8.7)="" (14.8.6),="" aj="f,φj" (14.8.8)="" becomes="" f,φi="" (x).="" (14.8.9)="" evidently,="" just="" finite="" series="" solution.="" 657="" finally,="" shall="" cite="" example="" show="" equivalence="" methods.="" consider="" poisson="" uxx="" uyy="f" (x,="" d="" ⊂="" r="" (14.8.10)="" homogeneous="" ∂d.="" equivalent="" finding="" minimum="" y="" 2fu="" dy.="" (14.8.11)="" trial="" y),="" (14.8.12)="" functions="" chosen="" they="" then="" use="" ∂i="" ∂ak="0," k="1," 2,="" ...,="" n="" ="" ∂un="" ∂x="" ∂φk="" ∂y="" fφk="" dy="0." (14.8.13)="" greens="" theorem="" leads="" ∇2un="" f="" φk="" (14.8.14)="" φk="0," ≡="" f.="" (14.8.15)="" undetermined="" ak.="" establishes="" 14.8.1.="" ∇2u="" :="" |x|="" <="" a,="" |y|="" b}="" ∂d="{(x," 658="" m,n="1" ="" 3,5,...="" amn="" φmn="" 4mπx="" 2a="" 5="" cos="" 4nπy="" 2b="" case="" +1="∇2uN" 1="" m="1" m2π="" 4a="" 2π="" 4b="" φmn3="" 1.="" φkl="" −a="" b="" −b="" kπx="" lπy="" akl="" 16ab="" π="" 2kl="" (−1){(k+l)="" 2}−1="" 8ab="" 2="" (−1)="" (k+l)−1="" (b="" 2k="" 2l="" 2)="" (m+n)−1="" 2m2="" 2n2)="" particular,="" derived="" square="" domain="" 1}.="" limit="" →="" ∞,="" these="" solutions="" perfect="" agreement="" those="" double="" series.="" 14.8.2.="" 14.8.1="" using="" algebraic="" polynomials="" functions.="" appropriate="" y="" a1="" a3y="" a4x="" ...="" obviously,="" conditions.="" approximation,="" assumes="" u1="" a1φ1="a1" 14.9="" kantorovich="" 659="" coefficient="" a1="" integral="" ∇2u1="" φ1dx="" 2a1="" 1!="" x="" evaluation="" −1="" hence,="" 1932,="" gave="" generalization="" rayleigh–="" ritz="" partial="" terms="" coefficients.="" essence="" reduce="" ordinary="" governed="" (14.7.1)–(14.7.2).="" it="" has="" been="" shown="" section="" 14.7="" (14.7.3).="" when="" problem,="" (14.7.6)="" ak="" constants.="" determine="" minimize="" (un).="" assume="" longer="" constants="" but="" one="" independent="" variables="" (14.9.1)="" products="" same="" minimizing="" (un="" (x))="I" 4n="" 5="" (14.9.2)="" 660="" perform="" integration="" respect="" all="" except="" ¯i="" (x),...,="" depending="" variable="" x.="" a2,...,an).="" under="" certain="" converges="" ∞.="" describe="" more="" precisely,="" dimensions:="" d,="" (14.9.3)="" ∂d,="" (14.9.4)="" closed="" bounded="" curves="" vertical="" lines="" (14.9.5)="" (14.9.6)="" condition,="" 1="" 2fun="" 3="" β(x)="" α(x)="" ⎧="" 1n="" 32="" 32="" 2f="" akφk="" ⎫="" ⎬="" ⎭="" (14.9.7)="" ak,="" a′="" )="" dx,="" (14.9.8)="" integrand="" assumed="" have="" performed="" result="" denoted="" ).="" reduced="" determining="" found="" euler="" equations:="" 661="" ∂f="" ∂a′="" 3,="" n.="" (14.9.9)="" solved="" (a)="ak" (b)="0," consequently,="" 14.9.1.="" torsion="" rectangle="" −a<x<a,="" −b<y<b}="" =="" +a,="" b.="" next="" (−a)="0." 4u="" yields="" (u1)="" "="" 4y="" 16="" 15="" 8="" dx.="" ′′="" constant="" coeffi-="" cients,="" general="" cosh="" kx="" sinh="" 2="" ka.="" ka="" 662="" torsional="" moment="" 2µα="" u1dx="" µαb3="" tanh="" (ak)="" 14.9.2.="" triangular="" 2u="" −x="" 1y="" 2xu1="" 02="" 135√="" 4="" 2x="" 5u="" 10x="" 4u1u="" 30x="" 3u="" 15x="" 3u1="" 5xu′="" 5u1="15." nonhomogeneous="" two.="" (r="" 1)="" 5)="0." particular="" bx−5="" solution,="" 0.="" implies="" therefore,="" final="" 4x="" 14.10="" element="" 663="" many="" problems="" mathematics,="" science="" engineering="" cannot="" closed-form="" analytical="" formulas.="" often="" asymptotic="" rather="" than="" solutions.="" evolved="" over="" years="" discrete="" easily="" computer.="" however,="" if="" carefully="" chosen,="" numerically="" computed="" anywhere="" close="" true="" another="" computation="" difficult="" take="" long="" impractical="" computer="" carry="" out.="" most="" commonly="" used="" differences="" give="" pointwise="" approximations="" governing="" equations.="" successfully="" fairly="" problems,="" their="" major="" weakness="" suitable="" irregular="" geometries,="" curved="" boundaries="" unusual="" example,="" difference="" particularly="" effective="" circular="" because="" circle="" accurately="" partitioned="" rectangles.="" there="" other="" including="" method.="" unlike="" methods,="" effectively="" accurate="" wide="" variety="" defined="" regions.="" entire="" modeled="" analytically="" approximated="" replacing="" small,="" interconnected="" elements="" (hence="" name="" element).="" extremely="" (linear="" functions)="" small="" such="" triangles.="" collected="" together="" requirements="" continuity="" equilibrium="" satisfied="" neighboring="" elements.="" nutshell,="" basic="" idea="" (fem)="" consists="" decomposing="" set="" arbitrary="" shape="" size.="" decomposition="" usually="" called="" mesh="" grid="" overlap="" nor="" leave="" any="" part="" uncovered.="" each="" element,="" number="" points="" located="" edges="" inside.="" nodes="" vertices="" triangles="" figure="" 14.10.1.="" consideration="" whole="" interpolation="" historically,="" was="" developed="" originally="" study="" stress="" fields="" complicated="" aircraft="" structures="" early="" 1960s.="" subsequently,="" extended="" widely="" science,="" engineering.="" richard="" courant="" (1888–1972)="" who="" piecewise="" continuous="" domains="" 1943;="" he="" combined="" potential="" en-="" 664="" ergy="" st.="" venant="" continuum="" mechanics.="" also="" described="" properties="" based="" principle.="" 1965,="" received="" even="" broader="" interpretation="" zienkiewicz="" cheung="" (1965)="" suggested="" applicable="" field="" cast="" form.="" during="" late="" 1960s="" 1970s,="" considerable="" attention="" errors,="" bounds="" convergence="" criteria="" various="" develop="" recall="" celebrated="" euler–lagrange="" (14.6.12)="" divide="" interval="" ≤="" parts="" rn+1="" set:="" x1="" x2="" xn="b." subinterval="" element.="" general,="" length="" equal,="" though="" simplicity,="" equal="" h="1" a).="" uk="u" (xk),="" 2,...,n="" (x0)="α" (xn)="β," while="" u1,="" u2,="" un−1="" quantities.="" rewrite="" (14.6.3)="" x0="" u,="" u′="" xn−1="" (14.10.1)="" define="" interpolating="" l(x)="" ui="" [a,="" b]="" whose="" graph="" straight="" line="" segments="" joining="" consecutive="" pairs="" (xk,="" uk),="" (xk+1,="" uk+1)="" (n="" 1),="" (uk+1="" uk)="" (x="" xk),="" xk="" xk+1,="" (14.10.2)="" 1).="" assuming="" integrals="" exactly="" in−1="In−1" (u1,="" u2,...,un−1).="" (14.10.3)="" ∂in−1="" ∂uk="0," 2,...,(n="" (14.10.4)="" substituted="" continuous,="" (the="" dirichlet="" plane).="" 665="" △="" (14.10.5)="" (14.10.6)="" region="" triangulated="" dn="" union="" figures="" 14.10.1="" (b).="" denote="" interior="" v1,="" v2,="" vn.="" choose="" v1="" v2="" vn="" vertex.="" vm="" its="" zero="" (c).="" c,="" b,="" c="" different="" triangle.="" requirement="" uniquely.="" indeed,="" simply="" pyramid="" unit="" height="" peak="" do="" touch="" vm.="" combination="" a2v2="" anvn="" amvm="" (14.10.7)="" a1,="" a2,="" multiply="" v="" green’s="" identity="" ∇u="" ·="" ∇v="" (14.10.8)="" valid="" only="" (a),="" (b),="" 666="" am="" (∇um="" ∇vk)="" vk="" (14.10.9)="" equations,="" am,="" rewritten="" αmk="" n,="" (14.10.10)="" dy,="" fk="" (14.10.11)="" (14.10.10).="" value="" (14.10.7).="" several="" comments="" order.="" first,="" depend="" geometry="" completely="" known.="" second,="" vanishes="" ∂dn.="" third,="" vertex="" vi="(xi" yi),="" (xi="" yi)="aivi" arvr="" vr="" yk)="⎧" fourth,="" yi).="" 14.10.2.="" extremes="" 6="" ′2="" 2xu="" (14.10.12)="" (0)="1" (6)="7." three="" x3="6." (xk)="" u3="u" (14.10.13)="" 667="" (14.10.14)="" ⎪⎪⎪⎪⎨="" ⎪⎪⎪⎪⎩="" (u1="" u0)="" (u2="" u1)="" 2),="" (u3="" 4),="" 6,="" (14.10.15)="" derivative="" u0),="" u1),="" 6.="" (14.10.16)="" (14.10.15)–(14.10.16)="" get="" ="" 0="" −2x="" 2)02="" −2="" 2)0="" 2)0dx="" 4)02="" 4)0="" 4)0dx.="" integrating="" u1u2="" 37="" 67="" 101="" ∂i2="" ∂u1="14" ∂u2="−" 668="" putting="" (14.10.17)="" (1="" x).="" (14.10.18)="" identical="" due="" simplicity="" problem.="" will="" different.="" adding="" some="" technique="" (boundary="" method).="" research="" solid="" mechanics,="" fluid="" theory="" electromagnetic="" theory.="" breakthrough="" came="" 1963="" classic="" papers="" were="" published="" jaswon="" (1963)="" symm="" (1963).="" mathematical="" aspect="" prescribed="" uses="" volume="" surface="" very="" useful="" especially="" dimensional="" rapidly="" changing="" fracture="" contact="" computationally="" less="" efficient="" industry.="" popular="" acoustic="" problems.="" 1970s="" continued="" fast="" pace="" include="" nonlinear="" 14.11="" exercises="" explicit="" utt="" 4uxx="0," <x<="" t=""> 0, u (0, t) = u (1, t)=0, t ≥ 0, u (x, 0) = sin 2πx, ut (x, 0) = 0, 0 ≤ x ≤ 1. Compare the numerical solution with the analytical solution u (x, t) = cos 4πtsin 2πx at several points. 14.11 Exercises 669 2. (a) Calculate an explicit finite difference solution of the wave equation uxx − utt = 0, 0 <x< 1,="" t=""> 0, satisfying the boundary conditions u (0, t) = u (1, t)=0, t ≥ 0, and the initial conditions u (x, 0) = 1 8 sin πx, ut (x, 0) = 0, 0 ≤ x ≤ 1. Show that the exact solution of the problem is u (x, t) = 1 8 cos πtsin πx. Compare the two solutions at several points. (b) Solve the wave equation in (a) with the same boundary data and the initial data u (x, 0) = sin πx, ut (x, 0) = 0, 0 ≤ x ≤ 1. 3. Use the Lax–Wendroff method to find a numerical solution of the problem ux + ut = 0, x > 0, t > 0, u (x, 0) = 2 + x, x > 0, u (0, t)=2 − t, t > 0. Show that the exact solution of the problem is u (x, t)=2+(x − t). Compare the two solutions at various points. 4. Show that the finite difference approximation to the equation aut + bux = f (x, t) is ui,j+1 − 1 2 (ui+1,j + ui−1,j ) +  εb 2a  (ui+1,j − ui−1,j ) − fi,j = 0, where a, b are constants and ε = k/h. 670 14 Numerical and Approximation Methods 5. Obtain a finite difference solution of the heat conduction problem ut = κ uxx, 0 < x < l, t > 0, with the boundary conditions u (0, t) = u (l, t)=0, t> 0, and the initial condition u (x, 0) = 4x l (l − x), 0 ≤ x ≤ l. 6. (a) Find an explicit finite difference solution of the parabolic system ut = uxx, 0 <x< 1,="" t=""> 0, u (0, t) = u (1, t)=0, t > 0, u (0, t) = sin xπ on 0 ≤ x ≤ 1. Compare the numerical results with the analytical solution u (x, t) = e −π 2 t sin πx, at t = 0.5 and t = 0.05. (b) Prove that the Richardson finite difference scheme for problem 6(a) is ui,j+1 = ui,j−1 + 2 ε δ2 x ui,j . Hence, show that the exact solution of this equation is ui,j = 4 A1α j 1 + A2α j 2 5 sin πhi, where α1 and α2 are the roots of the quadratic equation x 2 + 8εx sin2 (πh/2) − 1=0. 7. Using four internal grid points, find the explicit finite difference solution of the Dirichlet problem ∇2u ≡ uxx + uyy = 0, 0 <x< 1,="" 0="" <y<="" u="" (x,="" 0)="x" (1="" −="" x),="" 1)="0" on="" ≤="" x="" (0,="" y)="u" (1,="" y="" 1.="" compare="" the="" numerical="" solution="" with="" exact="" analytical="" π="" 3="" ∞="" n="0" 2="" (2n="" +="" 1)3="" sin="" nπx="" sinh="" nπ="" at="" point="" 1="" ,="" .="" 14.11="" exercises="" 671="" 8.="" solve="" dirichlet="" problem="" by="" explicit="" finite="" difference="" method="" uxx="" uyy="0," <x<="" πx,="" for="" and="" 9.="" using="" a="" square="" grid="" system="" h="1" find="" of="" laplace="" equation="" quarter-disk="" given="" x2="" <=""> 0, u (x, 0) = 0, −1 <x< 1,="" u="" (x,="" y)="102" ,="" x2="" +="" y="" 2="1,"> 0. 10. Find a finite difference solution of the wave problem utt − uxx = 0, 0 <x< 1,="" t=""> 0, u (0, t) = u (1, t)=0, t ≥ 0, u (x, 0) = 1 2 x (1 − x), ut (x, 0) = 0, 0 ≤ x ≤ 1. Compare the numerical results with the exact analytical solution u (x, t) = 2 π 3 ∞ r=1 1 r 3 {1 − (−1)r } cos πrtsin πrx, at various points. 11. Obtain a finite difference solution of the problem utt = c 2uxx, 0 <x< 1,="" t=""> 0, u (0, t) = sin πct, u (1, t)=0, t ≥ 0, u (x, 0) = ut (x, 0) = 0, 0 ≤ x ≤ 1. 12. Show that the transformation v = log u transforms the nonlinear system vt = vxx + v 2 x , 0 <x< 1,="" t=""> 0, vx (0, t)=1, v (1, t)=0, t ≥ 0, v (x, 0) = 0, 0 ≤ x ≤ 1 into the linear system ut = uxx, 0 <x< 1,="" t=""> 0, ux (0, t) = u (0, t), u (x, 1) = 1, t ≥ 0, u (x, 0) = 1, 0 ≤ x ≤ 1. Solve the linear system by the explicit finite difference method with the derivative boundary condition approximated by the central difference formula. 672 14 Numerical and Approximation Methods 13. Solve the following parabolic system by the Crank–Nicolson method ut = uxx, 0 <x< 1,="" t=""> 0, u (0, t) = u (1, t)=0, t ≥ 0, with the initial condition (a) u (x, 0) = 1, 0 ≤ x ≤ 1, (b) u (x, 0) = sin πx, 0 ≤ x ≤ 1. (c) u (x, 0) = sin πx, 0 ≤ x ≤ 1 with 0 ≤ t ≤ 0.2 and in formula (14.5.2) κ = 1, k = h 2 . 14. Use the Crank–Nicolson implicit method with the central difference formula for the boundary conditions to find a numerical solution of the differential system ut = uxx, 0 <x< 1,="" t=""> 0, ux (0, t) = ux (1, t) = −u, t ≥ 0, u (x, 0) = 1, 0 ≤ x ≤ 1. 15. Find a numerical solution of the wave equation utt = c 2uxx, 0 < x < l, t > 0, with the boundary and initial conditions u = 1 20 ux at x = 0 and x = l, t > 0, u (x, 0) = 0, ut (x, 0) = a sin 4πx l 5 0 ≤ x ≤ l. 16. Determine the function representing a curve which makes the following functional extremum: (a) I (y (x)) =  1 0  y ′2 + 12xy dx, y (0) = 0, y (1) = 1, (b) I (y (x)) =  π/2 0  y ′2 − y 2 dx, y (0) = 0, y 4π 2 5 = 1, (c) I (y (x)) =  x1 x0 1 x  1 + y ′2 1 2 dx. 17. In the problem of tautochroneous motion, find the equation of the curve joining the origin O and a point A in the vertical (x, y)-plane so that a particle sliding freely from A to O under the action of gravity reaches the origin O in the shortest time, friction and resistance of the medium being neglected. 14.11 Exercises 673 18. In the problem of minimum surface of revolution, determine a curve with given boundary points (x0, y0) and (x1, y1) such that rotation of the curve about the x-axis generates a surface of revolution of minimum area. 19. Show that the Euler equation of the variational principle δI [u (x, y)] = δ  D F (x, y, u, p, q, l, m, n) dx dy = 0 is Fu − ∂ ∂xFp − ∂ ∂yFq + ∂ 2 ∂x2 Fl + ∂ 2 ∂x∂yFm + ∂ 2 ∂y2 Fn = 0, where p = ux, q = uy, l = uxx, m = uxy, n = uyy. 20. Prove that the Euler–Lagrange equation for the functional I =  R F (x, y, z, u, p, q, r, l, m, n, a, b, c) dx dy dz is Fu − ∂ ∂xFp − ∂ ∂yFq + ∂ 2 ∂z Fr + ∂ 2 ∂x2 Fl + ∂ 2 ∂y2 Fm + ∂ 2 ∂z2 Fn + ∂ 2 ∂x∂yFa + ∂ 2 ∂y∂z Fb + ∂ 2 ∂z∂xFc = 0, where (p, q, r)=(ux, uy, uz), (l, m, n)=(uxx, uyy, uzz), and (a, b, c) = (uxy, uyz, uzx). 21. In each of the following cases apply the variational principle or its simple extension with appropriate boundary conditions to derive the corresponding equations: (a) F = u 2 x + u 2 y + 2u 2 xy. (b) F = 1 2 u 2 t − α  u 2 x + u 2 y − β 2u 2 ! , (c) F = 1 2  utux + αu2 x − βu2 xx , (d) F = 1 2  u 2 t − α 2u 2 xx , (e) F = p (x) u ′2 + d dx  q (x) u 2 − [r (x) + λs (x)] u 2 , where p, q, r, and s are given functions of x, and α, β are constants. 674 14 Numerical and Approximation Methods 22. Derive the Schr¨odinger equation from the variational principle δ  R  2 2m  ψ 2 x + ψ 2 y + ψ 2 z + (V − E) ψ 2 dx dy dz = 0, where h = 2π is the Planck constant, m is the mass of a particle moving under the action of a force field described by the potential V (x, y, z) and E is the total energy of the particle. 23. Derive the Poisson equation ∇2u = F (x, y) from the variational principle with the functional I (u) =  D u 2 x + u 2 y + 2uF (x, y) ! dx dy, where u = u (x, y) is given on the boundary ∂D of D. 24. Derive the equation of motion of a vibrating string of length l under the action of an external force F (x, t) from the variational principle δ  t2 t1  l 0 1 2 ρ u2 t − T ∗u 2 x  + ρ uF (x, t) dx dt = 0, where ρ is the line density and T ∗ is the constant tension of the string. 25. The kinetic and potential energies associated with the transverse vibration of a thin elastic plate of constant thickness h are T = 1 2 ρ  D u˙ 2 dx dy, V = 1 2 µ0  D " (∇u) 2 − 2 (1 − σ)  uxxuyy − u 2 xy # dx dy, where ρ is the surface density and µ0 = 2h 3E/3  1 − σ 2 . Use the variational principle δ  t2 t1  D [(T − V ) + fu] dx dy dt = 0 to derive the equation of motion of the plate ρu¨ + µ0∇4u = f (x, y, t), where f is the transverse force per unit area acting on the plate. 26. The kinetic and potential energies associated with the wave motion in elastic solids are 14.11 Exercises 675 T = 1 2  D ρ  u 2 t + v 2 t + w 2 t dx dy dz V = 1 2  D " λ (ux + vy + wz) 2 + 2µ  u 2 x + v 2 y + w 2 z +µ ( (vx + uy) 2 + (wy + vz) 2 + (uz + wx) 2 )# dx dy dz. Use the variational principle δ  t2 t1  D (T − V ) dx dy dz = 0 to derive the equation of wave motion in an elastic medium (λ + µ) grad divu + µ ∇2u = ρ utt, where u = (u, v, w) is the displacement vector. 27. From the variational principle δ  D L dx dt = 0 with L = −ρ  η −h  φt + 1 2 (∇φ) 2 + gz0 dz derive the basic equations of water waves ∇2φ = 0, −h (x, y) <z<η (x,="" y,="" t),="" t=""> 0, ηt + ∇φ · ∇η − φz = 0, on z = η, φz + 1 2 (∇φ) 2 + gz = 0, on z = η, φz = 0, on z = −h, where φ (x, y, z, t) is the velocity potential, and η (x, y, t) is the free surface displacement function in a fluid of depth h. 28. Derive the Boussinesq equation for water waves utt − c 2uxx − µ uxxtt = 1 2  u 2 xx from the variational principle δ  L dx dt = 0, where L ≡ 1 2 φ 2 t − 1 2 c 2 φ 2 x + 1 2 µ φ2 xt − 1 6 φ 3 x and φ is the potential for u (u = φx). 29. Determine an approximate solution of the problem of finding an extremum of the functional I (y (x)) =  1 0  y ′2 − y 2 − 2xy dx, y (0) = y (1) = 0. 676 14 Numerical and Approximation Methods 30. Find an approximate solution of the torsion problem of a cylinder with an elliptic base; the domain of integration D is the interior of the ellipse with the major and minor axes 2a and 2b respectively. The associated functional is I (u (x, y)) =  D 1 ∂u ∂x − y 2 u +  ∂u ∂y + x 2 u 3 dx dy. 31. Use the Rayleigh–Ritz method to find an approximate solution of the problem ∇2u = 0, 0 <x< 1,="" 0="" <y<="" u="" (0,="" y)="0=" (1,="" y),="" (x,="" 0)="x" (1="" −="" x).="" 32.="" find="" an="" approximate="" solution="" of="" the="" boundary-value="" problem="" ∇2u="0," x=""> 0, y> 0, x + 2y < 2, u (0, y)=0, u (x, 0) = x (2 − x), u (2 − 2y, y)=0. 33. In the torsion problem in elasticity, the Prandtl stress function Ψ (x, y) = ψ (x, y) − 1 2  x 2 + y 2 satisfies the boundary value problem ∇2Ψ = −2 in D Ψ = 0 on ∂D. Use the Galerkin method to find an approximate solution of the problem in a rectangular domain D = {(x, y) : −a ≤ x ≤ a, −b ≤ y ≤ b}. 34. Apply the Galerkin approximation method to find the first eigenvalue of the problem of a circular membrane of radius a governed by the equation ∇2u ≡ d 2u dr2 + 1 r du dr = λu in 0 <r 0, u (0) = u (l)=0, with given initial conditions. (a) Show that an appropriate requirement is that n i=1 A ′′ i (t)  l 0 vi (x) vj (x) dx + n i=1 Ai (t)  l 0 ∂vi ∂x · ∂vj ∂x dx = 0, where j = 1, 2, ..., n and that the approximate solution is given by un (x) = A1 (t) v1 (x) + ... + An (t) vn (t) = n i=1 Ai (t) vi (x). (b) Show that the finite element method leads to a system of ordinary differential equations B d 2A dt2 + C A (t)=0, A (0) = D where B and C are n × n matrices, A (t) is a n-vector function and D is an n-vector. 15 Tables of Integral Transforms In this chapter we provide a set of short tables of integral transforms of the functions that are either cited in the text or are in most common use in mathematical, physical, and engineering applications. For exhaustive lists of integral transforms, the reader is referred to Erd´elyi et al. (1954), Campbell and Foster (1948), Ditkin and Prudnikov (1965), Doetsch (1970), Marichev (1983), Debnath (1995), and Oberhettinger (1972). 15.1 Fourier Transforms f (x) F (k) = √ 1 2π  ∞ −∞ exp (−ikx) f (x) dx 1 exp (−a |x|), a > 0 4% 2 π 5 a  a 2 + k 2 −1 2 x exp (−a |x|), a > 0 4% 2 π 5 (−2aik)  a 2 + k 2 −2 3 exp  −ax2 , a > 0 √ 1 2a exp 4 − k 2 4a 5 4  x 2 + a 2 −1 , a > 0 π 2 exp(−a|k|) a 5 x  x 2 + a 2 −1 π 2  ik 2a exp (−a |k|) 682 15 Tables of Integral Transforms f (x) F (k) = √ 1 2π  ∞ −∞ exp (−ikx) f (x) dx 6  c, a ≤ x ≤ b 0, outside. √ ic 2π 1 k  e −ibk − e −iak 7 |x| exp (−a |x|), a > 0 % 2 π  a 2 − k 2 a 2 + k 2 −2 8 sin ax x π 2 H (a − |k|) 9 exp {−x (a − iω)} H (x) √ 1 2π i (ω−k+ia) 10  a 2 − x 2 − 1 2 H (a − |x|) π 2 J0 (ak) 11 sin b(x 2+a 2 ) 1 2 (x2+a2) 1 2 π 2 J0  a √ b 2 − k 2 H (b − |k|) 12 cos(b √ a2−x2) (a2−x2) 1 2 H (a − |x|) π 2 J0  a √ b 2 + k 2 13 e −ax H (x), a > 0 √ 1 2π (a − ik)  a 2 + k 2 −1 14 √ 1 |x| exp (−a |x|), a > 0  a 2 + k 2 − 1 2 " a +  a 2 + k 2 1 2 # 1 2 15 δ (n) (x − a), n = 0, 1, 2,... √ 1 2π (ik) n exp (−iak) 16 exp (iax) √ 2π δ (k − a) 15.2 Fourier Sine Transforms 683 15.2 Fourier Sine Transforms f (x) Fs (k) = % 2 π  ∞ 0 sin (kx) f (x) dx 1 exp (−ax), a> 0 % 2 π k  a 2 + k 2 −1 2 x exp (−ax), a> 0 % 2 π (2ak)  a 2 + k 2 −2 3 x α−1 , 0 <α< 1 % 2 π k −αΓ (α) sin  πα 2 4 √ 1 x √ 1 k , k> 0 5 x α−1 e −ax , α > −1, a > 0 % 2 π Γ (α) r −α sin (αθ), where r =  a 2 + k 2 1 2 , θ = tan−1  k a 6 x −1 e −ax, a> 0 % 2 π tan−1  k a , k> 0 7 x exp  −a 2x 2 2 −3/2  k a3 exp 4 − k 2 4a2 5 8 erfc (ax) % 2 π 1 k " 1 − exp 4 − k 2 4a2 5# 9 x  a 2 + x 2 −1 π 2 exp (−ak), a> 0 10 x  a 2 + x 2 −2 √ 1 2π  k a exp (−ak), (a > 0) 684 15 Tables of Integral Transforms f (x) Fs (k) = % 2 π  ∞ 0 sin (kx) f (x) dx 11 H (a − x), a> 0 % 2 π 1 k (1 − cos ak) 12 x −1J0 (ax) ⎧ ⎪⎨ ⎪⎩ % 2 π sin−1  k a , 0 <k 0, b> 0 π 2 e −akI0 (ab), a<k< ∞="" 14="" j0="" (a="" √="" x),="" a=""> 0 % 2 π 1 k cos 4 a 2 4k 5 15  x 2 − a 2 ν− 1 2 H (x − a), |ν| < 1 2 2 ν− 1 2  a k ν Γ  ν + 1 2 J−ν (ak) 16 x 1−ν  x 2 + a 2 −1 Jν (ax), ν > − 3 2 , a, b > 0 π 2 a −ν exp (−ak) Iν (ab), a<k< ∞="" 17="" x="" −νjν+1="" (ax),="" ν=""> − 1 2 k(a 2−k 2 ) ν− 1 2 2 ν− 1 2 aν+1Γ(ν+ 1 2 ) H (a − k) 18 erfc (ax) % 2 π 1 k " 1 − exp 4 − k 2 4a2 5# 19 x −α, 0 < Re α < 2 % 2 π Γ (1 − α) k α−1 cos  απ 2 20  ax − x 2 α− 1 2 H (a − x), α > − 1 2 √ 2 Γ  α + 1 2  a k α sin  ak 2 Jα  ak 2 15.3 Fourier Cosine Transforms 685 15.3 Fourier Cosine Transforms f (x) Fc (k) = % 2 π  ∞ 0 cos (kx) f (x) dx 1 exp (−ax), a> 0 4% 2 π 5 a  a 2 + k 2 −1 2 x exp (−ax), a > 0 4% 2 π 5  a 2 − k 2 a 2 + k 2 −2 3 exp  −a 2x 2 1 a √ 2 exp 4 − k 2 4a2 5 4 H (a − x) % 2 π  sin ak k 5 x a−1 , 0 <a< 1="" %="" 2="" π="" Γ="" (a)="" k="" −a="" cos="" ="" aπ="" 6="" ax2="" ,="" a=""> 0 1 2 √ a " cos 4 k 2 4a 5 + sin 4 k 2 4a 5# 7 sin  ax2 , a> 0 1 2 √ a " cos 4 k 2 4a 5 − sin 4 k 2 4a 5# 8  a 2 − x 2 ν− 1 2 H (a − x),ν> − 1 2 2 ν− 1 2 Γ  ν + 1 2  a k ν Jν (ak) 9  a 2 + x 2 −1 J0 (bx), a, b > 0 π 2 a −1 exp (−ak) I0 (ab), b<k< ∞="" 10="" x="" −νjν="" (ax),ν=""> − 1 2 (a 2−k 2 ) ν− 1 2 H(a−k) 2 ν− 1 2 aν Γ(ν+ 1 2 ) 686 15 Tables of Integral Transforms f (x) Fc (k) = % 2 π  ∞ 0 cos (kx) f (x) dx 11  x 2 + a 2 − 1 2 exp " −b  x 2 + a 2 1 2 # K0 " a  k 2 + b 2 1 2 # , a> 0, b> 0 12 x ν−1 e −ax, ν> 0, a> 0 % 2 π Γ (ν) r −ν cos nθ, where r =  a 2 + k 2 1 2 , θ = tan−1  k a 13 2 x e −x sin x % 2 π tan−1  2 k2 14 sin " a  b 2 − x 2 1 2 H (b − x) # π 2 (ab)  a 2 + k 2 − 1 2 × J1 " b  a 2 + k 2 1 2 # 15 (1−x 2 ) (1+x2) 2 π 2 k exp (−k) 16 x −α, 0 <α< 1 π 2 k α−1 Γ(α) sec  πα 2 17  1 a + x e −ax, a> 0 π 2 2a 2 (a2+k2) 2 18 log 4 1 + a 2 x2 5 , a> 0 √ 2π (1−e −ak) k 19 log 4 a 2+x 2 b 2+x2 5 , a, b > 0 √ 2π (e −bk−e −ak) k 20 a  x 2 + a 2 −1 , a> 0 π 2 exp (−ak), k> 0 15.4 Laplace Transforms 687 15.4 Laplace Transforms f (t) f (s) =  ∞ 0 exp (−st) f (t) dt 1 f (n) (t) s nf (s) − n−1 r=0 s n−r−1f (r) (0) 2  t 0 f (t − τ ) g (τ ) dτ f (s) g (s) 3 t nf (t) (−1)n d n dsn f (s) 4 f (t − a) H (t − a) exp (−as) f (s) 5 t n (n = 0, 1, 2, 3,...) n! sn+1 6 e at 1 s−a 7 t ne −at Γ(n+1) (s+a) n+1 8 t a (a > −1) Γ(a+1) s a+1 9 e at cos bt s−a (s−a) 2+b 2 10 e at sin bt b (s−a) 2+b 2 11 √ 1 t π s 12 2 √ t 1 s π s 688 15 Tables of Integral Transforms f (t) f (s) =  ∞ 0 exp (−st) f (t) dt 13 t −1/2 exp  − a t π s exp (−2 √ as) 14 t −3/2 exp  − a t π a exp (−2 √ as) 15 √ 1 πt (1 + 2at) e at s (s−a) √ s−a 16 1 2 √ πt3  e bt − e at √ s − a − √ s − b 17 exp  a 2 t erf  a √ t √ a s(s−a2) 18 exp  a 2 t erfc  a √ t 1 √ s( √ s+a) 19 √ 1 πt + a exp  a 2 t erf  a √ t √ s (s−a2) 20 √ 1 πt − a exp  a 2 t erfc  a √ t √ 1 s+a 21 exp(−at) √ b−a erf 4 (b − a)t 5 1 (s+a) √ s+b 22 1 2 e iωt " exp (−λz) erfc 4 ζ − √ iωt5 + exp (λz) erfc  ζ + √ iωt ! , where ζ = z/2 √ νt, λ = % iω ν (s − iω) −1 exp  −z s ν 23 1 2 " exp (−ab) erfc 4 b−2at 2 √ t 5 + exp (ab) erfc 4 b+2at 2 √ t 5# exp " −b  s + a 2 1 2 # 15.4 Laplace Transforms 689 f (t) f (s) =  ∞ 0 exp (−st) f (t) dt 24 J0 (at)  s 2 + a 2 − 1 2 25 I0 (at)  s 2 − a 2 − 1 2 26 t α−1 exp (−at), α > 0 Γ (α) (s + a) −α 27 t −1Jν (at) ν −1a ν √ s 2 + a 2 + s −ν , Re ν > − 1 2 28 J0  a √ t 1 s exp 4 − a 2 4s 5 29  2 a ν t ν/2Jν  a √ t s −(ν+1) exp 4 − a 2 4s 5 , Re ν > − 1 2 30 a 2t √ πt exp 4 − a 2 4t 5 exp (−a √ s), a> 0 31 √ 1 πt exp 4 − a 2 4t 5 √ 1 s exp (−a √ s), a ≥ 0 32 exp 4 − a 2 t 2 4 5 √ π a exp 4 s 2 a2 5 erfc  s a , a ≥ 0 33  t 2 − a 2 − 1 2 H (t − a) K0 (as), a> 0 34 δ (n) (t − a), n = 0, 1,... s n exp (−as) 690 15 Tables of Integral Transforms f (t) f (s) =  ∞ 0 exp (−st) f (t) dt 35 t mα+β−1E (m) α,β (+ at), m = 0, 1, 2,... m! s α−β (sα+ a) m+1 36 √ π Γ(ν+ 1 2 )  t 2a ν Jν (at)  s 2 + a 2 −(ν+ 1 2 ) , Re ν > − 1 2 37 1 2 e −ct" exp  −a √ b − c ×erfc( √a 4t −  (b − c)t ) − exp  a √ b − c ×erfc( √a 4t +  (b − c)t )# exp(−a √ s+b) (s+c) √ (s+b) 38 1 2 e −ct" exp  −a √ b − c ×erfc( √a 4t − t √ b − c ) − exp  a √ b − c ×erfc( √a 4t + t √ b − c )# exp(−a √ s+b) (s+c) 39 e −bt "% 4t π exp 4 − a 2 4t 5 −a erfc 4 √a 4t 5# exp(−a √ s+b) (s+b) 3/2 40 e −bt " t + 1 2 a 2 erfc 4 √a 4t 5 − % ta2 π exp 4 − a 2 4t 5 exp(−a √ s+b) (s+b) 2 15.5 Hankel Transforms 691 15.5 Hankel Transforms f (r) order n ˜fn (k) =  ∞ 0 r Jn (kr) f (r) dr 1 H (a − r) 0 a k J1 (ak) 2 exp (−ar) 0 a  a 2 + k 2 − 3 2 3 1 r exp (−ar) 0  a 2 + k 2 − 1 2 4  a 2 − r 2 H (a − r) 0 4a k3 J1 (ak) − 2a 2 k2 J0 (ak) 5 a  a 2 + r 2 − 3 2 0 exp (−ak) 6 1 r cos (ar) 0  k 2 − a 2 − 1 2 H (k − a) 7 1 r sin (ar) 0  a 2 − k 2 − 1 2 H (a − k) 8 1 r 2 (1 − cos ar) 0 cosh−1  a k H (a − k) 9 1 r J1 (ar) 0 1 a H (a − k), a> 0 10 Y0 (ar) 0  2 π a 2 − k 2 −1 11 K0 (ar) 0  a 2 + k 2 −1 692 15 Tables of Integral Transforms f (r) order n ˜fn (k) =  ∞ 0 r Jn (kr) f (r) dr 12 δ(r) r 0 1 13  r 2 + b 2 − 1 2 × exp ( −a  r 2 + b 2 1 2 ) 0  k 2 + a 2 − 1 2 exp ( −b  k 2 + a 2 1 2 ) 14  r 2 + a 2 − 1 2 0 1 k exp (−ak) 15 exp (−ar) 1 k  a 2 + k 2 −3/2 16 sin ar r 1 a H(k−a) k(k2−a2) 1 2 17 1 r exp (−ar) 1 1 k 1 − a (k2+a2) 1 2 18 1 r 2 exp (−ar) 1 1 k " k 2 + a 2 1 2 − a # 19 r n H (a − r) > −1 1 k a n+1 Jn+1 (ak) 20 r n exp (−ar), (Re a > 0) > −1 √ 1 π 2 n+1 Γ(n+ 3 2 ) a kn (a2+k2) n+ 3 2 21 r n exp  −ar2 > −1 k n (2a) n+1 exp 4 − k 2 4a 5 15.5 Hankel Transforms 693 f (r) order n ˜fn (k) =  ∞ 0 r Jn (kr) f (r) dr 22 r a−1 > −1 2a Γ[ 1 2 (a+n+1)] ka+1 Γ[ 1 2 (1−a+n)] 23 r n  a 2 − r 2 m−n−1 × H (a − r) > −1 2 m−n−1 Γ (m − n) a mk n−mJm (ak) 24 r m exp  −r 2/a2 > −1 k n am+n+2 2n+1 Γ(n+1) Γ  1 + m 2 + n 2 × 1F1  1 + m 2 + n 2 ; n + 1; − 1 4 a 2k 2 25 1 r Jn+1 (ar) > −1 k na −(n+1)H (a − k), a > 0 26 r n  a 2 − r 2 m H (a − r), m > −1 > −1 2 ma nΓ (m + 1)  a k m+1 × Jn+m+1 (ak) 27 1 r 2 Jn (ar) > 1 2 ⎧ ⎨ ⎩ 1 2n  k a n , 0 < k ≤ a 1 2n  a k n , a<k< ∞="" 28="" r="" n="" (a2+r="" 2)m+1="" ,="" a=""> 0 > −1  k 2 m a n−m Γ(m+1) Kn−m (ak) 29 exp  −p 2 r 2 Jn (ar) > −1  2p 2 −1 exp 4 − a 2+k 2 4p2 5 In 4 ak 2p2 5 30 1 r exp (−ar) > −1  (k 2+a 2 ) 1 2 −a n kn(k2+a2) 1 2 31 r n (r 2+a2) n+1 > −1  k 2 n K0(ak) Γ(n+1) 694 15 Tables of Integral Transforms f (r) order n ˜fn (k) =  ∞ 0 r Jn (kr) f (r) dr 32 r n (a2−r 2) n+ 1 2 H (a − r) < 1 √ 1 π  k 2 n Γ  1 2 − n  sin ak k 33 f (ar) n 1 a2 ˜fn  k a 34 r −1 exp  −ar2 1 1 k " 1 − exp 4 − k 2 4a 5# 35 r −1 sin  ar2 , a > 0 1 1 k sin 4 k 2 4a 5 36 r −1 cos  ar2 , a > 0 1 1 − cos 4 k 2 4a 5 37 exp (−ar), a > 0 > −1 (a+n √ k2+a2) (k2+a2) 3/2 4 k a+ √ a2+k2 5n 38 exp  −ar2 J0 (br) 0 a 2 exp 4 − k 2−b 2 4a 5 I0  bk 2a 39 H(a−r) √ a2−r 2 0 aπ 2k J 1 2 (ak), a > 0 40 r nH(a−r) √ a2−r 2 > −1  π 2k a n+ 1 2 Jn+1 (ak), a > 0 41 r −2 sin r 0 sin−1  1 k , (k > 1) 15.6 Finite Hankel Transforms 695 15.6 Finite Hankel Transforms f (r) order n ˜fn (ki) =  a 0 r Jn (r ki) f (r) dr 1 c, where c is a constant 0 4 ac ki 5 J1 (aki) 2  a 2 − r 2 0 4a k 3 i J1 (aki) 3  a 2 − r 2 − 1 2 0 k −1 i sin (aki) 4 J0(αr) J0(αa) 0 − aki (α2−k 2 i ) J1 (aki) 5 1 r 1 k −1 i {1 − J0 (aki)} 6 r −1  a 2 − r 2 − 1 2 1 (1−cos aki) (aki) 7 r n > −1 a n+1 ki Jn+1 (aki) 8 Jν (αr) Jν (αa) > −1 aki (α2−k 2 i ) J ′ ν (aki) 9 r −n  a 2 − r 2 − 1 2 > −1 π 2 & J n 2  aki 2 '2 10 r n  a 2 − r 2 −(n+ 1 2 ) < 1 2 Γ( 1 2 −n) √ π 2n k n−1 i sin (aki) 11 r n−1  a 2 − r 2 n− 1 2 > − 1 2 √ π 2 Γ  n + 1 2 4 2 ki 5n a 2nJ 2 n  aki 2 Answers and Hints to Selected Exercises 1.6 Exercises 1. (a) Linear, nonhomogeneous, second-order; (b) quasi-linear, first-order; (c) nonlinear, first-order; (d) linear, homogeneous, fourth-order; (e) linear, nonhomogeneous, second-order; (f) quasi-linear, third-order; (g) nonlinear, second-order; and (h) nonlinear, homogeneous. 5. u (x, y) = f (x) cos y + g (x) sin y. 6. u (x, y) = f (x) e −y + g (y). 7. u (x, y) = f (x + y) + g (3x + y). 8. u (x, y) = f (y + x) + g (y − x). 11. ux = vy ⇒ uxx = vxy, vx = −uy ⇒ vyx = −uyy. Thus, uxx + uyy = 0. Similarly, vxx + vyy = 0. 12. Since u (x, y) is a homogeneous function of degree n, u = x nf  y x . ux = n xn−1f  y x − x n−2y f′  y x , and uy = x n−1f ′  y x . Thus, x ux + y uy = n xnf  y x = n u. 23. ux = − 1 b exp  − x b f (ax − by) + exp  − x b d d(ax−by) f (ax − by) · d(ax−by) dx = − 1 b exp  − x b f + a exp  − x b f ′ (ax − by) uy = (−b) exp  − x b f ′ (ax − by). Thus, b ux + a uy + u = 0. 698 Answers and Hints to Selected Exercises 24. V ′′ (t)+2b V ′ (t) + k 2 c 2V (t)=0. 25. Differentiating with respect to r and t partially gives V ′′ (r) + n 2V (r) = 0. 2.8 Exercises 2. (a) xp − yq = x − y, (d) yp − xq = y 2 − x 2 . 3. (a) u = f (y), (b) u = f (bx − ay), (c) u = f (y e−x ), (d) u = f  y − tan−1 x , (e) u = f 4 x 2−y 2 x 5 , (f) Hint: dx y+u = dy y = du x−y = d(x+u) x+u = d(u+y) x , x dx = (u + y) d (u + y) ⇒ (u + y) 2 − x 2 = c1. d(u+x) u+x = dy y ⇒ u+x y = c2, f 4 u+x y ,(u + y) 2 − x 2 5 = 0. (g) dx y2 = dy −xy = du xu−2xy = d(u−y) x(u−y) . From the second and the fourth, (u − y) y = c1 and x 2 + y 2 = c2. Hence, (u − y) y = f  x 2 + y 2 . Thus, u = y + y −1f  x 2 + y 2 . (h) u + log x = f (xy), (i) f  x 2 + u 2 , y3 + u 3 = 0. 4. u (x, y) = f  x 2 + y −1 . Verify by differentiation that u satisfies the original equation. 5. (a) u = sin  x − 3 2 y , (b) u = exp  x 2 − y 2 , (c) u = xy + f  y x , u = xy + 2 −  y x 3 , (d) u = sin  y − 1 2 x 2 , (e) u = ⎧ ⎪⎨ ⎪⎩ 1 2 y 2 + exp −  x 2 − y 2 ! for x > y, 1 2 x 2 + exp −  y 2 − x 2 ! for x < y. (f) Hint: y = 1 2 x 2 + C1, u = C 2 1 x + C2, u = x  y − 1 2 x 2 2 +f  y − 1 2 x 2 , u = x  y − 1 2 x 2 2 + exp  y − 1 2 x 2 . (g) y x = C1 and u+1 y = C2, C2 = 1+ 1 C2 1 . Thus, u = y + x 2 y − 1, y = 0. (h) Hint: x + y = C1, dy −u = du u2+C2 1 , u 2 + C 2 1 = C2 exp (−2y). 2.8 Exercises 699 From the Cauchy data, it follows that 1 + C 2 1 = C2, and hence, u = "(1+(x + y) 2 ) e −2y − (x + y) 2 # 1 2 . (i) dy dx − y x = 1, d dx  y x = 1 x which implies that x = C1 exp  y x . u+1 x = C2. Hence, f  u+1 x , x exp  − y x = 0. Initial data imply x = C1 and x 2+1 x = C2. Hence C2 = C1 + 1 C1 . u+1 x = x exp  − y x + 1 x exp  y x . Thus, u = x 2 exp  − y x + exp  y x − 1. (j) √ dx x = dy u = du −u2 . The second and the third give y = − log (Au) and hence, A = 1 and u = exp (−y). The first and the third yield u −1 = 2√ x − B. At (x0, 0), x0 > 0, B = 2√ x0 − 1. Hence, u −1 = 2 √ x − √ x0 +1= 1 y . The solution along the characteristic is u = exp (−y) or u −1 = 2 √ x − √ x0 + 1. (k) dx ux2 = dy exp(−y) = du −u2 . The first and the third give x −1 = log u + A and hence, A = 1 x0 , x0 > 0. The second and third yield u = exp (−y). Or, eliminating u gives y =  x −1 0 − x −1 . 6. u 2 − 2ut + 2x = 0, and hence, u = t + √ t 2 − 2x. 7. u (x, y) = exp 4 x x2−y2 5 . 8. (a) u = f  y x , z x (b) Hint: u1 = x−y xy = C1, d(x−y) x2−y2 = dz z(x+y) gives u = x−z z = C2. Hence, u = f 4 x−y xy , x−y z 5 . (c) φ = (x + y + z) = C1. Hint: ( dx x ) y−z = ( dy y ) z−x = ( dz z ) x−y = dx x + dy y + dz z 0 = d log(xyz) 0 , ψ = xyz = C2, and hence, u = f (x + y + z, xyz) is the general solution. (d) Hint: x dx + y dy = 0, x 2 + y 2 = C1 z dz = −  x 2 + y 2 y dy = −C1 y dy, z 2 +  x 2 + y 2 y 2 = C2, u = f  x 2 + y 2 , z2 +  x 2 + y 2 y 2 . (e) x −1dx y2−z 2 = y −1dy z 2−x2 = z −1dz y2−x2 = d(log xyz) 0 . u = f  x 2 + y 2 + z 2 , xyz . 700 Answers and Hints to Selected Exercises 9. (a) Hint: y − x 2 2 = C1, u = xy − x 3 3 +C2, φ 4 u − xy + x 3 3 , y − x 2 2 5 = 0. u = xy − x 3 3 + f 4 y − x 2 2 5 , u = xy − x 3 3 + 4 y − x 2 2 52 . (b) u = xy − 1 3 x 3 + y − x 2 2 + 5 6 . 11. x+u y = C1, u 2 − (x − y) 2 = C2, u 2 − 2u y − (x − y) 2 − 2 y (x − y) = 0. u = 2 y + (x − y), y > 0. 12. (a) x = τ 2 2 + τs + s, y = τ + 2s, u = τ + s = (2x−2y+y 2 ) 2(y−1) (b) x = τ 2 2 + τs + s 2 , y = τ + 2s, u = τ + s, (y − s) 2 = 2x − s 2 , which is a set of parabolas. (c) x = 1 2 (τ + s) 2 , y = u = τ + s. 13. Hint: The initial curve is a characteristic, and hence, no solution exists. 14. (a) u = exp 4 xy x+y 5 , (b) u = sin 4 x 2−y 2+1 2 5 1 2 , (c) u = 2  xy 3 1 2 + 1 2 log  y 3x , (e) u = 1 2 x 2 − 1 4 y 2 + 1 2 x 2y + 1 4 . (f) Hint: dx 1 = dy 2 = du 1+u , y − 2x = c1 and (1 + u) e −x = c2, (1 + u) e −x = f (y − 2x), 1+u = exp (3x − y + 1) [1 + sin (y − 2x − 1)]. (g) Hint: dx 1 = dy 2 = du u , y−2x = c1, and u e−x = c2, u e−x = f (y − 2x), u = exp  y−x 2 cos  y−3x 2 . (h) dx 1 = dy 2x = du 2x u , (y − x) 2 = c1, and u e−x 2 = c2, u e−x 2 = f  y − x 2 , u (x, y) =  x 2 − y e y . (i) dx u = dy 1 = du u , u − x = c1, and u e−y = c2, f (u e−y , u − x) = 0, u ey = g (u − x), u = 2x ey 2e y−1 , dx dy = u, x = A (2e y − 1) is the family of characteristics. (j) dx 1 = dy 1 = du u2 , y − x = c1, and 1 u + x = c2, 1 u + x = f (y − x), f (x) = −  1−tanh x tanh x , u (x, y) = tanh(x−y) 1−y tanh(x−y) . 15. 3uy = u 2 + x 2 + y 2 . Hint: x dx+y dy+u du 0 , x 2 + y 2 + u 2 = c1, dy y = − du u gives uy = c2. x 2 + y 2 + u 2 = f (uy), and hence, 3u 2 = f  u 2 . 2.8 Exercises 701 16. (a) x (s, τ ) = τ , y (s, τ ) = τ 2 2 + aτs + s, u (s, τ ) = τ + as. τ = x, s = (1 + ax) −1  y − 1 2 x 2 a, and hence, u (x, y) = x + as = (1 + ax) −1 & x + a  y + 1 2 x 2 ', singular at x = − 1 a . (b) y = u 2 2 + f (u − x), 2y = u 2 + (u − x) 2 , u (0, y) = √y. 17. (a) Hint: d(x+y+u) 2(x+y+u) = d(y−u) −(y−u) = d(u−x) −(u−x) (x + y + u) (y − u) 2 = c1 and (x + y + u) (u − x) 2 = c2. (b) Hint: dx x = dy −y . Hence, xy = a. dx xu(u2+a) = du x4 . So, dx du = u(u 2+a) x3 giving x 4 = u 4 + 2au2 + b and, thus, x 4 − u 4 − 2u 2xy = b. (c) dx x+y = dy x−y = dy 0 (exact equation). u = f  x 2 − 2xy − y 2 . (d) f  x 2 − y 2 , u − 1 2 y 2  x 2 − y 2 = 0. (e) f  x 2 + y 2 + z 2 , ax + by + cz = 0. 18. Hint: dx x = dy y = dz z , and hence, x z = c, y z = d. x 2 + y 2 = a 2 and z = tan−1  y x give  c 2 + d 2 z 2 = a 2 and z = b tan−1  d c . c =  a z cos θ, d =  a z sin θ, and z = b tan−1 (tan θ) = bθ. Thus, the curves are xbθ = az cos θ and ybθ = az sin θ. 19. F ( x + y + u,(x − 2y) 2 + 3u 2 ) = 0. Hint: (dx−2dy) 9u = du −3(x−2y) . (x − 2y) 2 + 3u 2 = (x + y + u) 2 . 20. F  x 2 + y, yu = 0,  x 2 + y 4 = yu. 21. Hint: x − y + z = c1, dz −(x+y+z) = (dx+dy+dz) 8z , and hence, 8z 2 + (x + y + z) 2 = c2. F ( (x − y + z), 8z 2 + (x + y + z) 2 ) = 0. c 2 1 + c2 = 2a 2 , or (x − y + z) 2 + (x + y + z) 2 + 8z 2 = 2a 2 . 22. F  x 2 + y 2 + z 2 , y2 − 2yz − z 2 = 0. (a) y 2 − 2yz − z 2 = 0, two planes y =  1+√ 2 z. (b) x 2 + 2yz + 2z 2 = 0, a quadric cone with vertex at the origin. (c) x 2 − 2yz + 2y 2 = 0, a quadric cone with vertex at the origin. 702 Answers and Hints to Selected Exercises 23. Use the Hint of 17(c). dx dt = x + y, dy dt = x − y, d 2x dt2 = 2x.  dx dt 2 = 2x 2 + c. When x =0= y, dx dt = √ 2 x. √ 2 u = ln x + x 2 − 2xy + 2y. 24. (a) a = f  x + 3 2 y . (b) x = at + c1, y = bt, u = c2 e ct , c2 = f (c1), u (x, y) = f  x − a b y exp  cy b . (c) u = f 4 x 1−y 5 (1 − y) c . (d) x = 1 2 t 2 + αst + s, y = t; u = y + 1 2 α (αy + 1)−1  2x − y 2 . 26. (a) Hint: (f ′ ) 2 = 1 − (g ′ ) 2 = λ 2 ; f ′ (x) = λ and g ′ (y) = √ 1 − λ2. f (x) = λx + c1 and g (y) = y √ 1 − λ2 + c2. Hence, u (x, y) = λx + y √ 1 − λ2 + c. (b) Hint: (f ′ ) 2 + (g ′ ) 2 = f (x)+g (y) or (f ′ ) 2 −f (x) = g (y)−(g ′ ) 2 = λ. Hence, (f ′ ) 2 = f (x) + λ and g ′ =  g (y) − λ. Or, √ df f+λ = dx and √ dg g−λ = dy. f (x) + λ =  x+c1 2 2 and g (y) − λ =  y+c2 2 2 . u (x, y) =  x+c1 2 2 +  y+c2 2 2 . (c) Hint: (f ′ ) 2 + x 2 = −g ′ (y) = λ 2 . Or f ′ (x) = √ λ2 − x 2, and g (y) = −λ 2y + c2. Putting x = λ sin θ, we obtain f (x) = 1 2 λ 2 sin−1  x λ + x 2 √ λ2 − x 2 + c1, u (x, y) = 1 2 λ 2 sin−1  x λ + x 2 √ λ2 − x 2 − λ 2y + (c1 + c2). (d) Hint: x 2 (f ′ ) 2 = λ 2 and 1 − y 2 (g ′ ) 2 = λ 2 . Or, f (x) = λ ln x + c1 and g (y) = √ 1 − λ2 ln y + c2. 27. (a) Hint: v = ln u gives vx = 1 u · ux, and vy = 1 u · uy. x 2  ux u 2 + y 2  uy u 2 = 1. Or, x 2v 2 x + y 2v 2 y = 1 gives x 2 (f ′ ) 2 + y 2 (g ′ ) 2 = 1. 2.8 Exercises 703 x 2 {f ′ (x)} 2 = 1 − y 2 (g ′ ) 2 = λ 2 . Or, f (x) = λ ln x + c1 and g (y) = √ 1 − λ2 (ln y) + c2. Thus, v (x, y) = λ ln x + √ 1 − λ2 (ln y) + ln c, (c1 + c2 = ln c). u (x, y) = c xλ y √ 1−λ2 . (b) Hint: v = u 2 and v (x, y) = f (x) + g (y) may not work. Try u = u (s), s = λxy, so that ux = u ′ (y)·(λy) and uy = u ′ (s)·(λx). Consequently, 2λ 2  1 u du ds 2 = 1. Or, 1 u du ds = √ 1 2 1 λ . Hence, u (s) = c1 exp 4 s λ √ 2 5 . u (x, y) = c1 exp 4 √xy 2 5 . 28. Hint: vx = 1 2 √ux u , vy = 1 2 √ uy u . This gives x 4 (f ′ ) 2 + y 2 (g ′ ) 2 = 1. Or, x 4 (f ′ ) 2 = 1 − y 2 (g ′ ) 2 = λ 2 . Or, x 4 (f ′ ) 2 = λ 2 and y 2 (g ′ ) 2 = 1 − λ 2 . Hence, f (x) = − λ x + c1 and g (y) = √ 1 − λ2 ln y + c2 u (x, y) =  − λ x + √ 1 − λ2 ln y + c 2 . 29. Hint: vx = ux u , vy = uy u . v 2 x x2 + v 2 y y2 = 1, and v = f (x) + g (y). Or, (f ′ ) 2 x2 = 1 − 1 y2 (g ′ ) 2 = λ 2 . f ′ (x) = λx, and g ′ (y) = √ 1 − λ2 y. Or, f (x) = λ 2 x 2 + c1, and g (y) = 1 2 y 2 √ 1 − λ2 + c2. v (x, y) = λ 2 x 2 + y 2 2 √ 1 − λ2 + c = ln u. u (x, y) = c exp 4 λ 2 x 2 + y 2 2 √ 1 − λ2 5 , c1 + c2 = ln c. e x 2 = u (x, 0) = c e λ 2 x 2 , which gives c = 1 and λ = 2. 30. (a) Hint: ξ = x − y, η = y; u (x, y) = e yf (x − y), (b) ξ = x, η = y − x 2 2 , uξ = η + 1 2 ξ 2 , u = ξη + 1 6 ξ 3 + f (η). u (x, y) = xy − 1 3 x 3 + f 4 y − x 2 2 5 . (c) ξ = y exp  −x 2 , η = y, and e 2uf (ξ) = η, e 2uf 4 y e−x 2 5 = y, (d) dx 1 = dy −y = du 1+u , ξ = y ex , η = y. Thus, (1 + u) f (ξ) = 1 η . Or, (1 + u) f (y ex ) = y −1 . 31. (c) u (x, y) = α exp  βx − a b βy . 704 Answers and Hints to Selected Exercises 32. (a) v (x, t) = x + ct, u (x, t) = (6x+3ct2+5ct3 ) 6(1+2t) . (b) v (x, t) = x + ct, u (x, t) = (6x+3ct2+4ct3 ) 6(1+2t) . 33. (a) v (x, t) = e x+at , u − 1 a e at = c1, and u − 1 a e at = f  x − ut + t a e at − 1 a2 e at . u (x, t) = (1 + t) −1 & (x − ut) +  1 a + t a − 1 a2 e at +  1 a2 − 1 a '. (b) v = x − ct, u (x, t) = (6x−3ct2+4ct3 ) 6(1−2t) . 34. dt 1 = dy −x = du u , t + ln x = c1, and xu = c2. g (xu, t + ln x) = 0. Or, u = 1 x h (t + ln x). u (x, t) = e t ln (xet ), where g and h are arbitrary functions. 3.9 Exercises 11. Hint: Differentiate the first equation with respect to t to obtain ρtt + ρ0divut = 0. Take gradient of the last equation to get ∇ρ = −  ρ0/c2 0 ut. We next combine these two equations to obtain ρtt = c 2 0 ∇2ρ. Application of ∇2 to p − p0 = c 2 0 (ρ − ρ0) leads to ∇2p = c 2 0 ∇2ρ. Also ptt = c 2 0 ρtt = c 4 0 ∇2ρ = c 2 0 ∇2p. Using u = ∇φ in the first equation gives ρt + ρ0∇2φ = 0, and differenting the last equation with respect to t yields ρt = −  ρ0/c2 0 φtt. Combining these two equations produces the wave equation for φ. Finally, we take gradient of the first and the last equations to obtain ∇ρt + ρ0∇2u = 0 and ∇ρ = −  ρ0/c2 0 ut that leads to the wave equation for ut. 14. (a) Differentiate the first equation with respect to t and the second equation with respect to x. Then eliminate Vxt and Vx to obtain the desired telegraph equation. (e) (i) ∂ 2 ∂x2 (I,V ) = 1 c 2 ∂ 2 ∂t2 (I,V ), c2 = 1 LC . 3.9 Exercises 705 (ii) ∂ ∂t (I,V ) = κ ∂ 2 ∂x2 (I,V ), κ = 1 RC . (iii) 4 ∂ 2 ∂t2 + 2k ∂ ∂t + k 2 5 (I,V ) = c 2 ∂ 2 ∂x2 (I,V ). 17. (a) The two-dimensional unsteady Euler equations are du dt = − 1 ρ ∂p ∂x , dv dt = − 1 ρ ∂p ∂y , where d dt = ∂ ∂t + u · ∇ = ∂ ∂t + u ∂ ∂x + v ∂ ∂y , and u = (u, v). (b) For two-dimensional steady flow, the Euler equations are u ux + v uy = − 1 ρ px, u vx + v vy = − 1 ρ py. Using dp dρ = c 2 , these equations become u ux + v uy = −c 2 (ρx/ρ), u vx + v vy = −c 2 (ρy/ρ). Multiply the first equation by u and the second by v and add to obtain u 2 ux + uv (uy + vx) + v 2vy = − 4 c 2 ρ 5 (uρx + vρy). Using the continuity equation (ρu)x + (ρv)y = 0, the right hand of this equation becomes c 2 (ux + vy). Hence is the desired equation. (c) Using u = ∇φ = (φx, φy), the result follows. (d) Substitute ρx and ρy from 17(b) into the continuity equation uρx + vρy + ρ (ux + vy) = 0 to obtain  c 2 − u 2 φxx − 2uvφxy +  c 2 − v 2 φyy = 0. Also du = ux dx + uy dy = −φxxdx − φxydy, dv = vx dx + vy dy = −φxydx − φyydy. Denoting D for the coefficient determinant of the above equations for φxx, φxy and φyy gives the solutions φxx = − D1 D , φxy = D2 D , φyy = −D3 D . D = 0 gives a quadratic equation for the slope of the characteristic C, that is,  c 2 − u 2 4 dy dx 52 + 2uv 4 dy dx 5 +  c 2 − v 2 = 0. Thus, directions are real and distinct provided 706 Answers and Hints to Selected Exercises 4u 2v 2 − 4  c 2 − u 2 c 2 − v 2 > 0, or  u 2 + v 2 > c2 . D2 = 0 gives − dy dx = (c 2−v 2 ) (c 2−u2)  dv du . Substitute into the quadratic equation to obtain  c 2 − v 2  dv du 2 − 2uv  dv du +  c 2 − u 2 = 0. Note that when D1 = D2 = D3 = D = 0, any one of the second order φ derivatives can be discontinuous. 18. (a) Hint: Use ∇ × ∂u ∂t = ∂ω ∂t , ∇ × (u · ∇u) u · ∇ω − ω∇u, where we have used ∇ · u = 0 and ∇ · ω = 0. Since ∇×∇f = 0 for any scalar function f, these lead to the vorticity equation in this simplified model. (b) The rate of change of vorticity as we follow the fluid is given by the term ω · ∇u. (c) u = i u (x, y) + j v (x, y) and ω = ω (x, y) k and hence, ω · ∇u = ω (x, y) ∂ ∂z [i u (x, y) + j v (x, y)] = 0. This gives the result. 20. We differentiate the first equation partially with respect to t to find Ett = c curl Ht. We then substitute Ht from the second equation to obtain Ett = −c 2 curl (curl E). Using the vector identity curl (curl E) = grad (div E)−∇2E with div E = 0 gives the desired equation. A similar procedure shows that H satisfies the same equation. 21. When Hooke’s law is used to the rod of variable cross section, the tension at point P is given by TP = λ A (x) ux, where λ is a constant. A longitudinal vibration would displace each cross sectional element of the rod along the x-axis of the rod. An element P Q of length δx and mass m = ρ A (x) δx will be displaced to P ′Q′ with length (δx + δu) with the same mass m. The acceleration of the element P ′Q′ is utt so that the difference of the tensions at P ′ and Q′ must be equal to the product m utt. Hence, m utt = TQ′ − TP ′ =  ∂ ∂t TP ′ δx = ∂ ∂x (λ A (x) ux) δx. This gives the equation. 4.6 Exercises 707 4.6 Exercises 1. (a) x < 0, hyperbolic; uξη = 1 4 4 ξ−η 4 54 − 1 2 4 1 ξ−η 5 (uξ − uη), x = 0, parabolic, the given equation is then in canonical form; x > 0, elliptic and the canonical form is uαα + uββ = 1 β uβ + β 4 16 . (b) y = 0, parabolic; y = 0, elliptic, and hence, uαα + uββ = uα + e α. (d) Parabolic everywhere and hence, uηη = 2ξ η2 uξ + 1 η2 e ξ/η . (f) Elliptic everywhere for finite values of x and y, then uαα + uββ = u − 1 α uα − 1 β uβ. (g) Parabolic everywhere uηη = 1 1−e 2(η−ξ) sin−1  e η−ξ − uξ ! . (h) B2 − 4AC = y − 4x. Equation is hyperbolic if y > 4x, parabolic if y = 4x and elliptic if y < 4x. (i) y = 0, parabolic; and y = 0, hyperbolic, uξη = (1+ξ−ln η) η uξ + uη + 1 η u. 2. (i) u (x, y) = f (y/x) + g (y/x) e −y , (ii) Hint: ϕ = ru and check the solution by substitution. u (r, t) = (1/r) f (r + ct) + (1/r) g (r − ct); (iii) A = 4, B = 12, C = 9. Hence, B2 − 4AC = 0. Parabolic at every point in (x, y)-plane. dy dx = 3 2 or y = 3 2 x+c ⇒ 2y−3x = c1, ξ = 2y−3x, η = y. The canonical form is uηη − u = 1 ⇒ u (ξ,η) = f (ξ) cosh η + g (ξ) sinh η−1. Or, u (x, y) = f (2y − 3x) cosh y+g (2y − 3x) sinh y−1. (iv) Hyperbolic at all points in the (x, y)-plane. ξ = y − 2x, η = y + x. Thus, uξη + uη = ξ, u (ξ,η) = η (ξ − 1) + f (ξ) e −ξ + g (ξ). 708 Answers and Hints to Selected Exercises u (x, y)=(x + y) (y − 2x − 1) + f (x + y) exp (2x − y) + g (y − 2x). (v) Hyperbolic. ξ = y, η = y−3x. ξuξη+uη = 0, u (ξ,η) = 1 ξ f (η)+g (ξ). u (x, y) = 1 y fu (y − 3x) + g (y). (vi) A = 1, B = 0, C = 1, B2 − 4AC = −4 < 0. So, this equation is elliptic. dy dx = + i or dx dy = + i or ξ = x + i y = c1 and η = x − i y = c2. The general solution is u = φ (ξ) + ψ (η) = φ (x + iy) + ψ (x − iy). (vii) u = φ (x + 2iy) + ψ (x − 2iy). (viii) B2 − 4AC = 0. Equation is parabolic. The general solution is given by (4.3.16), where λ =  B 2A = −1 and hence, the general solution becomes u = φ (y + x) + y ψ (y + x). (xi) B2 − 4AC = 25 > 0. Hyperbolic. The general solution is u = φ  y − 3 2 x + ψ  y − 1 6 x . 3. (a) ξ = (y − x) + i √ 2 x, η = (y − x) − i √ 2 x, α = y − x, β = √ 2 x, uαα + uββ = − 1 2 uα − 2 √ 2 uβ − 1 2 u + 1 2 exp  β/√ 2 . (b) ξ = y + x, η = y; uηη = − 3 2 u. (c) ξ = y − x, η = y − 4x; uξη = 7 9 (uξ + uη) − 1 9 sin [(ξ − η) /3]. (d) ξ = y + ix, η = y − ix. Thus, α = y, β = x. The given equation is already in canonical form. (e) ξ = x, η = x − (y/2); uξη = 18uξ + 17uη − 4. (f) ξ = y + (x/6), η = y; uξη = 6u − 6η 2 . (g) ξ = x, η = y; the given equation is already in canonical form. (h) ξ = x, η = y; the given equation is already in canonical form. (i) Hyperbolic in the (x, y) plane except the axes x = y = 0. ξ = xy, η = (y/x); y = √ ξη, x =  ξ/η; uξη = 1 2 4 1 + 1 2 %η ξ 5 uη − 1 4 √ 1 ξη − 1 4ξη − 1 2 . (j) Elliptic when y > 0; dy dx = + i √y, α = 2√y and β = −x; uαα + uββ = α 2uβ. Parabolic when y = 0; uxx + 1 2 uy = 0. 4.6 Exercises 709 Hyperbolic when y < 0; ξ = x − 2 √ −y, η = −x − 2 √ −y. The canonical form is uξη = 1 16 (ξ + η) 2 (uη − uξ). (k) Parabolic, dy dx = (xy) −1 . Integrating gives 1 2 y 2 = ln x+ln ξ, where ξ is an integrating constant. Hence, ξ = 1 x exp  1 2 y 2 , η = x. uxx = x −4 e y 2 uξξ − 2x −2 exp  1 2 y 2 uξη + uηη + 2x −2 exp  1 2 y 2 uξ, uxy = −yx−3 exp  y 2 uξξ + yx−1 exp  1 2 y 2 uξη − yx−2 exp  1 2 y 2 uξ, uyy =  y 2x −2 e y 2 uξξ +  y 2x −2 exp  1 2 y 2 uξ. uηη +  ξ/η2 uξ = 0. (l) Elliptic if y > 0, ξ = x + 2i √y, η = x − 2i √y, α = 1 2 (ξ + η) = x, β = 1 2i (ξ − η)=2√y; uαα + uββ = 1 β uβ. Hyperbolic y < 0, ξ = x + 2i √y, η = x − 2i √y, ξ − η = 4i √y; uξη + 1 2 4 uξ−uη ξ−η 5 = 0. The equations of the characteristic curves are dy dx = + i √y that gives 2√y = + i(x − c), or y = + 1 4 (x − c) 2 , where c is an integrating constant. Two branches of parabolas with positive or negative slopes. 4. (i) u (x, y) = f (x + cy)+g (x − cy); (ii) u (x, y) = f (x + iy)+f (x − iy); (iii) Use z = x + iy. Hence, u (x, y)=(x − iy) f1 (x + iy) + f2 (x + iy) + (x + iy) + f3 (x − iy) + f4 (x − iy) (iv) u (x, y) = f (y + x)+g (y + 2x); (v) u (x, y) = f (y)+g (y − x); (vi) u (x, y)=(−y/128) (y − x) (y − 9x) + f (y − 9x) + g (y − x). 5. (i) vξη = − (1/16) v, (ii) vξη = (84/625) v. 7. (ii) Use α = 3y 2 , β = −x 3/2. 8. x = r cos θ, y = r sin θ; r =  x 2 + y 2, θ = tan−1  y x . ∂u ∂x = ∂u ∂r · ∂r ∂x + ∂u ∂θ · ∂θ ∂x = ur · x r + uθ ·  − y r 2 . uxx = (ux)x = (ux) r · ∂r ∂x + (ux) θ ∂θ ∂x =  x r ur − y r 2 uθ r =  x r +  x r ur − y r 2 uθ − y r 2 =  x r urr − x r 2 ur + 1 r ur ∂x ∂r x r − 4 y r 2 urθ − 2y r 3 uθ + 1 r 2 ∂y ∂r uθ 5 x r +  x r urθ + 1 r ur · ∂x ∂θ − y r 2 + 4 y r 2 uθθ + 1 r 2 uθ · ∂y ∂θ 5 y r 2 710 Answers and Hints to Selected Exercises = x 2 r 2 urr − 2xy r 3 urθ + y 2 r 4 uθθ + y 2 r 3 ur + 2xy r 4 uθ. Similarly, uyy = y 2 r 2 urr +  2xy r 3 urθ + x 2 r 4 uθθ + x 2 r 3 ur − 2xy r 4 uθ. Adding gives the result: ∇2u = uxx + uyy = urr + 1 r ur + 1 r 2 uθθ = 0. 9. (c) Use Exercise 8. 10. (a) ux = uξξx + uηηx = a uξ + c uη = 4 a ∂ ∂ξ + c ∂ ∂η5 u, uy = uξξy + uηηy = b uξ + d uη = 4 b ∂ ∂ξ + d ∂ ∂η5 u. uxx = (ux)x = 4 a ∂ ∂ξ + c ∂ ∂η54a ∂ ∂ξ + c ∂ ∂η5 u =  a 2uξξ + 2ac uξη + c 2uηη . uyy = (uy) y = 4 b ∂ ∂ξ + d ∂ ∂η54b ∂ ∂ξ + d ∂ ∂η5 u = b 2uξξ + 2bd uξη + d 2uηη. uxy = (uy)x = 4 a ∂ ∂ξ + c ∂ ∂η54b ∂ ∂ξ + d ∂ ∂η5 u = ab uξξ + (ad + bc) uξη + cd uηη. Consequently, 0 = A uxx + 2B uxy + C uyy =  A a2 + 2Bab + C b2 uξξ+2 [ac A + (ad + bc) B + bd C] uξη +  A c2 + 2Bcd + C d2 uηη. Choose arbitrary constants a, b, c and d such that a = c = 1 and such that b and d are the two roots of the equation Cλ2 + 2Bλ + A = 0, and λ = −B + √ D C = b, d, D = B2 − AC. Thus, the transformed equation with a = c = 1 is given by [A + (b + d) B + bd C] uξη = 0. Or,  2 C AC − B2 uξη = 0. If B2 −AC > 0, the equation is hyperbolic, and the equation uξη = 0 is in the canonical form. The general solution of this canonical equation 4.6 Exercises 711 is u = φ (ξ) + ψ (η), where φ and ψ are arbitrary functions and the transformation becomes ξ = x + by and η = x + dy, where b, d are real and distinct. If B2 −AC < 0, the equation is elliptic, and b and d are complex conjugate numbers  d = b . With a = c = 1, the transformation is given by ξ = x+by and η = x+b y. Then α = 1 2 (ξ + η) and β = 1 2i (ξ − η) can be used to transform the equation into the canonical form uαα + uββ = 0. If B2 − AC = 0, the equation is parabolic, here b = − B C , a, c and d are arbitrary, but c and d are not both zero. Choose a = c = 1, d = 0 so that ξ = x − B C y and η = y are used to transform the equation into the form uηη = 0. The general solution is u = φ (ξ)+ηψ (η), where φ and ψ are arbitrary functions, and b is the double root of Cλ2 + 2Bλ+A = 0, and ξ = x + by. 11. Seek a trial solution u (x, y) = f (x + my) so that uxx = f ′′ , uyy = m2f ′′. Substituting into the Laplace equation yields  m2 + 1 f ′′ = 0 which gives that either f ′′ = 0 or m2 + 1 = 0. Thus, m = + i. The general solution is u (x, y) = F (x + iy) + G (x − iy). Identifying c with i gives the d’Alembert solution u (x, y) = 1 2 [f (x + iy) + f (x − iy)] + 1 2i  x+iy x−iy g (α) dα. 12. (a) Hyperbolic. (ξ,η) = 2 3  y 3/2 + x 3/2 , 3  ξ 2 − η 2 uξη = ηuη − ξuη. (b) Elliptic. dy dx = + isech2 x, ξ = y + itanh x, η = y − itanh x; α = y, β = tanh x. Thus, uαα + uββ = 2β (1−β2) uβ. (d) Hyperbolic. ξ = y + tanh x, η = y − tanh x. uξη = " 4 − (ξ − η) 2 #−1 (η − ξ) (uξ − uη) in the domain (η − ξ) 2 < 4. (e) Parabolic. ξ = y − 3x, η = y; uηη = − η 3 (uξ + uη). (f) Elliptic. α = 1 2  y 2 − x 2 , β = 1 2 x 2 . The canonical form is 712 Answers and Hints to Selected Exercises uαα + uββ = [2β (α + β)]−1 [αuα − (α + 2β) uβ]. (g) Elliptic. α = sin x + y, β = x, uαα + uββ = (sin β) uα − u. (h) Parabolic. ξ = x + cos y, η = y. Thus, uηη =  sin2 η cos η uξ. 13. The general solution is u (x, y) = e x  x 0 e −α cos (α + y eα−x ) dα + e x f (y e−x ) + g (x), where f and g are arbitrary functions. 5.12 Exercises 1. (a) u (x, t) = t, (b) u (x, t) = sin x cos ct + x 2 t + 1 3 c 2 t 3 , (c) u (x, t) = x 3 + 3c 2xt2 + xt, (d) u (x, t) = cos x cos ct + (t/e), (e) u (x, t)=2t + 1 2 log  1 + x 2 + 2cxt + c 2 t 2 + log  1 + x 2 − 2cxt + c 2 t 2 !, (f) u (x, t) = x + (1/c) sin x sin ct. 2. (a) u (x, t)=3t + 1 2 xt2 . (c) u (x, t)=5+ x 2 t + 1 3 c 2 t 3 +  1/2c 2 (e x+ct + e x−ct − 2e x ), (e) u (x, t) = sin x cos ct + (e t − 1) (xt + x) − xtet , (f) u (x, t) = x 2 + t 2  1 + c 2 + (1/c) cos x sin ct. 3. s (r, t) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 0, 0 ≤ t ct. 30. (a) When ω = ck, u (x, t) = 1 (k2c 2−ω2) sin (kx − ωt) − (ω−kc) 2kc(ω2−c 2k2) sin [k (x + ct)] + (ω+kc) 2kc(ω2−c 2k2) sin [k (x − ct)] . This solution represents three harmonic waves which propagate with different amplitudes and with speeds + c and the phase velocity (ω/k). (b) When ω = ck, u (x, t) = 1 4 sin (x − t) − 1 4 sin (x + t) + 1 2 t cos (x − t). This solution represents two harmonic waves with constant amplitude and another harmonic wave whose amplitude grows linearly with time. 31. (a) u (x, t) = 1 2 [cos (x − 3t) + cos (x + 3t)] + 1 6  x+3t x−3t sin (2α) dα = cos x cos (3t) + 1 6 sin (2x) sin (6t). (c) u (x, t) = cos (3x) cos (21t) + tx. (e) u (x, t) = x 3 + 27xt2 + 1 6 [cos (x + 3t) − cos (x − 3t)] + 1 6 [(x + 3t) sin (x + 3t) − (x − 3t) sin (x − 3t)]. 714 Answers and Hints to Selected Exercises (f) u (x, t) = 1 2 [cos (x − 4t) + cos (x + 4t)] + 1 8 e −x (x + 1 − 4t) e 4t − (x +1+4t) e −4t ! . 32. Verify that u (x, t) =  t 0 v (x, t; τ ) dτ satisfies the Cauchy problem. ut (x, t) = v (x, t;t) +  t 0 vt (x, t; τ ) dτ =  t 0 vt (x, t; τ ) dτ utt (x, t) = vt (x, t;t)+ t 0 vtt (x, t; τ ) dτ = p (x, t)+ t 0 vtt (x, t; τ ) dτ uxx (x, t) =  t 0 vxx (x, t; τ ) dτ. Thus, utt − c 2uxx = p (x, t) +  t 0  vtt − c 2vxx dτ = p (x, t). 33. ut = v (x, t;t) +  t 0 vt (x, t; τ ) dτ = p (x, t) +  t 0 vt (x, t; τ ) dτ uxx =  t 0 vxx (x, t; τ ) dτ . Hence, ut − κ uxx = p (x, t) +  t 0 (vt − κ vxx) dτ = p (x, t). 34. According to the Duhamel principle u (x, t) =  t 0 v (x, t; τ ) dτ is the solution of the problem where v (x, t; τ ) satisfies vt = κ vxx, 0 ≤ x < 1, t> 0, v (0, t; τ )=0= v (1, t; τ ), v (x, τ ; τ ) = e −τ sin πx, 0 ≤ x ≤ 1. Using the separation of variables, the solution is given by v (x, t) = X (x) T (t) so that X′′ + λ 2X = 0 and T ′ + κλ2T = 0. The solution is v (x, t; τ ) = ∞ n=1 an (τ ) e −λ 2 nκt sin λnx, when λn = nπ, n = 1, 2, 3,.... 6.14 Exercises 715 Since v (x, τ ; τ ) = e −τ sin πx, e −τ sin πx = ∞ n=1 an (τ ) exp  −n 2π 2κτ sin (πnx). Equating the coefficients gives e −τ = a1 (τ ) exp  −π 2κτ , an (τ )=0, n = 2, 3,.... Consequently, v (x, t; τ ) = exp π 2κ − 1 τ ! exp  −π 2κt sin πx. Thus, u (x, t) = exp  −π 2κt sin πx t 0 exp π 2κ − 1 τ ! dτ = e −t−exp(−π 2κt) (π2κ−1) · sin πx. 36. (a) The solution is u (x, t) = 1 n e nx sin  2n 2 t + nx + e −nx sin  2n 2 t − nx !, and u (x, t) → ∞ as n → ∞ for certain values of x and t. (b) un (x, y) = 1 n exp (− √ n ) sin nx sinh ny is the solution. For y = 0, un (x, y) → ∞ as n → ∞. But (un)y (x, 0) = exp (− √ n ) sin nx → 0 as n → ∞. 6.14 Exercises 1. (a) f (x) = − π 4 + h 2 + ∞ k=1 ( 1 πk2 " 1+(−1)k+1# cos kx + 1 πk " h + (h + π) (−1)k+1# sin kx) . (c) f (x) = sin x + ∞ k=1 2(−1)k+1 k sin kx. (e) f (x) = sinh π π 1 1 +∞ k=1 2(−1)k 1+k2 (cos kx − k sin kx) 3 . 2. (a) f (x) = ∞ k=1 2 k sin kx (b) f (x) = ∞ k=1  2 πk " 1 − 2 (−1)k + cos kπ 2 # sin kx. 716 Answers and Hints to Selected Exercises (c) f (x) = ∞ k=1 " 2 (−1)k+1 π k + 4 πk3 4 (−1)k − 1 5# sin kx. (d) f (x) = ∞ k=2 2k π " 1+(−1)k k2−1 # sin kx. 3. (a) f (x) = 3 2 π + ∞ k=1 2 πk2 " (−1)k − 1 # cos kx. (b) f (x) = π 2 + ∞ k=1 2 πk2 " (−1)k − 1 # cos kx. (c) f (x) = π 2 3 + ∞ k=1 4(−1)k k2 cos kx. (d) f (x) = 2 3π + ∞ k=1,2,4,... 6 π " 1+(−1)k 9−k2 # cos kx, k = 3. 4. (b) f (x) = ∞ k=1  2 kπ sin kπ 2 cos  kπx 6 . (c) f (x) = 2 π + ∞ k=2  2 kπ " 1+(−1)k 1−k2 # cos  kπx l . (f) f (x) = ∞ k=1 kπ 1+k2π2 (−1)k+1  e − e −1 sin (kπx). 5. (a) f (x) = ∞ k=−∞ 1 π 4 2+ik 4+k2 5 (−1)k sinh 2π eikx . (b) f (x) = ∞ k=−∞ (−1)k π(1+k2) sinh π eikx . (d) f (x) = ∞ k=−∞ (−1)k  i kπ e ikπx . 6. (a) f (x) = π 8 + ∞ k=1 " 1 2πk2 ( (−1)k − 1 ) cos kx + (−1)k+1 2k sin kx# . 7. (a) f (x) = l 2 3 + ∞ k=1 4 (−1)k  1 kπ 2 cos  kπx l . 8. (a) sin2 x = ∞ k=1,3,4,... 4(1−cos kπ) kπ(4−k2) sin kx. (b) cos2 x = ∞ k=1,3,4,... 2 kπ 4 1−k 2 4−k2 5 (1 − cos kπ) sin kx. (d) sin x cos x = ∞ k=1,3,4,... 2 π 4 1−cos kπ 4−k2 5 cos kx. 6.14 Exercises 717 9. (a) x 2 4 = π 2 12 − ∞ k=1 (−1)k+1 k2 cos kx. (c)  ∞ 0 ln  2 cos x 2 dx = ∞ k=1 (−1)k+1 sin kx k2 . (e) π 2 − 4 π ∞ k=1 cos(2k−1)x (2k−1)2 = ⎧ ⎪⎨ ⎪⎩ −x, −π<x< 0="" x,="" <="" x="" π.="" 10.="" (a)="" f="" (x,="" y)="16" π2="" ∞="" m="1,3,..." n="1,3,..." ="" 1="" mn="" sin="" mx="" ny="" (c)="" 4="" 9="" +="" 2="" 8="" 3="" π="" (−1)m="" m2="" cos="" n2="" 16(−1)m+n="" m2n2="" ny.="" (e)="" 2(−1)m+1="" y.="" (g)="" dmn="" mπx="" nπy="" ,="" where="" 1.2="" ="" xy="" (mπx)="" 4nπy="" 5="" dx="" dy="2" m2π="" −="" mπ="" y="" ="" −4="" (−1)n="" nπ="" ="8" (−1)m+n="" 2mn="" .="" (h)="" 16="" [(2m="" 1)="" (2n="" 1)]−1="" "="" (2m−1)πx="" a="" #="" ×="" (2n−1)πy="" b="" (double="" fourier="" sine="" series).="" (i)="" (−1)m+1="" πmx="" 718="" answers="" and="" hints="" to="" selected="" exercises="" (−1)m+n+1="" m(4−π2n2)="" πny="" (j)="" 2∞="" mx+="" 8(−1)m+n+1="" (k)="" 4="" 4π="" 1∞="" ny3="" +16∞="" 20.="" b2n="0," b2n+1="8" π(2n+1)3="" (x)="x" (π="" x)="8" 3x="" 5x="" 5="" ...="" (b)="" put="" find="" the="" sum="" of="" series.="" 21.="" bn="2" nx="" n2π2="" 1,="" 2,....="" π2(2n+1)2="" 22.="" nπx="" (1="" nπ)="2" [1="" ]="⎧" ⎪⎨="" ⎪⎩="" for="" odd="" n,="" 0,="" even="" n.="" ∼="" πx="" 3πx="" 5πx="" ...!="" a0="" an="" (sin="" 0)="0," ·="" 2nπ="" (−1)n+1="" 2a="" ((−1)n="" 4a="" 6.14="" 719="" 23.="" −a="" π2n2="" !a="" (cos="" (−nπ))="0" ="" (−nπ)="(−1)n+1" 2π="" all="" [−1+(−1)n="" is="" odd.="" k="0" sin(2k+1)x="" (2k+1)="" 24.="" 2nx="" (4n2−1)="" 26.="" hint:="" argument="" similar="" that="" used="" in="" section="" 6.5="" can="" be="" employed="" prove="" this="" general="" parseval="" relation.="" more="" precisely,="" use="" (6.5.10)="" (f="" g)="" obtain="" −π="" (a0="" α0)="" (ak="" αk)="" (bk="" βk)="" subtracting="" later="" equality="" from="" former="" gives="" g="" (akαk="" bkβk).="" 27.="" ∞="" α="" αx="" aα="" dα,="" πα="" (1−α2)="" dα.="" 720="" 33.="" ck="1" e−ikxdx="1" 1="" e−ikx="" −ik="" ik="" e="" −ikxdx3="1" e−ikπ="" eikπ="" (−ik)="" −ikπ="" 1="" ikπ="" 3="i" kπ="" i="" c0="1" 35.="" cke="" ikx="" ibk)="1" −ikxdx,="1" i(a−k)x="" −i(a+k)x="" dx,="1" 4πi(a="" k)="" #π="" (a="" real="" quantity="" hence,="" bk="0" 2,="" 3,="" ...,="" ak="2" (−1)k="" (πa)="" 2)="" thus,="" (ax)="2a" aπ="" 2x="" ...="" since="" ax="" even,="" above="" series="" continuous,="" at="" nπ.="" putting="" treating="" variable="" cot="" or,="" (n2="" convergence="" uniform="" any="" interval="" x-axis="" does="" not="" contain="" integers,="" term-by-term="" integration="" <a<x<="" 721="" πt="" πt="" dt="ln" ln="" πa="" limit="" as="" →="" we="" or="" 3∞="" product="" representation="" πx.="" (d)="" wallis="" formula="" infinite="" (2n)="" 6="" 7="" ····="" 36.="" (x="" 2π)="f" (x).="" kx="" dx.="" evaluating="" these="" integrals="" (k="" kx)="" case,="" (−x)="−f" odd,="" periodic="" period="" 2π.="" given="" by="" 6x="" 10x="" with="" 1.="" 722="" (2πkx)="" corresponding="" represent="" value="" ....="" [f="" (n+)="" (n−)]="1" 37.="" represents="" square="" wave="" function.="" l="" −l="" πkx="" πk="" πk)="⎧" ⎨="" ⎩="" 2l="" sin="" 4πx="" ∞="" (2k="" !="" nl,="" terms="" after="" first="" vanish="" (l="" 2).="" graphs="" partial="" sums="" sn="" (="" 5)="" drawn.="" show="" how="" converges="" points="" continuity="" (x),="" ∞.="" however,="" discontinuity="" l,="" converge="" mean="" value.="" just="" beyond="" discontinuities="" sums,="" overshoot="" |l|.="" behavior="" known="" gibbs="" phenomenon.="" 723="" (πk)="" 1),="" 4l="" cos="" 1)2="" )="" [ak="" (πkx)="" (πkx)]="" 2k="" π3k3="" 38.="" have="" summing="" n="" n="" dividing="" result="" adding="" both="" sides="" result.="" kx),="" (t)="" kt="" (t="" dt.="" 724="" &n="" '="" which="" is,="" t="" =="" π−x="" −π−x="" ξ)="" ξ="" dξ,="" integrand="" has="" 2π,="" replace="" interval−π−x,="" length="" (−π,="" π).="" 7.9="" u="" t)="∞" (nπ)="" (nπct)="" (nπx).="" ctsin="" x.="" 2.="" 32[(−1)n−1]="" πcn2(n2−4)="" (nct)="" (nx).="" 5.="" antn="" tn="" ⎪⎪⎪⎪⎨="" ⎪⎪⎪⎪⎩="" −at="" cosh="" αt="" 2α="" sinh=""> 0 e −at/2  1 + at 2 , for α = 0 e −at/2 4 cos βt + a 2β sin βt5 , for α 2 < 0, in which α = 1 2 " a 2 − 4 4 b + n 2π 2 c 2 l 2 5# 1 2 , β = 1 2 " 4 4 b + n 2π 2 c 2 l 2 5 − a 2 # 1 2 . 7.9 Exercises 725 6. u (x, t) = ∞ n=1 anTn (t) sin  nπx l , an = 2 l  l 0 g (x) sin nπx l dx, and Tn (t) = ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ 2e −at/2 √ (a2−α) sinh √ (a2−α) 2 t  , for a 2 > α, t e−at/2 , for a 2 = α, 2e −at/2 √ (α−a2) sin √ (α−a2) 2 t  , for a 2 < α. 7. θ (x, t) = ∞ n=1 an cos (aαnt) sin (αnx + φn), where an = 2(α 2 n+h 2 ) 2h+(α2 n+h2)l  l 0 f (x) sin (αnx + φn) dx and φn = tan−1  αn h ; αn are the roots of the equation tan αl = 4 2hα α2−h2 5 . 11. u (x, t) = v (x, t) + U (x), where v (x, t) = ∞ n=1 1 −  2 l  l 0 U (τ ) sin  nπτ l dτ3 cos  nπct l sin  nπx l and U (x) = − A c 2 sinh x + A c 2 sinh (l + k − h) x l + h. 12. u (x, t) = A 6c 2 x 2 (1 − x) + ∞ n=1 12 (nπ) 3 (−1)n cos (nπct) sin (nπx). 14. (a) u (x, t) = − hx2 2k +  2u0 + h 2k x − 4h kπ e −kπ2 t sin (πx) + ∞ n=2 ane −kn2π 2 t sin (nπx), where an = 2u0 nπ [1 + (−1)n ] + 2u0n (n2−1)π [1 + (−1)n ] + 2h kπ3n3 [(−1)n − 1]. (b) Hint: v (x, t) = e −htu (x, t). u (x, t) = e −ht 1 1 2 a0 + ∞ n=1 an cos  nπx l exp  −n 2π 2kt/l2 3 , where an = 2 l  l 0 f (ξ) cos 4 nπξ l 5 dξ. 15. (a) u (x, t) = ∞ n=1 4 n3π3 " 2 (−1)n+1 − 1 # e −4n 2π 2 t sin (nπx). (b) u (x, t) = ∞ n=1,3,4,... [(−1)n − 1] " n π(4−n2) − 1 nπ # e −n 2kt sin (nx). 726 Answers and Hints to Selected Exercises 16. u (x, t) = ∞ n=1 2l 2 n3π3 [1 − (−1)n ] e −(nπ/l) 2 t cos  nπx l . 18. v (x, t) = Ct  1 − x l − Cl2 6k " x l 3 − 3  x l 2 + 2  x l # + 4 2Cl2 π3k 5∞ k=1 e −n2π2kt/l2 n3 sin  nπx l . 21. u (x, t) = v (x, t) + w (x), where v (x, t) = e −kt sin x + ∞ n=1 an e −n 2kt sin (nx), and an = −n (n2+a2) [(−1) e −n−ax − 1] + 2A a2kπ 1 n {(−1)n − 1} ! + (−1)n n [e −aπ − 1] w (x) = A a2k 1 − e −ax + x π (e −aπ − 1)! . 36. Hint: Suppose R is the rectangle 0 ≤ x ≤ a, 0 ≤ y ≤ b and ∂R is its boundary positively oriented. Suppose that u1 and u2 are solutions of the problem, and put v = (u1 − u2). Then v satisfies the Laplace equation with v = (x, 0) = 0 = v (x, b), vx (0, y)=0= vx (a, y). 8.14 Exercises 1. (a) λn = n 2 , φn (x) = sin nx for n = 1, 2, 3,... (b) λn = ((2n − 1) /2)2 , φn (x) = sin ((2n − 1) /2) πx for n = 1, 2, 3,... (c) λn = n 2 , φn (x) = cos nx for n = 1, 2, 3,.... 2. (a) λn = 0, n 2π 2 , φn (x) = 1, sin nπx, cos nπx for n = 1, 2, 3,... (b) λn = 0, n 2 , φn (x) = 1, sin nx, cos nx for n = 1, 2, 3,... (c) λn = 0, 4n 2 , φn (x) = 1, sin 2nx, cos 2nx for n = 1, 2, 3,.... 3. (a) λn = −  3/4 + n 2π 2 , φn (x) = e −x/2 sin nπx, n = 1, 2, 3,.... 4. (a) λn =1+ n 2π 2 , φn (x) = (1/x) sin (nπ ln x), n = 1, 2, 3,.... (b) λn = 1 4+(nπ/ ln 3)2 , φn (x) = " 1/ (x + 2) 1 2 # sin [(nπ/ ln 3) ln (x + 2)], n = 1, 2, 3,.... 9.10 Exercises 727 (c) λn = 1 12 " 1 + (2nπ/ ln 2)2 # , φn (x) = " 1/ (1 + x) 1 2 # sin [(nπ/ ln 2) ln (1 + x)], n = 1, 2, 3,.... 5. (a) φ (x) = sin 4√ λ ln x 5 , λ > 0. (b) φ (x) = sin 4√ λ x5 , λ > 0. 7. f (x) ∼ ∞ n=1 2 π " (−1)n−1 n2 # cos nx. 11. (a) G (x, ξ) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ x, x ≤ ξ ξ, x > ξ. 12. (a) u (x) = − cos x +  cos 1−1 sin 1 sin x + 1, (b) u (x) = − 2 5 cos 2x − 1 10  1+2 sin 2 cos 2 sin 2x + 1 5 e x . 16. G (x, ξ) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩  x 3 ξ/2 +  xξ3/2 − (9xξ/5) + x, for 0 ≤ x<ξ  x 3 ξ/2 +  xξ3/2 − (9xξ/5) + ξ, for ξ ≤ x ≤ 1. 24. (a) Hint: Differentiate cot θ = py′ y with respect to x to find −cosec2 θ dθ dx = 1 y (py′ ) ′ − 1 y 2  py′2 = −  λr + q + 1 p cot2 θ  . dθ dx = (q + λr) sin2 θ + 1 p cos2 θ, dr dx = r 2  1 p − q − λr sin 2θ. (b) At θ = nπ, dθ dx = 1 p , and at θ =  n + 1 2 π, dθ dx = (q + λp). 9.10 Exercises 8. (a) u (r, θ) = 4 3  1 r − r 4 sin θ. (c) u (r, θ) = ∞ n=1 an sinh (nπ/ ln 3)  θ − π 2 ! sin [(nπ/ ln 3) ln r], where 728 Answers and Hints to Selected Exercises an = 2 ln 3 sinh(nπ2/2 ln 3) ( nπ ln 3 n2π2+4(ln 3)2 [9 (−1)n − 1] − 4nπ ln 3 n2π2+(ln 3)2 [3 (−1)n − 1] + 3 ln 3 nπ [(−1)n − 1]) . 9. u (r, θ) = ∞ n=1 an  r −nπ/α − b −2nπ/α r nπ/α sin  nπθ α + ∞ n=1 bn sinh " nπ ln(b/a) (θ − α) # sin " nπ ln(b/a) (ln r − ln a) # , where an = 2 α  a −nπ/α − b −2nπ/αa nπ/α !−1  α 0 f (θ) sin  nπθ α dθ, bn = −2 ln  b a sinh {αnπ/ ln (b/a)} !−1 +  b a f (r) sin [(nπ/ ln [b/a]) ln (ra)] dr r . 12. u (r, θ) = ∞ n=1 2 αJν (a)  α 0 f (θ) sin  nπτ α dτ Jν (r) sin  nπθ α , ν = nπ/α. 13. u (r, θ) = 1 2  a 2 − r 2 . 14. (a) u (r, θ) = − 1 3  r + 4 r sin θ + constant. 16. u (r, θ) = ∞ n=1 1 nRn−1 r n sin nθ. 18. u (r, θ) = a0 2 + ∞ n=1 r n (an cos nθ + bn sin nθ), where an = R 1−n (n+Rh)π  2π 0 f (θ) cos nθ dθ, n = 0, 1, 2,.... bn = R 1−n (n+Rh)π  2π 0 f (θ) sin nθ dθ, n = 1, 2, 3,.... 20. u (r, θ) = c − r 4 12 sin 2θ + 1 6 4 r 6 1−r 6 2 r 4 1−r 4 2 5 r 2 sin 2θ + 1 6 4 r 2 1−r 2 2 r 4 1−r 4 2 5 r 4 1 r 4 2 r −2 sin 2θ. 21. u (r, θ) = ∞ n=1 an  r −nπ/α − b −2nπ/αr nπ/α sin  nπθ α + ∞ n=1 bn sinh " nπ ln(b/a) (θ − α) # sin " nπ ln(b/a) (ln r − ln a) # . 9.10 Exercises 729 22. (a) u (x, y) = ∞ n=1 4[1−(−1)n] (nπ) 3 sinh nπ sin nπx sinh {nπ (y − 1)} (c) u (x, y) = ∞ n=1 an (sinh nπx − tanh nπ cosh nπx) sin nπy, where an = 1 tanh nπ " 2nπ3 n2π4−4 + 1−(−1)n nπ # . 23. (a) u (x, y) = c + ∞ n=1 an (cosh nx − tanh nπ sinh nx) cos ny, where an = 2 [1 − (−1)n ] =n 3π tanh nπ . (c) u (x, y) = − 1 tanh π [cosh y − tanh π sinh y] cos x + C. 25. u (x, y) = xy (1 − x) + ∞ n=1 4(−1)n (nπ) 3 sinh nπ sin (nπx) sinh (nπy). 27. u (x, y) = c +  x 2/2 4 x 2 3 − y 2 5 + ∞ n=1 8a 4 (−1)n+1 (nπ) 3 sinh nπ cosh  nπx a cos  nπy a . 29. u (x, y) = x [(x/2) − π] + ∞ n=1 an sin & 2n−1 2 x ' cosh & 2n−1 2 y ' , where an = 2 Aπ  π 0 " f (x) − h 4 x 2 2 − πx5# sin  2n−1 2 x ! dx with A =  2n−1 2 sinh  2n−1 2 π + h cosh  2n−1 2 π. 32. Hint: The solution is given by (9.5.3) and the boundary conditions require sin2 θ = 1 2 a0 + ∞ n=1 [(an + bn) cos nθ + (cn + dn) sin nθ] , 0 = 1 4 b0 + ∞ n=1 n an 2 n−1 − bn 2 −n−1 cos nθ +  cn 2 n−1 − dn 2 −n−1 sin nθ! . Using sin2 θ = 1 2 (1 − cos 2θ), we equate coefficients to obtain a0 = 1, b0 = 0; a2 + b2 = − 1 2 , 2a2 − 1 8 b2 = 0; an + bn = 0 2 n−1 an − 2 −n−1 bn = 0 ⎫ ⎪⎬ ⎪⎭ n = 1, 3, 4, 5,.... 730 Answers and Hints to Selected Exercises cn + dn = 0 2 n−1 cn − 2 −n−1 dn = 0 ⎫ ⎪⎬ ⎪⎭ n = 1, 2, 3,.... Thus, a0 = 1, b0 = 1, a2 = − 1 34 , b2 = − 8 17 , and the remaining coeffi- cients are zero; finally u (r, θ) = 1 2 − 1 34  r 2 + 16 r 2 cos 2θ. 33. (a) Hint: Seek a separable solution u (r, z) = R (r)Z (z) so that r 2R′′ + r R′ − λr2R = 0, and Z ′′ + λZ = 0, with Z (0) = 0 = Z (h). The solution of this eigenvalue problem is λn =  nπ h 2 , Zn (z) = sin  nπz h , n = 1, 2, 3,.... The solution of the radial equation is Rn (r) = anI0  nπr h + bnK0  nπr h , where I0 and K0 are modified Bessel functions. Since K0 is unbounded at r = 0, all bn ≡ 0. Thus, u (r, z) = ∞ n=1 an I0  nπr h sin  nπz h . f (z) = u (1, z) = ∞ n=1 an I0  nπ h sin  nπz h . This is a Fourier sine series for f (z) and hence, anI0  nπ h = 2 h  h 0 f (z) sin  nπz h dz. (d) u (r, z) = a I0  3πr h sin  3πz h =I0  3π h . 35. (a) u (r, z)=8∞ n=1 sinh z kn z kn sinh kn J0(knr) J0(kn) . (b) u (r, z) =  4a π ∞ n=1 1 (2n−1) I0[ 1 2 (2n−1)r] I0[ 1 2 (2n−1)] sin &1 2 (2n − 1) z ' . (c) u (r, z) = ∞ n=1 anI0  nπr h sin  nπz h , where anI0  nπa h = 2 h  h 0 f (z) sin  nπz h dz. 10.13 Exercises 731 10.13 Exercises 1. u (x, y, z) = sinh[(π/b) 2+(π/c) 2 ] 1 2 (a−x) sinh[(π/b) 2+(π/c) 2 ] 1 2 a sin  πy b sin  πz c . 2. u (x, y, z) = sinh( √ 2 πz) √ 2 π − cosh( √ 2 πz) √ 2 π tanh √ 2 π cos πx cos πy. 4. (a) u (r, θ, z) = ∞ m=0 ∞ n=1 (amn cos mθ + bmn sin mθ) Jmn (amnr/a) × sinh αmn(l−z)/a sinh αmnl/a , where amn = 2 a 2πεn [Jm+1 (αmn)]2  2π 0  a 0 f (r, θ) Jm (αmnr/a) cos mθ r dr dθ bmn = 2 a 2π [Jm+1 (αmn)]2  2π 0  a 0 f (r, θ) Jm (αmnr/a) sin mθ r dr dθ with εn = ⎧ ⎪⎨ ⎪⎩ 1, for m = 0 2, for m = 0 and αmn is the nth root of the equation Jm (αmn) = 0. 5. u (r, θ) = 1 3 +  2/3a 2 r 2P2 (cos θ). 7. u (r, z) = ∞ n=1 an sinh αn(l−z)/a cosh αnl/a J0 (αnr/a), where an = 2qu kα2 nJ0(αn) and αn is the root of J0 (αn) = 0 and k is the coefficient of heat conduction. 8. u (r, z) = 4u0 π ∞ n=1 [I0(2n+1)(πr/l)] [I0(2n+1)(πa/l)] sin(2n+1)πz/l (2n+1) . 9. u (r, θ) = u2 +  u1−u2 2 ∞ n=1 4 2n+1 n+1 5 Pn−1 (0)  r a n Pn (cos θ). 11. u (r, θ, φ) = C + ∞ n=1 ∞ m=0 r nP m n (cos θ) [anm cos mφ + bnm sin nφ], where 732 Answers and Hints to Selected Exercises anm = (2n + 1) (n − m)! 2nπ (n + m)!  2π 0  π 0 f (θ, ϕ) P m n (cos θ) cos mϕ sin θ dθ dϕ bnm = (2n + 1) (n − m)! 2nπ (n + m)!  2π 0  π 0 f (θ, ϕ) P m n (cos θ) sin mϕ sin θ dθ dϕ an0 = (2n + 1) 4nπ  2π 0  π 0 f (θ, ϕ) Pn (cos θ) sin θ dθ dϕ. 12. u (x, y, t) = ∞ n=1,3,4,...  − 4 π [1−(−1)n] n(n2−4) cos 4 (n2 + 1)πct5 (sin nπx sin nπy). 13. u (r, θ, t) = ∞ n=0 ∞ m=1 Jn (αmnr/a) cos (αmnct/a) [amn cos nθ + bmn sin nθ] + ∞ n=0 ∞ m=1 Jn (αmnr/a) sin (αmnct/a) [cmn cos nθ + dmn sin nθ], where amn = 2 πa2εn [J ′ n (αmn)]2  2π 0  a 0 f (r, θ) Jn (αmnr/a) cos nθ r dr dθ bmn = 2 πa2 [J ′ n (αmn)]2  2π 0  a 0 f (r, θ) Jn (αmnr/a) sin nθ r dr dθ cmn = 2 πacαmnεn [J ′ n (αmn)]2  2π 0  a 0 g (r, θ) Jn (αmnr/a) cos nθ r dr dθ dmn = 2 πacαmn [J ′ n (αmn)]2  2π 0  a 0 g (r, θ) Jn (αmnr/a) sin nθ r dr dθ in which αmn is the root of the equation Jn (αmn) = 0 and εn = ⎧ ⎪⎨ ⎪⎩ 2 n = 0 1 n = 0 . 10.13 Exercises 733 15. u (r, θ, t) = ∞ n=0 ∞ m=1 Jn (αmnr) exp (−αmnkt) [amn cos nθ + bmn sin nθ], where anm = 2 πεn [J ′ n (αmn)]2  2π 0  1 0 f (r, θ) Jn (αmnr) cos nθ r dr dθ, bnm = 2 π [J ′ n (αmn)]2  2π 0  1 0 f (r, θ) Jn (αmnr) sin nθ r dr dθ, where αmn is the root of the equation Jn (αmn) = 0 and εn = ⎧ ⎪⎨ ⎪⎩ 1 for n = 0 2 for n = 0. 16. u (x, y, z, t) = sin πx sin πy sin πz cos √ 3 πct . 18. u (r, θ, z, t) = ∞ n=0 ∞ m=1 ∞ l=1 Jn (αmnr/a) sin (mπz/l) cos (ωct) × [anml cos nθ + bnml sin nθ] + ∞ n=0 ∞ m=1 ∞ l=1 Jn (αmnr/a) sin (mπz/l) sin (ωct) × [cnml cos nθ + dnml sin nθ], where anml = 4 πa2lεn [J ′ n (αmn)]2  a 0  2π 0  l 0 f (r, θ, z) Jn (αmnr/a) × sin (mπz/l) cos nθ r dr dθ dz, bnml = 4 πa2l [J ′ n (αmn)]2  a 0  2π 0  l 0 f (r, θ, z) Jn (αmnr/a) × sin (mπz/l) sin nθ r dr dθ dz, cnml = 4 ω −1 πa2lεn [J ′ n (αmn)]2  a 0  2π 0  l 0 g (r, θ, z) Jn (αmnr/a) × sin (mπz/l) cos nθ r dr dθ dz, 734 Answers and Hints to Selected Exercises dnml = 4 ω −1 πa2l [J ′ n (αmn)]2  a 0  2π 0  l 0 g (r, θ, z) Jn (αmnr/a) × sin (mπz/l) sin nθ r dr dθ dz, where αmn is the root of the equation Jn (αmn) = 0 and ω = " (mπ/l) 2 + (αmn/a) 2 # 1 2 , εn = ⎧ ⎪⎨ ⎪⎩ 1; for n = 0 2; for n = 0. 20. u (r, θ, z, t) = ∞ n=0 ∞ m=1 ∞ p=1 (anmp cos nθ + bnmp sin nθ) ×Jn (αmnr/a) sin (pπz/l) e −ωt , where anmp = 4 πa2lεn [J ′ n (αmn)]2  a 0  2π 0  l 0 f (r, θ, z) Jn (αmnr/a) × sin (pπz/l) cos nθ r dr dθ dz bnmp = 4 πa2l [J ′ n (αmn)]2  a 0  2π 0  l 0 f (r, θ, z) Jn (αmnr/a) × sin (pπz/l) sin nθ r dr dθ dz, in which εn = ⎧ ⎪⎨ ⎪⎩ 1 for n = 0 2 for n = 0 and ω = " (pπ/l) 2 + (αmn/a) 2 # . 23. u (x, y, t) = ∞ m=1 ∞ n=1 umn (t) sin mx sin ny, where 10.13 Exercises 735 umn (t) = 4 (−1)m+n+1 mn αmnc sin (αmnct)  cos (1 − αmnc)t − 1 2 (1 − αmnc) + cos (1 + αmnc)t − 1 2 (1 + αmnc) 0 + cos (αmnct)  sin (1 − αmnc)t 2 (1 − αmnc) + sin (1 + αmnc)t 2 (1 + αmnc) 0 , and αmn =  m2 + n 2 1 2 . 25. u (x, y, t) = ∞ n=1 ∞ m=1 4A mnπ2 [(−1)n−1][(−1)n−1] k(n2+m2) " 1 − e −k(n 2+m2 )t # × sin nx (sin my − m cos my). 27. u (x, y, t) = x (x − π)  1 − y π sin t + ∞ n=1 ∞ m=1 vmn (t) sin nx sin my, where α 2 mn =  m2 + n 2 and vmn (t) = 8 exp  −c 2α 2 tα2 mn [1 − (−1)n ] π 2mn (1 + c 4α4 mn) c 2 n2  α 2 mn − n 2 × & cost exp  −c 2α 2 tα2 mn − 1 ' +  1 n2 + c 4α 2 mn sin t exp  −c 2α 2 tα2 mn . 30. u (x, y, t) = 4 4qb4 π5D 5 ∞ n=1,3,... 1 n5 " 1 − vn(x) 1+cosh(nπa/b) # sin (nπy/b), where vn (x) = 2 cosh 4nπa 2b 5 cosh 4nπx b 5 + 4nπa 2b 5 sinh 4nπa 2b 5 cosh 4nπx b 5 − 4nπx b 5 sinh 4nπx b 5 cosh 4nπa 2b 5 . 32. Hint: In region 1, x ≤ −a, the solution of the Schr¨odinger equation d 2ψ dx2 = κ 2ψ, κ2 = 2M 2 (V0 − E), is ψ1 (x) = A eκx + B e−κx , where A and B are constants. For boundedness of the solution as 736 Answers and Hints to Selected Exercises x → −∞, B ≡ 0, and hence, ψ1 (x) = A eκx . In region 2, x ≥ a, the solution of the Schr¨odinger equation is ψ2 (x) = C eκx + D e−κx . For boundedness as x → ∞, C ≡ 0. The solution is ψ2 (x) = D e−κx . In region 3, −a ≤ x ≤ a, the potential is zero and hence, the equation takes the simple form ψxx + k 2ψ = 0, where k 2 =  2M 2 E. The solution is ψ3 (x) = E sin kx + F cos kx. For matching conditions at x = a, ψ2 (a) = ψ3 (a), or, De−aκ = E sin ka+ F cos ak. (1) Similarly, matching conditions at x = −a gives ψ1 (−a) = ψ3 (−a), or A e−aκ = −E sin ak + F cos ak. (2) Further, matching the derivatives ψ ′ (a), ψ ′ (−a) gives ψ ′ 2 (a) = ψ ′ 3 (a) and ψ ′ 1 (−a) = ψ ′ 3 (−a), or −κDe−aκ = k (E cos ak − F sin ak), (3) κAe−κa = k (E cos ak − F sin ak). (4) Adding and subtracting (1) and (2) gives 2F cos ak = (A + D) e −aκ , 2E sin ak = − (A − D) e −aκ . Adding and subtracting (3) and (4) gives 2k E cos ak = −κ (A − D) e −aκ , 2k F sin ak = κ (A + D) e −aκ . Setting A − D = −A1 and A + D = A2, the last two sets of equations can be combined and rewritten as 2E sin ak − A1e −aκ = 0 2k E cos ak + κA1e −aκ = 0 ⎫ ⎪⎬ ⎪⎭ (5) and 2F cos ak − A2e −aκ = 0 2k F sin ak − κA2e −aκ = 0 ⎫ ⎪⎬ ⎪⎭ . (6) The set (5) has nontrivial solutions for E and A1 only if 10.13 Exercises 737           2 sin ak −e −aκ 2k cos ak κe−aκ           = 0 which gives k cot ak = −κ. Similarly, the set (6) has nontrivial solutions for F and A2 only if k tan ak = κ. Note that it is impossible to satisfy both k cot ak = −κ and k tan ak = κ simultaneously. Hence, there are two classes of solutions, and solution is possible in quantum mechanics only if the energy satisfies certain conditions. 1. Odd solutions: k cot ak = −κ. In this case, F = A2 = 0. In terms of dimensionless variables, ξ = ak and η = aκ with definitions of k and κ, it follows that ξ 2 + η 2 = a 2  k 2 + κ 2 = a 2 2M 2 (V0 − E) + 2M 2 E ! = 2M V0a 2 2 . (7) This represents a circle. In terms of ξ and η, we write k cot ak = −κ as ak cot ak = −aκ, or ξ cot ξ = −η. (8) The simultaneous solutions of equations (7) and (8) can be determined from graphs of these functions at their point of intersection. It turns out that both ξ and η assume the positive values in the first quadrant only. Clearly, in the range 0 ≤ α = 2M V0a 2 2 < π 2 4 , there is no solution. For  π 2 2 ≤ α ≤  3π 2 2 , there is one solution. Thus, the existence of solutions depends on the parameters, M, V0 and the range of the potential. A simultaneous solution determines the allowed energy for which the quantum mechanical motion is described by an odd solution. 2. Even solutions: k tan ak = κ. In this case, E = A1 = 0, and (5) still holds. We can write the above condition in terms of nondimensional variables as 738 Answers and Hints to Selected Exercises ξ tan ξ = η. (9) The simultaneous solutions of (7) and (9) can be found graphically as before. It follows from the graphical representation that (7) and (9) intersect once if 0 ≤ α<π2 in the first quadrant. There are two points of intersection if π 2 ≤ α < (2π) 2 . The number of intersections (solutions) increases with the value of the parameter α. For each such allowed value of the energy determined from the points of intersection, there is an even solution in the present case. Note also that, for both even and odd solutions, ψ (x) is nonzero outside the finite square well so that there exists a nonzero probability for finding the particle there. This result is different from what is expected in classical mechanics. Finally, if V0 → ∞, it is easy to see that the intersections occur at ξ = nπ,  n + 1 2 π which are in agreement with the analysis of the infinite square well potential discussed in Example 10.10.1. 33. Hint: The boundary conditions at x = −a yields the matching conditions A e−ika + B eika = C eaκ + D e−aκ , A e−ika − B eika =  iκ k   C eaκ − D e−aκ . These results give the desired solution. Similarly, matching conditions at x = a gives the desired answer. Combining the matching relations leads to the final matrix equation. 11.11 Exercises 739 11.11 Exercises 3. u (ρ, θ) = 1 2π  2π 0 (ρ 2−1)f(β)dβ [1−ρ2−2ρ cos(β−θ)] . 7. u (x, y) = −  2 b ∞ n=1 sin(nπy/b) sinh(nπa/b) sinh & nπ b (a − x) '  x 0 f (ξ) sinh nπξ b dξ + sinh  nπx b  a 0 f (ξ) sinh & nπ b (a − ξ) ' dξ . 8. u (r, θ) = − ∞ n=0 ∞ k=1 (R/αnk) 2 Jn (αnkr/R) (Ank cos nθ + Bnk sin nθ), where A0k = 1 πR2J 2 1 (α0k)  R 0  2π 0 rf (r, θ) J0 (α0kr/R) dr dθ Ank = 2 πR2J 2 n+1(αnk)  R 0  2π 0 rf (r, θ) Jn (αnkr/R) cos nθ dr dθ Bnk = 2 πR2J 2 n+1(αnk)  R 0  2π 0 rf (r, θ) Jn (αnkr/R) sin nθ dr dθ n = 1, 2, 3,...; k = 1, 2, 3,... and αnk are the roots of J (αnk) = 0. 9. G (r, r′ ) = e ik|r−r ′ | |r−r ′ | − e ik|ρ−r ′ | |ρ−r ′ | , where r = (ξ, η, ζ), r ′ = (x, y, z), and ρ = (ξ, η, −ζ). 10. G (r, r′ ) = e ik|r−r ′ | |r−r ′ | + e ik|ρ−r ′ | |ρ−r ′ | . 14. G = − 4a π ∞ n=1  ∞ 0 1 (α2a2+n2π2) sin  nπx a sin 4 nπξ a 5 sin αy sin αη dα. 16. u (r, z) = 2C π  ∞ 0  ∞ 0 1 (κ2−λ2−β2) J0 (βr) J1 (βa) cos λz dβ dλ. 17. u (r, θ) = A r 1 2 sin (θ/2). 18. G = − 2 a ∞ n=1 sinh σy′ sinh σ(y−b) σ sinh σb sin  nπx a sin 4 mπx′ a 5 , σ =  (κ 2 + (n2π 2) /α2), 0 < x′ < x < a, 0 < y′ <y x/c 1 2 [f (x + ct) + f (x − ct)] for t < x/c. 29. u (x, t) = ⎧ ⎪⎨ ⎪⎩ 0, for t < x/c f (t − x/c), for x/c < t ≤ (2 − x) /c. 30. u (x, t) = f0 + (f1 − f0) erfc 4 (x 2/4κt) 5 . 31. u (x, t) = x − x erfc  x/√ 4κt . 32. u (x, t)=2 t 0  η 0 erfc  x/√ 4κξ dη dξ. 33. u (x, t) = f0 e −ht 1 − erfc  x/√ 4κt !. 34. u (x, t) = f0 erfc  x/√ 4κt . 35. u (x, t) = ⎧ ⎪⎨ ⎪⎩ f0t, for t < x/c f0x/c, for t > x/c. 36. u (x, t) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 1 2 [f (x + ct) − f (ct − x)] , t < x/c 1 2 [f (x + ct) + f (x − ct)] , t > x/c. 744 Answers and Hints to Selected Exercises 37. V (x, t) = V0  t − x c H  t − x c . (i) V = V0H  t − x c , (ii) V = V0 cos & ω  t − x c ' H  t − x c . 38. u (z, t) = U t " 1+2ζ 2 erfc (ζ) − √ 2ζ π e −ζ 2 # , where ζ = z 2 √ νt . 41. V (x, t) = V0erfc 4 x 2 √ κt5 . 42. q (z, t) = a 2 e iωt " e −λ1z erfc( ζ − [it(2Ω + ω)] 1 2 ) +e λ1z erfc( ζ + [it(2Ω + ω)] 1 2 )# + b 2 e −iωt " e −λ2z erfc( ζ − [it(2Ω − ω)] 1 2 ) +e λ2z erfc( ζ + [it(2Ω − ω)] 1 2 )# , where λ1,2 =  i(2Ω + ω) ν 0 . q (z, t) ∼ a exp (iωt − λ1z) + b exp (−iωt − λ2z), δ1,2 =  ν |2Ω + ω| 01 2 . 43.  ν 2Ω 1 2 . 45. f (t) = f (0) + 1 Γ(α)Γ(1−α)  t 0 g (x) (t − x) α−1 dx. 46. x = a (θ − sin θ), y = a (1 − cos θ). 55. u (x, t) = ∞ n=1 sin nx t 0 e −n 2 (t−τ)an (τ ) dτ + ∞ n=1 bn (0) sin nx e−n 2 t , where an (t) = 2 π  π 0 g (x, t) sin nx dx, bn (0) = 2 π  π 0 f (x) sin nx dx. 57. u (x, t) = ∞ n=1 2 π sin &n − 1 2 x '  t 0  π 0 e −(2n−1)2 (t−τ)/4 × sin &n − 1 2 ξ ' g (ξ, τ ) dξ dτ . 61. u (x, t) = c 2 sinh x √ 1/c sinh π √ 1/c − sin x √ 1/c sin π √ 1/c sin t +  2 πc ∞ n=1 (−1)n+1 n n4−(1/c) 2 sin n 2 ctsin nx, in which  1/c is not an integer. 14.11 Exercises 745 65. f (x) =  ∞ 0 g (t) h (xt) dt, where h (x) = M−1 " 1 K(1−p) # . 14.11 Exercises 13. (c) With h = 0.2, the initial values are ui,0 (ih, 0) = sin π (ih). u1,0 = sin 0.2π = 0.5878, u2,0 = sin 0.4π = 0.9511. Also, u2,0 = u3,0 and u1,0 = u4,0. In each time step, there are 4 internal mesh points. We have to solve 4 equations with 4 unknowns. However, the initial temperature distribution is symmetric about x = 0.5, and u = 0 at the endpoints for all time t. We have u3,1 = u2,1 and u4,1 = u1,1 in the first time row and similarly for the other time rows. This gives two equations with two unknowns. 16. (a) y = x 3 , (b) y = sin x, (c) x 2 + (y − β) 2 = r 2 , where β and r are constants. 17. x = a (θ − sin θ), y = a (1 − cos θ). 18. Hint: I (y (x)) = 2π  x1 x0 y  1 + y ′2 1 2 dx. x = c1t + c2, y = c1 cosh t = c1 cosh 4 x−c2 c1 5 . (A surface generated by rotation of a catenary is called a catenoid). 21. (a) ∇4u = 0 (Biharmonic equation) (b) utt − α 2∇2u + β 2u = 0 (Klein–Gordon equation) (c) φt + αφx + βφxxx = 0, (φ = ux) (KdV equation) (d) utt + α 2uxxxx = 0 (Elastic beam equation) (e) d dx (pu′ )+(r + λs) u = 0 (Sturm–Liouville equation). 22.  2 2m ∇2ψ + (E − V ) ψ = 0 (Schr¨odinger equation). 24. utt − c 2uxx = F (x, t), where c 2 = T ∗/ρ. 746 Answers and Hints to Selected Exercises 29. Hint: yn = x (1 − x) n r=1 arx r−1 . Find the solution for n = 1 and n = 2. n =1: a1 = 5 18 , y1 = a1x (1 − x). n =2: a1 = 71 369 , a2 = 7 41 , y2 = x (1 − x) (a1 + a2x). 30. Hint: u1 (x, y) = a1xy. I (u1) = πab 4 " (a1 + 1)2 a 2 + (a1 − 1)2 b 2 # , a1 = 4 b 2−a 2 b 2+a2 5 . 31. Hint: u3 = x (1 − x) (1 − y) + x (1 − x) y (1 − y) (a2 + a3y). 32. Hint: u2 = x (2 − x − 2y) + a2 xy (2 − x − 2y). 33. Hint: ΨN =  N m,n=1 amn φmn =  N m,n=1 amn cos  mπx 2a cos  nπy 2b . 34. Hint: φn = cos (2n − 1) πr 2a ! . 35. Hint: I (u) =  a −a  a −a " ∇2u 2 −  4α a2 u # dx dy = min, and un =  x 2 − a 2 2  y 2 − a 2 2  a1 + a2x 2 + a3y 3 + ... . 36. Hint: Ψ1 =  b 2 − y 2 U (x). 37. (a) Introduce two functions φ and ψ and two parameters α and β such that U = u + αφ (x) and V = v + βψ (x). Then ∂I ∂α = 0 and ∂I ∂β = 0. 40. (a) uy − uxy ′ = uy′′ 1+y′2 . (b) y ′2 = 1−A 2 (y1−y) A2(y1−y) , (d) 2y ′′ − 3y + 3xy2 = 0. 42. Seek an approximate solution un (x, y) =  a 2 − x 2 b 2 − y 2 a1 + a2x 2 + a3y 2 + ... + anx 2ry 2s . For n = 1, f = 2 0 =  R (−u1xx − u1yy − 2)  a 2 − x 2 b 2 − y 2 dx dy = 2 a −a  b −b 1 − a1  a 2 − x 2 − a1  b 2 − y 2 ! a 2 − x 2 b 2 − y 2 dx dy = 32 9 a 2 b 3 − 128 45 (ab) 3  a 2 + b 2 a1, a1 = 5 4  a 2 + b 2 −1 and u1 = 5 4 (a 2−x 2 )(b 2−y 2 ) (a2+b 2) . 14.11 Exercises 747 The torsional moment M = 2Gθ a −a  b −b u1dx dy =  40 9 (Gθ) 4 a 3 b 3 a2+b 2 5 , where G is the shear modulus and θ is the angle of twist per unit length. When a = b, M =  20 9 Gθa4 ∼ 0.1388 (2a) 4 Gθ. The tangential stresses are τzx = Gθ 4 ∂u1 ∂y 5 , τzy = Gθ  ∂u1 ∂x . (b) The exact solution is u (x, y) = x (a − x) − 8a 2 π3 ∞ n=1 cosh{(2n−1) πy 2a } sin{(2n−1) πx a } (2n−1)3 cosh{(2n−1) πb 2a } . M = 2Gθ 1 a 3 b 6 − 32a 4 π5 ∞ n=1 1 (2n−1)5 tanh & (2n − 1) πb 2a ' 3 . For a = b, M = 0.1406 (2a) 4 Gθ. 43. This problem deals with the expansion of a rectangular plate under tensile forces. Make the boundary conditions homogeneous. Integrating the boundary conditions gives u0 = 1 2 c y2 4 1 − y 2 6b 2 5 . Set u = u0+u5 so that ∇4u5 =  2c b 2 and the boundary conditions become u5xy =0= u5yy for x = + a, u5xy =0= u5xx for y = + b. These boundary conditions hold if u5 = 0, u5x = 0 for x = + a, u5 = 0, u5y = 0 for y = + b. By the Rayleigh–Ritz method  R  ∇4un − f φkdx dy = 0, k = 1, 2, . . . , n, where the nth approximate solution un (x, y) has the form un (x, y) =  x 2 − a 2 2  y 2 − b 2 2  a1 + a2x 2 + a3y 2 + ... . For n = 1, 0 =  a −a  b −b 24a1  y 2 − b 2 2 + 16a1  3x 2 − a 2 3y 2 − b 2 +24a1  x 2 − a 2 2 −  2c b #  x 2 − a 2 2  y 2 − b 2 2 dx dy, 748 Answers and Hints to Selected Exercises or 4 54 7 + 256 49 b 2 a2 + 64 7 b 4 a4 5 a1 = c a6b 2 . When a = b, a1 = (0.04325) c a6 . u1 ∼ u0 + u51 = 1 2 c y2 4 1 − y 2 6b 2 5 + (0.04325)  c a−6 ×  x 2 − a 2 2  y 2 − b 2 2 . 44. dF dx = ∂F ∂x + ∂F ∂u du dx + ∂F ∂u′ · du′ dx = ∂F ∂x + u ′ ∂F ∂u + u ′′ ∂F ∂u′ d dx  u ′ ∂F ∂u′ = u ′ d dx  ∂F ∂u′ + ∂F ∂u′ · u ′′ . Subtracting the latter from the former with (14.6.12) we obtain d dx  F − u ′ ∂F ∂u′ = ∂F ∂x + u ′ ∂F ∂u − d dx  ∂F ∂u′ ! = ∂F ∂x . 45. H = I − λJ =  b a F (x, y, y′ ) dx =  b 0 p (x) y ′2 − q (x) y 2 − λ r (x) y 2 ! dx. The extremum of H leads to the Euler–Lagrange equation d dx 4 ∂F ∂y′ 5 − ∂F ∂y = 0. This leads to the answer. 46. (a) For simplicity, we assume that l is an integer and partition the interval into l equal subintervals. Each of the l − 1 = n interior vertices has the trial function vj (x) defined by vj (x) = ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ 1 − j + x for j − 1 ≤ x ≤ j, 1 + j − x for j ≤ x ≤ j + 1, 0 otherwise. vj (x) is continuous and piecewise linear with vj (j) = 1 and vj (k)=0 for all integers k = j. Appendix: Some Special Functions and Their Properties “One of the properties inherent in mathematics is that any real progress is accompanied by the discovery and development of new methods and simplifications of previous procedures ... The unified character of mathematics lies in its very nature; indeed, mathematics is the foundation of all exact natural sciences.” David Hilbert This appendix is a short introduction to some special functions used in the book. These functions include gamma, beta, error, and Airy functions and their main properties. Also included are Hermite and Webber–Hermite functions and their properties. Our discussion is brief since we assume that the reader is already familiar with this material. For more details, the reader is referred to appropriate books listed in the bibliography. A-1 Gamma, Beta, Error, and Airy Functions The Gamma function (also called the factorial function) is defined by a definite integral in which a variable appears as a parameter Γ (x) =  ∞ 0 e −t t x−1 dt, x > 0. (A-1.1) In view of the fact that the integral (A-1.1) is uniformly convergent for all x in [a, b] where 0 < a ≤ b < ∞, Γ (x) is a continuous function for all x > 0. Integrating (A-1.1) by parts, we obtain the fundamental property of Γ (x) 750 Appendix: Some Special Functions and Their Properties Γ (x) = −e −t t x−1 !∞ 0 + (x − 1)  ∞ 0 e −t t x−2 dt = (x − 1) Γ (x − 1), for x − 1 > 0. Then we replace x by x + 1 to obtain the fundamental result Γ (x + 1) = x Γ (x). (A-1.2) In particular, when x = n is a positive integer, we make repeated use of (A-1.2) to obtain Γ (n + 1) = n Γ (n) = n (n − 1) Γ (n − 1) = ··· = n (n − 1) (n − 2)··· 3 · 2 · 1 Γ (1) = n!, (A-1.3) where Γ (1) = 1. We put t = u 2 in (A-1.1) to obtain Γ (x)=2  ∞ 0 exp  −u 2 u 2x−1 du, x > 0. (A-1.4) Letting x = 1 2 , we find Γ  1 2  = 2  ∞ 0 exp  −u 2 du = 2 √ π 2 = √ π. (A-1.5) Using (A-1.2), we deduce Γ  3 2  = 1 2 Γ  1 2  = √ π 2 . (A-1.6) Similarly, we can obtain the values of Γ  5 2 , Γ  7 2 ,...,Γ  2n+1 2 . The gamma function can also be defined for negative values of x by rewriting (A-1.2) as Γ (x) = Γ (x + 1) x , x = 0, −1, −2,... (A-1.7) For example Γ  − 1 2  = Γ  1 2 − 1 2 = −2 Γ  1 2  = −2 √ π, (A-1.8) Γ  − 3 2  = Γ  − 1 2 − 3 2 = 4 3 √ π. (A-1.9) We differentiate (A-1.1) with respect to x to obtain A-1 Gamma, Beta, Error, and Airy Functions 751 Figure A-1.1 The gamma function. d dxΓ (x) = Γ ′ (x) =  ∞ 0 d dx (t x ) e −t t dt =  ∞ 0 d dx [exp (x log t)] e −t t dt =  ∞ 0 t x−1 (log t) e −t dt. (A-1.10) At x = 1, this gives Γ ′ (1) =  ∞ 0 e −t log t dt = −γ, (A-1.11) where γ is called the Euler constant and has the value 0.5772. The graph of the gamma function is shown in Figure A-1.1. Several useful properties of the gamma function are recorded below without proof for reference. Legendre Duplication Formula 2 2x−1 Γ (x) Γ  x + 1 2  = √ π Γ (2x), (A-1.12) 752 Appendix: Some Special Functions and Their Properties In particular, when x = n (n = 0, 1, 2,...) Γ  n + 1 2  = √ π (2n)! 2 2n n! . (A-1.13) The following properties also hold for Γ (x): Γ (x) Γ (1 − x) = π cosec πx, x is a noninteger, (A-1.14) Γ (x) = p x  ∞ 0 exp (−pt)t x−1 dt, (A-1.15) Γ (x) =  ∞ −∞ exp  xt − e t dt. (A-1.16) Γ (x + 1) ∼ √ 2π exp (−x) x x+ 1 2 for large x, (A-1.17) n! ∼ √ 2π exp (−n) x n+ 1 2 for large n. (A-1.18) The incomplete gamma function, γ (x, a), is defined by the integral γ (a, x) =  x 0 e −t t a−1 dt, a > 0. (A-1.19) The complementary incomplete gamma function, Γ (a, x), is defined by the integral Γ (a, x) =  ∞ x e −t t a−1 dt, a > 0. (A-1.20) Thus, it follows that γ (a, x) + Γ (a, x) = Γ (a). (A-1.21) The beta function, denoted by B (x, y) is defined by the integral B (x, y) =  t 0 t x−1 (1 − t) y−1 dt, x > 0, y> 0. (A-1.22) The beta function B (x, y) is symmetric with respect to its arguments x and y, that is, B (x, y) = B (y, x). (A-1.23) This follows from (A-1.22) by the change of variable 1 − t = u, that is, B (x, y) =  1 0 u y−1 (1 − u) x−1 du = B (y, x). If we make the change of variable t = u /(1 + u) in (A-1.22), we obtain another integral representation of the beta function A-1 Gamma, Beta, Error, and Airy Functions 753 B (x, y) =  ∞ 0 u x−1 (1 + u) −(x+y) du =  ∞ 0 u y−1 (1 + u) −(x+y) du, (A-1.24) Putting t = cos2 θ in (A-1.22), we derive B (x, y)=2  π/2 0 cos2x−1 θ sin2y−1 θ dθ. (A-1.25) Several important results are recorded below without proof for ready reference. B (1, 1) = 1, B  1 2 , 1 2  = π, (A-1.26) B (x, y) =  x − 1 x + y − 1  B (x − 1, y), (A-1.27) B (x, y) = Γ (x) Γ (y) Γ (x + y) , (A-1.28) B  1 + x 2 , 1 − x 2  = π sec 4πx 2 5 , 0 </y</x<></k<></k<></a<></k<></k<></k</r</x<></z<η></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></p<></x<></x<></x<></x<></x<></x<></x<></x<></x</x<></x<></x<></x<></r<></r<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></y<></x<></x<></x<></x<></y<></x<></x<></x<></x<></x<></x<></r<></r<></z<></z<></x<></x<π,></y<></y<></r<></r<></r<></k<></x<></z<a,></t<></t∗></x<></x<></k<b,></x</x</x<></x<></x<></x<></x<></x<></x<></x<></x<l,></x<></x<π,></x<></x<></x<></x<></x<π,></x<π,></x<π,></x<></x<></x<π,></x<π,></x<></x<></x<></x<π,></x<></x<></x<></x</x<></x<xj+1></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></x<></t<t0,></x<></x<></x<></x<></x<l,>

W C A T A L O G U E 2011 - 2012 WOOSTER, OHIO CONTENTS THE COLLEGE OF WOOSTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ABOUT THE COLLEGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 INDEPENDENT STUDY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 THE ACADEMIC PROGRAM THE CURRICULUM: A WOOSTER EDUCATION . . . . . . . . . . . . . . . . . . . . . . . . . 14 ACADEMIC RESOURCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 DEPARTMENTS, PROGRAMS, AND COURSES OF INSTRUCTION . . . . . . . . . . 20 Africana Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Ancient Mediterranean Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Anthropology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Archaeology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Art and Art History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Biochemistry and Molecular Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Business Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chemical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Chinese Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Classical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Comparative Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Computer Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 East Asian Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Environmental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102 Film Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 French . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Geology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 German Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Interdepartmental Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 International Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Latin American Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Neuroscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Physical Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Political Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Religious Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Russian Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Sociology and Anthropology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 South Asian Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Spanish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Theatre and Dance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Urban Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Women’s, Gender, and Sexuality Studies . . . . . . . . . . . . . . . . . . . . . . . . . 202 PRE-PROFESSIONAL AND DUAL DEGREE PROGRAMS . . . . . . . . . . . . . . . . . 206 OFF-CAMPUS STUDY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 SUMMER ACADEMIC PROGRAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 DEGREE REQUIREMENTS Bachelor of Arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Bachelor of Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Bachelor of Music Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Music Double Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Graduate and Professional Dual Degree . . . . . . . . . . . . . . . . . . . . . . . . . 226 ACADEMIC POLICIES Requirements for All Degree Programs and Commencement . . . . . . . 227 Majors and Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Registration, Courses, and Grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Academic Standing, Withdrawal, and Readmission . . . . . . . . . . . . . . . 236 Petitions for Exceptions to Academic Policies . . . . . . . . . . . . . . . . . . . . . 238 CODES OF COMMUNITY AND INDIVIDUAL RESPONSIBILTY . . . . . . . . . 238 STUDENT ACADEMIC CENTERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 STUDENT LIFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 STUDENT SERVICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 ADMISSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 EXPENSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 FINANCIAL AID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 HONORS AND PRIZES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 ENDOWED RESOURCES Endowed Chairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Endowed Funds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Endowed Scholarships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 BUILDINGS AND FACILITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 THE DIRECTORIES Presidents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Board of Trustees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 Faculty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Committees of the Faculty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Women‘s Advisory Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 ACADEMIC CALENDARS — 2010-2011 and Three-Year Calendar . . . . . . . . 360 TRAVEL DIRECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 CAMPUS MAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 CONTACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .inside back cover Photo by Scott Jones 5 THE COLLEGE OF WOOSTER Wooster was founded in 1866 by Presbyterians who wanted to do “their proper part in the great work of educating those who are to mold society and give shape to all its institutions.” The goal of the first Board of Trustees was to “establish an institution with broad foundations and facilities equal to the best in the land, capable of preparing men and women for every department of life, for the highest walks of science and all its forms.” A citizen of Wooster, Ephraim Quinby, donated a venerable oak grove set on twenty-two acres on a hill overlooking the Killbuck Valley, and the Trustees of the fledgling institution spent the next four years raising funds so that the school might open with buildings, books, a laboratory, scientific equipment, experienced faculty members, and an adequate endowment. On September 8, 1870, Wooster opened its doors as a university, with a faculty of five and a student body of thirty men and four women. By 1915 there were eight divisions, including a medical school whose faculty outnumbered those in the college of arts and sciences. Gradually, however, the institution’s definition as a liberal arts college had been evolving. In 1915 a traumatic episode occurred: there was a bitter fight over whether Wooster should establish yet another division within its structure. At first, the Trustees sided with the minority of the faculty which favored the new division, and then, after the resignation of President Holden, reversed themselves and supported the majority of the faculty which wished to devote itself entirely to undergraduates in the liberal arts. It was an angry struggle in which friends and colleagues of thirty years parted company. Speaking in Chapel in 1930, Howard Lowry, who was to become Wooster’s seventh President, gave some sense of the conflict which had occurred. As he recalled it, those who had triumphed in 1915 had told his entering class in 1919 that Wooster was “not a university nor a vocational school but a college of the liberal arts. . . .They told us to postpone for four years all training which would be directly useful and assured us that upon graduation we should be quite good for nothing. They summoned us to a way that was long and hard and full of grief. For ours was the impatience of youth and we could scarcely wait to give the world our impress. There were fortunes to be made, bridges to be built, and marriages to be contracted. We were in a frenzy to go places and do things. For many of us it meant entering seriously into debt and accepting questionable sacrifices from our loved ones, but down in our hearts we knew somehow that, if the world had in it truly educated men and women, here they were and they were worth attending to.” Thus, after the great conflict, Wooster, in the words of Dean Elias Compton, gradually “lopped off one appendage after another” and became a college of the liberal arts devoting itself exclusively to undergraduates. An aspiration for excellence marked the College from its inception. Jonas Notestein, a student in Wooster’s first graduating class, wrote that “a kind of prophetic feeling possessed us all that this was to be a great institution after a time, that we were starting ideals and setting standards and that it became us to do our The College of Wooster 6 very best so that the after generations of students would have something to be proud of.” The refrain of “something to be proud of” echoes through the years: the “habit of mastery” which became the trademark of the early faculty; the rebuilding of the College after the great fire of 1901, five buildings replacing one within a year’s time; President Wishart’s vigorous defense of the freedom of inquiry in a clash with William Jennings Bryan over the examination at Wooster of the subject of evolution; the practice of student research projects which led Karl Compton to work with George Bacon on x-rays in the early 1900s; Arthur Compton’s receipt of a Nobel Prize in 1927; and the establishment by Howard Lowry of Independent Study and the faculty leave program in the 1940s. Another important dimension of Wooster’s history is its early dedication to the education of women. Willis Lord, the first President, made a strong commitment to coeducation, warning the early classes that Wooster had the same expectations of its women as it had of its men and that men and women would be taught in the same classes and pursue the same curriculum. In 1870 this was a controversial policy, and a diary of one of the students who heard the announcement on the first day recorded the following observation: “Coeducation is announced as a feature of the institution. I think favorably of it myself but hear a great many saying that it will be a failure. I have heard ten reasons this afternoon why it must fail.” It did not fail, however, and women quickly assumed positions of leadership in the student body. The first Ph.D. granted by Wooster was given to a woman, Annie Irish, in 1882, and many of the early women graduates made careers for themselves in foreign missions, doing abroad what they could not easily do in this country — founding colleges, administering hospitals, and managing printing houses. Wooster’s concern for the education of women has remained unabated, and more recent women graduates have entered path-breaking careers in business, higher education, and the diplo matic corps. Likewise, on the matter of race, Wooster was clear from the beginning. The first President declared that Wooster should be a place of studies for all: “The sameness of our origin as men and women carries with it our original and essential equality. Had our national life been the true expression of our national creed, slavery would have been forever impossible. Caste, in whatever name, strikes at the soul of our humanity and liberty.” The first African-American student, Clarence Allen, entered the College in the 1880s, and the promise of the early vision still inspires the College. Today approximately seven percent of Wooster’s student body is African-American. In 1988, Wooster’s Board of Trustees created The Clarence Allen Scholarships to be awarded on the basis of academic merit. These scholarships commemorate the achievements of Wooster’s first African-American graduate a century ago. Wooster has long emphasized international education. An unusually high percentage of its early graduates went overseas as missionaries, and soon not only their sons and daughters but also the students from their schools were enrolling at Wooster as students. There were special houses for these students where every occupant spoke two or three languages and where friendships developed among students from Asia, Africa, and Latin America. A student living in one of these houses observed: “For much of the time, we were as far removed from the ordinary atmosphere of the surrounding Ohio farm country as if we had actually been transplanted to Asia.” This international presence affected the entire campus, establishing a tradition which continues to influence the College. Today approximately six percent of the student body is international in origin, representing more than 32 different countries. The College supports Modern Foreign Language and Cultural Studies in Chinese, French, German, Hebrew, Russian, and Spanish. In addition, there are programs in East Asian, South Asian, and Latin American Studies. The Comparative Literature and International Relations majors facilitate students’ global understanding through The College of Wooster 7 the study of literature, culture, history, economics, and politics. The College supports faculty and curriculum on global issues through the Hales Fund, which has recently funded faculty trips to India, China, Iceland, Cuba, Ghana, and Mexico. Off-campus study provides students with the opportunity to study in more than 60 countries. The recently opened Center for Diversity and Global Engagement, housed in Babcock Hall, provides an array of resources and helps students obtain an integrated view of issues relating to diversity and global understanding. Religion also played a vital part in the creation of the College. The Articles of Incorporation specify that the purpose of the institution is “the promotion of sound learning and education under religious influences.” Moreover, the College’s motto — Scientia et religio ex uno fonte (Science and religion from one source) — emphasizes the integrated life. For its first hundred years, the College was owned by the Synod of Ohio. In 1969, the Synod of Ohio voted to release ownership of the College and its assets to Wooster’s Board of Trustees, and thus today the College is a fully independent institution which, however, has voluntarily chosen to continue its relationship with The Presbyterian Church (USA) through a Memorandum of Understanding with The Synod of the Covenant. Wooster was a college born of a faith, a faith that education ought to be concerned with the total implication of things, both with those questions which may be empirically tested and those for which there are no definitive answers. Wooster has always possessed a strong Department of Religious Studies as well as the conviction that there is something beyond men and women which may confer a sense of proportion and worth on their lives and give them purpose and direction, a faith which Arthur Compton defined as “the best we know, on which we would willingly bet our lives.” The expressions of this religious spirit have been many and varied, and in each decade there have been student projects which express the ethical concerns of the time. In the midst of the Depression, Wooster students raised funds to send a graduating senior to India to teach, a tradition which continued until the 1970s. There were rice meals to raise money to assist international students and to bring refugees to this country from Nazi Germany. Today, approximately two-thirds of the College’s students are involved in volunteer service through the Wooster Volunteer Network, an umbrella organization that links College of Wooster students to volunteer organizations in the Wooster, Ohio, national, and international communities. Wooster’s graduates have continued the tradition of being oriented toward service and finding the purpose of their lives in fields through which they can enrich the lives of others. The aspiration to join the ability to think logically with the ability to act morally, to link science with service, to educate the heart as well as the mind, was present from the beginning and continues to inform the College and its graduates today. From the beginning, science was given a prominent place at the College because it was believed that scientific discovery could only lend greater weight to moral truth; science could, in President Lord’s words, give “silent but eloquent witness to the uncreated and the infinite.” There could be no conflict between reason and faith because of their common source, and whatever the unfettered mind found to be true would be in tune with the infinite harmony of the cosmos; the physical sciences should, therefore, be strong at Wooster. It is extraordinary, given the fierce religious convictions of the women and men who shaped Wooster and the conflict between science and religion in the late nineteenth century, to find the intensity with which these same religious convictions supported a scientific establishment at the College. There was nothing backward about Wooster’s physical sciences whose early graduates included Nobel laureate Arthur Compton and his brother Karl, who became President of Massachusetts Institute of Technology. This commitment to the sciences has endured in the progressive programs of quality in the departments of Biology, The College of Wooster 8 Geology, Physics, Mathematical Sciences, and Chemistry, which, for example, ranks in the top ten in the nation in the percentage of its graduates who eventually receive Ph.D.s. These are the memories of the past to which the College is entitled: “the habit of mastery,” the faith in liberal learning, the commitment to “put its students in the way of great things,” the commitment to offer studies for all regardless of gender or race, the international and religious dimensions of the College, and the strong commitment to the physical sciences. As Jonas Notestein understood more than a century ago, “It is our glory to dwell, to make a home and to become a part of an order which will go on after our time is finished.” Wooster and its more than 30,000 graduates have inherited this inspiring tradition. In a visit to Wooster, Robert Frost once said that if you had to love something, you could do worse than to give your heart to a college, and that those who attend Wooster have a sense of belonging to a succession of generations originating in the past and stretching into the future. INSTITUTIONAL ACCREDITATION AND MEMBERSHIPS The College is accredited by national, regional, and state agencies for academic excellence. It is accredited by the Higher Learning Commission and is a member of the North Central Association (www.ncahlc.org). Individuals may contact the Commission at: The Higher Learning Commission 30 North LaSalle Street, Suite 2400 Chicago, Illinois 60602-2504 Phone: 1-800-621-7440 / 312-263-0456 Fax: 312-263-7462 The State of Ohio Department of Education, the American Chemical Society, the National Association of Schools of Music, and the American Association of University Women have, for their various purposes, officially approved the academic standards of the College. The College is an institutional member of the American Council on Education, the Association of American Colleges and Universities, the Association of Independent Colleges and Universities of Ohio, the Association of Presbyterian Colleges and Universities, Council for the Advancement and Support of Education, the Great Lakes Colleges Association, Inc. (GLCA), the National Association of Independent Colleges and Universities, the Ohio College Association, and the Ohio Foundation of Independent Colleges, Incorporated. LOCATION AND ASSETS Wooster is in north-central Ohio. Cleveland is about 60 miles northeast, Columbus 90 miles southwest, and Pittsburgh 120 miles east. Five principal highways run through Wooster — U.S. Routes 30 and 250, and State Routes 3, 585, and 83. Bus service connects Wooster with all parts of the country. By air, Wooster may be reached through either the Cleveland or Akron-Canton airports. Cleveland-Hopkins Airport is about 50 miles due north of the campus, while Akron-Canton is about 35 miles east and north. The Wayne County Airport is about 5 miles northeast of Wooster and has a 5,200-foot paved east-west runway. A city of 26,000, Wooster is the county seat of Wayne County. It has representative industrial activity and is the business center for a rich agricultural district. The College grounds, comprising some 240 acres, are in a residential section about a mile north and east of the public square. On the south side of town is the Ohio Agricultural Research and Development Center, an integral part of The Ohio State University. About The College 9 As of June 30, 2010, the assets of the College were valued at $435 million. Investment in buildings, equipment, and grounds at the time amounted to approximately $126 million. The Endowment Funds at current market value, including trustee-designated endowment funds, totaled $229 million. ABOUT THE COLLEGE The College of Wooster draws together approximately 2,000 students and 160 faculty from diverse cultural backgrounds into an academic community committed to intellectual achievement, personal integrity, and respect for others. The liberal arts involve the study of human achievements in extending the boundaries of knowledge — of efforts to comprehend the unknown, to formulate values, to evolve and express a sense of human understanding. Wooster believes that such study will provide the best means of acquiring the capacity and perspective necessary in our complex and ever-changing world and the insight and vision to shape the future. The College believes, moreover, that all liberal education must be a continuing education that offers increase and renewal to the end of life. It does not assume that everything can and must be taught. It seeks, rather, a liberal education that will truly free undergraduates for a lifetime of intellectual adventure, one that will help them meet new situations as they arise, one that will allow them to develop harmoniously and independently. Students should expect to discover new worlds, both in courses and in the experiences they will have on the campus and in off-campus study. They will be expected to explore the intellectual life beyond the course work and experiences described elsewhere in this catalogue. They will discover the necessity of submitting their own patterns of thought to the rigors of analysis so that they are aware of identifiable criteria of growth. From their origin the liberal arts have been the essential preparation for the professions and for roles of leadership in society. They remain so. Wooster students who discover they are fascinated by chemistry or geology may pursue their work in medical school or in graduate study leading to a career in industry. The painter, the writer, the actor, or the musician may go on to a lifetime of performance and creation. Others will enter law, business, social work, teaching, the ministry, or foreign service. Whatever their choices, students will gain a deepened awareness of the possibilities available to them; Wooster’s educational program is designed to give flexibility in pursuing differing paths toward competence and achievement. Wooster has chosen to remain a small and predominantly residential college because its primary educational purpose is the intellectual fulfillment of the individual. We believe that the easy and informal association between students and faculty possible in this kind of institution fosters intellectual growth. A number of interdependent groups enhance the educational aims of the College. While students have the greatest share in the regulation of life within the residence halls and in matters relating to student government, members of the faculty and administrative staff, through the Campus Council, also participate in the governance of the social life of the College. Students in turn have a significant influence on the academic program through membership on faculty committees dealing with the structure of the curriculum and the educational life of the College. About The College 10 Wooster values its religious heritage and is committed to exploring its meaning for today’s world. The College’s commitment to the spiritual development and religious understanding of students is embodied in a religious perspectives requirement for all students, active student religious groups, and a covenantal relationship with the regional synod of the Presbyterian Church, USA. Westminster Presbyterian Church is the congregation-in-residence on the campus and assists in encouraging students to continue active participation in congregational life. Other congregations, the local Synagogue and Unitarian Fellowship also welcome students. Annual programs like the Clergy Academy of Religion, Theologian-in-Residence and Lay Academy of Religion provide opportunities for students to participate in discussion and exploration of important issues with members of the wider religious community. Active student groups like the Fellowship of Christian Athletes, Hillel at The College of Wooster, the Muslim Student Association, the Newman Catholic Student Association, Sisters in Spirit, and Wooster Christian Fellowship encourage both a fuller appreciation of one’s own religious heritage and a better understanding of the traditions and beliefs of others, as do courses in the Department of Religious Studies. This religious dimension lends an important tone to campus life and provides students an opportunity to make their own informed choices and to experience religion as a vital option for the creative person. As partners in liberal learning, Wooster students and faculty attempt in their individual pursuit of knowledge to acquire a sense of the relatedness of its parts, a perspective on its past, a basis for critical judgment, and an ability to bring informed and rigorous reflection to bear on contemporary problems. Through all their work students attempt to identify those values that give direction to human conduct. They grow in mind and spirit as they become increasingly aware of the complexity of human existence and as they learn to cope with ambiguity. They learn to ask the important question, to cut through irrelevance to the heart of issues, and ultimately to shape knowledge into vision and action. These observations and those which follow are an expression of the Mission Statement of the College as adopted by its Board of Trustees. Mission Statement Our institutional purpose – Why we exist and what we seek to accomplish: The College of Wooster is a community of independent minds, working together to prepare students to become leaders of character and influence in an interdependent global community. We engage motivated students in a rigorous and dynamic liberal education. Mentored by a faculty nationally recognized for excellence in teaching, Wooster graduates are creative and independent thinkers with exceptional abilities to ask important questions, research complex issues, solve problems, and communicate new knowledge and insight. Wooster’s Core Values The values that govern our shared pursuits and the ideas that we hold true: • Education in the Liberal Arts Tradition We believe that the most valuable approach to undergraduate education engages each student in a course of study that cultivates curiosity and develops independent judgment, creativity, breadth, depth, integration of knowledge, and intellec - tual skills in the tradition of liberal education tuned for the contemporary era. About The College 11 • A Focus on Research and Collaboration At Wooster, faculty and students are co-learners, collaborating in liberal inquiry. Our faculty’s commitment to excellence in teaching is nationally recognized for enabling students to realize their full potential as engaged scholars. We embrace unique pedagogical principles at Wooster: that research and teaching are integrated forms of inquiry, and that faculty and students share a common purpose in their pursuits of knowledge, insight, and creative expression. • A Community of Learners Wooster is a residential liberal arts college. As such, we believe the learning process unfolds on our campus and beyond, in conversations in classrooms and residence halls, libraries and studios, laboratories and on playing fields, and through the relationships that develop between and among students, faculty and staff and which endure long after graduation. We recognize that the very process of living together educates, and that much of the learning that is part of our mission takes place through artistic expression, the performance of music, theater, and dance, athletics, community involvement, and in the myriad student organizations that infuse vitality in campus life. We embrace a holistic philosophy of education and seek to nurture the physical, social, and spiritual well-being of our students. • Independence of Thought We are a community of independent minds, working together. We place the highest value on collegiality, collaboration, openness to persons and ideas in all of their variety, and the free exchange of different points of view. We vigorously champion academic freedom, and seek to sustain a campus culture where the understanding of each is made more complete through an on-going process of dialogue with others who think differently. • Social and Intellectual Responsibility As a community of learners, we hold ourselves to high standards of sound evidence, careful reasoning, proper attribution, and intellectual and personal integrity in all activities of teaching, learning, research, and governance. We recognize the privilege of being able, collectively, to pursue the mission of the College. We therefore seek to extend the benefits of learning beyond the campus and beyond ourselves, endeavoring to analyze problems, create solutions, exercise civic and intellectual leadership, and contribute to the welfare of humanity and the environment. • Diversity and Inclusivity Wooster actively seeks students, faculty, and staff from a wide variety of backgrounds, starting places, experiences, and beliefs. We believe that achieving our educational purpose is only possible in a diverse community of learners. Therefore, we value members who bring a diversity of identities and beliefs to our common purpose, and who reflect a diversity of voices as varied as those our students will engage upon graduation. Graduate Qualities Graduates of the College should demonstrate the following personal and intellectual qualities: • Independent Thinking, through the ability to: Engage in critical and creative thinking Devise, formulate, research, and bring to fruition a complex and creative project Embody the intellectual curiosity, passion, and self-confidence necessary for life-long learning Independent Study 12 • Integrative and Collaborative Inquiry, through the ability to: Synthesize knowledge from multiple disciplines Actively integrate theory and practice Engage in effective intellectual collaboration • Dynamic Understanding of the Liberal Arts, through the ability to: Understand disciplinary knowledge in arts, humanities, social sciences, mathematics, and physical and natural sciences Evaluate evidence using methodologies from multiple disciplines Demonstrate quantitative, textual, visual, and digital literacy Employ deep knowledge, insight, and judgment to solve real world problems • Effective Communication, through the ability to: Exhibit skill in oral, written, and digital communication Engage in effective discourse through active listening, questioning, and reasoning • Global Engagement and Respect for Diversity, through the ability to: Understand the histories, causes, and implications of global processes Engage with the global community through knowledge of a second language Understand and respect diverse cultural and religious traditions Display self-reflective awareness of their role as citizens in a diverse local, national, and global community • Civic and Social Responsibility, through the ability to: Appreciate and critique values and beliefs including their own Demonstrate ethical citizenship and leadership and embody a concern for social justice Exhibit a commitment to community and serving others INDEPENDENT STUDY The College of Wooster is nationally recognized for mentored undergraduate research, and for more than sixty years the Independent Study program has required that every graduate engage in mentored research and create an original scholarly work. The capacity for individual inquiry and expression marks the liberally educated person, and the Independent Study program at Wooster provides an opportunity through which this capacity may be nurtured. Describing the challenge of the program, President Lowry, out of whose vision the program was established, said, “it invites all students to come to their best in terms of their own talents.” Independent Study provides all students the opportunity to engage in an activity both personally meaningful and appropriate to their individual fields and interests. It is not reserved for the few. Independent Study is an integral part of a Wooster education and provides the basis for a lifetime of independent learning. Students begin in their first year to develop their abilities in writing, reading, and critical thinking required for the project and explore various areas of intellectual interest. Ideas for Independent Study are stimulated not only by course work in the major but also by courses in other areas, informal exchanges with faculty and students, visiting lectures and arts events, off-campus study, volunteer work, and internship experiences. Students beginning Independent Study are assigned a faculty adviser to serve as mentor, guide, and critic. Department or curriculum committee chairpersons will Independent Study 13 assign advisers after consultation with the student and appropriate faculty and consideration of the topic the student wishes to investigate. Each student works closely with his or her adviser through regularly scheduled meetings designed to assist, encourage, and challenge the student. Learning is approached as an exploration shared by student and adviser, each enjoying the opportunity to collaboratively search for solutions. Specific format and procedures vary from program to program. The Handbook for Independent Study provides general information on the program, and the Departmental/Program Independent Study Handbook for each major gives more specific details. Students should request a current copy of the Departmental/Program Independent Study Handbook when declaring a major. The first unit of Independent Study often consists of a seminar or a tutorial program, designed to explore the possible range of research and creative projects in the chosen field and to initiate the student into a methodology of research or the techniques necessary for creative work. Usually elected during the junior year, this introduction stresses the development of the student’s confidence and ability to carry out a more substantial project in the senior year. During the latter part of the first unit of Independent Study, a preliminary survey of exploration of the subject of the senior project may be undertaken. In the senior year the student spends two semesters working on a major investigative or creative project which culminates in the writing of a thesis or the production of a substantial creative work. Attention is given to the method, form, and content of intellectual activity, and there is an emphasis on the communication of the results of the individual’s own intellectual and creative achievement. Competitive grants from the Henry J. Copeland Fund for Independent Study make available funds to assist students with unusual expenses associated with their projects and to complete projects of exceptional distinction. Examples of Independent Study projects over the last few years include: — in Africana Studies, God and the Gods: Two Black Thelogies of Liberation — in Archaeology, “Wait, Are We Related?” A Critical Analysis of the Neanderthals and the Ancestry of Modern Humans With Regards to the Child from Abrigo do Lagar Velho, Portugal — in Art and Art History, An Exploration of Memory: A Grandfather’s Past, A Grandchild’s Present — in Biochemistry & Molecular Biology, An Investigation Into the Role of Creatine in Muscle Toxicity Associated with Statin-Use — in Chemistry, The Photophysical Behavior of Sunscreen Active Ingredients: A Combined Computational and Spectroscopic Study — in Comparative Literature, Resisting the Faust Myth: A Study of Two Post-Goethe Faustian Texts — in Communication Studies, To Entertain or to Inform: Mainland Chinese Audiences’ Perceptions of Entertainment Oriented Television News Programs — in Economics, The Effect of Foreign Direct Investment on Income Inequality — in English, Looking for Watts and Other Stories — in French, L’abecedaire de trois documentaires d’Agnes Varda — in Geology, Decline in Alaskan Yellow-Cedar: Tree-Ring Investigations into Climate Responses and Possible Causes, Glacier Bay, Alaska — in History, From “Big Jack” to “Bugsy”: Jewish Gangsters and the Jewish Immigrant Communities of Chicago and New York City, 1900-1933 The Academic Program 14 — in Mathematics, Using Math to Play Like a Champion: How Game Theory Can Be Used to Predict Behavior in International Relations, Biology, Politics and Economics — in Music Education/History, To Boldly Go Where No Country Has Gone Before: Star Trek and American Values in the Late Twentieth Century — in Philosophy, Environmental Ethics in the Arctic: Investigating the Source of Moral Obligations to Wilderness — in Physics, Percolation Via Electrical Conduction — in Political Science, Refugees and the Liklihood of Conflict: Does a High Influx of Refugees Increase the Likelihood of Conflict in a Host Country? — in Psychology, The Effects of Motivational Orientation in the Context of Competitive and Non-Competitive Environments — in Religious Studies, Lutheran, Methodist, Presbyterian . . . Does It Really Matter? A Study of the Changing Face of Mainline Protestant Denominationalism in the United States — in Sociology, Colorblind Youth: An Analysis of Black Youth’s Understanding of Racism in Contemporary American Society — in Spanish, La torre de Bable: Un estudio sobre la political inguistica de Franco y sus efectos nacionales y mindiales — in Theatre and Dance, Creating Authentic Fiction: An Examination of and Exercise in Mockumentary — in Women’s Studies, It Is A Wild Thing: Using the Connection Between Women and Horses to Ride Into Myth A full list of Independent Study titles for the current year is available on the College website. Each student is required to submit to The College of Wooster a digital copy of his or her thesis for archiving, granting to the College and its employees a nonexclusive, royalty-free license to archive it and make it accessible, in whole or in part, in any medium. The student retains all other ownership rights to the copyright of the thesis. THE ACADEMIC PROGRAM THE CURRICULUM: A WOOSTER EDUCATION A liberal arts education is not for four years but for a lifetime. As such, it should provide an intellectual experience that is both inherently valuable and also provides the resources necessary for a lifetime of inquiry, discovery, and responsible citizenship. These resources involve not the study of any particular discipline but the acquiring of certain intellectual abilities, including a critical disposition, an understanding of the nature of academic knowledge and the different ways of knowing that are reflected in the disciplines, the necessary skills to communicate effectively, an openness to inquiry in all its forms, and an appreciation of cultures and perspectives The Academic Program 15 that are different from one’s own. These abilities will help students to become independent learners for whom education is a life-long process and whose lives are marked by their commitment to knowledge and their ability to contribute meaningfully to their communities. The College of Wooster seeks to create such independent learners. A Wooster education can be characterized by how it identifies the goals of a liberal arts curriculum and how these goals relate to the process of creating engaged and independent learners and informed and involved citizens. • A liberal arts education should be rich in content and intellectually rigorous, to engage the minds and the imaginations of students and faculty alike. It should enable students to respond critically and creatively to the range of human inquiry into the nature of the physical world, society, and the human self, and to share their ideas orally, in writing, and through the forms of artistic expression. • A liberal arts education should help students to appreciate the nature of the academic disciplines—as intellectual tools that enable us to think in structured and systematic ways, and for the depth of inquiry they allow. By study in a number of disciplines, students should come to understand the different ways of knowing that are embodied in the disciplines, and by coming to know at least one discipline in depth, students should equip themselves to become scholars engaged in the creation of knowledge. By reflecting on the connections among the disciplines, students should appreciate how the understanding of a subject may be enlarged by different disciplinary approaches, how different kinds of knowledge are interrelated, and how work in one field is affected by developments in others. • A liberal arts education should prepare students for lives of responsibility in a pluralistic society and instill a breadth of understanding, concern, and commitment. It should provide opportunities to examine values, to reflect upon the richness and diversity of human experience, and to develop the necessary skills to contribute to the discussion of contemporary issues and to communicate effectively to individuals and across cultural differences. The kind of independence which Wooster seeks to inspire is epitomized in the program of Independent Study, in which students are required to demonstrate their capacity for critical inquiry, their ability to create new knowledge in a disciplinary context, and the necessary skills to share their learning with a larger community. While Independent Study represents the culmination of one’s learning in a discipline, the goals of the program go beyond disciplinary training. By engaging in the process of Independent Study, students come to regard learning as a process that requires a strong commitment, painstaking research, and the careful development of one’s approach to a subject. Through I.S., students come to understand not only their chosen subject but also the nature of learning itself, and they can bring this approach to other situations in their lives and careers. Because they have developed the resources necessary for independent learning, they can become effective citizens able to respond to the needs of their societies. These curricular goals find expression in the graduation requirements for each of the degrees the College offers: Bachelor of Arts, Bachelor of Music, and Bachelor of Music Education — see Degree Requirements. The College has emphasized its expectation that all students will complete academic coursework in a number of areas: First-Year Seminar, writing, global and cultural perspectives, religious perspectives, quantitative reasoning, learning across the disciplines, learning in the major, and Independent Study. The Academic Program 16 In addition to its departments and interdepartmental programs and courses, curricular opportunities are available through two College-wide programs, the Program in Writing and the Program in Interdisciplinary Studies. THE PROGRAM IN WRITING The College of Wooster has achieved a national reputation for its program of writing instruction, which extends from a student’s first year at the College through the senior year. This regimen, focused on the student as both an individual and a member of an academic community, is predicated upon the understanding that to write well involves a life-long learning process and that all students can improve their writing. A college education can enhance a student’s journey toward good writing, serving as a stage in that journey rather than an endpoint. The Program in Writing emphasizes the understanding that throughout this journey many forms of writing are possible and that writing can serve many different purposes. In keeping with this philosophy, the Program encourages students to use writing as a learning tool and to view their efforts through the complementary processes of writing-tolearn and learning-to-write. Specifically, the Program in Writing is designed to help students learn and practice the following characteristics of effective writing: • Range — Students should learn to write well in a variety of forms for a range of different kinds of readers. • Audience — Student writers should learn rhetorical strategies appropriate for the audience and purpose. • Argument — If the rhetorical strategy involves an argument, it should contain a thesis and develop that thesis with coherence, logic, and evidence. • Coherence — Whatever the purpose, the parts of a paper should contribute to a greater, connected whole. • Editing — Writing should be edited to address surface error, including irregularities in grammar, syntax, diction, and punctuation. To achieve these goals, the Program in Writing features four major components: • First-Year Seminar in Critical Inquiry (IDPT 10100), a writing-intensive course required of all entering students; • The College Writing Course (IDPT 11000), small group instruction to address problems in basic writing and required of some first-year students based on College assessment procedures; • The Writing-Intensive Course, a course offered by all departments and a requirement that students complete at least one such course before beginning Junior Independent Study; • Independent Study, a junior and senior year capstone experience requiring significant writing from students within their majors. For further information, contact the Dean for Curriculum and Academic Engagement. TEAM-TAUGHT INTERDISCIPLINARY COURSES Wooster has a long and proud tradition of courses and programs that are inter - departmental, interdisciplinary, and collaborative in nature. These courses and programs give students a window into the extent to which different disciplines cross-fertilize each other, incorporating materials, methods, and perspectives from each other. Each year the College will aim to offer six courses on a range of topics that will benefit from or require an interdisciplinary approach. Courses are typically team-taught by two faculty members from different departments or programs and provide opportunities for both students and faculty to experiment with new ideas, materials, and The Academic Program 17 pedagogies. These courses provide opportunities for students across the College to do course work that is integrative in nature and which can serve to model the making of such connections elsewhere in their academic programs. This kind of work also encourages students to think creatively and ambitiously as they plan for Independent Study, taking them into areas where exciting and even original projects can be undertaken. Enrollment in courses will be limited to 20. For courses to be offered annually through the Program, see Interdepartmental Courses. For further information, contact the Dean for Curriculum and Academic Engagement. EDUCATIONAL ASSESSMENT The College of Wooster assesses each student’s learning as they progress through the curriculum. Senior Independent Study is a particular focus for assessment as it represents the culmination of a student’s undergraduate academic journey. In addition to the individual-level assessment of learning, the College also has a formal program of systematic assessment of student learning. The inception of this program coincided with faculty approval and adoption of the academic curriculum, A Wooster Education. Out of an initial focus on the general education curriculum has grown an evolving program of assessment of student learning and development that includes general education, graduate qualities, high-impact educational practices, majors and minors, courses, and co-curricular and extra-curricular activities and functions. Departments and programs use their assessment findings to improve pedagogy, enhance programs, and shape Wooster’s curricular and co-curricular offerings. Ultimately, The College of Wooster is committed to continual improvement of student learning and development through assessment as it relates to the educational mission of the College. Wooster’s program of assessment is a shared experience, characterized by collaborative engagement by faculty, staff, and administration. The College’s Assessment Committee, whose membership is comprised of faculty, staff, administrators, and students, has an advisory and resource role. The College shares its assessment practices and findings externally as well as internally, and has contributed to the national conversation on assessment in higher education. The president of the College has joined the President’s Alliance of the New Leadership Alliance for Student Learning and Accountability and is a signatory to the Consortium on Financing Higher Education’s (COFHE) statement on assessment in higher education. By joining the Presidents’ Alliance, he has committed the College to improving significantly its assessment of, and accountability for, student learning. Faculty and staff have published promising practices and findings about assessment in teaching, assessment, and research journals, and have presented innovations in teaching, learning and assessment at professional conferences. The College has been awarded multi-institutional grants to assess several aspects of a liberal education, and faculty and staff have further participated in other college and university’s multi-institutional grants to assess student learning in the liberal arts. The College has also been a partner campus to the Association for American Colleges and Universities (AAC&U) for its Valid Assessment of Learning in Undergraduate Education (VALUE) project, as part of its Liberal Education and America’s Promise (LEAP) initiative. The educational assessment weblog of the College is an important source of assessment news, practices, findings, and resources for both internal and external audiences (http://assessment.voices.wooster.edu). Academic Resources 18 ACADEMIC RESOURCES The College’s commitment to an academic program of the highest quality and to the program in Independent Study is reflected in the excellence of the resources that are available to students and faculty. These resources include a talented and dedicated staff, facilities and equipment that incorporate the most modern technologies, and a traditional campus of exceptional beauty. FACULTY A strong teaching faculty is Wooster’s paramount asset. All courses are taught by regular faculty members, with senior faculty often teaching introductory courses. The faculty numbers approximately 160 members holding advanced degrees from institutions across the United States and abroad. While teaching is the pre-eminent commitment of the faculty, the College regards continuing education as a necessity for its faculty no less than its graduates. The benefits students derive from studying with faculty who are committed to developing as teachers and scholars, growing in their respective fields and often exploring new areas in and out of their disciplines, are an essential element of a Wooster education. Wooster’s faculty is professionally active and productive, as reflected in an outstanding record of publications, papers, performances, and other measures of scholarly accomplishment. To support the intellectual life of the faculty, the College has established a generous program of research and study leaves that recognizes the importance of the faculty’s ability to employ new materials, concepts, and technologies in directing student research. EDUCATIONAL PLANNING AND ACADEMIC ADVISING At The College of Wooster, all academic advising is done by members of the faculty, and the adviser-advisee relationship is among the most important relationships a student will form. The adviser assists the advisee in the construction of his or her academic program in a number of important ways: by providing information about requirements, policies, procedures, and educational options; by assisting students in planning a program that is consistent with their interests and abilities; and by helping students to integrate the resources of the College to meet their educational needs and aspirations. Although decisions about policies, course selection, and construction of an educational program are ultimately the responsibility of the student, the adviser provides an essential resource. For entering students, the student’s instructor in First-Year Seminar in Critical Inquiry is also his or her faculty adviser; as such, the adviser will have special insight into the student’s background and interests, goals and needs, strengths and weaknesses. When the student declares a major, a new adviser is assigned who will help to introduce the student into the discipline as a professional, socialize the student into the culture of the department or program, mentor the student closely in the development of an appropriate academic program, and look for special opportunities that will help the student to grow both as a major in the discipline and as a liberally educated person. When the student undertakes the senior project in Independent Study, the adviser will work extremely closely with the student and mentor his or her final development as a student-scholar in the discipline and prepare the student for graduation. The adviser will also counsel the student and offer assistance as he or she plans for life and a career after Wooster. In Fall 2010, the College established a new Educational Planning and Advising Center (EPAC) to supplement faculty advising of first-years and sophomores. For Academic Resources 19 more information, please contact Alison Schmidt, Department of Education, Associate Dean for Educational Planning and Advising, or Karen Parthemore, Administrative Coordinator, at 330-263-2428. THE COLLEGE OF WOOSTER ART MUSEUM IN EBERT ART CENTER The College of Wooster Art Museum has been located in the Ebert Art Center since 1998, and presents six to eight exhibitions each academic year in two galleries—the Sussel Gallery and the Burton D. Morgan Gallery. The museum’s permanent collections are comprised of over 8,500 objects, and although the facility operates much like a kunsthalle (art hall) by mounting temporary exhibitions, at least one exhibition each year is dedicated to presenting collection materials. Additionally, collection materials are available for study, classroom, and other teaching and research purposes. The art museum supports and enhances the College’s goals of teaching, research, and service through exhibitions, scholarship, collection preservation, and public engagement. Because artists play a crucial role in all aspects of culture and society, direct experiences with original works of art actively supports the teaching of critical thinking and visual literacy through engagement with art forms—from ancient to contemporary—presented within a social and historical context. The museum program also promotes campus-wide collaborations and interdisciplinary dialogue, and acts a catalyst for creative engagement both on campus and between the College and regional and national audiences. For more information about The College of Wooster Art Museum and its program visit: artmuseum.wooster.edu or contact Kitty Zurko, Director, or the Administrative Coordinator at 330-263-2388. INFORMATION TECHNOLOGY Information Technology (IT) at Wooster facilitates access to and use of information, communication, and collaboration technologies. IT strives to provide technology resources that are appropriate in the context of Wooster’s liberal arts tradition, its mission, and its core values. The use of information technology resources is integral to students’ development in each of Wooster’s Graduate Qualities. Students, faculty, and staff have access to information resources, communications and multimedia tools, software applications, and specialized computing environments. They are supported in their endeavors by a team of professional staff and a team of Student Technology Assistants. Wooster’s campus network provides access to campus technology and Internet resources. Pervasive wired and wireless networks make it possible for students to use their notebook computers anywhere on campus for research, study, work, communications, and entertainment. For additional information about Information Technology at Wooster, please visit the Information Technology section of the College’s website. LIBRARIES The College of Wooster Libraries consist of the Andrews Library (1962), made possible largely through a gift from the late Mabel Shields (Mrs. Matthew) Andrews of Cleveland; the Flo K. Gault Library for Independent Study (1995), made possible by a major gift from Stanley and Flo K. Gault of Wooster; and the Timken Science Library in Frick Hall, the original University of Wooster Library (1900-62), the gift of Henry Clay Frick of Pittsburgh, and renovated in 1998 largely through the gift of the Timken Foundation of Canton, Ohio. The libraries provide seating for nearly 800 library users, including over 350 carrels for seniors engaged in Independent Study. Eight group study rooms allow small groups of students to work collaboratively. All libraries have secure wireless access to the Internet. Academic Resources 20 The libraries contain approximately one million items including books, periodicals, microforms, recorded materials, newspapers, and government publications. The libraries are a selective depository for United States government publications. There are several special collections. Most notable is the Wallace Notestein Library of English History; others include the McGregor Collection of Americana, the Homer E. McMaster Lincoln Collection, the Paul O. Peters Collection on rightist American politics, the Gregg D. Wolfe Memorial Library of the Theatre, and the Josephine Long Wishart Collection of women’s advice literature, “Mother, Home, and Heaven.” The extensive microtext collections include the Atlanta University-Bell & Howell Black Culture Collection, the Library of American Civilization, Herstory, and the Greenwood Science Fiction Collection. Wooster’s library catalog is part of CONSORT, an electronic catalog shared with Denison University, Kenyon College, and Ohio Wesleyan University. CONSORT, in turn, is part of OhioLINK, a network of 88 academic and public libraries throughout the state. Wooster faculty and students may order any of over 48 million books and other materials directly from any CONSORT or OhioLINK library via the online catalog and receive them within 2–3 working days. Interlibrary loan of books from out-of-state libraries or periodical articles is also available. The CONSORT and OhioLINK catalogs, as well as more than 200 other electronic reference databases and more than 60,000 electronic journals, are available in residence halls and faculty offices via the campus computer network. The campus’ Virtual Private Network provides Wooster faculty, staff, and students with worldwide access to electronic library resources. The libraries also include classrooms, computer labs, and the Media Library, which houses the libraries’ collection of recorded materials and listening and viewing stations. Librarians are available to assist users in locating information. Aid is given at the reference desk, in course-related presentations, or in individual consultations. An active information fluency program equips students at all levels for independent research. For more information, please contact Mark Christel, Director, or Sharon Bodle, Administrative Coordinator, at 330-263-2152. DEPARTMENTS, PROGRAMS, AND COURSES OF INSTRUCTION COURSE NUMBERING The College of Wooster uses a five-digit course numbering system. The first three digits indicate the primary course number. The next two digits are the secondary course number and indicate whether there is a special focus for the course. For example: H I S T 1 0 1 7 6 . T H E H I S T O R Y O F I S L A M Department Course Title Primary Course Number Secondary Course Number Academic Resources 21 The first letters are the department or program abbreviation. The next three digits are the primary course number (101 is the primary course number for all Introduction to Historial Study courses). The last two digits are the secondary course number. These two digits indicate that the special focus for this HIST 101 course is The History of Islam. A course with a given three-digit primary course number can only be taken once for credit unless specifically indicated otherwise by the department. The following policy has been used in assigning primary course numbers: • 100-level courses are usually introductory courses; some 100-level courses do have prerequisites, and students are advised to consult the description for each course. • 200-level courses are usually beyond the introductory level, although many 200- level courses are open to first-year students and to majors and non-majors. • 300-level courses are seminars and courses primarily for majors but open to other students with the consent of the instructor. • The following numbers are for Independent Study: I.S. 40100 (Junior Independent Study), I.S. 45100 and I.S. 45200 (Senior Independent Study). In addition to the regular course offerings, many departments offer individual tutorials under the number 40000 and internships under 40700-40800. On occasion, departments will offer a course on a special topic as approved by the Educational Policy Committee, designated 19900, 29900, or 39900. ABBREVIATIONS In keeping with the general education requirements of the College’s curriculum (see Degree Requirements), course listings employ the following abbreviations: W Writing Intensive (W† indicates that not all sections are Writing Intensive) C Studies in Cultural Difference R Religious Perspectives Q Quantitative Reasoning AH Learning Across the Disciplines: Arts and Humanities HSS Learning Across the Disciplines: History and Social Sciences MNS Learning Across the Disciplines: Mathematical and Natural Sciences Except where otherwise noted, all courses carry one course credit. Africana Studies 22 AFRICANA STUDIES Josephine Wright, Chair Boubacar N’Diaye Charles Peterson Black Studies began at The College of Wooster in 1968 as an interdisciplinary Program, examining the history and culture of peoples of the African diaspora from an African-centered perspective. It moved to departmental status in 2000. The fundamental mission of Africana Studies is critical study of peoples of African ancestry from social, historical, and cultural perspectives not covered by traditional disciplines. It seeks to provide students comprehensive exposure to the experiences of Black people, wherever they reside, from multiple theoretical and methodological approaches designed to help them think critically about issues related to the African diaspora and educate global citizens who understand the intellectual history, origin, purpose, and challenges of Africana Studies as a distinct discipline within the liberal arts. By the end of the senior year, a Wooster Africana Studies graduate should be able to: identify and articulate the intellectual history, origin, purposes, and challenges of Africana Studies within the academy; identify and explain the connections between Africana Studies to historic Africa and the contemporary experiences of people of African descent around the world; identify and explain major historical events crucial to Africana people and their experiences in Africa, North America, the Caribbean, Europe, and other parts of the world; identify, articulate, and apply relevant Africancentered theories and methodologies to the investigation or critical analysis of topics, texts, artistic productions, events, or phenomena related to the African diaspora; and conceptualize, research, organize, and write an independent study project that meets the learning outcomes of the department. Major in Africana Studies Consists of twelve courses: • AFST 10000 • Four 200-level Africana Studies courses taken within the Department of Africana Studies • One 300-level Africana Studies course taken within the Department of Africana Studies • Three electives from the Department of Africana Studies or cross-listed courses accepted for AFST credit • Junior Independent Study: AFST 40100 • Senior Independent Study: AFST 45100 • Senior Independent Study: AFST 45200 Minor in Africana Studies Consists of six courses: • AFST 10000 • Three 200-level Africana Studies courses taken within the Department of Africana Studies • One 300-level Africana Studies course taken within the Department of Africana Studies • One elective from the Department of Africana Studies or cross-listed courses accepted for AFST credit Africana Studies 23 Special Notes • Course sequence suggestions for majors: First Year: AFST 10000, 20000-20019 Sophomore Year: AFST 21300 One AFST 200-level course One elective from AFST or cross-listed courses Junior Year: One AFST 200-level course One AFST 300-level course One elective from AFST or cross-listed courses AFST 40100 Senior Year: AFST 45100, 45200 One elective from AFST or cross-listed courses • S/NC courses are not permitted in either the major or minor. • Only grades of C- or better are accepted for the major or minor. AFRICANA STUDIES COURSES AFST 10000. INTRODUCTION TO AFRICANA STUDIES (Education) Interdisciplinary foundation course presents overview of the historical, social, psychological, political, economic, and cultural experiences of all the major branches of people of African descent. Course focuses on the contributions and achievements of Africana people, with some emphasis on African Americans, and it explores the concerns as well as the challenges they face. Students are introduced to African-centered perspectives of prominent continental and diasporic scholars, artists, and activists, who mostly challenge the tenets and assumptions of the dominant cultural and intellectual paradigms. Annually. Fall and Spring. [C, AH, or HSS] AFST 20000-20019. SPECIAL TOPICS IN AFRICANA STUDIES (some sections cross-listed with Women’s, Gender, and Sexuality Studies) An in-depth examination of an issue or topic relevant to the Black experience. Possible topics include Black biography and autobiography, post-colonial struggles, Maroon communities, civil rights, anti-colonial resistance movements, and Blacks in science and society. Topics vary and will be designated to meet the Learning Across the Disciplines requirement as appropriate. Annually. Fall and Spring. [Depending on the topic, W†, C, AH, or HSS] AFST 21200. SURVEY OF AFRICAN-AMERICAN FOLKLORE: THE CREATIVE AND PERFORMING ARTS (Music) Study of African American folklore in the United States. Focuses on the contextual and historical framework in which folk music, tales, religious practices, and the visual arts evolved. Examines the impact of these traditions on contemporary American society. Open to non-music majors. No technical knowledge required. Prerequisite: AFST 10000 or permission of the instructor. Not offered 2011-2012. [C, AH] AFST 21300. RACISM 101 Americans have historically found it difficult to discuss issues of racism openly. This course examines from historical perspectives the foundations of racism towards Blacks as a vestige of chattel slavery in the United States, and it explores various manifestations of racism in Black-White relationships in contemporary American society. Prerequisite: AFST 10000 or permission of the instructor. Annually. Fall 2011 and Spring 2012. [W†, C, HSS] AFST 23100. SURVEY OF CONTEMPORARY AFRICA Course surveys the major areas and issues in contemporary Africa using an interdisciplinary approach. It explores in some detail the major post-colonial cultural, economic, political, and societal structures, dynamics, ideas, and trends that depict contemporary Africa as shaped by its recent colonial history and international environment. The course critically examines these realities, the potential, and the challenges facing African soci- Africana Studies 24 eties in this new century. Connections are made between these features and the globalizing world community, with special focus on U.S.-African multiform relations. Prerequisite: AFST 10000 or permission of the instructor. Fall 2011. [C, HSS] AFST 24000. AFRICANA WOMEN IN NORTH AMERICA: EARLIEST TIMES THROUGH THE CIVIL RIGHTS MOVEMENT (Women’s, Gender, and Sexuality Studies) Africana women in North America have historically suffered from racial, class, and gender oppression. Historically their oppression in American society resulted from exploitation of their labor, historical patterns of disenfranchisement from institutions controlled by the dominant society, as well as persistent stereotyping by mainstream U.S. society as justification for this exclusion. Course critically examines historic Africana women from colonial times to the Civil Rights Movement through the lens of black feminist theory, investigates their responses to such oppression, and explores the contributions these women made to social, intellectual, and cultural history of the United States. Emphasizes primary readings. Prerequisite: AFST 10000 or permission of the instructor. Not offered 2011-2012. [C, HSS] AFST 24100. AFRICANA WOMEN IN CONTEMPORARY SOCIETY (Women’s, Gender, and Sexuality Studies) Course examines the ways in which contemporary society since the 1960s has shaped the lives of Africana women and how these women have influenced U.S. society. Examines such issues as family life, education, career opportunities, political activities, Africana male/female relationships, societal constraints on their lives, and Africana women’s roles in civil rights and feminist movements. Prerequisite: AFST 10000 or permission of the instructor. Alternate years. Not offered 2011-2012. [C, HSS] AFST 24200. MARTIN, MALCOLM, AND MANDELA Course examines the lives, philosophies, contributions, and legacies of three outstanding leaders to the struggles of people of African descent for civil and human rights in the 20th century. The course will focus on comparing and contrasting their lives, ideas, and actions while situating these in the historical and socio-political contexts that shaped them. Prerequisite: AFST 10000 or permission of the instructor. Alternate years. Spring 2012. [C, HSS] AFST 24400. CINEMA OF AFRICA AND THE AFRICAN DIASPORA (Film Studies) This Course explores issues of race, class, culture, the colonial, and the anti-colonial thought through an examination of cinema created within and focusing on continental and diasporic African life. Accompanying the cinematic texts will be an array of written texts that contribute to the class discussion across the fields of history, post-colonial theory, and film theory. Prerequisite: AFST 10000 or permission of the instructor. Alternate years. Not offered 2011-2012. [C, AH] AFST 24500. PAN-AFRICANISM Course focuses on the political, cultural, and social articulations of the Pan-African idea by major scholars, leaders, and activists who sought to create global unity among peoples of African ancestry, where ever they resided, as well as the various attempts to implement their theories in practice in Africa, the Caribbean, and the United States throughout the twentieth century. The successes and failures of the unfolding experiments on the African continent since independence and the similar efforts in the diaspora are examined with an eye toward identifying their implications for the future of Pan-Africanism. Prerequisite: AFST 10000 or permission of the instructor. Alternate years. Spring 2012. [C, HSS] AFST 24600. SURVEY OF AFRICANA POPULAR CULTURE Course surveys the historical evolution and cultural consequences of Africana popular culture. The antecedents of “Gangsta Rap,” “Hip Hop,” “Reggae-Rastas,” and contemporary modes of “attitude,” behavior, dress, speech, and public representation are part of a self-reinforcing African-Black New World dialectic. In this course contemporary Black urban youth culture is analyzed as aesthetic and socio-cultural vehicles for personal and public critique and transformation. Africa and its various diasporas have created a world-view that transcends global boundaries of class, culture, gender, race, and society. This culture, now universally recognized, has been marketed for global public consumption. The course explores this phenomenon. Prerequisite: AFST 10000 or permission of the instructor. Alternate years. Not offered 2011-2012. [C, AH] AFST 24700. BLACK NATIONALISM Course examines from a geographic-specific context the political, cultural, and theoretical aspects of historic and contemporary African diasporic nationalist movements. By examining major figures, texts, and movements, the course investigates the ways in which race, class, and culture inform Black nationalist theory and practice. Prerequisite: AFST 10000 or permission of the instructor. Alternate years. Spring 2012. [C, HSS] Africana Studies 25 AFST 30000. CRITICAL READINGS IN AFRICANA STUDIES Advanced special topics seminar focuses on critical issues in a variety of locations and time periods crucial to understanding Africana Studies. Possible readings include the works of John Bracey, W.E.B. Du Bois, Frantz Fanon, C.L.R. James, John Hope Franklin, Fannie Lou Hammer, Vincent Harding, Benjamin Mays, August Meier, Joanne Robinson, Carter G. Woodson, C. Van Woodward, etc. Prerequisite: AFST 10000, one 200-level Africana Studies course, or permission of the instructor. Alternate years. Not offered 2011-2012. AFST 30100. AFRICANA RESISTANCES Surveys social, cultural, and political movements, individual and group thoughts, and actions in the historical and ongoing struggle against oppression. Examines multiform resistances by Africana people against enslavement, colonization, and other forms of oppression and discrimination. Explores these rich traditions as unique expressions and illustrations of the human spirit that inexorably strives for freedom, justice, and dignity. Examines and critically analyzes resistances to enslavement in Africa, the Americas, and Europe, as well as the fight for emancipation and civil rights in these regions throughout the 20th century. Prerequisite: AFST 10000, one 200-level Africana Studies course, or permission of the instructor. Alternate years. Not offered 2011-2012. [C, HSS] AFST 30200. MARXISM AND AFRICANA RADICAL THOUGHT Course offers students an in-depth opportunity to read and examine major thinkers and works of the “Black Radical Tradition.” More specifically, the course will contrast and examine the ways African continental and diasporic thinkers and activists engage, borrow from, contribute to, and expand the theories of Karl Marx, Friedrich Engels, and Vladimir L. Lenin. The goal is to show the various ways in which Africana radical thought has re-calibrated Marxian thought and activism through the particularities of the Africana experience. Prerequisite: AFST 10000, one 200-level Africana Studies course, or permission of the instructor. Alternate years. Fall 2011. [HSS] AFST 40000. TUTORIAL Offered to individual students under the supervision of an Africana Studies faculty member on a selected topic. Permission of the chair of Africana Studies is required. Arrangements must be made with the supervising faculty member before registration. Prerequisite: The approval of both the supervising faculty member and the chairperson are required prior to registration. AFST 40100. INDEPENDENT STUDY Group tutorial taken during one semester of the junior year includes bibliographic and methodological instruction and a written essay/project designed by the student. Special attention will be given to the disciplinary concerns in the humanities and social science areas of Africana Studies. Prerequisite: AFST 10000 and three 200- level Africana Studies courses. AFST 45100. INDEPENDENT STUDY THESIS – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: AFST 40100. AFST 45200. INDEPENDENT STUDY THESIS – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: AFST 45100. CROSS-LISTED COURSES ACCEPTED FOR AFRICANA STUDIES CREDIT ART AND ART HISTORY ARTD 22000. AFRICAN ART [C] ARTD 23000. AFRICAN-AMERICAN ART [C, AH] ENGLISH ENGL 21007. BLACK LITERATURE AND CULTURE (this course only) [AH] ENGL 23002. SURVEY OF AFRICAN AMERICAN LITERATURE (this course only) [AH] FRENCH FREN 23500. LITERATURE AND CULTURE OF FRANCOPHONE AFRICA [C] Archaeology 26 HISTORY HIST 11500. HISTORY OF BLACK AMERICA: FROM WEST AFRICAN ORIGINS TO THE PRESENT [C, HSS] HIST 10173. SOCIAL HISTORY OF HIP HOP HIST 23100. AFRICA BEFORE 1800 [C, HSS] HIST 23200. AFRICA SINCE 1800 [C, HSS] HIST 24600. UNITED STATES URBAN HISTORY [HSS] MUSIC MUSC 16500. GOSPEL CHOIR (.125 CREDIT) MUSC 21400. HISTORY OF AFRICAN AMERICAN MUSIC [C, AH] MUSC 21700. SURVEY OF JAZZ [C, AH] PHILOSOPHY PHIL 23400. AFRICAN PHILOSOPHY [C, AH] POLITICAL SCIENCE PSCI 20800. RACE AND POLITICS [C, HSS] PSCI 21300. THE CONSTITUTIONAL LAW OF CIVIL RIGHTS [C, HSS] PSCI 24900. THE GOVERNMENT AND POLITICS OF AFRICA [C, HSS] RELIGIOUS STUDIES RELS 26100. BLACK RELIGIOUS EXPERIENCE IN AMERICA [C, R] SOCIOLOGY SOCI 20900. INEQUALITY IN AMERICA [HSS] SOCI 21400. RACIAL AND ETHNIC GROUPS IN AMERICAN SOCIETY [C, HSS] SOCI 21700. BLACKS IN CONTEMPORARY AMERICAN SOCIETY [C, HSS] SPANISH SPAN 21200. LITERATURE AND CULTURE OF THE HISPANIC CARIBBEAN [C, AH] THEATRE AND DANCE THTD 24200. AFRICAN AMERICAN THEATRE HISTORY [W, C, AH] ANCIENT MEDITERRANEAN STUDIES The concentration in Ancient Mediterranean Studies is one of two within the major of Classical Studies. Through this concentration, students comparatively study multiple cultures in the Near East and Mediterranean basin, including ancient Mesopotamia, Israel, Egypt, Greece and Rome. (see CLASSICAL STUDIES) ANTHROPOLOGY Anthropology explores the variety of human groups and cultures that have developed across the globe and throughout time. Anthropologists hope that by seeing ourselves in the mirror of alternative cultural and historical possibilities, we can come to a better understanding of our own assumptions, values and patterns of behavior. (see SOCIOLOGY AND ANTHROPOLOGY) Archaeology 27 ARCHAEOLOGY CURRICULUM COMMITTEE: P. Nick Kardulias (Archaeology), Chair J. Heath Anderson (Archaeology) Josephine Shaya (Classical Studies) Gregory Wiles (Geology) Archaeology is an interdisciplinary field of study that investigates the past by finding and analyzing evidence from material culture and the natural environment. Its history as an academic field in this country began in 1879, when scholars from a number of established academic disciplines – especially history, classical studies, anthropology, and art – founded the Archaeological Institute of America. Archaeologists draw on the humanities, history and the social sciences, and the physical sciences in their research to identify the unique achievements and common elements of past societies around the world. Wooster’s archaeology curriculum has been designed to reflect the interrelatedness of the participating fields and to promote appreciation of human diversity. The program is designed both for majors and for students with a more general interest in archaeology. Majors may view the degree in archaeology as partial preparation for a career in teaching, museum curatorship, or field archaeology. If so, they should secure as broad a background as possible in the liberal arts and plan to pursue their studies on a graduate level. Major in Archaeology Consists of fourteen courses: • ARCH 10300 • ANTH 11000 • GEOL 10500 • ARCH 21900-21906 • ARCH 35000 • Four electives in one area of emphasis taken from cross-listed courses accepted for ARCH credit • Two electives in a second area of emphasis taken from cross-listed courses accepted for ARCH credit • Junior Independent Study: ARCH 40100 • Senior Independent Study: ARCH 45100 • Senior Independent Study: ARCH 45200 Minor in Archaeology Consists of six courses: • ARCH 10300 • ARCH 35000 • Four of the following courses: ANTH 11000, 20500, 21000, ARCH 21900-21906, SOAN 24000, GEOL 10500, 20000, 21000, 30000, GRK 20000-20001, HIST 20200- 20201, 20300, LAT 20000, IDPT 24000 or 24100 Special Notes • The chairperson of Archaeology will approve a substitute for ARCH 21900- 21906 for majors and minors unable to schedule the course. • Only grades of C- or better are accepted for the major or minor. Archaeology 28 ARCHAEOLOGY COURSES ARCH 10300. INTRODUCTION TO ARCHAEOLOGY As an overview of the discipline, this includes study of historical development of archaeology, consideration of basic field and analytical methods, and a review of world prehistory beginning with the emergence of the first humans to the rise of civilization. Emphasis is on how archaeologists reconstruct past societies out of fragmentary evidence. Required prior to ARCH 35000 and recommended prior to other courses listed under Archaeological Perspectives and Methods, which best serve as specialized case studies. Annually. Fall. [HSS] ARCH 21900-21906. TOPICS IN ARCHAEOLOGY The course material will vary. Examples include models of explanation and the nature and biases of evidence in interpreting the past; North American archaeology; recent excavations of specific sites; spatial analysis; Old World prehistory. May be taken more than once with permission of the chairperson. Prerequisite: a minimum of two courses in the major or permission of the instructor(s). Alternate years. Spring 2012. [C; depending on the topic, AH or HSS] ARCH 35000. ARCHAEOLOGICAL METHODS AND THEORY This course is an in-depth study of the methodological and theoretical foundations of archaeology. The student becomes familiar with the process of archaeological reasoning — the assumptions, models, and techniques scholars use to analyze and interpret the material record. Topics include dating techniques, systems of classification, research design, and central debates in modern theory. Students work with materials in the Archaeology Lab. Students are strongly encouraged to complete ARCH 35000 prior to enrolling in ARCH 40100. Prerequisite: ARCH 10300. Alternate years. Not offered 2011-2012. [HSS] ARCH 40100. JUNIOR INDEPENDENT STUDY A one-semester course that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. ARCH 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: ARCH 40100. ARCH 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: ARCH 45100. CROSS-LISTED COURSES ACCEPTED FOR ARCHAEOLOGY CREDIT BY AREA EMPHASIS ART AND ART HISTORY ARTD 12000. INTRODUCTION TO ART HISTORY [AH] ARTD 15100. INTRODUCTION TO DRAWING [AH] ARTD 15900. INTRODUCTION TO PHOTOGRAPHY [AH] ARTD 20100. THE BRONZE AGE [AH] ARTD 20600. EARLY MEDIEVAL ART [R, AH] ARTD 22300. ARCHITECTURE I: STONEHENGE TO BEAUX-ARTS [R, AH] IDPT 24000. GREEK ARCHAEOLOGY AND ART [AH] IDPT 24100. ROMAN ARCHAEOLOGY AND ART [AH] CLASSICAL STUDIES GRK 20000. SEMINAR IN GREEK LITERATURE (INTERMEDIATE LEVEL I) [AH] GRK 25000. SEMINAR IN GREEK LITERATURE (INTERMEDIATE LEVEL II) [AH] HIST 20200. GREEK CIVILIZATION [HSS] HIST 20300. ROMAN CIVILIZATION [HSS] IDPT 24000. GREEK ARCHAEOLOGY AND ART [AH] IDPT 24100. ROMAN ARCHAEOLOGY AND ART [AH] LAT 20000. SEMINAR IN LATIN LITERATURE (INTERMEDIATE LEVEL I) [AH] LAT 25000. SEMINAR IN LATIN LITERATURE (INTERMEDIATE LEVEL II) [AH] Art and Art History 29 GEOLOGY GEOL 10000. HISTORY OF LIFE [MNS] GEOL 10500. GEOLOGY OF NATURAL HAZARDS [MNS] GEOL 20000. PROCESSES AND CONCEPTS OF GEOLOGY [MNS] GEOL 20800. MINERALOGY [MNS] GEOL 21000. CLIMATE CHANGE [Q] GEOL 22000. INTRODUCTION TO GEOGRAPHIC INFORMATION SYSTEMS (GIS) GEOL 26000. SEDIMENTOLOGY AND STRATIGRAPHY [W, MNS] GEOL 30000. GEOMORPHOLOGY AND HYDROGEOLOGY GEOL 30800. IGNEOUS AND METAMORPHIC PETROLOGY HISTORY HIST 23400. TRADITIONAL CHINA [C, HSS] HIST 20400. GREEK CIVILIZATION [HSS] HIST 20500. ROMAN CIVILIZATION [HSS] HIST 20601. MEDIEVAL EUROPE, 500-1350 [HSS] SOCIOLOGY AND ANTHROPOLOGY ANTH 11000. INTRODUCTION TO ANTHROPOLOGY [C, HSS] ANTH 20500. POLITICAL ANTHROPOLOGY [W†, C, HSS] ANTH 21000. PHYSICAL ANTHROPOLOGY [C, HSS, MNS] ANTH 22000. LINGUISTIC ANTHROPOLOGY [C, HSS] ANTH 23100-23112. PEOPLES AND CULTURES [C, HSS] ANTH 35200. CONTEMPORARY ANTHROPOLOGICAL THEORY [C, HSS] SOAN 24000. RESEARCH METHODS [HSS] SOCI 35000. CLASSICAL SOCIAL THEORY [HSS] ART AND ART HISTORY Marina Mangubi, Chair Bridget Milligan Kara Morrow Diana Bullen Presciutti John Siewert Walter Zurko The Department of Art and Art History offers majors in Studio Art and in Art History. Courses in both majors are designed to allow the student to develop a sensitive understanding of the visual arts past and present. In studio courses, students learn to conceive and express ideas in two- and three-dimensional media, to evaluate the aesthetic character of works of art, and to become more alert to their sociopolitical implications. Art history courses are concerned with the production and reception of the visual arts within their social, religious, cultural, and political contexts. The Department of Art and Art History strongly urges students interested in offcampus experiences to take advantage of the New York Arts Program administered by the Great Lakes Colleges Association (GLCA). Other off-campus study programs, both in this country and abroad, are available to majors with adequate course preparation for advanced study. More information on such programs is available through the office of Off-Campus Studies. Architecture: Students interested in a career in architecture should consider one of the following options, bearing in mind that Wooster does not offer a major in architecture: 1) a pre-architecture program of recommended coursework, preparing students for graduate study in architecture upon completion of the B.A. in any discipline Art and Art History 30 at Wooster; or 2) the Cooperative Program between The College of Wooster and Washington University in St. Louis, providing an opportunity to earn both a bachelor’s degree from Wooster and, upon acceptance into the graduate program, a Master’s of Architecture degree at Washington University. For either option, interested students should meet with one of the Pre-Architecture co-advisers, Professor John Siewert and Professor Walter Zurko, early in their undergraduate education (see Pre-Professional and Dual Degree Programs for additional information). STUDIO ART The program in Studio Art is designed to engage students in the creative process and to provide training necessary for graduate study and/or a professional career in visual art. Majors normally choose upper-level courses that lead to an emphasis in one of the following areas: drawing, painting, printmaking, sculpture, ceramics, and photography. Studio art courses usually are restricted to fewer than twenty students so that the instructor may spend sufficient time with each member of the class. These classes will be organized by collective experiences — slide lectures, field trips to galleries and museums, group critiques — and for personal, creative work and individualized suggestions and criticism offered by the instructor. Major in Studio Art Consists of eleven courses: • ARTD 12000 • ARTD 15100 • One of the following 100-level courses: ARTD 16100, 16300, or 16500 • One elective 100-level Studio Art course • ARTD 25100 • One of the following courses: ARTD 21600, 22200, or 36000 • Two elective Studio Art courses at the 200-level or above • Junior Independent Study: ARTD 40100 • Senior Independent Study: ARTD 45100 • Senior Independent Study: ARTD 45200 Minor in Studio Art Consists of six courses: • ARTD 15100 • One of the following 100-level courses: ATRD 16100, 16300, or 16500 • One elective 100-level Studio Art course • One of the following courses: ATRD 12000, 21600, 22200, or 36000 • Two elective Studio Art courses at the 200-level or above Special Notes • AP credit in studio art is granted with a grade of 4 or 5 on the Studio Art General Portfolio or the Studio Art Drawing Portfolio, and a faculty portfolio review of artwork submitted to the AP Board. • To declare a major in Studio Art, a student should have completed at least three courses in art, two of which must have been studio courses. • Junior Independent Study in Studio Art (ARTD 40100) is a one-semester course that offers majors an opportunity to integrate techniques with creative concepts and serves as a preparatory experience for the two-semester Senior Independent Study (ARTD 45100 and 45200). ARTD 40100 is offered only in the Spring semester. Students must plan off-campus study so that it does not conflict with this course. Art and Art History 31 • Courses taken S/NC will not fulfill requirements for a major or a minor in Studio Art. • Only grades of C- or better are accepted for the major or minor. STUDIO ART COURSES ARTD 15100. INTRODUCTION TO DRAWING (Archaeology) This course introduces students to the various media and methods of freehand drawing. In order to advance their understanding of the visual and verbal language of drawing, students engage in a series of topical exercises, each combining a slide presentation, a group discussion, and a drawing assignment. Various approaches to representational drawing, including figure drawing, are explored. The course is required for the studio art major and is strongly recommended as the first course in studio art. There are six hours of weekly class time. Preference given to art majors. Annually. Fall and Spring. [AH] ARTD 15300. INTRODUCTION TO PAINTING Students are introduced to the fundamental painting techniques and principles of color. In the process, they explore issues of subject matter and content as well as the role of painting today. The course is organized around a schedule of studio work, critiques, and discussion of artists’ works. Six hours of weekly class time. Preference given to art majors. Prerequisite: ARTD 15100. Annually. Spring. [AH] ARTD 15500. INTRODUCTION TO PRINTMAKING The course is organized around a schedule of technical demonstrations, studio work, critiques, and discussions of artists’ works. Although intaglio techniques are emphasized in the course, students are also acquainted with monotype, relief and non-toxic printmaking media, including photo-etching. Six hours of weekly class time. Preference given to art majors. Annually. Fall. [AH] ARTD 15700. INTRODUCTION TO TWO-DIMENSIONAL DESIGN AND COLOR The course introduces students to the fundamentals of two-dimensional design and color theory. In it, we explore the properties and the interaction of formal elements in a composition and discuss their function in the works of artists, designers, and architects. Six hours of weekly class time that include lecture, demonstrations, slide presentations, group critiques, and in-class work time. Preference given to art majors. Alternate years. Not offered 2011-2012. [AH] ARTD 15900. INTRODUCTION TO PHOTOGRAPHY (Archaeology, Film Studies) This course introduces the student to the technical and aesthetic issues of basic black and white photography. The class assignments are designed to emphasize the versatility of the medium and to promote individual expression. Basic camera operation, black and white processing and printing techniques will be covered. Group critiques are scheduled regularly to develop analytical skills and to provide an arena for the photographer to discuss his or her intent. Six hours of weekly class time that include lecture, demonstrations, digital slide presentations, group critiques, and in-class work time. Preference given to art majors. Annually. Fall and Spring. [AH] ARTD 16100. INTRODUCTION TO THREE-DIMENSIONAL DESIGN This course is designed to explore the elements of visual organization as they apply to three-dimensional forms. The goal of this course is to familiarize students with a shared vocabulary, both visual and verbal. This language will serve as the basis for engaging in constructive criticism and the exchange of ideas. We will explore the properties and the interaction of formal elements in a three-dimensional structure and discuss their role in the works of artists, designers, and architects from around the globe. Six hours of weekly class time that include lecture, demonstrations, slide presentations, group critiques, and in-class work time. Preference given to art majors. Alternate years. Not offered 2011-2012. [AH] ARTD 16300. INTRODUCTION TO SCULPTURE This course investigates the concepts and practices of organizing three-dimensional form through such techniques as casting and mold making, assemblage, and carving. The range of materials in the course could include wood, plaster, stone, metal, paper products, and found objects. Six hours of weekly class time that include lecture, demonstrations, slide presentations, and in-class work time. Preference given to art majors. Annually. Fall and Spring. [AH] ARTD 16500. INTRODUCTION TO CERAMICS This course introduces clay as an art medium through a variety of fundamental forming, surface decoration, and firing techniques commonly used by potters and sculptors. Six hours of weekly class time that include lec- Art and Art History 32 ture, demonstrations, slide presentations, group critiques, and in-class work time. Preference given to art majors. Annually. Spring. [AH] ARTD 17100. INTRODUCTION TO DIGITAL IMAGING This course is designed to examine the concepts and practices of digital imaging as an art form. As part of that process, students will explore various techniques that include digital manipulation, digital collage, animation, and interactive website authoring. A special emphasis will be placed on understanding the practice of Adobe Creative Suite, specifically Adobe Photoshop in order to generate, collage and manipulate still images and text. Flash will be utilized to create interactive websites and animations that may incorporate video and sound effects. Six hours of weekly class time that include lecture, digital demonstrations, slide presentations, critiques, and in-class work time. Preference given to art majors. Prerequisite: any ARTD 100-level studio art course. Annually. Spring. [AH] THTD 10000. ARTS AND ENTREPRENEURSHIP [AH] THTD 10400. THE IMPULSE TO CREATE [AH] ARTD 25100. INTERMEDIATE DRAWING This course is designed to develop a more expressive visual vocabulary through the continued exploration of media, methods, and a wide range of subjects including life drawing. Seminars and visits to exhibitions will stress visual concepts and the role of drawing in contemporary art. Prerequisite: ARTD 15100. Annually. Spring. ARTD 25300. INTERMEDIATE PAINTING Advanced study in oil painting including representational and abstract subject matter. Students engage in conceptual problems, which characterize contemporary painting practices Additional study through individual projects and field trips. Prerequisite: ARTD 15300. Alternate years. Not offered 2011-2012. ARTD 25500. INTERMEDIATE PRINTMAKING Advanced study in the media of printmaking and continued investigation of the ideas encountered in the initial printmaking course. Exhibitions, discussions, and field trips to museums are designed to acquaint the student with the role of printmaking in the world of contemporary art. Prerequisite: ARTD 15500. Annually. Fall. ARTD 25900. INTERMEDIATE PHOTOGRAPHY Continued study in the medium of photography, including an introduction to digital imaging that will include color images, fine-art digital prints, and outputting negatives for non-silver antiquated processes. The course may also incorporate camera formats and book arts. Running parallel to these technical investigations, assigned readings and discussions will address contemporary issues surrounding photography and digital imaging. Emphasis will be placed on developing creative projects and generating a cohesive body of work for each student. Prerequisite: ARTD 15900. Annually. Fall. ARTD 26300. INTERMEDIATE SCULPTURE Continued study of the medium of sculpture, including the study of theory and the creation of three-dimensional forms encountered in the initial sculpture course. Consideration of the possibilities of contemporary processes for creating and transforming three-dimensional forms and spaces. Prerequisite: ARTD 16300. Alternate years. Not offered 2011-2012. ARTD 26500. INTERMEDIATE CERAMICS Upper-level problems in creative ceramics, continuing the approaches of the initial course in ceramics with emphasis on throwing and instruction in glaze formulation. Prerequisite: ARTD 16500. Alternate years. Fall. ARTD 27000-27005. SPECIAL TOPICS IN STUDIO ART A course for students who have taken at least one ARTD 200-level course in studio art. It provides faculty and students opportunities to study and to create in a medium not regularly taught, or to enable faculty and students to focus on an issue in creative art that is not adequately addressed in listed courses. Prerequisite: any ARTD 200-level studio art course. Fall 2011. ARTD 35100. ADVANCED DRAWING Advanced exercises in traditional drawing media as well as experimental techniques not covered in earlier classes. There will be structured assignments along with numerous independent projects. Prerequisite: ARTD 25100. Annually. Spring. ARTD 35300. ADVANCED PAINTING Advanced study in various painting media to include structured assignments and independent work. Art and Art History 33 Continued investigation of contemporary issues in painting through field trips and readings of art criticism. Topics range from approaches to figure painting to open-ended conceptual problems. Prerequisite: ARTD 25300. Alternate years. Not offered 2011-2012. ARTD 35500. ADVANCED PRINTMAKING In this course students will explore further conventional and experimental printmaking techniques. Students may concentrate on editioning, or they may develop a portfolio of individual prints. Prerequisite: ARTD 25500. Annually. Fall. ARTD 35900. ADVANCED PHOTOGRAPHY This course is designed to develop an advanced understanding of the theory and practice of photography and digital imaging. A focus on advanced techniques will involve both structured projects with an emphasis on the development of an individual portfolio. Prerequisite: ARTD 25900. Annually. Fall. ARTD 36300. ADVANCED SCULPTURE This course will be comprised of both individually arranged and structured projects in advanced sculptural concepts and techniques. There will be an investigation of critical attitudes applicable to sculpture. Individual experimentation is encouraged. Prerequisite: ARTD 26300. Alternate years. Not offered 2011-2012. ARTD 36500. ADVANCED CERAMICS Concentration on advanced problems in both functional and sculptural ceramic design and techniques. A portion of the course will focus on plaster mold-making and slip-casting. Continued instruction in glaze formulation. Individual experimentation is encouraged. Prerequisite: ARTD 26500. Alternate years. Fall 2011. ARTD 38500. MEDIA AND METHODS IN ART EDUCATION Study of the creative and mental growth of children through art experience in various media. This course may not count toward a major in art but does count as a course for Visual Art licensure in education. Prerequisite: PSYC 11000 and EDUC 10000. Alternate years. Not offered 2011-2012. ARTD 40000. TUTORIAL Advanced work in an area in preparation for doing Independent Study. Permission must be obtained from the instructor offering an advanced course in the special area. The student must schedule the same instructor and class hours as the advanced course. Prerequisite: previous coursework in the requested area; the approval of both the supervising faculty member and the chairperson are required prior to registration. ARTD 40100. INDEPENDENT STUDY A creative, individual program, organized within a classroom structure to integrate techniques and artistic concepts as a preparatory experience for the senior project. Students and professors meet weekly in a seminar to discuss problems and to critique projects. Prerequisite: two ARTD 100-level courses, two advanced-level courses, and one art history course. Annually. Spring. ARTD 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in the creation of a body of artwork and independent research guided by a faculty mentor, and which culminates in the presentation of a one- or two-person exhibition, a thesis, and an oral examination in the second semester. Prerequisite: ARTD 40100. ARTD 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the presentation of a one- or two-person exhibition, the thesis, and an oral examination. Prerequisite: ARTD 45100. ART HISTORY The major in Art History exposes students to a wide variety of perspectives and academic fields and provides a good liberal arts foundation for careers in many different areas. It can also provide undergraduate preparation for graduate degrees leading to careers in teaching, research, criticism, library science, visual resources curating, museum or gallery work, art conservation (with substantial background in chemistry), community art programs, architecture, or historic preservation. Art history courses are usually lecture-discussion classes primarily concerned with art’s cultural and historical contexts; art as a revelation of human intelligence, Art and Art History 34 imagination, and skill; and the tools—vocabulary, methods, approaches—used to study cultures through their artistic achievements. Major in Art History Consists of twelve courses: • ARTD 12000 • One of the following courses: ARTD 20100, IDPT 24000 or 24100 • One of the following courses: ARTD 20600 or 20700 • One of the following courses: ARTD 20800 or 21200 • One of the following courses: ARTD 20400, 21400, or 22200 • One of the following courses: ARTD 21600, 22000, 22100, 22300, 22400, or 23000 • Two elective 300-level Art History courses • One elective course in Studio Art • Junior Independent Study: ARTD 40100 • Senior Independent Study: ARTD 45100 • Senior Independent Study: ARTD 45200 Minor in Art History Consists of six courses: • ARTD 12000 • Four 200-level courses, one each in four of the following five areas: ARTD 20100, IDPT 24000 or 24100 ARTD 20600 or 20700 ARTD 20800 or 21200 ARTD 20400, 21400, or 22200 ARTD 21600, 22000, 22100, 22300, 22400, or 23000 • One elective 300-level Art History course Special Notes • AP credit for ARTD 12000 is granted with a score of 4 or 5 on the AP Examination in Art History. • Prospective majors are strongly encouraged to take ARTD 12000 as their first Art History course. • At least one of the student’s 200-level courses must be Writing-Intensive. • Two courses in Studio Art are strongly recommended for the major. • Junior Independent Study in Art History (ARTD 40100) is a one-semester seminar course, taught only in the Fall. Art History majors must plan off-campus study so that it does not conflict with this course. • Courses taken S/NC will not fulfill requirements for a major or minor in Art History. • Only grades of C- or better are accepted for the major or minor. ART HISTORY COURSES ARTD 12000. INTRODUCTION TO ART HISTORY (Archaeology) This course introduces the student to the discipline of art history by focusing on several case studies, explored in chronological order and in depth. A cluster of readings from both primary and secondary sources will be utilized for each unit of the course material. Students will gain experience in viewing art objects and architecture, as well as an understanding of how art and architecture function in their historical contexts, both as expressions and instruments of the social forces operating in those contexts. Taught by lecture and discussion, this course is primarily for first-year students and sophomores, and is strongly recommended as the first course in art history. Annually. Fall and Spring. [AH] Art and Art History 35 THTD 10000. ARTS AND ENTREPRENEURIALISM [AH] THTD 10400. THE IMPULSE TO CREATE [AH] ARTD 20100. THE BRONZE AGE (Archaeology) Explores the artistic and architectural achievements of the early civilizations of Mesopotamia, Egypt, and the Aegean prior to the rise of Greco-Roman culture (3500-500 BCE). Particular focus will be given to the role of intercultural exchange in the region. Students will be introduced to a variety of art historical and archaeological methods including traditional formal (stylistic, iconographic, structural) analysis of monuments as well as contextual (social, economic, gendered) approaches to material culture. ARTD 12000 and/or ARCH 10300 are recommended as prior courses. Alternate years. Not offered 2011-2012. [AH] ARTD 20400. AMERICAN ART This course examines social, ideological, and economic forces that shaped American painting, sculpture, and architecture from the colonial period to around 1940. Issues considered include representing “nation” in portrait, landscape, and genre painting; constructions of race in ante- and post-bellum America; the expatriation of American artists after the Civil War; the identification of an abstract style with political ascendance in the U.S.; and tensions between the ideal and the real in American cultural expression. ARTD 12000 is recommended as a prior course. Alternate years. Not offered 2011-2012. [AH] ARTD 20600. EARLY MEDIEVAL ART (Archaeology) This course will trace the development of art and architecture in the Mediterranean basin and on the European continent, 200-1000 CE — a period that saw the fragmentation of the late Roman Empire, the rise of Christianity, and the migration and settlement of the Germanic peoples. Frequently characterized by the so-called “demise” of Greco-Roman visual culture, the period is best understood in terms of the dynamic intermingling of artistic styles and religious beliefs. Monuments such as the catacombs of early Christian Rome, the ship burials of the North Sea littoral, and the Celtic manuscripts of Ireland will be explored in depth. ARTD 12000 is recommended as a prior course. Alternate years. Not offered 2011-2012. [R, AH] ARTD 20700. LATE MEDIEVAL ART This course will introduce students to the art and architecture of the period c.1000-1400 CE in western Europe and the Byzantine Empire. Each week, lectures and discussion — focusing on a particular region, culture, or discrete chronological period — will consider a variety of art historical approaches toward the study of objects (style, iconography, technique, etc.) and their cultural context. Key socio-historical themes and their impact on the arts will be addressed including pilgrimage, the Crusades, monasticism, feudalism, the role of women as artists and patrons, and cross-cultural artistic exchange. The course will cover a wide range of monuments (monasteries, cathedrals, castles and palaces) and a variety of artistic media (manuscripts, textiles, mosaics, frescoes, ivory, and metalwork). ARTD 12000 is recommended as a prior course. Alternate years. Fall 2011. [R, AH] ARTD 20800. RENAISSANCE ART, 1400-1550 This course introduces the student to the art and architecture of Italy and northern Europe during the fifteenth and early sixteenth centuries. Although “Renaissance” connotes the revival of Greco-Roman antiquity, classical culture was assimilated into a Christian context emphasizing an individualized and humanized spirituality that was manifested in various artistic forms, such as the altarpiece, the private devotional picture, the narrative fresco cycle, and the devotional print. The Renaissance intensification of individual piety culminated in the Reformation, which confronted the issues of how one is saved as well as the role of religious art. ARTD 12000 is recommended as a prior course. Alternate years. Not offered 2011-2012. [W†, R, AH] ARTD 21200. BAROQUE ART, 1600-1700 The course will explore the art and architecture of the Baroque era, primarily in Italy, Spain, Flanders, and Holland. This includes such masters as Caravaggio, Bernini, Velázquez, Rubens, Rembrandt, and Vermeer. The works will be studied in the context of the social, political, and religious milieu of the Baroque period, an era of dynamic change and violent conflicts. ARTD 12000 is recommended as a prior course. Alternate years. Spring 2012. [W†, AH] ARTD 21400. NINETEENTH-CENTURY ART Surveys major movements and figures in painting, approximately 1789-1885, focusing primarily on France. Changing social and political conditions provide the context for investigating themes such as art’s engagement with history, nature, and urban experience; the place of gender and class in the formulation of artistic subjects; institutions of art exhibition and criticism; and the relationship between painting and other media such as sculpture, printmaking, and photography. ARTD 12000 is recommended as a prior course. Alternate years. Fall 2011. [W, AH] Art and Art History 36 ARTD 21600. GENDER IN TWENTIETH-CENTURY ART (Women’s, Gender, and Sexuality Studies) Explores the ideologies and implications of significant gender issues in Western visual culture since the early twentieth century. The goal of the course is to examine social, historical and visual constructions — femininity and masculinity, sexuality and the body, domesticity and the family — by focusing on the place of artistic representation in the modern and current debates about such theoretical and material categories. ARTD 12000 or WGSS 12000 is recommended as a prior course. Alternate years. Not offered 2011-2012. [AH] ARTD 22000. AFRICAN ART (Africana Studies) This course will introduce by region the art and architecture of the African continent from the prehistoric to early modern periods. Representative groups will be explored in depth by considering the impact of historical, geopolitical and social development on traditional art forms/visual culture. Emphasis will be placed on ubiquitous themes such as rulership/social status, gender, performance/ritual, and belief systems. ARTD 12000, AFST 10000, or HIST 23100 is recommended as a prior course. Alternate years. Not offered 2011-2012. [C] ARTD 22100. ISLAMIC ART This course will introduce students to the art and architecture of historical Islam from its rise following the death of Mohammed to the imperial age of the Ottomans, Persians, and Mughals, c. 650-1650. Particular attention will be given to the evolution of a distinctive Islamic material culture (calligraphy, textiles, mosques, and palaces), and the development of regional styles that resulted from artistic exchange with indigenous European, African, and Asian traditions. Alternate Years. Not offered 2011-2012. [C, R, AH] ARTD 22200. MODERN ART Examines developments in European painting and sculpture between approxi mately 1885 and 1945, including selected moments in American art after the turn of the twentieth century. The course will consider major modernist artists and movements that sought to revolutionize and renew vision and experience, from Symbolism to Surrealism. Issues include modernism’s interest in primitivism and mass culture, theoretical rationales for abstraction, and the impact of industrial production and two world wars on the production and reception of art. ARTD 12000 is recommended as a prior course. Annually. Spring. [AH] ARTD 22300. ARCHITECTURE I: STONEHENGE TO BEAUX-ARTS (Archaeology) A chronological and contextual study of world architecture from its origins among Neolithic peoples to the revival-style architecture of nineteenth-century Europe and America. Themes addressed include: the definition of sacred space and the structure of worship in various traditions of religious architecture; the classical tradition and its permutations through Renaissance, Baroque, and nineteenth-century architecture; the medieval tradition and medievalism in nineteenth-century architecture; and architecture as it shapes and expresses political ideas. ARTD 12000 is recommended as a prior course. Alternate years. Spring 2012. [R, AH] ARTD 22400. ARCHITECTURE II: CHICAGO SCHOOL TO POSTMODERNISM A survey of developments in European and American architecture from the late nineteenth century to postmodernism. The course will examine structural innovations, the impact of the machine on theory and practice, the death and rebirth of ornament, the challenge of urban problems, and the responses of particular architects to the challenges facing designers in the twentieth century. Prerequisite: ARTD 12400 or ARTD 12000 or permission of instructor (ARTD 12400 is the preferred prerequisite for students interested in graduate training in architecture). Alternate years. Not offered 2011-2012. ARTD 23000. AFRICAN-AMERICAN ART (Africana Studies) Explores artistic production by and about peoples of African descent living in the United States, from emancipation to the present. Emphasis on the Harlem Renaissance, expatriate black experience in Paris, art and the New Deal, the civil rights movement and black nationalism, and recent identity politics. The course also considers the idea of a “black aesthetic” and its impact on American art. ARTD 12000 or AFST 10000 is recommended as a prior course. Alternate years. Spring 2012. [C, AH] IDPT 24000. GREEK ARCHAEOLOGY AND ART [AH] IDPT 24100. ROMAN ARCHAEOLOGY AND ART [AH] ARTD 31000-31018. SEMINAR: SPECIAL TOPICS IN THE HISTORY OF ART (some sections cross-listed with Women’s, Gender, and Sexuality Studies WGSS 3200) A seminar on a specific artist or a limited number of artists, on a theme, problem, or methodological approach offered periodically for students who have taken at least one ARTD 200-level course in the history of art and who wish to concentrate on a defined issue in a collaborative effort by students and faculty. Fall and Spring. Art and Art History 37 ARTD 31800. HISTORY OF PRINTS From their inception around 1400 in Europe, the graphic media have established social functions and aesthetic criteria that differ considerably from those of painting, sculpture, and architecture. This course surveys the techniques and development of printmaking, explores the various implications of the multiplied image on paper, and makes use of the College’s print collection to give students firsthand experience in viewing and interpreting prints. Offered occasionally in conjunction with The College of Wooster Art Museum. Prerequisite: any of the following art courses — ARTD 15500, 20800, 21200, 21400, 22200 — or permission of instructor. Not offered 2011-2012. [AH] ARTD 36000. CONTEMPORARY ART (Film Studies) Examines practice and theory in American and European art since approximately 1945, from abstract expressionism to current trends. Topics include the critique of modernism and representation, the emergence of new media and multimedia art forms, and the questioning of agency, identity, and audience in the contemporary art world. Readings range from contemporary criticism to historical analysis from a variety of perspectives (e.g., formal, feminist, multicultural, deconstructive). Prerequisite: any of the following — ARTD 21600, ARTD 22200, junior/senior studio major status, or permission of instructor. Alternate years. Fall. ARTD 40000. TUTORIAL Independent research and writing under the direction of a faculty member of the department. For advanced students. Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. ARTD 40100. INDEPENDENT STUDY This seminar will focus on current methods used in art historical research, various approaches historians have employed in studying works of art, use of library resources, and writing about art. Coursework includes substantial reading and a variety of research and writing projects. Annually. Fall. ARTD 40700, 40800. INTERNSHIP IN ART HISTORY/ARCHITECTURE Supervised participation for art majors at an art museum or gallery, or with organizations providing prag matic experience in architectural history, urban planning, or historic preservation. This experience may be studentdesigned, with the consultation of an art history faculty member and a site supervisor, or arranged in the context of an existing program, such as the Harvard Graduate School of Design Summer Career Discovery Program or Habitat for Humanity. Coursework includes a journal and regular communication with the supervising faculty member, and may culminate with a written analysis of the student’s experience. S/NC course. Prerequisite: Art History majors must have completed ARTD 12000 and at least two ARTD 200-level art history courses. Studio Art majors must have ARTD 12000 and one upper-level art history course. Prior consultation with the supervising faculty member or the Pre-architecture adviser is required. ARTD 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: ARTD 40100. ARTD 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: ARTD 45100. Biology 38 BIOCHEMISTRY AND MOLECULAR BIOLOGY CURRICULUM COMMITTEE: William Morgan (Biology), Chair Paul Edmiston (Chemistry) Dean Fraga (Biology) Mark Snider (Chemistry) Stephanie Strand (Biology) James West (Biochemistry and Molecular Biology) This interdisciplinary program, jointly administered by faculty from the Depart - ments of Biology and Chemistry, enables students to ask and explore fundamental questions concerning the molecular events that occur in organisms. Students who complete this program will possess an understanding of the structures of biological molecules, the reactions involved in biological energy conversions, the formation and organization of complex cellular structures, and the communication of biological information spatially and temporally. Major in Biochemistry and Molecular Biology Consists of sixteen courses: • CHEM 11000 (see note below) • CHEM 12000 • MATH 11100 (see note below) • BIOL 20000 • BIOL 20100 • CHEM 21100 • CHEM 21200 • PHYS 20300 or 10100 • BIOL 30500 • BIOL 30600 • BCMB 30300 • BCMB 33100 • One of the following courses: BCMB 33200 or 33300 • Junior Independent Study: BCMB 40100 • Senior Independent Study: BCMB 45100 • Senior Independent Study: BCMB 45200 Special Notes • Refer to the catalogue section for Chemistry for information concerning CHEM 11000/12000 placement exams. Students who place out of CHEM 11000 must take one elective from the following: BCMB 33200, 33300, BIOL 30400, 30700, 33500, CHEM 21500, IDPT 20013, or NEUR 38000. • The MATH 11100 requirement may be fulfilled by successful completion of both MATH 10700 and 10800. • There is no minor in Biochemistry and Molecular Biology. • A student may not double major in Biochemistry and Molecular Biology with Biology, Chemistry, or Neuroscience. • To complete the Biochemistry and Molecular Biology major, students should follow the sequence below: Biochemistry and Molecular Biology 39 First Year: CHEM 12000 (and 11000, if needed) BIOL 20000, 20100 Sophomore Year: CHEM 21100, 21200 BIOL 30500, 30600 MATH 11100 (or 10700 and 10800) PHYS 20300 (or 10100) Junior Year: BCMB 30300, 33100, and either 33200 or 33300 BCMB 40100 Senior Year: BCMB 45100, 45200 One required elective for students placing out of CHEM 11000 • A student who desires to replace a course listed above with a different course to count toward the major can petition the BCMB Curriculum Committee. • A BCMB major who desires an American Chemical Society-Certified Biochem - istry Degree is required to take the following courses in addition to the course requirements for the BCMB major: CHEM 21500, CHEM 31800, MATH 11200, and PHYS 20400. The A.C.S.-certified degree is encouraged for those students who plan to enter a graduate program in a biochemical discipline. • Required courses in the major, including Physics and Mathematics, must be passed with a grade of C– or higher. All courses must be taken concurrently with the corresponding laboratory. BIOCHEMISTRY AND MOLECULAR BIOLOGY COURSES BCMB 30300. TECHNIQUES IN BIOCHEMISTRY AND MOLECULAR BIOLOGY (Biology, Chemistry) This laboratory-based course gives students hands-on experience with experimental methods used in biochemistry and molecular biology. It is organized around a semester-long project in which students design and work toward specific research goals. This course counts for major credit in Biology and Chemistry. BCMB majors are encouraged to have prior or concurrent enrollment in BCMB 33100. Prerequisites: C- or better in CHEM 12000 and BIOL 20100. Annually. Fall. BCMB 33100. PRINCIPLES OF BIOCHEMISTRY (Biology, Chemistry) This course focuses on the structural and chemical properties of the four main categories of biological molecules — amino acids, nucleic acids, carbohydrates, lipids — as a means of critically analyzing the functions of complex biological macromolecules and cellular processes at the molecular level. Structure, equilibria, thermodynamics, kinetics and reactivity of biological macromolecules, with emphasis on proteins and enzymes, are the course cornerstones. Principles of bioenergetics and intermediary metabolism (glycolysis, citric acid cycle, and oxidative phosphorylation) also discussed. Critical thinking and inquiry encouraged by analysis and discussion of current research literature. This course counts for major credit in Biology and Chemistry. Concurrent enrollment in BCMB 30300 highly recommended. Suggested previous courses: BIOL 20100, 30500 and 30600. Prerequisite: C- or better in CHEM 21200 and BIOL 20000 or by permission of instructor. Annually. Fall. [MNS] BCMB 33200. BIOCHEMISTRY OF METABOLISM (Biology, Chemistry) A continuation of BCMB 331 with molecular and mechanistic emphasis on advanced cellular metabolism, metabolomics, signal transduction, as well as DNA, RNA and protein metabolism. Critical thinking and inquiry encouraged by analysis and discussion of current research literature. This course counts for major credit in Biology and Chemistry. Prerequisite: C- or better in BCMB 33100 or permission of instructor. Annually. Spring. [MNS] BCMB 33300. CHEMICAL BIOLOGY (Biology, Chemistry) This course explores how chemistry can be utilized to examine and manipulate molecular events in biological systems. Specifically, the course is divided into different units, including proteomic profiling, enzyme activity profiling, metabolic engineering, and protein engineering. Critical thinking and inquiry encouraged by analysis and discussion of current research literature. This course counts for major credit in Biology and Chemistry. Prerequisite: C- or better in BCMB 33100 or permission of instructor. Alternate years. Spring 2012. Biology 40 BCMB 40000. TUTORIAL Special and advanced topics in Biochemistry & Molecular Biology. Evaluation of the student’s accomplishment will be based on a contract with the supervising professor. Students apply to the program chairperson for this option. This course does not count toward a major in Biochemistry and Molecular Biology. (.5 - 1 course credit) Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. BCMB 40100. INTRODUCTION TO INDEPENDENT STUDY This course focuses on scientific writing, experimental design, and informational retrieval systems, including accessing and evaluating the growing collection of molecular databases. Students explore the literature related to their proposed senior I.S. thesis through a series of structured writing assignments that culminate in a research proposal for the senior project. In addition, students learn the mechanics of scientific presentations and give a brief seminar on their proposed project. Annually. Spring. BCMB 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE An original investigation is conducted, culminating in a thesis and oral defense of the thesis in the second semester. During the year each student gives at least one research poster and oral presentation on the research topic. A student normally has one research advisor. Prerequisite: C- or better in BCMB 40100. BCMB 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The thesis is evaluated by the research advisor and one other professor from the BCMB Curriculum Committee, in consultation with the other members of the BCMB Curriculum Committee. Prerequisite: BCMB 45100. BIOLOGY Dean Fraga, Chair Patrick Crittenden Julie Heck Richard Lehtinen Marilyn Loveless Sharon Lynn William Morgan Sheryl Petersen Laura Sirot Stephanie Strand James West Biologists seek to understand the living world in all of its complexity through scientific methods of inquiry. Biology can be studied at different organizational levels, including cell biology, organismal biology, and population biology. The Department of Biology includes a group of committed faculty with expertise in diverse fields and sub-disciplines. Our curriculum provides majors with opportunities to explore the full breadth of biological organization and provides experiential learning opportunities that enhance students’ understanding of content and techniques, as well as the limitations of scientific methods of inquiry. The Biology curriculum is designed to give students a strong background in fundamental concepts of biology at the cellular, organismal, and population levels of organization. Student-generated investigations are built into the structure of courses throughout the Biology curriculum beginning in the Gateway courses and continuing through Independent Study. Students collaborate and communicate with peers and faculty as they progress through their courses and Independent Study. These opportunities develop students’ oral and written communication skills as well as their capacity for self-education and problem-solving. These abilities, combined with a liberal arts education, are essential for remaining competitive in the rapidly developing life sciences. Biology 41 Through its curriculum, the Biology Department seeks to develop students who: • comprehend foundational and unifying biological principles and their implications; • retain the knowledge essential to a broad understanding of Biology; • can explain scientific methods of inquiry and the philosophy of science, including methodologies for distilling biological information; • utilize scientific knowledge and methods of inquiry to make reasoned decisions and to critically evaluate the work of others; • can articulate how new knowledge continues to alter pre-existing understandings and paradigms; and • think, study and learn independently. Major in Biology Consists of fifteen courses: • CHEM 12000 • One of the following courses: MATH 10700, 11100, CHEM 21100, PHYS 10100, or 20300 • BIOL 20000 • BIOL 20100 • BIOL 20200 • Six elective 300-level Biology courses • One elective 300-level Biology course to satisfy the Breadth Requirement (see note below) • Junior Independent Study: BIOL 40100 • Senior Independent Study: BIOL 45100 • Senior Independent Study: BIOL 45200 Minor in Biology Consists of six courses: • BIOL 20000 • BIOL 20100 • BIOL 20200 • Three elective 300-level Biology courses Special Notes • The Breadth Requirement: The Department of Biology feels that Biology majors should appreciate and understand a range of topics studied in the field of biology. Students are introduced to the range of biological topics in our Gateway course sequence (BIOL 20100 and 20200) and then develop additional depth in each subdivision by completing at least one course from each of the two major subdivisions, as organized below. Molecular and Cellular Biology Ecology, Evolution, and Organismal Biology BIOL 30400. Human Physiology BIOL 31100. Natural History of Vertebrates BIOL 30500. Cell Physiology BIOL 32300. Natural History of Invertebrates BIOL 30600. Genes and Genomes BIOL 34000. Field Botany BIOL 30700. Development BIOL 34400. Comparative Animal Physiology BIOL 33500. Microbiology BIOL 35000. Population and Community Ecology BIOL 36600. Immunology BIOL 35200. Animal Behavior BCMB 30300. Techniques in BCMB BIOL 35600. Conservation Biology BCMB 33100. Principles of Biochemistry BIOL 37700. Behavioral Endocrinology BCMB 33200. Biochemistry of Metabolism BCMB 33300. Chemical Biology NEUR 38000. Neurobiology Biology 42 • The course BIOL 36000 Evolution synthesizes the major organizational levels in biology for a deeper understanding of this essential biological principle. Thus, BIOL 36000 is not applicable to either subdivision but does count for credit towards the major. Students are strongly encouraged to complete the breadth requirement before beginning BIOL 45100 so that they can incorporate a range of biological concepts into their Independent Study thesis project. • The Foundations course, BIOL 20000, must be taken as the first course by all Biology majors (unless the student has received advanced placement credit). The Gateway courses (BIOL 20100, 20200) may be taken in any order but should be completed by the end of the sophomore year and before enrolling in Junior Independent Study. One or both of the Gateway courses is a prerequisite to each upper-level course, although a student may be admitted to an upper-level course by permission of the instructor without having completed the prerequisite, when justifiable. • CHEM 12000 must be taken before or with BIOL 20100 and is a prerequisite to several 300-level Biology courses; it should therefore be completed in the first year. Students should complete as many Biology courses as possible before beginning Junior Independent Study. • BIOL 40100 must be completed before the student enrolls in BIOL 45100 and is normally taken in the second semester of the junior year. Students planning a semester off campus should consult with a Biology adviser early in the planning stage. • Course sequence suggestions for majors: First Year: BIOL 20000, 20200 CHEM 11000, 12000 Sophomore Year: BIOL 20100 Two 300-level electives Junior Year: BIOL 40100 Two 300-level electives Senior Year: BIOL 45100, 45200 Two 300-level electives • The Biochemistry and Molecular Biology courses (BCMB 30300, 33100, 33200, 33300) count toward the Biology major and minor and are considered Biology courses for purposes of determining departmental honors. BIOL 10000, 39500, 40000, 40200 and 40300 courses do not count toward the major or minor, nor do they apply to Honors calculations. • Biology majors contemplating graduate or professional school are strongly encouraged to take a full year of Organic Chemistry (CHEM 21100, 21200), a full year of general physics (PHYS 10100, 10200 or 20300, 20400), AND at least one course in calculus. • Laboratory Grade Policy: Biology courses with a laboratory will receive one grade that reflects performance in the classroom and laboratory components; the relative weight of the two components will be stated in each course syllabus. Because the Registrar requires a grade for both the course and the laboratory, the course grade and the laboratory grade recorded on student transcripts will be identical. Biology 43 • Advanced Placement: Students receiving a score of 5 on the Advanced Placement Examination in Biology will receive credit for Foundations of Biology (BIOL 20000). With a score of 4 on the Biology AP exam a student can receive one course credit in BIOL 10000 Topics in Biology, or upon successful completion of the Biology Placement exam may receive one course credit for Foundations of Biology (BIOL 20000). Advanced placement credit cannot be substituted for any other Biology courses than those specified above. To receive appropriate Biology credit for AP scores, please contact the Chairperson of the Biology department. The advanced placement policy of the College is explained in the section on Admissions. • Off-Campus Study: Off-campus study can be a valuable and enriching part of the college curriculum, and we encourage our students to consider off-campus study as a means of augmenting and enriching their study of biology. Students who would like to include this in their program of study are encouraged to talk with a departmental faculty member in their first year, and to think about scheduling choices that would make this possible. Biology courses taken at other institutions may count toward the major for up to two 300-level courses. Students should discuss their proposed course electives with the department chair prior to their study-abroad experience (or prior to enrolling in courses at other institutions), to determine whether the courses are equivalent to Wooster courses, and whether they will count toward the major. • Non-Science Majors: Biological information has become increasingly important as citizens face crucial decisions on such issues as the environment, emerging diseases, genetic engineering, and our aging population as well as debate ethical questions rooted in science. To gain an appreciation of how biologists approach and understand life processes, non-science majors may enroll in either Topics in Biology (BIOL 10000) or Foundations of Biology (BIOL 20000). Topics in Biology (BIOL 10000) courses address specific topical issues in applied biology on a rotating basis (see catalogue description). Foundations of Biology (BIOL 20000) is intended as an entry course for students considering a major in one of the Biological Sciences, and focuses on a serious study of the conceptual underpinnings of genetics and evolution as they relate to the field of biology. For students interested in a more extensive laboratory experience, BIOL 20100 or 20200 would be appropriate after first completing BIOL 20000. • A maximum of fifteen courses (including BCMB 30300, 33100, 33200 and 33300) from the Department of Biology may count toward the College’s thirty-two course graduation requirement. • Students are not permitted to count any courses taken for S/NC credit towards the major or minor. • A student must earn a grade of C- or higher for a course to count toward the major or minor. BIOLOGY COURSES BIOLOGY FOR THE NON-SCIENCE MAJOR BIOL 10000. TOPICS IN BIOLOGY (some sections cross-listed with: Communication, Environmental Studies, Neuroscience) The course focuses on a selected topic in biology in order to demonstrate fundamental principles of biology and/or how biology influences human society. The precise nature of the topic will vary from year to year, but in general will focus on a clearly defined topic in biology, often with some discussion of how the topic intersects with human society. Topics taught in the past have included the following: human inheritance, disease, tropical biology, neuroscience, human ecology, animal behavior, and insect biology. All sections of the course are suitable for non-science majors and will feature discussion and lecture formats. Annually. Fall and Spring. [MNS] Biology 44 BIOLOGY FOR THE SCIENCE MAJOR BIOL 14200. TROPICAL FIELD BIOLOGY This course is an introduction to the ecology and conservation of tropical environments and their biota. Through lectures, field experiences and an independent research project, students will learn about such topics as ecological interactions, the natural history of locally important plant and animal species, biodiversity dynamics and human impacts on tropical ecosystems. Students will also receive instruction in data analysis and methodology in field biology. The course is taught in a tropical location during the summer for three intensive weeks. Note: Biology majors seeking major credit will be required to complete additional assignments. Annually. Summer. [MNS] BIOL 20000. FOUNDATIONS OF BIOLOGY (Biochemistry and Molecular Biology, Environmental Studies, Neuroscience) This introductory course focuses on concepts considered central to understanding biology, including the nature of science, inheritance, gene expression, descent with modification and evolution by natural selection. This course is designed to provide potential biology majors with the fundamental concepts required for the study of biology. The course serves as a prerequisite for all biology courses numbered higher than 20000. Three class hours weekly. The course is also open to non-majors. Annually. Fall and Spring. [MNS] BIOL 20100. GATEWAY TO MOLECULAR AND CELLULAR BIOLOGY (Biochemistry and Molecular Biology, Neuroscience) This course serves as an introduction to the major concepts in the fields of molecular and cellular biology. Topics include cellular structure, bioenergetics, metabolism, biosynthesis, photosynthesis, cell division and growth, and molecular genetics. In laboratory, students will learn specific laboratory techniques and will gain experience interpreting and communicating experimental results. This course is a pre-requisite for many upper level biology courses and must be completed with a C- or better before enrolling in BIOL 40100. This course is open to non-biology majors. (1.25 course credits) Prerequisite: C- or better in BIOL 20000 and previous or concurrent registration in CHEM 12000. Annually. Fall and Spring. [Q, MNS] BIOL 20200. GATEWAY TO ECOLOGY, EVOLUTION, AND ORGANISMAL BIOLOGY (Environmental Studies) An introduction to the major concepts in the fields of ecology, evolution, behavior and physiology. These biological disciplines are approached from the population and individual levels of biological organization. Through lecture, laboratory, in-class exercises and readings, this course focuses on the structure and function of individual organisms, as well as their behavior, interactions, origination and conservation. This course is a prerequisite for many upper level biology courses and must be completed with a C- or better before enrolling in BIOL 40100. This course is open to non-biology majors. Three class hours and one laboratory period weekly. (1.25 course credits) Prerequisite: C- or better in BIOL 20000. Annually. Spring. [W, Q, MNS] BIOL 30400. HUMAN PHYSIOLOGY (Biochemistry and Molecular Biology, Neuroscience) This course focuses on human physiology at the cellular and organ system levels. An emphasis is placed on neural control of movement, metabolism and organ system function. Laboratory investigations include studies of nerves and muscle excitability, regulation of heart rate and blood pressure, respiration, and renal control of salt and volume. While the course will focus on human physiology, non-human vertebrates and amphibians will be used as subjects for laboratory investigations. This course is also an elective for the Neuroscience major. (1.25 course credits) Prerequisite: C- or better in BIOL 20100 and CHEM 12000. Annually. Fall 2011. BIOL 30500. CELL PHYSIOLOGY (Biochemistry and Molecular Biology, Neuroscience) This course focuses on the cellular and molecular basis for complex physiological processes such as aging, disease pathologies, tissue formation and maintenance, and intracellular communication. Specific concepts covered include, signal transduction, membrane biology, cell division, maintaining cellular organization, and motility. The laboratory will include student-led investigations, using model organisms to explore complex cellular processes. Three lectures and one laboratory/discussion section a week. This course is also an elective for the Neuroscience major. (1.25 course credits) Prerequisite: C- or better in BIOL 20100 and CHEM 12000 or permission of the instructor. Annually. Fall and Spring. [W†] BIOL 30600. GENES AND GENOMES (Biochemistry and Molecular Biology, Neuroscience) Genetic analysis has been transformed by the ability to investigate not only single genes, but also complete genomes. This course examines the structure, function, and variation of genes and genomes and provides an introduction to the fundamental methodologies for the modern analysis of genes and genomes. Three classroom meetings and one laboratory/recitation period weekly. This course is also an elective for the Neuroscience Biology 45 major. (1.25 course credits) Prerequisite: C- or better in BIOL 20100 and CHEM 12000 or permission of instructor. Annually. Fall and Spring. BIOL 30700. DEVELOPMENT (Biochemistry and Molecular Biology, Neuroscience) Consideration of selected developmental programs, especially those of multicellular animals, with particular reference to molecular and cellular phenomena involved in determination, morphogenesis and differentiation. Descriptive and analytical laboratory experience. Three lectures and laboratory/recitation period weekly. This course is also an elective for the Neuroscience major. Prerequisite: C- or better in CHEM 12000, BIOL 20100 and 30600, or permission of instructor. Alternate years. Not offered 2011-2012. BIOL 31100. NATURAL HISTORY OF THE VERTEBRATES This course covers the major lineages of extinct and extant vertebrates. Emphasis in lecture is on ecology, behavior, conservation and the evolutionary history of each clade. The laboratory component has two foci: field based experiences (accommodated through numerous field trips) and identification. Students will learn to identify many common vertebrates of Ohio by sight and sound. Three classroom meetings and one laboratory period weekly. One and one-fourth course credits. (1.25 course credits) Prerequisite: C- or better in BIOL 20200 or permission of instructor. Annually. Not offered 2011-2012. BIOL 32300. NATURAL HISTORY OF THE INVERTEBRATES More than 1,000,000 species of invertebrates swim, crawl, fly, and float upon the earth. What explains this incredible diversity? In this course, we will investigate the diverse and fascinating world of invertebrates with emphases on ecology, behavior, evolutionary history, and conservation. The laboratory-field period of the course will emphasize identification of taxonomic groups and exploring the rich ecology and behavior of invertebrates in their natural environment. Three classroom meetings and one laboratory-field period weekly. Prerequisite: C- or better in BIOL 20200 or permission of instructor. Alternate years. Fall 2011. BIOL 33500. MICROBIOLOGY (Biochemistry and Molecular Biology) Study of the morphology, classification, physiology, biochemistry, and genetics of bacteria and viruses, and resistance to diseases caused by these organisms. The laboratory provides training in current technology using bacteria and viruses. Three classroom meetings and two laboratory periods. Recommended: Organic Chemistry. (1.25 course credits) Prerequisite: C- or better in CHEM 12000 and BIOL 20100 or permission of instructor. Annually. Spring. BIOL 34000. FIELD BOTANY AND SYSTEMATICS Introduction to the principles of field botany and plant systematics. Topics covered include floral and vegetative morphology, plant family characteristics, the use of keys, and basic collecting techniques. We will discuss current methods of biological systematics, traits useful for making phylogenetic inferences, and the evolutionary history of vascular plant groups, especially angiosperms. Topics will include floral biology and pollination, hybridization and speciation, molecular phylogenetics, ethnobotany, and biogeography. Three classroom meetings and one laboratory weekly. (1.25 course credits) Prerequisite: C- or better in BIOL 20200. Alternate years. Not offered 2011-2012. BIOL 34400. COMPARATIVE ANIMAL PHYSIOLOGY (Neuroscience) A detailed study of selected aspects of the physiological ecology of vertebrates and invertebrates, with emphasis on circulatory systems, respiratory systems, energetics, thermoregulation, salt and water balance, and chemical regulation. The laboratory component emphasizes techniques in organismal physiology and experimental design. Three classroom meetings and one lab meeting weekly. (1.25 course credits) Prerequisite: C- or better in BIOL 20100 and 20200 and in CHEM 12000 or permission of instructor. Annually. Spring. BIOL 35000. POPULATION AND COMMUNITY ECOLOGY (Environmental Studies) A study of ecological principles as they apply to populations, communities, and ecosystems. Topics include physiological ecology, population growth, competition, predation, community structure, patterns of energy and nutrient cycling, and species diversity. Laboratory exercises emphasize experimental techniques used to investigate ecological questions. Three classroom meetings and one laboratory weekly. (1.25 course credits) Prerequisite: C- or better in BIOL 20200 or permission of the instructor. Alternate years. Fall 2011. BIOL 35200. ANIMAL BEHAVIOR (formally Behavioral Ecology) (Environmental Studies, Neuroscience) Why do animals behave the way they do? In this course, we will study this question from a variety of angles including: development, mechanistic causes, functional significance, and evolution. We will draw examples from a wide taxonomic spectrum of animals. The laboratory-field period of the course will emphasize how to address animal behavior questions by involving students in studies in which they learn techniques and tools Biology 46 used for observation, experimental design, conducting experiments, and analyzing and presenting results. This course is also an elective for the Neuroscience major. Two classroom meetings and one laboratory-field period weekly. Prerequisite: C- or better in BIOL 20002 or PSYC 32300, or permission of the instructor. Alternate years. Not offered 2011-2012. BIOL 35600. CONSERVATION BIOLOGY (Environmental Studies) This course examines the theory, methods, and tools by which biologists attempt to understand and to protect biological habitats and their attendant natural populations of organisms. Topics included demographic and genetic conservation, invasive species, fragmentation and habitat loss, design of nature reserves, management for conservation, and sustainable development within a conservation context. We also examine economic, social, and political pressures that influence conservation decision-making. Laboratory exercises include computer simulations, field trips, and group projects. Three classroom meetings and one three-hour laboratory weekly. (1.25 course credits) Prerequisite: C- or better in BIOL 20200, and C- or better in one 300-level class in ecology or organismal biology prior to enrolling. Alternate years. Fall 2011. BIOL 36000. EVOLUTION (Geology) This course provides an in-depth introduction to evolutionary theory using both molecular and organismal approaches. Topics include: natural and sexual selection, population genetics, speciation, phylogenetics, and adaptation. The history of evolutionary thought and its place in human tradition will also receive emphasis. Three classroom meetings weekly. Prerequisite: C- or better in BIOL 20100 and 20200 or GEOL 25000 and BIOL 20200 or permission of instructor. Annually. Fall. BIOL 36600. IMMUNOLOGY This course will investigate concepts in immunology from a physiological and molecular perspective. Topics to be covered include the lymphatic system and the lymphoid organs, immune cell development and function, antibody structure and function, specific and nonspecific response to infections, allergy, hypersensitivity and other immunological disorders, transplantation immunology, vaccination, and immunological applications in biotechnology. Laboratory exercises will focus on basic immunological techniques such as antibody-antigen interactions, antibody production, and cellular response to infection. Recommended: Organic Chemistry. (1.25 course credits) Prerequisite: C- or better in CHEM 12000 and BIOL 20100 or permission of instructor. Annually. Not offered 2011-2012. BIOL 37700. BEHAVIORAL ENDOCRINOLOGY (Neuroscience) A study of the interrelationships of the endocrine system and behavior of animals. Topics include reproduction, parental behavior, aggression, biological rhythms, mood, and stress. Special emphasis will be placed on endocrine and neuroendocrine mechanisms of behavior. Laboratory exercises include an introduction to endocrine techniques, experimental investigations of hormones and behavior, and comparative anatomy of the endocrine system. Three hours of lecture and three hours of laboratory weekly. This course is also an elective for the Neuroscience major. (1.25 course credits) Prerequisite: C- or better in BIOL 20100 and BIOL 20200 or NEUR 38000/BIOL 38000, and in CHEM 12000, or permission of instructor. Annually. Fall. BIOL 39500-39503. SPECIAL TOPICS IN BIOLOGY A seminar for advanced students in the life sciences to further explore interdisciplinary topics in biology, such as Biological Rhythms, Bioinformatics, Plant-Animal Interactions, and Biogeography. Prerequisites: Junior or senior standing with significant coursework in biology, as determined by the course instructor. This course does not count toward a major or minor in Biology. (.5 course credits) Offered occasionally as needed. Spring 2012. BCMB 30300. TECHNIQUES IN BIOCHEMISTRY AND MOLECULAR BIOLOGY BCMB 33100. PRINCIPLES OF BIOCHEMISTRY BCMB 33200. BIOCHEMISTRY OF METABOLISM BCMB 33300. CHEMICAL BIOLOGY NEUR 38000. NEUROBIOLOGY (Biochemistry and Molecular Biology) This course focuses on cellular and molecular aspects of nervous system function. Topics include functional implications and physiological basis of neuronal impulse conduction and neurotransmission, sensation and perception (e.g. pain and vision), neuronal plasticity, and the cellular and molecular basis of common neurological diseases. Three lecture periods and one laboratory period weekly. Recommended: one upper-level Biology course or NEUR 32300. Prerequisite: C- or better in BIOL 20100 and in CHEM 12000, or permission of instructor. Annually. Not offered 2011-2012. Business Economics 47 BIOL 40000. TUTORIAL Evaluation of the student’s accomplishment will be based on a contract with the supervising professor. Normally, laboratory exercises constitute at least one-quarter of the tutorial. Students will apply to the departmental chair for this option. This course does not count toward a major or minor in Biology. (.5 - 1.0 course credits) Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. BIOL 40100. INDEPENDENT STUDY AND BIOSTATISTICS An introduction to the techniques and practices of biological research. One classroom meeting weekly will focus specifically on the design of experiments and the analysis of biological data. An additional weekly meeting with the student’s advisor will focus on project design and exploration of the literature related to the proposed I.S. thesis. A written I.S. thesis proposal is due at the end of the semester. Prerequisite: A grade of C- or better in BIOL 20000, 20100, 20200, and one 300-level Biology course. Annually. Spring 2012. BIOL 40200, 40300. INDEPENDENT STUDY These courses allow a student to pursue a special interest on an independent basis and usually require laboratory or field work as well as examination of pertinent literature. The work will be supervised and evaluated by one faculty member. This course does not count toward a major or a minor in Biology. BIOL 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The thesis in Biology is based on a laboratory or field investigation in which data are collected and analyzed in comparison with the literature related to the project. A student should devote the same amount of time to the research and the subsequent thesis in BIOL 45100 and 45200 as that required for two major laboratory courses. The work is ordinarily done in two terms, one of which may be completed in the summer session. Data may be collected off campus if suitable supervision can be arranged. Normally, a student will have one research adviser. Prerequisite: BIOL 40100. BIOL 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The research adviser, together with a second professor, reads the thesis and conducts an oral examination of the student on the field of research. The evaluation of the thesis will be determined by these two readers in consultation with the department as a whole. Prerequisite: BIOL 45100. BIOLOGY SEMINAR The seminar series provides group experiences in oral communication and criticism. In addition to student presentations, guest speakers and departmental staff present their recent research activities. All students pursuing thesis research or enrolled in Independent Study courses are required to attend a weekly departmental seminar. Biology majors are urged to attend these seminars in anticipation of thesis research and as a means of broadening their perspectives. BUSINESS ECONOMICS James Burnell, Chair Barbara Burnell Phillip Mellizo Amyaz Moledina John Sell Lisa Verdon James Warner Jingjing Yang Affirming the mission of the college, the Economics department enables students and faculty to collaboratively research and understand complex questions from a diversity of economic perspectives. The department uses appropriate theories and empirical methods to foster an active engagement with local and global communities. The Business Economics major provides an academically challenging program within the context of the liberal arts for those who desire a sophisticated understanding of business operation and an appreciation for the social and economic Business Economics 48 complexities of the world in which firms operate. The major is intended for students who plan to enter the business world directly after graduation, but it is sufficiently flexible to accommodate those who may choose graduate study. Those interested in international business should consider the special recommendations that pertain to them below. Students who desire a more policy-oriented major should consider the Economics major also offered by the Department of Economics. The requirements for the major are formulated to acquaint the student with the structure and organization of the business firm, and to provide a framework of theoretical and quantitative analysis necessary for business decision-making. For students who qualify, the Business Intern program provides the opportunity to gain experience in working for a business firm as part of the academic program. Major in Business Economics Consists of fifteen courses: • BUEC 11900 • ECON 10100 • ECON 11000(see note below) • One of the following courses: MATH 10400, 10800, or 11100 • ECON 20100 • ECON 20200 • ECON 21000 (see note below) • Three elective Business Economics courses, one of which must be at the 300-level • Two elective Economics courses • Junior Independent Study: BUEC 40100 • Senior Independent Study: BUEC 45100 • Senior Independent Study: BUEC 45200 The Interdisciplinary Minor in International Business Economics • BUEC 11900 • ECON 10100 • ECON 11000 • Two elective Business Economics courses at the 200-level or above, excluding BUEC 40700, 40800 • One of the following courses: ECON 25100, 25400, or 35000 Special Notes • Majors may substitute MATH 24100 for ECON 11000 and MATH 24200 for ECON 21000. • Majors who do not place into MATH 10400 or 11100 on the Mathematics placement test should take MATH 10300 or MATH 10700 as soon as possible in their College career to prepare them for MATH 10400 or 10800 and to provide a basis for their Economics courses. • ECON 10100, ECON 11000, and MATH 10400 should be completed no later than the end of the student’s fifth semester. The department recommends that students considering graduate study in Economics enroll in MATH 11100 rather than MATH 10400 and that they also take calculus through MATH 11200. • The department requires that ECON 20100 or 20200 be taken prior to enrolling in BUEC 40100. • The minor in International Business Economics must be taken in conjunction with a language major (currently French, German, or Spanish) selected by the student. Business Economics 49 • There is no general Business Economics minor. The non-major who desires a background in business economics is urged to take BUEC 11900, ECON 10100 and 11000, MATH 10400, and other elective Business Economics courses according to his or her interests. • Business Economics majors are not permitted to take courses in the major on an S/NC basis. • A grade of C- or better is required for all courses counting toward the major, including the Mathematics course. Students receiving a grade below C- in ECON 10100 must retake that course before proceeding to the other Economics or Business Economics courses. BUSINESS ECONOMICS COURSES BUEC 11900. FINANCIAL ACCOUNTING The study of basic accounting concepts and principles used in the preparation and interpretation of financial statements. Annually. Fall and Spring. [Q] BUEC 22700. MONEY AND CAPITAL MARKETS An analysis of financial intermediaries, why they exist, and how they function. Topics include money market theory and practice, primary and secondary stock and bond markets, mortgage markets, insurance markets, and the markets for derivative securities. Prerequisite: ECON 10100. Fall 2011. [HSS] BUEC 23000. MARKETING An analysis of the entrepreneurial aspects of establishing mutually beneficial exchange relationships. Topics include market research and segmentation strategies as well as product development, promotion, pricing, and distribution. Prerequisite: ECON 10100. Not offered 2011-2012. [HSS] BUEC 25000. CORPORATE FINANCE Study of the firm’s investing and financing decision-making process and its relationship to the firm’s internal and external economic environment. Particular attention is paid to the firm’s stakeholders in the financial markets and to a discussion of the advantages and disadvantages of the methods used in capital budgeting, capital structure, and dividend policy decisions. Prerequisite: ECON 10100 and BUEC 11900. Annually. Spring 2012. [HSS] BUEC 29900. SPECIAL TOPICS IN BUSINESS ECONOMICS A course designed to explain an application of business economic analysis to contemporary issues. Prerequisite: ECON 10100 and BUEC 11900. BUEC 32500. AGENCY ECONOMICS This course surveys how economistshave studied and conceptualized individual and group agency—or the capacity for human beings to make choices and to impose those choices on the world around them. Topics examining the main insights from Classical, Evolutionary, Behavioral, and Experimental Game Theory are explored. Additional topics survey the principle findings and implications of Behavioral Economics, Neuroeconomics, and Behavioral Finance for Economics and related social sciences. Prerequisite: ECON 20100. Alternate years. Fall 2011. BUEC 35500. ORGANIZATION OF THE FIRM Study of the internal structure of the firm, examining the incentives of the firm’s various constituencies (owners, managers, suppliers, employees). Particular emphasis is placed on the separation of ownership and control in the public corporation. Prerequisite: ECON 10100 and 20100. Fall 2011. [W] BUEC 36500. PORTFOLIO THEORY AND ANALYSIS A study of alternative types of investments, including a discussion of the methods utilized in selecting and evaluating security portfolios. Prerequisites: ECON 10100, 11000, 21000; BUEC 11900; MATH 10400 or 10800. Not offered 2011-2012. BUEC 37000. STRATEGIC MANAGEMENT This course analyzes business problem-solving from the perspective of various functional areas within a complex external environment. The approach is a mix of theory and case study designed to give students an oppor- Chemical Physics 50 tunity to apply previous course work while forcing them to consider tradeoffs and balance alternatives. Prerequisite: BUEC 11900, ECON 11000 and 20100, and one other Business Economics course. Annually. Spring. BUEC 39900. SPECIAL TOPICS IN ADVANCED BUSINESS ECONOMICS A seminar designed for the advanced business economics major. Topics will reflect new developments in business economics. Prerequisite: BUEC 11900, ECON 20100. BUEC 40100. INDEPENDENT STUDY A one-semester course that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. Prerequisite: ECON 11000 and 21000 (ECON 21000 may be taken concurrently) and either ECON 20100 or 20200. Annually. Spring. BUEC 40700, 40800. BUSINESS ECONOMICS INTERNSHIP Qualified students will be placed with a firm selected in accordance with their goals and interests. Placement will be for 22 weeks. (2 course credits) S/NC course. Prerequisite: junior standing, 2.75 cumulative grade point average, ECON 10100 and 20200, BUEC 11900, and permission of intern coordinator. Annually. Fall. BUEC 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Spring semester enrollment in BUEC 45100 is by permission only. Prerequisite: Successful completion of all components of BUEC 40100. BUEC 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: BUEC 45100. CHEMICAL PHYSICS CURRICULUM COMMITTEE: Paul Edmiston (Chemistry Department Chair) John Lindner (Physics Department Chair) Chemical physics provides an interdisciplinary approach to the fields of chemistry and physics using mathematical techniques. The major allows students to explore the interface between chemistry and physics by studying structure, surfaces, bonding, atoms and molecules. By combining the methodologies and knowledge of physics and chemistry, many intriguing scientific questions can be addressed by a student with a strong predilection for mathematics and the physical sciences. Major in Chemical Physics Consists of sixteen courses: • CHEM 12000 • MATH 11100 • MATH 11200 • PHYS 20300 • PHYS 20400 • PHYS 20500 • MATH 22100 • CHEM 31800 • CHEM 31900 • PHYS 35000 • One elective Chemistry course at the 200-level or above • Two elective Chemistry or Physics courses at the 200-level or above Chemistry 51 • Junior Independent Study: CHEM 40100 or PHYS 40100 (see note below) • Senior Independent Study: CHEM 45100 or PHYS 45100 • Senior Independent Study: CHEM 45200 or PHYS 45200 Special Notes • The Junior and Senior Independent Study courses must be in the same department. • For students who begin in CHEM 11000, the required CHEM 31900 will count as their upper-level Chemistry elective. • Examples of courses commonly taken as the elective Chemistry and/or Physics courses are: Organic Chemistry (CHEM 21100, 21200), Analytical Chemistry (CHEM 21500), Inorganic Chemistry (CHEM 34000), Principles of Biochemistry (BCMB 33100), Mathematical Methods for the Physical Sciences (PHYS 20800), Mechanics (PHYS 30100), Thermal Physics (PHYS 30200), and Electricity and Magnetism (PHYS 30400). • No minor is offered in Chemical Physics. • Students may not double major in Chemical Physics and in any of the participating departments of Chemistry, Physics, or Mathematics. Any student who anticipates attending graduate school in chemistry or physics should also take additional courses in those disciplines. • Interested students should discuss plans with the chairperson of the Department of Chemistry or Physics. • The S/NC grading option may not be used for courses required for this major. • All courses and associated labs must be completed with a C- or better. CHEMISTRY Paul Edmiston, Chair Judith Amburgey-Peters Paul Bonvallet Sibrina Collins Karl Feierabend Sarah Schmidtke Melissa Schultz Nicholas Shaw Mark Snider Robert Woodward Chemistry is broadly defined as the study of the properties of matter and how matter is transformed. The faculty and staff of the Department of Chemistry work to maintain a student-centered curriculum, a supportive environment, and a vibrant intellectual community for Chemistry majors and non-majors alike. Students are guided to become ethical, productive members of society who apply their scientific knowledge and skills in a broad range of endeavors. Instruction in the discipline integrates learning through coursework, laboratory, and research. Consequently, students develop a variety of skills including laboratory methods, use of instrumentation, information literacy, problem solving, oral and written communication, and research design necessary to succeed in their future endeavors. The curriculum is influenced by the guidelines from the American Chemical Society Committee on Professional Training (ACS CPT) and is comprised of courses Chemistry 52 covering the major sub-disciplines of chemistry. Feedback from alumni indicates that their Wooster education has prepared them well for a range of careers and life pursuits. Feedback from graduate and professional schools and employers indicate that students are well prepared in chemistry knowledge, techniques, instrumentation, and have the capabilities necessary to learn, adapt, and lead. Major in Chemistry Consists of sixteen courses: • CHEM 12000 (see note below) • MATH 11100 • MATH 11200 • CHEM 21100 • CHEM 21200 • CHEM 21500 • PHYS 20300 (or 10100) • PHYS 20400 (or 10200) • CHEM 31800 • CHEM 31900 • CHEM 34000 • Two of the following courses: CHEM 21600, 31300, 31600, 32000, 34100, 39900, BCMB 30300, 33100, 33200, or 33300 (see note below) • Junior Independent Study: CHEM 40100 • Senior Independent Study: CHEM 45100 • Senior Independent Study: CHEM 45200 Minor in Chemistry Consists of six courses: • CHEM 12000 (see note below) • CHEM 21100 • CHEM 21500 • Three Chemistry courses at the 200-level or above (see note below) Special Notes • Students who intend to take Chemistry courses at Wooster should take the Chemistry placement exam. Enrollment into CHEM 12000 requires satisfactory performance on the Chemistry Department placement exam, completion of CHEM 11000 with a C- or better, or AP Chemistry credit. Students who test out of CHEM 11000 without AP Chemistry credit do not receive credit for CHEM 11000. • For the major, students who begin in CHEM 11000 are only required to take one elective, and CHEM 31900 will count as their second elective for the major. • For the minor, students who begin with CHEM 11000 are only required to take two Chemistry courses at or above the 200-level. • The MATH 11100 requirement may be fulfilled by the successful completion of both MATH 10700 and 10800. • Concurrent enrollment in both class and laboratory is required for students taking a course with a laboratory component. Students who do not complete the class or laboratory component of a course with a C– or better must repeat both the class and the laboratory. • A student may not take CHEM 10100 concurrent with or after CHEM 12000. Chemistry 53 • A student who presents a score of 4 or 5 on the Advanced Placement Examination in Chemistry automatically receives credit for CHEM 11000. Students who take the Chemistry Department placement exam will be placed into Principles of Chemistry (CHEM 12000) or Organic Chemistry I (CHEM 21100) depending upon the exam results. • International students with a certificate from a foreign Baccalaureate program may receive either one or two Chemistry course credits. Students who take the Chemistry Department placement exam will be placed into Principles of Chemistry (CHEM 12000) or Organic Chemistry I (CHEM 21100) depending upon the department placement exam results. If the student places into Principles of Chemistry (CHEM 12000), 1.0 credit will be awarded for CHEM 11000. If the student places into Organic Chemistry I (CHEM 21100), 2.25 credits will be awarded for CHEM 11000, CHEM 12000, and CHEM 12000L. • Chemistry majors who plan to attend graduate school are strongly encouraged to pursue an ACS-certified degree. The requirements for an American Chemical Society Certified Degree are summarized below: (a) Chemistry: CHEM 12000, 21100, 21200, 21500, 31800, 31900, 34000, 40100, 45100, 45200, BIOL 20000, BCMB 33100, MATH 11100, 11200, PHYS 20300, 20400. This differs from the minimal Wooster major by two courses: Principles of Biochemistry (BCMB 33100) and its prerequisite Foundations of Biology (BIOL 20000). (b) Chemistry/Chemical Physics: CHEM 12000, 21100, 21200, 21500, 31800, 31900, 34000, BIOL 20000, BCMB 33100, MATH 11100, 11200, PHYS 20300, 20400, two Physics courses beyond PHYS 20400, two advanced courses in theoretical chemistry, physics, or math; CHEM or PHYS 40100, 45100, 45200. • Students considering a Chemistry major should consider one of the sequences below: (i) beginning in CHEM 12000 with sufficient math preparation: Fall Spring First Year CHEM 12000 CHEM 21500 MATH 11100 MATH 11200 Sophomore Year CHEM 21100 CHEM 21200 PHYS 20300 (or 10100) PHYS 20400 (or 10200) (ii) beginning in CHEM 11000 with additional math preparation needed: Fall Spring First Year CHEM 11000 CHEM 12000 MATH 10700 MATH 10800 Sophomore Year CHEM 21100 CHEM 21200 PHYS 20300 (or 10100) PHYS 20400 (or 10200) MATH 11200 CHEM 21500 Chemistry 54 • All courses counting towards the Chemistry major must be passed with a C– or better and may not be taken S/NC; this applies to classroom and laboratory components. CHEMISTRY COURSES CHEMISTRY FOR THE NON-SCIENCE MAJOR CHEM 10100-10103. CHEMISTRY AND THE WORLD IN WHICH WE LIVE A study of chemistry is undertaken using the world around us as a starting point in developing an understanding of the facts, theories, and methodology of the chemical sciences. Topics may include environmental chemistry, food chemistry, forensics, and science in society. Topics will be announced in advance; past are listed below. Not open to students who have received credit for or are concurrently enrolled in CHEM 12000. Students with CHEM 12000 credit may apply to serve as a Teaching Apprentice. Three class hours per week. Annually. Fall and Spring. [Q, MNS] 10102. FORENSIC SCIENCE Law enforcement techniques such as DNA typing, fingerprint identification, drug/explosives detection, and fiber analysis are covered in class and short laboratory experiments. The underlying principles of forensic techniques are discussed, drawing on examples from true crime investigations. The broader impact of such methods and investigations is covered in the course. Not Offered 2011-2012. 10103. CHEMISTRY AND THE ENVIRONMENT - WATER A study of chemistry is undertaken using the world around us as a starting point in developing an understanding of the facts, theories, and methodology of the chemical sciences. The chemistry involved in environmental topics such as climate change, ozone depletion, acid rain, and water pollution will be emphasized in this course offering. The central importance of water’s physical and chemical properties to these topics will be highlighted as a theme throughout the semester. The popular media’s role in covering environmental chemistry issues will also be addressed, with an emphasis on critical thinking. Spring 2012. CHEMISTRY FOR THE SCIENCE MAJOR CHEM 11000. INTRODUCTORY CHEMISTRY (Biochemistry and Molecular Biology) Fundamental facts, concepts, and theories of chemistry and mathematical skills are emphasized. Topics include matter, measurements, calculations, elements, atomic theory, atomic mass, the mole, ionic and molecular compounds, types of bonding, mole calculations, types of reactions, limiting reagents, percent yield, solutions, gases, quantum mechanics, orbitals and electrons, electronic structure, atomic periodicity, and Lewis theory. Emphasis will be placed on problem-solving and the development of critical thinking skills. The course is intended for students with limited chemistry and math preparation in high school. Three class hours per week. Annually. Fall and Spring. [Q, MNS] CHEM 12000. PRINCIPLES OF CHEMISTRY (Biochemistry and Molecular Biology) Fundamental facts, concepts, and theories central to chemistry are examined. The topics include VSEPR, valence bond, and molecular orbital theories, intermolecular forces, solutions and colligative properties, chemical kinetics, reaction mechanisms, equilibria (chemical, acid-base, aqueous, ionic), thermodynamics (enthalpy, entropy, free energy), and electrochemistry. The laboratory focuses on fundamental techniques, data manipulation, notebook and reporting skills. Three class hours and one three-hour laboratory period per week. (1.25 course credits) Prerequisite: CHEM 11000 with a C- or better, or satisfactory performance on the Chemistry Department placement exam. Annually. Fall and Spring. [Q, MNS] CHEM 21100. ORGANIC CHEMISTRY I (Biochemistry and Molecular Biology) The fundamental principles of structure, bonding, and reactivity of organic compounds are introduced. Content focuses on functional groups, reaction mechanisms, spectroscopic techniques, data interpretation, and introductory synthetic methods. Critical thinking, application of general concepts to new examples, and problemsolving skills are emphasized. Laboratory experiments incorporate key synthetic organic laboratory skills, reactions, techniques, and instrumentation. The experiments promote independence, information literacy, safety, writing skills, and laboratory competency. Three class hours and one three-hour laboratory period per week. (1.25 course credits) Prerequisite: CHEM 12000 with a C- or better. Annually. Fall. [MNS] Chemistry 55 CHEM 21200. ORGANIC CHEMISTRY II (Biochemistry and Molecular Biology) The study of organic structure, bonding, and reactivity continues with more complex molecules including aromatics, carbonyl compounds, amino acids, and carbohydrates. Advanced spectroscopic data analysis and multi-step syntheses challenge students to be creative, critical thinkers. In the laboratory, students apply skills from CHEM 21100, increase independence, and learn new techniques through research-based projects involving synthesis and spectroscopic identification. Information literacy, safety, and writing (notebooks, technical reports, summaries, and experimental plans) are emphasized. Three class hours and one three-hour laboratory period per week. (1.25 course credits) Prerequisite: CHEM 21100, C- or better. Annually. Spring. [W, MNS] CHEM 21500. ANALYTICAL CHEMISTRY (Biochemistry and Molecular Biology) The fundamental principles and methodology of chemical analysis are examined with examples from biochemistry and organic and inorganic chemistry. Topics include discussion of errors and statistical treatment of data, a review of equilibria, and introduction to spectroscopy, electrochemistry, and analytical separations. The laboratory emphasizes experimental design, using library resources, and methods for obtaining and evaluating quantitative data. Methods employed include spectroscopy, potentiometry, chromatography, mass spectrometry, and titrimetry. Three class hours and one three-hour laboratory period per week. Recommended previous course: CHEM 21200. (1.25 course credit) Prerequisite: CHEM 12000, C- or better. Annually. Spring. [Q, MNS] CHEM 21600. ENVIRONMENTAL CHEMISTRY (Environmental Studies) Various aspects of the chemistry of the environment, both unpolluted and polluted, are discussed. Emphasis is placed on chemical reactions in the atmospheric and aquatic realms, the relationship between chemical structure and environmental transport, and the toxicity and effects of common environmental pollutants. Case studies are used from the literature to further explore the course material. Three class hours per week. Suggested previous course: CHEM 211. Prerequisite: CHEM 120, C- or better. Alternate years. Prerequisite: CHEM 12000, Cor better. Alternate years. Not offered 2011-2012. CHEM 31300. ADVANCED ORGANIC CHEMISTRY The course focuses on the experimental, instrumental, and theoretical methods by which the structure, reactivity, and electronic properties of organic compounds are determined. Various aspects of modern organic chemistry, including synthesis, mechanism, advanced spectroscopic methods, and computational chemistry may be covered. Historic and current case studies are taken from the chemical literature. Three class hours per week. Prerequisite: CHEM 21200, C- or better. Alternate years. Spring 2012. CHEM 31600. INSTRUMENTAL ANALYSIS Modern methods of chemical analysis are covered with an emphasis on spectroscopy, mass spectrometry, separations, and surface analytical techniques. Particular focus is placed on the use of instruments in chemical industry, clinical analysis, and environmental monitoring. Laboratory work involves multi-week independent projects. Three class hours and one three-hour laboratory period per week. Suggested previous course: CHEM 31800. (1.25 course credits) Prerequisite: CHEM 21500, C- or better. Fall 2011. CHEM 31800. PHYSICAL CHEMISTRY I Chemical thermodynamics and kinetics. Topics include chemical kinetics, rate laws, laws of thermodynamics, free energy and chemical equilibrium. Additional topics may include chemical dynamic models, X-ray diffraction, solid-state structure, and structure determination. Three class hours and one three-hour laboratory per week. (1.25 course credits) Prerequisite: CHEM 12000 with a C- or better, and MATH 11100 with a C- or better. Annually. Fall. [MNS] CHEM 31900. PHYSICAL CHEMISTRY II Quantum and statistical mechanics. Topics include quantum mechanical theory, quantum mechanical models for motion, the structure of atoms and molecules, molecular symmetry, molecular spectroscopy, and statistical mechanics and thermodynamics. Three class hours and one three-hour laboratory per week. (1.25 course credits) Prerequisite: CHEM 12000 with a C- or better, and MATH 11200 with a C- or better. Annually. Spring. [MNS] CHEM 32000. TOPICS IN PHYSICAL CHEMISTRY Advanced topics in physical chemistry are examined. Topics may include: computational chemistry, advanced spectroscopic methods, chemical modeling, atmospheric or condensed phase kinetics. Three class hours per week. Prerequisite: CHEM 31800 or CHEM 31900 with a C- or better. Not offered 2011-2012. [MNS] CHEM 34000. INORGANIC CHEMISTRY The details of the chemistries of selected elements and their compounds are studied. For each class of substances studied, the topics of structure, bonding, and reactivity are linked, with some discussion of mechanism, in order to give an overall survey of the chemistry of elements from various portions of the periodic table. Three class hours and one three-hour laboratory period per week. (1.25 course credits) Prerequisite: CHEM 21200, C- or better, or permission of the instructor. Annually. Fall. [MNS] Chinese Studies 56 CHEM 34100. ADVANCED INORGANIC CHEMISTRY Advanced aspects of inorganic chemistry are treated, including the organometallic chemistry of transition metal compounds and the chemistry of catalysis. The course is designed to emphasize structure, bonding, and spectroscopy, as well as syntheses and reaction mechanisms. Three class hours per week. Prerequisites: CHEM 31800 and 34000, C- or better, or permission of the instructor. Alternate years. Not offered 2011-2012. [MNS] CHEM 39900. BIOPHYSICAL CHEMISTRY The underlying physical principles and laws that govern the behavior of biological systems and biochemical reactions are examined. The fundamental principles of molecular structure, chemical kinetics, and thermodynamics are explored in relationship to biological phenomena. Three class hours per week. Prerequisite: MATH 11100 or 10800 and either BCMB 33100 or CHEM 31800, with a C- or better. Not offered 2011-2012. BCMB 30300. TECHNIQUES IN BIOCHEMISTRY AND MOLECULAR BIOLOGY BCMB 33100. PRINCIPLES OF BIOCHEMISTRY BCMB 33200. BIOCHEMISTRY OF METABOLISM BCMB 33300. CHEMICAL BIOLOGY CHEM 40000. TUTORIAL Advanced library and laboratory research problems in analytical, inorganic, organic, and physical chemistry and biochemistry. (.5 - 1.0 course credit) Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. CHEM 40100. INTRODUCTION TO INDEPENDENT STUDY This course builds background knowledge and practical skills for independent scientific work. Activities in searching the literature, experimental design, drafting and revising scientific writing, and oral presentation culminate in a written research proposal for the Senior Independent Study project. Prerequisite: CHEM 21200 with a C or better or Departmental approval. CHEM 40700, 40800. CHEMICAL RESEARCH INTERNSHIP Students are placed in research positions in non-academic laboratories. The normal schedule involves work during the summer months, in addition to either the fall or spring semester, on a research problem related to the function of the employing laboratory. The work is directed by scientists at the laboratory. Liaison is established by regularly-scheduled consultations with one or more faculty members of the Department of Chemistry. The student’s schedule is arranged only after consultation with the Chemistry chairperson. (1 - 3 course credits) S/NC course. Prerequisite: junior standing; CHEM 21200, 21500, 31800, and 40100. CHEM 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE An original investigation is conducted, culminating in a thesis and an oral defense of the thesis in CHEM 45200. During the Fall each student gives a research seminar on the Independent Study research topic. Projects are offered in selected areas of analytical, inorganic, organic, physical chemistry, and biochemistry. Prerequisite: CHEM 21200 and 40100, C- or better, or approval of the Department. CHEM 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis, the presentation of a poster, and an oral defense. Prerequisite: CHEM 45100. CHINESE STUDIES Rujie Wang, Chair Dan Liu Tingting Lu (Chinese Language Assistant) The Program in Chinese Studies introduces students to both Chinese language and Chinese literature. Its objective is to teach students the basic skills of reading, listening, speaking, and writing Chinese. Besides providing practical training in Chinese for career purposes, the program is also designed to familiarize students with non-Western conceptual schemes and modes of thought. The courses in Chinese Chinese Studies 57 language and literature, together with related courses from the departments of Anthropology, History, Philosophy, and Religious Studies, will give students a strong background in China. Major in Chinese Studies Consists of eleven courses: • CHIN 20100 • CHIN 20200 • CHIN 30100 • CHIN 30200 • One of the following courses: CHIN 31100 or 31200 • Two of the following courses, in two different departments: HIST 10105 or 10150 (when China-focused), 20000, 20100, 23700; PHIL 23200; RELS 21600; SOCI 21100-21105 (when China-focused) • One of the following courses: CHIN 22000, 22200, 22300, 40000; HIST 10105 or 10150 (when China-focused), 20000, 20100, 23700; PHIL 23200; RELS 21600; SOCI 21100-21105 (when China-focused) • Junior Independent Study: CHIN 40100 • Senior Independent Study: CHIN 45100 • Senior Independent Study: CHIN 45200 Minor in Chinese Studies • Consists of six courses: • CHIN 20100 • CHIN 20200 • CHIN 31100 • Three of the following courses: CHIN 22000, 22200, 22300, HIST 20000, 23501, 23700, or PHIL 23200 Special Notes • Overseas Study: Majors in Chinese are required to complete an approved offcampus study program in China. Approved transfer credit from participation in this program can count toward the major requirements. • Minors in Chinese may satisfy the CHIN 20100, 20200, and 31100 requirements by taking the equivalent courses from endorsed off-campus programs. • Students who wish to take the maximum number of courses for this major are encouraged to complete additional courses from the options offered in the major requirements. • Majors and minor are not permitted to take any courses within the department for S/NC credit, nor will classes taken for audit count. • Only grades of C- or better are accepted for the major or minor. CHINESE STUDIES COURSES CHIN 10100. BEGINNING CHINESE LEVEL I Introduces the fundamentals of modern Chinese. Objectives are attainment of proper pronunciation, with special emphasis on tones, basic grammatical patterns, and mastery of approximately 100 characters and compounds. Students are expected to memorize short skits. Five hours per week. Annually. Fall. CHIN 10200. BEGINNING CHINESE LEVEL II A continuation of CHIN 101, the course further develops the four basic skills of reading, writing, speaking, and listening comprehension; it introduces Chinese calligraphy, but the main emphases are oral proficiency and comprehension skills. Students are expected to memorize short skits. Five hours per week. Annually. Spring. Chinese Studies 58 CHIN 20100. INTERMEDIATE CHINESE LEVEL I (East Asian Studies) A continuation of beginning Chinese, with more emphasis on vocabulary-building (over 400 characters and compounds) and reading comprehension. Students are expected to memorize short skits and to write short character essays regularly to express their thoughts. In addition, students are also reading short stories from outside the regular textbooks. Prerequisite: CHIN 10200 or equivalent. Annually. Fall. [C] CHIN 20200. INTERMEDIATE CHINESE LEVEL II (East Asian Studies) A continuation of CHIN 20100 or the equivalent; in addition to textbooks, students will do exercises on language CDs and software applications such as Chinese e-mail or Chinese word processor. The syntactical and grammatical patterns are more complex than those taught in the first year. Students are expected to write and present their essays in Chinese weekly; in addition, students are also reading short stories from outside the regular textbooks. Required of minors. Prerequisite: CHIN 20100 or equivalent. Annually. Spring. [C] CHIN 22000. REBELS, ROMANTICS, AND REFORMERS: BEING YOUNG IN CHINA (Comparative Literature, East Asian Studies) Taught in English. This course introduces the lived experiences of modern Chinese youth as represented in twentieth-century fiction and film. Readings include narrative works by Lu Xun, Lao She, Ba Jin, Mao Dun, Ding Ling, Zhang Ailing, Zhang Jie, Wang Meng, Liu Heng, Wang Shuo, and Xi Xi, as well as poems by Bei Dao, Gu Cheng. The pain, frustration, loneliness, fear and aspiration of the fictional hero shall be understood in relation to social changes in China. We will study many fictional heroes as the shadows of modern man becoming a fragment of his primitive self under the pressures of a progressive civilization. Alternate years. Spring 2012. [C, AH] CHIN 22200. WOMEN IN CHINESE LITERATURE (Comparative Literature, East Asian Studies, Women’s, Gender, and Sexuality Studies) Taught in English. A survey of women’s experience as represented in Chinese literature, ranging from philosophical texts, poetry, song lyrics, short narrative works, music and biographies to films from both pre-modern and modern periods, written about and by women. The course examines how women are depicted and how men and women define womanhood differently in various works of imagination. The primary texts and secondary readings that establish connections and comparisons among the different works include: The Red Brush: Writing Women of Imperial China, Teachers of the Inner Chambers: Women and Culture in Seventeenth-Century China, and Precious Records: Women in China’s Long Eighteenth Century. The theoretical focus is on the construction of femininity in a patrilineal society. Alternate years. Spring 2012. [C, AH] CHIN 22300. CHINESE CINEMA AS TRANSLATION OF CULTURES (Comparative Literature, East Asian Studies, Film Studies) Taught in English. What do Chinese people think of the social transformation of the past 30 years? What are their views and attitudes towards these changes that have affected their lives in profound ways? What are their dreams and fantasies about modernizations? What are their fear and hope when they look into the future? Divided into four groups: historical, rural, urban and Hong Kong and Taiwan, 24 narrative films are studied as auto-ethnographic texts in which the people in the PRC, Taiwan, Hong Kong, and the Chinese diaspora try to negotiate their cultural identity and achieve a translated modernity. In these films of self-representation, China, its people, and its past all get reinvented. Annually. Fall 2011. [C, AH] CHIN 30100. ADVANCED CHINESE I (East Asian Studies) Practice in listening, speaking, reading, and writing at an advanced level. Review of grammatical patterns and expansion of vocabulary for practical use outside the classroom setting. Use of multi-media resources (audio recordings, film, screenplays, newspapers, expository prose) to achieve proficiency. Introduction to cultural topics and intellectual currents most pertinent to contemporary China. Prerequisite: CHIN 20200 or equivalent. Annually. Fall. CHIN 30200. ADVANCED CHINESE II (East Asian Studies) Continuation of CHIN 30100. Additional, more intensive and extensive practice in listening, speaking, reading, and writing at an advanced level. Continued review of grammatical patterns and expansion of vocabulary for practical use outside the classroom setting. Use of multi-media resources (audio recordings, film, screenplays, newspapers, expository prose) to achieve greater proficiency. Continued discussion of cultural topics and intellectual currents most pertinent to contemporary China. Prerequisite: CHIN 30100 or equivalent. Annually. Spring. CHIN 31100. CHINESE MODERNITY AND FILM (East Asian Studies) Development of advanced skills in listening, speaking, reading, and writing. Study of language usage and the acquisition of common popular expressions, newly coined terms, slang, proverbs, and idioms as presented in Chinese film. Use of film as the imagistic representation of modern China. Discussion of current events and introduction to textual analysis. Prerequisite: CHIN 30200 or equivalent. Annually. Fall. Classical Studies 59 CHIN 31200. CHINA: A CULTURAL PANORAMA (East Asian Studies) Study of key issues in Chinese society through the exposure to authentic materials (novella, commercial manuals, classified ads, travel and tourist literature, rental and real estate documents, legal proceedings, job descriptions). Extensive use of audio and video materials to simulate a variety of real life situations to improve oral and written proficiency and deepen cultural knowledge. Prerequisite: CHIN 30200 or equivalent. Annually. Spring. CHIN 40000. TUTORIAL (East Asian Studies) Individually supervised language learning. By prior arrangement with the department only. Prerequisite: CHIN 20200 or equivalent; the approval of both the supervising faculty member and the chairperson is required prior to registration. CHIN 40100. JUNIOR INDEPENDENT STUDY A one-semester course that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. CHIN 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: CHIN 40100. CHIN 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: CHIN 45100. CROSS-LISTED COURSES ACCEPTED FOR CHINESE STUDIES CREDIT HISTORY HIST 10105 or 10150. INTRODUCTION TO HISTORICAL INVESTIGATION (when China-focused) [W, some sections count toward C, HSS] HIST 20000. TRADITIONAL CHINA [C, HSS] HIST 20100. MODERN CHINA [C, HSS] HIST 23700. THE UNITED STATES AND CHINA [C, HSS] PHILOSOPHY PHIL 23200. CHINESE PHILOSOPHY [C, AH] RELIGIOUS STUDIES RELS 21600. CHINESE RELIGIONS [C, R, AH] SOCIOLOGY AND ANTHROPOLOGY SOCI 21105. ADVANCED TOPICS IN SOCIOLOGY (when China-focused) [HSS] CLASSICAL STUDIES Monica Florence, Chair Josephine Shaya Wendy Teo The Department of Classical Studies provides students with opportunities to explore the ancient Mediterranean world with a special focus on the period from the eighth century BCE through the fourth century CE. Through the comparative study of ancient languages and cultures, Classics students acquire additional cultural literacy, becoming better critical thinkers and more engaged global citizens. Our primary goal is for students to understand and examine critically the ancient beliefs, values, and traditions that have shaped modern cultures. The study of the Ancient Mediterranean is inherently interdisciplinary and intercultural. Students are encouraged to learn Ancient Greek, Latin, and Hebrew. They Classical Studies 60 will study the ancient literature, archaeology, history, religion, philosophy, and art produced in the Near East and Mediterranean basin, including ancient Mesopotamia, Israel, Egypt, Greece, and Rome. The Department accommodates and encourages a semester’s study abroad in the Mediterranean region. ANCIENT MEDITERRANEAN STUDIES The concentration in Ancient Mediterranean Studies is one of two within the major of Classical Studies. Through this concentration, students comparatively study multiple cultures in the Near East and Mediterranean basin, including ancient Mesopotamia, Israel, Egypt, Greece and Rome. Our period of study, from the eighth century BCE through the fourth century CE, allows for a particular focus on Greece and Rome, but the approach to the ancient Mediterranean region is interdisciplinary and intercultural. Students in Ancient Mediterranean Studies will examine the ancient literature, archaeology, history, religion, philosophy, and art produced by the network of ancient cultures that relied upon the Mediterranean Sea. The primary goal of this concentration is to examine critically the ancient beliefs, values, and traditions that have shaped modern cultures. Students in Ancient Mediterranean Studies will acquire additional cultural literacy, becoming better critical thinkers and more engaged global citizens. Major in Classical Studies, Concentration: Ancient Mediterranean Studies Consists of eleven courses: • Two courses in either GRK or LAT, at least one at the 200-level • Two of the following courses: AMST 22000, 22100, 22300, 22600, 26000, 26100, HIST 20200 or 20300 • One elective from cross-listed courses accepted for CLST credit • Three electives from Classical Studies or cross-listed courses accepted for CLST credit • Junior Independent Study: CLST 40100 • Senior Independent Study: CLST 45100 • Senior Independent Study: CLST 45200 Minor in Classical Studies, Concentration: Ancient Mediterranean Studies Consists of six courses: • Two of the following courses: GRK 10100, 10200, 20000, 25000, 30000, 35000, LAT 10100, 10200, 20000, 25000, 30000, or 35000 • Two of the following 200-level course: AMST 22000, 22100, 22300, 22600, 26100, HIST 20200 or 20300 • Two elective courses from Classical Studies or cross-listed courses accepted for CLST credit Special Notes • Language Requirement and Courses: The concentration in Ancient Medi - terranean Studies requires a minimum of one semester of ancient Greek or Latin at the 200-level or higher. Most students will need to take GRK 10100 and 10200 or LAT 10100 and 10200 as well as GRK 20000 or LAT 20000. Incoming students who have previously studied ancient Greek or Latin will be placed in the appropriate languages courses through the College’s foreign language placement exams, which are administered during Summer registration for first year students. Students may satisfy the College’s language requirement, and the requirement of introductory ancient Greek or Latin, by testing out of GRK 10100 Classical Studies 61 and 10200 or LAT 10100 and 10200. Majors, however, must take a minimum of one upper-divisional language course at The College of Wooster or an equivalent university during a semester abroad. If a student completes a language course below the level recommended by the placement exam, the student will not receive credit toward graduation for that course without prior permission of the Department Chair. The College’s advanced placement policy is explained in the section on Admission. • Majors who intend to pursue graduate studies in Classics are strongly urged to complete four years of Ancient Greek and four years of Latin. • S/NC courses are not permitted in the major or minor. • Only grades of C- or better are accepted for the major or minor. CLASSICAL LANGUAGES The concentration in Classical Languages is one of two concentrations within the major of Classical Studies. Students of Classical Languages study ancient Greek, Latin, and/or Hebrew, as well as the rich cultural traditions of Greece, Rome, Israel, ancient Mesopotamia, and Egypt. Through the comparative study of these ancient languages and literatures, students in Classical languages acquire additional cultural literacy as they examine critically the ancient beliefs, values, and traditions that have shaped modern cultures. The concentration in Classical Languages best prepares students for graduate school in the discipline of Classics or the fields of Ancient History and Ancient Philosophy. Students in Classical Languages pursue successfully careers in law, medicine, and publishing, as well as graduate school in Linguistics, Comparative Literature, and Classical Archaeology. Major in Classical Studies, Concentration: Classical Languages Consists of eleven courses: • GRK 10100 (see note below) • GRK 10200 • LAT 10100 • LAT 10200 • Three of the following courses: GRK 20000, 25000, 30000, 35000, LAT 20000, 25000, 30000, or 35000 • One of the following courses: AMST 22000, 22100, 22300, 22600, 26100, HIST 20200 or 20300 • Junior Independent Study: CLST 40100 • Senior Independent Study: CLST 45100 • Senior Independent Study: CLST 45200 Minor in Classical Studies, Concentration: Classical Languages Consists of six courses: • GRK 10100 and 10200, or LAT 10100 and 10200 • Four of the following courses: GRK 20000, 25000, 30000, 35000, LAT 20000, 25000, 30000, or 35000 Special Notes • Language Requirement and Courses: Incoming students who have previously studied Latin or Ancient Greek will be placed in the appropriate languages courses through the College’s foreign language placement exams, which are administered during Summer registration for first year students. Students may satisfy the College’s language requirement by testing out of GRK 10100 and Classical Studies 62 10200 or LAT 10100 and 10200. If a student completes a language course below the level recommended by the placement exam, the student will not receive credit toward graduation for that course without prior permission of the Department Chair. The College’s advanced placement policy is explained in the section on Admission. • Majors who intend to pursue graduate studies in Classics are strongly urged to complete four years of Ancient Greek and four years of Latin. • S/NC courses are not permitted in the major or minor. • Only grades of C- or better are accepted for the major or minor. CLASSICAL STUDIES COURSES GREEK GRK 10100. BEGINNING GREEK LEVEL I An introduction to the grammar, syntax, and vocabulary of classical Attic Greek. Emphasis on reading continuous passages in ancient Greek and appreciation of their cultural context. Annually. Fall. GRK 10200. BEGINNING GREEK LEVEL II Continued work in Attic Greek grammar and readings, including selections from prose authors, such as Herodotus, Thucydides, and Plato. Prerequisite: GRK 10100 or placement. Annually. Spring. GRK 20000. SEMINAR IN GREEK LITERATURE (INTERMEDIATE LEVEL I) (Archaeology, Comparative Literature) Offered in conjunction with GRK 30000. Translation and careful study of continuous passages selected from several representative Greek texts — for instance, works of Homer, Hesiod, selected Greek lyric poets, Sophocles, Euripides, Thucydides, Herodotus, Plato, Aristotle, Attic orators, and occasionally non-literary materials (e.g., inscriptions or papyrus). A review of basic grammar; instruction in the use of commentaries, lexicon, reference works, and scholarly literature; an introduction to textual analysis, both literary and historical, and the Major in Classical Studies. Readings will change from year to year. Prerequisite: GRK 10200 or placement. Annually. Not offered 2011-2012. [AH] GRK 25000. SEMINAR IN GREEK LITERATURE (INTERMEDIATE LEVEL II) (Archaeology, Comparative Literature) Intensive readings in and critical study of significant Greek texts. Course may be arranged around a particular author, genre, period, or topic. Readings will change from year to year. Offerings may include Homer and the Epic Tradition; Greek Historians: Herodotus and Thucydides; Greek Lyric Poetry; The Dialogues of Plato; Greek Tragedy: Sophocles and Euripides; The Greek New Testament; The Greek Novel; and The Biography in Greek. Prerequisite: GRK 20000 or placement. Annually. Not offered for 2011-2012. [AH] GRK 30000. SEMINAR IN GREEK LITERATURE (ADVANCED LEVEL I) (Comparative Literature) Offered in conjunction with GREK 200. An in-depth translation and examination of representative texts—for instance, Homer, Hesiod, selected Greek lyric poets, Sophocles, Euripides, Thucydides, Herodotus, Plato, Aristotle, Attic orators, and occasionally non-literary materials (e.g., inscriptions or papyrus). Peer teaching of Greek grammar; active engagement with commentaries, reference works, and the scholarly literature; textual analysis, both literary and historical, as well as theoretical approaches to Greek history and Greek literature. Readings will change from year to year. May be repeated once for credit. Prerequisite: GRK 20000 or placement. Annually. Not offered 2011-2012. [AH] GRK 35000. SEMINAR IN GREEK LITERATURE (ADVANCED LEVEL II) (Comparative Literature) Offered in conjunction with GRK 25000. Intensive readings in and critical study of significant Greek texts. Course may be arranged around a particular author, genre, period, or topic. Readings will change from year to year. May be repeated once for credit. Annually. Not offered 2011-2012. [AH] GRK 40000. TUTORIAL Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. LATIN LAT 10100. BEGINNING LATIN LEVEL I An introduction to the Latin language with emphasis on vocabulary, morphology, syntax and the mastery of basic grammar. This course is designed for students who have had no previous work in Latin or who, based on performance on the placement examination, place in LAT 10100. Annually. Fall. Classical Studies 63 LAT 10200. BEGINNING LATIN LEVEL II Continued work in the basics of the Latin language, with emphasis on reading selections from a variety of Latin authors, whose work we situate in proper cultural context. Prerequisite: LAT 10100 or placement. Annually. Spring. LAT 20000. SEMINAR IN LATIN LITERATURE (INTERMEDIATE LEVEL I) (Archaeology, Comparative Literature) Offered in conjunction with LATN 300. Translation and careful study of continuous passages selected from several representative Latin texts — for instance, Cicero, Sallust, Catullus, Ovid, Vergil, Petronius, Pliny, and occasionally non-literary materials (e.g., inscriptions or papyrus). A review of basic grammar; instruction in the use of commentaries, reference works, and scholarly literature; and an introduction to textual analysis, both literary and historical, and the Major in Classical Studies. Readings will change from year to year. Prerequisite: LAT 10200 or placement. Annually. Fall. [AH] LAT 25000. SEMINAR IN LATIN LITERATURE (INTERMEDIATE LEVEL II) (Archaeology, Comparative Literature) Intensive readings in and critical study of significant Latin texts. Course may be arranged around a particular author, genre, period, or topic. Readings will change from year to year. Offerings include The World of Cicero; Vergil and the Epic Tradition; Roman Historians: Sallust, Livy, and Tacitus; Roman Comedy: Plautus and Terence; Roman Satire: Horace and Juvenal; Roman Erotic Poetry: Catullus, Horace, and Ovid; Petronius and Roman Novel; Medieval Latin. Prerequisite: LAT 20000 or placement. Annually. Spring. [AH] LAT 30000. SEMINAR IN LATIN LITERATURE (ADVANCED LEVEL I) (Comparative Literature) Offered in conjunction with LAT 20000. Translation and careful study of extended passages selected from several representative Latin texts—for instance, Cicero, Sallust, Catallus, Ovid, Vergil, Petronius, Pliny, and occasionally non-literary materials (e.g., inscriptions or papyrus). Peer teaching of basic grammar; active engagement with commentaries, reference works, and scholarly literature; textual analysis, both literary and historical, and an introduction to theoretical approaches to Roman history and Latin literature. Readings will change from year to year. May be repeated once for credit. Prerequisite: LAT 20000 or placement. Annually. Fall. [AH] LAT 35000. SEMINAR IN LATIN LITERATURE (ADVANCED LEVEL II) (Comparative Literature) Offered in conjunction with LAT25000. Intensive readings in and critical study of significant Latin texts. Course may be arranged around a particular author, genre, period, or topic. Readings will change from year to year. May be repeated for credit. Annually. Spring. [AH] LAT 40000. TUTORIAL Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. ANCIENT MEDITERRANEAN STUDIES Knowledge of Greek or Latin is not required for the following courses: AMST 22000. MYTHOLOGY OF THE ANCIENT WORLD (Comparative Literature) A comparative approach to ancient myths with particular regard to how these narrative patterns and religious beliefs recur in other cultures and time periods. Texts vary but may include the Mesopotamian Epic of Gilgamesh, selected ancient Egyptian fairy tales, the Hindu Ramayana, and classical Greek, Roman, and Italian works such as Homer’s Iliad and Odyssey, Hesiod’s Theogony, Sophocles’ Oedipus, Vergil’s Aeneid, Ovid’s Metamorphoses, and Dante’s Inferno. Spring 2012. [AH] AMST 22100. ANCIENT THEATER: TRAGEDY AND COMEDY (Comparative Literature) An examination of the drama of the ancient world. Particular attention may be paid to Greek and Roman representations of Persia, Egypt, and other ancient cultures. Other themes may include the origins of comedy and tragedy, theories of drama, stagecraft, costuming, and the classical tradition. Plays vary but may include Aeschylus’ Persians, Sophocles’ Oedipus, Euripides’ Medea and Bacchae, Aristophanes’ Lysistrata, and the Roman comedies of Plautus and Terence. Fall 2011. [AH] AMST 22300. GENDER & SEXUALITY IN CLASSICAL ANTIQUITY (Comparative Literature, Women’s, Gender, and Sexuality Studies) An exploration of gender and sexuality in ancient popular literature and drama. An examination of the complex representations of masculinity, femininity, and transgender in classical antiquity, paying particular attention to images in popular literature, drama, and art. An introduction to theories of gender by Aristotle, Freud, Foucault, Butler, and others, and an analysis of how representations of gender and sexuality reinforced Classical Studies 64 cultural beliefs in ancient Mediterranean cultures. Topics of inquiry may include gender and the gods, the visual representation of actors on stage, costuming the body, and the relationship between gender roles and political ideology, desire, religion, democracy, and cultural change. Not offered 2011-2012. [AH] AMST 22600. HISTORY OF ANCIENT MEDICINE A survey of medical practices and the cultural implications of these practices in the ancient world. An examination of medical writings and material evidence in ancient Egypt, India, China, Greece, Rome, and Europe. Topics of inquiry include medicine and gender, class ideologies, shamanism and magical practices, surgical instruments and artifacts, and theories of medical treatments. Students are required to attend several extra lectures by practicing physicians and scientists on subjects such as Chinese medicine and acupuncture, alternative healing therapies, the intersection of modern and ancient healing practices, and theories of gynecology and obstetrics. Spring 2012. [AH] AMST 26000-26001. SPECIAL TOPICS IN CLASSICAL LITERATURE Not offered 2011-2012. AMST 26100. SPECIAL TOPICS IN ANCIENT HISTORY An intensive examination of a specific topic in the history and civilizations of the ancient Mediterranean world. Course titles vary but may include: Religion in the Ancient Mediterranean World, Science and Engineering in the Ancient World, Travel in the Ancient World, Food and Famine in the Ancient World, Late Antiquity, Alexander the Great and the Hellenistic World. Not offered 2011-2012. AMST 40000. TUTORIAL Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. INDEPENDENT STUDY CLST 40100. JUNIOR INDEPENDENT STUDY SEMINAR This writing-intensive seminar offers Classical Studies majors a firm grounding in the discipline, with an emphasis on the diverse materials, methods, and approaches that can be brought to bear on the study of GrecoRoman antiquity. Each student produces a junior thesis on the topic of his or her choice. That topic may be in Latin, Greek, or Classical Civilization. Annually. Fall. CLST 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. The main fields of choice for a major with a concentration in Classical Languages are the literature, philosophy, religion, or history of Greece or Rome. Suggested fields of specialization for a major with a concentration in Ancient Mediterranean Studies are archaeology, ancient history, mythology, classical or comparative literary criticism, philosophy. Prerequisite: CLST 40100. CLST 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: CLST 45100. CROSS-LISTED COURSES ACCEPTED FOR CLASSICAL STUDIES CREDIT COMPARATIVE LITERATURE CMLT 22200. CLASSICAL TRADITION IN MODERN DRAMA, FICTION, AND FILM [W, AH] CMLT 29000. SEMINAR IN COMPARATIVE STUDIES (Approval of Chair, when topic is appropriate to the concentration) HEBREW HEBR 10100. BIBLICAL HEBREW I HEBR 10200. BIBLICAL HEBREW II HISTORY HIST 20200. GREEK CIVILIZATION [HSS] HIST 20300. ROMAN CIVILIZATION [HSS] Communication 65 INTERDEPARTMENTAL IDPT 24000. GREEK ARCHAEOLOGY AND ART [AH] IDPT 24100. ROMAN ARCHAEOLOGY AND ART [AH] PHILOSOPHY PHIL 25000. ANCIENT PHILOSOPHY: PLATO AND ARISTOTLE [AH] RELIGIOUS STUDIES RELS 12000. INTRODUCTION TO BIBLICAL STUDIES: INTERPRETATION AND CULTURE [C,R, AH] RELS 22400. HEBREW PROPHECY AS RELIGIOUS IMAGINATION [R, AH] RELS 22500. THE LIFE AND TEACHINGS OF JESUS [W, C, R, AH] RELS 26700-26722. TOPICS IN RELIGIOUS TRADITIONS AND HISTORIES (Approval of Chair, when topic is appropriate to the concentration) [R] RELS 26900-26929. TOPICS IN THEORIES AND ISSUES IN THE STUDY OF RELIGION (Approval of Chair, when topic is appropriate to the concentration) [R] COMMUNICATION Michelle Johnson, Chair Ahmet Atay Denise Bostdorff Joan Furey Donald Goldberg Margaret Wick Communication is the study of the innate human ability to use symbols and create meaning. The Department of Communication contains within it two tracks: Communication Studies and Communication Sciences and Disorders. COMMUNICATION STUDIES Research and instruction in Communication Studies focus on the study of how messages in various media (spoken, written, printed, photographic, electronic) are produced, used, and interpreted within and across different contexts, channels, and cultures. Communication Studies focuses on how people arrive at shared meanings through an interchange of messages or, in other words, the symbolic processes through which meaning and social reality are created. The origin of Communication Studies goes back to the ancient Greeks and, in its infancy, the discipline emphasized public speaking alone. Today the discipline studies symbolic processes — whether oral, written, or nonverbal — in a variety of contexts: intrapersonal communication, interpersonal communication, group communication, organizational communication, public address, and the mass media. Majors in the track of Communication Studies learn how to be more effective communicators and how to be critical analysts of communication, thereby preparing them for life as enlightened citizens and professionals in a variety of career paths such as business, education, law, politics, media, and the ministry. Major in Communication Studies Consists of eleven courses: • One of the following courses: COMM 14500, 22000, 22100, or 22500 • One of the following courses: COMM 15200, 25000, 25200, or 25400 • One of the following courses: COMM 22900, 23100, or 33200 Communication 66 • Three of the following courses (cannot be the same courses as taken for the above requirements): COMM 11100, 14500, 15200, 20000-20003, 22000, 22100, 22500, 22900, 23100, 25000, 25200, 25400, 33200, or 35000-35002 • COMM 31100 • One of the following courses: COMM 35200 or 35300 • Junior Independent Study: COMM 40100 • Senior Independent Study: COMM 45100 • Senior Independent Study: COMM 45200 Minor in Communication Studies Consists of six courses: • One course from the Human Dynamics category • One course from the Rhetorical Studies category • One course from the Media Studies category (except 13000, Radio Workshop) • Three course credits of electives chosen from (cannot be the same courses as taken for the above requirements): COMM 11100, 14500, 15200, 20000, 22100, 22500, 22900, 23100, 25000, 25200, 25400, 33200, 35000 Special Notes • Majors in the Communication Studies track must complete their methods course (COMM 35200 or 35300) no later than the spring of their junior year. Students are strongly encouraged to take their methods course in the sophomore year. Majors should also complete the theory course (COMM 31100) prior to the first semester of Senior Independent Study (COMM 45100). • In addition to demonstrating proficiency in research and writing through Independent Study, a major in the Communication Studies track must demonstrate proficiency in public speaking, as certified by all faculty members in the Department of Communication, based upon the student’s oral presentation of his/her Senior Independent Study proposal. These public presentations will typically be scheduled in the fall, and students will be provided with specific guidelines to follow. The faculty also encourages majors to seek the help of their advisers in preparing their presentations. • No more than two Communication Studies and/or Communication Sciences and Disorders courses can be applied toward the general education requirements. • No courses may be taken on an S/NC basis — with the exception of COMM 13000. • Only grades of C- or better are accepted for the major or minor. COMMUNICATION STUDIES COURSES COMM 11100. INTRODUCTION TO COMMUNICATION STUDIES This course examines the significance of communication in human life and introduces students to funda mental principles and processes of communication in a variety of contexts: intrapersonal, interpersonal relationships, small groups, public settings, and the mass media. Students will learn to think critically about communication and will apply the knowledge they gain through a variety of means: class exercises, a group project of limited scope, message analysis, and a public speech. Annually. Fall and Spring. [HSS] COMM 20000-20003. SPECIAL TOPICS IN COMMUNICATION A topical seminar that focuses on special issues within communication studies or communication sciences and disorders. Annually. Fall. [W] COMM 31100. THEORIES OF HUMAN COMMUNICATION The goal of this advanced course is to provide students with in-depth knowledge of theories of human com- Communication 67 munication in order to provide a more coherent understanding of Communication Studies as a discipline. Course topics include, but are not limited to, system theory; theories of signs and language; rules approach and speech act theory; theories of message production; theories of message reception and processing; symbolic interactionism, dramatism, and narrative; theories of social and cultural reality; theories of experience and interpretation; critical theories. Prerequisites: Two courses from the categories of Human Dynamics, Rhetorical Studies, or Media Studies with each course representing a different category — or permission of instructor. Annually. Spring. COMM 35000. ADVANCED SEMINAR IN COMMUNICATION STUDIES Selected topics or issues for advanced study in human dynamics, rhetorical studies, or media studies. May be taken more than once. Prerequisite: Sophomore standing. Fall. Not offered 2011-2012. COMM 35200. RHETORICAL CRITICISM This course examines the nature and methods of rhetorical criticism, with the goal of teaching students how to write rhetorical criticisms of their own and how to critique the work of others. Topics include Neo-Aristotelian criticism, narrative criticism, Burkean criticism, generic criticism, cultural (metaphor, value, myth, fantasy theme) analysis, and ideological (feminist, Neo-Marxist, and deconstructionist) criticism. Prerequisite: One of the following — COMM 25000, 25200, or 25400 — or permission of instructor. Annually. Spring. COMM 35300. QUANTITATIVE METHODS This course examines experimental and field research methods as they apply to research in Communication Studies and Communication Sciences and Disorders. The goal of this course is to provide students with a working knowledge of quantitative methods so that they can make informed choices when conducting their own research studies and can critique research studies conducted by others. Course topics include, but are not limited to, measurement techniques (surveys, survey interviews, focus groups, content analysis) and related concerns such as creating research questions, reliability, validity, and coding; sampling; experimental design; data entry; data analysis; writing research results. Prerequisite: One completed course in Communication Studies or Communication Sciences and Disorders, or permission of the instructor. Annually. Spring. [Q] COMM 39000. COMMUNICATING COMMON GROUND This service-learning course is part of a national program sponsored by the American Association for Higher Education, the National Communication Association, the Southern Poverty Law Center, and Campus Compact. Students create and teach lessons in communication, conflict management, and diversity to local preschoolers. (.25 course credits) Prerequisite: Application to the program and permission of the instructor are required. Fall and Spring. Not offered 2011-2012. COMM 40000. TUTORIAL A tutorial course on a special topic may be offered to an individual student under the supervision of a faculty member. Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. COMM 40100. JUNIOR INDEPENDENT STUDY This course examines how scholars conduct communication research and culminates with students writing a Junior Independent Study thesis under the direction of a faculty adviser. Topics include the selection of a research question or purpose; the use of the library for scholarly communication research; a broad overview of humanistic and social scientific methods; the evaluation of scholarly research; and guidelines for scholarly writing. The course involves a number of writing assignments, as well as the draft and revision of thesis chapters, in order to help students clarify their goals and articulate their research findings in a coherent way. Prerequisites: COMM 35200 or 35300 completed or taken concurrently and completion of a W course. Fall and Spring by assignment. COMM 40700, 40800. INTERNSHIP Internships are negotiated with the Dean for Curriculum and Academic Engagement and the faculty of the department. For information on the Washington Semester, the Philadelphia Center, and the New York Arts Program — off-campus programs that offer internships especially pertinent to Communication Studies majors — see Off-Campus Study and Internships. (Variable course credit) S/NC course. COMM 45100. SENIOR INDEPENDENT STUDY THESIS — SEMESTER ONE An original communication research investigation is required. An oral presentation is given to the department. Prerequisite: COMM 40100. COMM 45200. SENIOR INDEPENDENT STUDY THESIS – SEMESTER TWO An original communication research investigation is required, culminating in the I.S. thesis and an oral examination. Prerequisite: COMM 45100. Communication 68 HUMAN DYNAMICS Through courses in human dynamics, students learn how antecedent influences contribute to the formation and exchange of messages within and among dyads, groups, and organizations, and how these message exchanges, in turn, relate those involved cognitively, affectively, and behaviorally. COMM 22000. INTRAPERSONAL DIMENSIONS OF HUMAN COMMUNICATION The course focuses on the interdependence of perception and the construction of meaning in human communication. The focus is on the internal generation and regulation of meaning through perceptual systems which link the individual to the environment. Topics include selective perception in human communication, verbal and visual thought, and both the private and social constructions of self and social reality as related to the contexts of human communication. Alternate years. Spring. Not offered 2011-2012. COMM 22100. INTERPERSONAL COMMUNICATION This course examines the form, content, and consequences of communication between two people, primarily focusing upon informal contexts, such as the communication between parent and child, siblings, romantic partners, and friends. Topics include communication rules, self-disclosure, cultural and intercultural influences, gender similarities and differences, nonverbal communication, compliance-gaining, relational stages and strategies, relational conflict, and ethics and power in interpersonal communication. Alternate years. Fall 2011. [HSS] COMM 22500. GROUP AND ORGANIZATIONAL COMMUNICATION This course analyzes the form, content, and consequences of communication within both small groups and larger organizations, primarily focusing on the dynamics of communication exchanges within such contexts. Topics include roles, norms, culture, decision-making, conflict management, identification, leadership, recruitment/ indoctrination, and ethics and power in group/organizational communication. Alternate years. Fall. Not offered 2011-2012. RHETORICAL STUDIES Through courses in rhetorical studies, students learn how antecedent influences contribute to the formation and exchange of messages in public contexts, and how these messages encourage members of the public to relate to one another cognitively, affectively, and behaviorally. COMM 15200. PUBLIC SPEAKING (Education) The course involves the study of public address and the performance of various types of speeches. The course examines public speaking theories from classical to contemporary times and makes use of model speeches to help students learn to write and deliver better public presentations. Senior majors may enroll only with the permission of the instructor and department chair. Annually. Fall and Spring. [AH] COMM 25000. PRINCIPLES OF RHETORIC The course surveys basic concepts of rhetoric or persuasive symbol use. Topics include the nature of rhetoric, rhetoric as a response to and/or reconstruction of situation, rhetoric and motive, meaning and context, metaphor, doublespeak, rhetoric and perceptions of self, legitimation and delegitimation, moral arguments and the assessment of ethics. Theorists whose works are considered include Plato, Aristotle, Hugh Blair, I. A. Richards, Richard Weaver, Edwin Black, and Kenneth Burke, among others. Alternate years. Fall 2011. [W†, AH] COMM 25200. ARGUMENTATION AND PERSUASION The course examines both the theoretical and pragmatic aspects of argumentation as they relate to decisionmaking and the persuasion of both self and others. The goals of the course are to familiarize students with the basic concepts of argumentation and reasoning, to teach students how to articulate cogent arguments in both written and oral form, and to improve students’ abilities to analyze the arguments of others. Prerequisite: One of the following — COMM 11100, 15200, or 25000 — or permission of instructor. Alternate years. Spring 2012. [AH] COMM 25400. POLITICAL RHETORIC This course examines the role that rhetoric plays in constructing and shaping our political realities. Topics include the nature of political rhetoric, rhetoric and issue construction, campaign discourse, political rhetoric and the news, domestic issue management, foreign policy rhetoric, issue advocacy and the disenfranchised, and the ethics of political discourse. The course aims to sharpen students’ critical skills in analyzing and evaluating political rhetoric, and to provide students with a greater awareness of both the artistry and potential manipulation of political discourse. Alternate years. Fall. Not offered 2011-2012. Communication 69 MEDIA STUDIES Through courses in media studies, students learn how antecedent influences contribute to the formation and exchange of media messages, and how such messages then relate media audiences cognitively, affectively, and behaviorally. COMM 13001. RADIO WORKSHOP This course provides training in radio broadcasting and station management associated with the activities of WCWS-FM, the College radio station. (.25 course credit) S/NC course. Annually. Fall and Spring. COMM 13002. RADIO WORKSHOP This course provides the opportunity to work in radio broadcasting and station management associated with the activities of WCWS-FM, the College of Wooster radio station. (.25 course credit) S/NC course. May be taken more than once. Prerequisite: COMM 13001. Annually. Fall and Spring. COMM 13003. RADIO WORKSHOP This course provides training in radio broadcasting and station management associated with the activities of WCWS-FM, the College radio station. (.5 course credit) S/NC course. Prerequisite: Must be a member of the WCWS management staff and have the permission of the course instructor. Annually. Fall and Spring. COMM 22900. MASS COMMUNICATION PROCESSES AND EFFECTS The course examines the form, content, and consequences of mass communication as it applies to human interaction. The focus of this course is the influence of mass communication on human behavior. Topics include communication and culture, mass persuasion, mass entertainment, diffusion of innovations, social learning theory, and models of mass communication effects as they relate to the issues of gender, sex, race, and violence in the media. Alternate years. Fall 2011. [HSS] COMM 23100. RADIO, TELEVISION, AND FILM IN AMERICA (Film Studies) This course examines the dynamic influences of American political and economic thought on the development of radio, television, and film in America, and emphasizes how present-day media owe much of their current structure and function to the social, technological, and regulatory decisions made years ago. Topics include the structure of broadcasting, comparative broadcast systems, the technological limitations and potentials of the mass media, and the mass media as forces of social and cultural influence. Alternate years. Fall. Not offered 2011- 2012. COMM 33200. VISUAL COMMUNICATION (Film Studies) This course introduces students to the form, content, and consequences of visual literacy as they relate to screen composition, photographic design, and applied media aesthetics. Students will develop the ability to understand and interpret screen language, and will construct their own visual statements using video production techniques. Topics include spatial and temporal continuity, movement, cutting, camera angles, lighting, pacing, and the basics of production and editing equipment. Prerequisite: One of the following — COMM 23100 or 22900 — or permission of instructor. Alternate years. Spring 2012. COMMUNICATION SCIENCES AND DISORDERS Communication Sciences and Disorders is a discipline that has evolved from hearing, speech, and language sciences research and the clinical endeavor of assessing, diagnosing, and treating those with communicative disorders. Knowledge, theories, and tools have been integrated from those sciences as well as the life sciences (human anatomy and physiology), linguistics, physics (acoustics and psychoacoustics), psychology (developmental and clinical psychology), and sociology/anthropology (sociolinguistics). The major in the Communication Sciences and Disorders track includes courses in the discipline itself, cognate courses that are selected from related disciplines, and the clinic practicum. The curriculum provides the student with an understanding of normal and abnormal human speech and language communication. The curriculum and supervised clinic practica of the major contribute to this understanding, and courses in the major are taught from these perspectives: 1) the evolutionary biolinguistic capacity of humans for using language for communication; 2) the principles of human development and maturation from biologic, Communication 70 anthropologic, psycho-social, and communicative perspectives; 3) the causes, effects, assessment, and treatment of those with communicative disorders; 4) the opportunities for service education through participation in the activities of the Freedlander Speech and Hearing Clinic. When combined with the required research methodology course, the major prepares the qualified student for graduate or professional study. Major in Communication Sciences and Disorders Consists of thirteen courses: • COMM 14000 (Four semesters at one-fourth credit each) • COMM 14100 • COMM 14300 • COMM 14500 • COMM 24400 • COMM 31600 • COMM 35300 • COMM 37000 • One elective from cross-listed courses accepted for COMM credit • One elective from COMM 20000, 25000, 22100, 34500 or cross-listed courses accepted for COMM credit • Junior Independent Study: COMM 40100 • Senior Independent Study: COMM 45100 • Senior Independent Study: COMM 45200 Minor in Communication Sciences and Disorders Consists of six courses: • COMM 14000 (Four semesters at .25 credit each) • COMM 14100 • COMM 14300 • COMM 14500 • COMM 37000 • One elective Communication Sciences and Disorders course Special Notes • Majors in the Communication Sciences and Disorders track must complete their methods course (COMM 35300) before the end of the junior year. Students are strongly encouraged to take their methods course in the sophomore year. • In addition to demonstrating proficiency in research and writing through Independent Study, a major in the Communication Sciences and Disorders track must demonstrate proficiency in public speaking, as certified by all faculty members in the Department of Communication, based upon the student’s oral presentation of his/her Senior Independent Study proposal. These public presentations will typically be scheduled in the fall, and students will be provided with specific guidelines to follow. The faculty also encourages majors to seek the help of their advisers in preparing their presentations. • Some nationally certified professional clinicians are employed in the public schools. This usually requires additional certification controlled by state departments of education, requiring completion of courses in education. The student should consult with the faculty in Communication Sciences and Disorders and the Department of Education about this certification. • No more than two Communication Studies and/or Communication Sciences Communication 71 and Disorders courses can be applied toward the general education requirements. • Majors and minors in Communication Sciences and Disorders may not take any courses within the department for S/NC credit except the first enrollment of COMM 14000. • Only grades of C- or better are accepted for the major or minor. COMMUNICATION SCIENCES AND DISORDERS COURSES COMM 14000. SPEECH AND HEARING CLINIC PRACTICUM Procedures and practices in the assessment and management of persons who are speech, language, and/or hearing impaired as applied under the direct supervision of ASHA certified and state-licensed speech-language pathologists and/or audiologists in the Freedlander Speech and Hearing Clinic. Four semesters required by majors and minors for credit toward graduation. (.25 course credit) First semester of enrollment is graded S/NC. Following semesters are graded with letter grades. Prerequisite: COMM 14100, 14300, and 14500 or permission of the instructor. Annually. Fall and Spring. COMM 14100. INTRODUCTION TO COMMUNICATION SCIENCES AND DISORDERS At the completion of this course, the student will possess a knowledge of a host of speech, language, and hearing disorders (including stuttering, voice, developmental language, aphasia, other neurogenic disorders, articulation/phonology, cleft palate, and hearing disorders). The study of speech-language pathology and audiology and the nature of the clinical practices of these professions will also be addressed. Annually. Fall 2011. [HSS] COMM 14300. PHONETIC TRANSCRIPTION AND PHONOLOGY Content areas to be addressed include anatomy and physiology of the speech mechanisms; speech acoustics and speech science basics; introduction to articulation, phonological, and speech intelligibility testing; spoken language and communication differences (multicultural aspects of spoken language, including dialects of American English); and disordered speech. In addition, the course will prepare the student to be a skilled practitioner in phonetic transcription using the International Phonetic Alphabet. Prerequisite: COMM 14100 or permission of the instructor. Annually. Fall 2011. COMM 14500. LANGUAGE DEVELOPMENT IN CHILDREN (Education) At the completion of this course, the student will have a comprehensive knowledge of the developmental process of children learning spoken language. Annually. Spring 2012. [HSS] COMM 24400. AUDIOLOGY At the completion of this course, the student will have comprehensive knowledge, skills, and abilities in the areas of both diagnostic and rehabilitative audiology. Prerequisite: COMM 14100 or permission of the instructor. Alternate years. Fall 2011. COMM 31600 ANATOMY AND PHYSIOLOGY OF THE SPEECH AND HEARING MECHANISM This course will provide students with an understanding of the anatomy and physiology of the speech and hearing mechanisms. Systems to be covered include respiration, laryngeal, articulatory, nervous, auditory, and circulatory. Prerequisite: Completed or enrolled in COMM 14100, or permission of instructor. Alternate years. Spring. Not offered 2011-2012. COMM 34500. ADVANCED SEMINAR IN COMMUNICATION SCIENCES AND DISORDERS A series of courses to focus on current topics of interest in the fields of speech, language, and hearing sciences and disorders. Prerequisite: COMM 14100 or permission of the instructor. Spring 2012. COMM 37000. AUDIOLOGICAL REHABILITATION This course will address the implications of hearing loss in children and adults including educational, vocational, social, and legislative concerns of children and adults with hearing impairments; hearing aid orientation approaches; and assessment tools and intervention techniques used in order to maximize the communication skills of people with hearing impairment and their communication partners. Prerequisite: COMM 24400 or permission of the instructor. Alternate years. Spring 2012. COMM 40000. TUTORIAL A tutorial course on a special topic may be offered to an individual student under the supervision of a faculty member. Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. Communication 72 COMM 40100. JUNIOR INDEPENDENT STUDY The course examines how scholars conduct communication research and culminates with students writing a Junior Independent Study thesis under the direction of a faculty adviser. Topics include the selection of a research question or purpose; the use of the library for scholarly communication research; a broad overview of humanistic and social scientific methods; the evaluation of scholarly research; and guidelines for scholarly writing. The course involves a number of writing assignments as well as the drafting and revision of thesis chapters, in order to help students clarify their goals and articulate their research findings in a coherent way. Prerequisite: COMM 35300 completed or taken concurrently and completion of a W course. Fall and Spring by assignment. COMM 40700, 40800. COMMUNICATION SCIENCES AND DISORDERS INTERNSHIP (Variable course credit) S/NC course. Prerequisite: Approved by the Dean for Curriculum and Academic Engagement and the faculty of the department. COMM 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: COMM 40100. COMM 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: COMM 45100. CROSS-LISTED COURSES ACCEPTED FOR COMMUNICATION CREDIT BIOLOGY BIOL 10000. TOPICS IN BIOLOGY [MNS] EDUCATION EDUC 11000. USING PHONICS TO TEACH READING AND DEVELOP LITERACY EDUC 20000. TEACHING CHILDREN WITH SPECIAL NEEDS ENGLISH ENGL 25000. READERS’ RESPONSES TO TEXTS NEUROSCIENCE NEUR 32300. BEHAVIORAL NEUROSCIENCE [W] PHYSICS PHYS 10100. GENERAL PHYSICS PSYCHOLOGY PSYC 11000. CHILD AND ADOLESCENT DEVELOPMENT PSYC 21100. MATURITY AND OLD AGE PSYC 23000. HUMAN NEUROPSYCHOLOGY [HSS] PSYC 32200. MEMORY AND COGNITION [W] PSYC 33500. PERCEPTION AND ACTION [W] SOCIOLOGY AND ANTHROPOLOGY ANTH 22000. LINGUISTIC ANTHROPOLOGY [C, HSS] Comparative Literature 73 COMPARATIVE LITERATURE CURRICULUM COMMITTEE: Mary Addis (Spanish), Chair Yuri Corrigan (Russian Studies) Carolyn Durham (French) Monica Florence (Classical Studies) Beth Ann Muellner (German Studies) Mazen Naous (English) Rujie Wang (Chinese) The discipline of Comparative Literature promotes the study of intercultural relations across linguistic and cultural boundaries. Appealing to the desire to transcend a merely national point of view, it enables the student to develop a uniquely transnational perspective on imaginative works from antiquity to post modernity. The program at The College of Wooster is interdepartmental in character and includes both explicitly comparative courses and courses that focus on a particular national literature, both in the original and in translation. Major in Comparative Literature Consists of fifteen courses: • ENGL 12000-12012 • Two courses from Group I • Four courses beyond the 10200-level in a foreign language department. (One of the four courses may be in translation. When department offerings allow, at least two should be from Group II or III.) • Two courses from national literatures other than the four courses beyond the 10200-level above. (These may be selected from Groups II and III.) • Three electives, selected from Groups I, II, and III and/or from beyond the 10200-level in a foreign language. • Junior Independent Study: CMLT 40100 • Senior Independent Study: CMLT 45100 • Senior Independent Study: CMLT 45200 Minor in Comparative Literature Consists of six courses: • Three courses from Group I • Three courses from Groups II and III Special Notes • CMLT 40100, 45100, and 45200 will all involve projects of a comparative character; the student may select an adviser from any of the departments that participate in the program; the project must be approved by the chair of the Comparative Literature Curriculum Committee by the end of the second week of the semester in which the student is enrolled in CMLT 40100 and 45100. • Courses applied toward a Comparative Literature major may not be taken on an S/NC basis. • Only grades of C- or better are accepted for the major or minor. COMPARATIVE LITERATURE COURSES Group I: COMPARATIVE COURSES ENGL 12000-12012. INVESTIGATIONS IN LITERARY AND CULTURAL STUDIES (Comparative Emphasis) [AH] Comparative Literature 74 CMLT 22000. THEORY AND PRACTICE OF TRANSLATION Taught in English. This is primarily a theory course in translation studies with practical application. We will think carefully and critically about both the process and reception of translation, and cover a variety of approaches to help us better analyze, read, and perform translations. The class will examine major approaches to translation, including: the process of translation, especially literary translation; the science of translation, including functional approaches; descriptive translation studies; deconstruction; and postcolonial theories of both cultural and linguistic translation. We will read, analyze, and respond to a variety of translation theories and experiment with short translation assignments in which we apply the theoretical frameworks. The final project for this course is a paper in which the students develop their own translation method based on the theories we have read during the semester. Alternate years. Not offered 2011-2012. [C, AH] CMLT 22200. CLASSICAL TRADITION IN MODERN DRAMA, FICTION, AND FILM (Classical Studies, Film Studies) What do we mean when we say that one work “influences” another, or that a later work is “derived from” an earlier one? This course will study a number of twentieth-century works that draw on the classical tradition (myth, literature, history, ritual) for their content, form, or thematic concerns within the framework of contemporary critical theory: e.g., narrative analysis, anthropological criticism, theories of intertextuality. The course will attempt to appreciate how these modern works function as readings of their ancient models, and how these models are fundamentally rewritten in being translated into a different social, historical, and intellectual context. Modern works will be drawn from a range of national and ethnic traditions, and may include drama by O’Neill, Elliot, Sartre, Anouilh, Albee, Stoppard; fiction by Joyce, Gide, Camus, Kafka, Kazantzakis, Renault, Wolf; films by Cocteau, Camus, Pasolini, Fellini, and Cacoyannis. Alternate Years. Not offered 2011-2012. [W, AH] CMLT 23000. COMPARATIVE SEXUAL POETICS (Women’s, Gender and Sexuality Studies) An exploration within the framework of contemporary feminist theory of notions of gender-specific culture, aesthetics, and language. Extensive comparison of similar texts of men and women writers to test the validity of key theoretical assumptions. Readings from multiple genres and national literatures to allow analysis of differences attributable to gender, culture, and textual context. Comparative pairings may include the following: García Márquez and Allende; Sartre and Beauvoir; Whitman and Dickinson; Von Trotta and Schloendorff; Bâ and Laye; Montaigne and Woolf; Miller and Nin; Hellman and Hammett. Alternate years. Not offered 2011-2012. [W, AH] CMLT 23200. MODERN COMPARATIVE DRAMA A presentation of modern dramatic theories and their implications in the form, themes, and techniques of modern dramatic literature. Inquiries into the specific philosophical, literary, and thematic issues of modern dramaturgy as evidenced in metatheatre; epic theatre; existentialist theatre; the theatre of the absurd; social, political, and feminist theatre. Readings from multiple national literatures to include England, France, Germany, Italy, Spain, and the USA. Alternate years. Not offered 2011-2012. [C, AH] CMLT 23600. COMPARATIVE FILM STUDIES (Film Studies) A special topics course focusing on various aspects of film history, theory, or analysis. Introduction to basic concepts and skills necessary for the exploration of technical, stylistic, narrative, and ideological articulation in cinema. Possible categories of inquiry include national cinemas, genres (film noir, melodrama, etc.), representation and spectatorship, feminist cinema, African American film; documentary, political cinema, the avant garde, experimental film, etc. Extensive readings of theory and criticism as well as regular film screenings. Alternate years. Spring 2012. [C, AH] CMLT 24800: THE PERILS OF ROMANTICISM: NINETEENTH CENTURY EUROPEAN LITERATURE (German Studies, Russian Studies) This course will examine some of the major issues that arose from European Romanticism (German, French, English and Russian) – the rebellion against rationalism, new notions of selfhood and individuality, the rejection of traditional morality and models of authority, and the longing for a reintegration with nature. We will study these questions in the works of major nineteenth-century authors, and we will consider the commentaries of some twentieth-century artists, philosophers and critics on this period. The goal will be to understand how European writers engaged in a complex cross-cultural intellectual dialogue not simply on a discursive level but through the use of symbolic, dramatic and formal paradigms. Authors include Goethe, Hoffman, Shelley, Flaubert, Maupassant, Dostoevsky, Tolstoy, Chekhov, and Thomas Mann. Supplementary selections of philosophy will be provided – from Rousseau and Schlegel to Simone Weil and Hannah Arendt. Every three years. Not offered 2011-2012. [C, AH] Comparative Literature 75 CMLT 29000. SEMINAR IN COMPARATIVE STUDIES (some sections cross-listed with Classical Studies) An advanced seminar offering in-depth study of selected issues in comparative literature. Although the topic will vary, the course will include an exploration of current theories and methodologies of textual and contextual comparison. Focus may involve comparative studies of particular texts, genres, or historical periods, or address broader questions of ideology, aesthetics, influence, or language within a comparative framework. Topics announced in advance by faculty member teaching the course. Prerequisite: ENGL 12000-12012 or permission of the instructor. Fall 2011. Group II: NATIONAL LITERATURE IN TRANSLATION CHINESE STUDIES CHIN 22000. REBELS, ROMANTICS, AND REFORMERS: BEING YOUNG IN CHINA [C, AH] CHIN 22200. WOMEN IN CHINESE LITERATURE [C, AH] CHIN 22300. CHINESE CINEMA AS TRANSLATION OF CULTURES [C, AH] CLASSICAL STUDIES AMST 22000. MYTHOLOGY OF THE ANCIENT WORLD [AH] AMST 22100. ANCIENT THEATER: TRAGEDY AND COMEDY [AH] AMST 22300. GENDER AND SEXUALITY IN CLASSICAL ANTIQUITY [AH] FRENCH FREN 25300. TOPICS IN FRANCOPHONE LITERATURE AND SOCIETY: FRANCOPHONE FILM GERMAN STUDIES GRMN 22700. GERMAN LITERATURE IN TRANSLATION GRMN 22800. TOPICS IN GERMAN SOCIETY AND CULTURE (GERMAN FILM AND SOCIETY) [C] RUSSIAN STUDIES RUSS 21000. RUSSIAN CIVILIZATION: FROM FOLKLORE TO PHILOSOPHY [W, C, AH] RUSS 22000. RUSSIAN CULTURE THROUGH FILM [C, AH] RUSS 23000. RUSSIAN DRAMA PRACTICUM [C, AH] RUSS 25000. RUSSIAN LITERATURE IN THE AGE OF DOSTOEVSKY AND TOLSTOY [C, AH] RUSS 26000. THE ARTIST AND THE TYRANT: TWENTIETH-CENTURY RUSSIAN LITERATURE [C, AH] SPANISH SPAN 21200. LITERATURE AND CULTURE OF THE HISPANIC CARIBBEAN [C, AH] SPAN 21300. U.S. LATINO LITERATURES AND CULTURES [C, AH] SPAN 28000. HISPANIC FILM [C, AH] SPAN 39900. DON QUIXOTE: METAFICTION AND THE DAWNING OF THE MODERN NOVEL [C, AH] Group III: NATIONAL LITERATURE IN THE ORIGINAL CLASSICAL STUDIES GRK 20000. SEMINAR IN GREEK LITERATURE (INTERMEDIATE I) [AH] GRK 25000. SEMINAR IN GREEK LITERATURE (INTERMEDIATE II) [AH] GRK 30000. SEMINAR IN GREEK LITERATURE (ADVANCED I) [AH] GRK 35000. SEMINAR IN GREEK LITERATURE (ADVANCED II) [AH] LAT 20000. SEMINAR IN LATIN LITERATURE (INTERMEDIATE I) [AH] LAT 25000. SEMINAR IN LATIN LITERATURE (INTERMEDIATE II) [AH] LAT 30000. SEMINAR IN LATIN LITERATURE (ADVANCED I) [AH] LAT 35000. SEMINAR IN LATIN LITERATURE (ADVANCED II) [AH] ENGLISH ENGL 21000. GENDER, RACE, AND ETHNICITY [AH] ENGL 22000. WRITERS [AH] ENGL 23000. HISTORY [AH] Comparative Literature 76 ENGL 24000. TEXTUAL FORMATION [AH] ENGL 25000. READERS’RESPONSES TO TEXTS [AH] ENGL 30000-30007. SPECIAL TOPICS IN LITERARY STUDIES [AH] FRENCH FREN 22000. INTRODUCTION TO FRANCOPHONE TEXTS [C, AH] FREN 23000-23004. TOPICS IN FRANCOPHONE LITERATURE AND SOCIETY FREN 23500. LITERATURE AND CULTURE OF FRANCOPHONE AFRICA [C] FREN 32000. STUDIES IN THE MIDDLE AGES AND THE RENAISSANCE FREN 32200. STUDIES IN THE SEVENTEENTH CENTURY FREN 32400. STUDIES IN THE EIGHTEENTH CENTURY [C, AH] FREN 32800. STUDIES IN THE NINETEENTH CENTURY FREN 32900. STUDIES IN THE TWENTIETH CENTURY [C, AH] GERMAN STUDIES GRMN 26000. KULTURKUNDE: INTRODUCTION TO GERMAN STUDIES [W, C, AH] GRMN 30000. MAJOR EPOCHS OF GERMAN LITERATURE AND CULTURE GRMN 32000. MAJOR AUTHORS IN GERMAN LITERATURE AND CULTURE GRMN 33000. GENRES OF GERMAN LITERATURE AND CULTURE [C] GRMN 34000. MAJOR THEMES IN GERMAN LITERATURE AND CULTURE [Depending on the topic, C, AH] SPANISH SPAN 21100. INTERMEDIATE SEMINAR: SPECIAL TOPICS IN HISPANIC LANGUAGE, LITERATURE, & CULTURE [Depending on the topic, C, AH] SPAN 24700. TWENTIETH AND TWENTY-FIRST CENTURY SPANISH PENINSULAR WRITERS [C, AH] SPAN 24800. TWENTIETH AND TWENTY-FIRST CENTURY SPANISH AMERICAN WRITERS [C, AH] SPAN 30100. CERVANTES: DON QUIXOTE [C, AH] SPAN 30200. GOLDEN AGE LITERATURE [C, AH] SPAN 30500. THE CONTEMPORARY LATIN AMERICAN NOVEL [C, AH] SPAN 30900. TRENDS IN SPANISH AMERICAN LITERATURE [C, AH] INDEPENDENT STUDY CMLT 40000. TUTORIAL Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. CMLT 40100. JUNIOR INDEPENDENT STUDY A one-semester course that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. CMLT 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: CMLT 40100. CMLT 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: CMLT 45100. Computer Science 77 COMPUTER SCIENCE Pamela Pierce (Mathematics), Chair Denise Byrnes James Daehn Sofia Visa Computer Science is more than just programming. It is the study of computer programs, abstract models of computers, and the many applications of computing. Computer Science combines elements of mathematics, philosophy, languages, and natural science. Although computing technology is continuously changing, the core skills required to practice Computer Science remain the same: problem solving, abstract thinking, and independent learning. The mission of the Computer Science program is to educate students in the theoretical foundation of the discipline and its creative application to the solution of complex problems, and to prepare students to learn independently in a discipline that is constantly changing. Supported by a liberal arts education, the program seeks to develop students who are sensitive to the wide range of social concerns influenced by the discipline and are articulate in expression of their ideas and actions. Students successfully completing the Computer Science major should have the computer science background and the mathematical maturity needed to enter a graduate program in Computer Science or to take an entry-level position in a computing-related field. As computing is increasingly applied to other fields, students in the natural sciences, business and economics, and other majors may benefit from a minor or double major in Computer Science. Major in Computer Science Consists of thirteen courses: • CSCI 15100 • CSCI 15200 • One of the following courses: MATH 10800 or 11100 • One of the following courses: MATH 12300 or 22300 • CSCI 25100 • CSCI 25200 • CSCI 25300 • One of the following courses: MATH 21100 or 24100 • CSCI 35100 • Two elective full-credit Computer Science courses numbered above 35100 • Junior Independent Study: See note below • Senior Independent Study: CSCI 45100 • Senior Independent Study: CSCI 45200 Minor in Computer Science Consists of eight courses: • CSCI 15100 • CSCI 15200 • One of the following courses: MATH 10800 or 11100 • One of the following courses: MATH 12300 or 22300 • CSCI 25100 • Three elective full-credit Computer Science courses at the 200-level or above Computer Science 78 Special Notes • Junior Independent Study: The College requirement of a third unit of Independent Study is satisfied through the independent work done as part of the courses numbered above 20000, which are taken to fulfill the requirements of the major. • Advanced Placement: At most two courses of advanced placement may be counted toward a major or minor. Advanced placement of one or two courses in Computer Science is available to students who have taken the Advanced Placement Examination or an equivalent furnished by the Department of Mathematics and Computer Science. Students are urged to take the AP Examination for this purpose when possible. The decision about granting such placement and its amount is made by the Department of Mathematics and Computer Science after the student has consulted with the chairperson. Normally a minimum score of 4 on the examination is necessary, but such a score alone does not guarantee advanced placement. A student placed in CSCI 15200 will receive one course credit; two course credits will be granted if the student is placed in a course above the level of CSCI 15200. The advanced placement policy of the College is explained in the section on Admission. • Students are given a recommended placement in Computer Science based upon their high school record, their performance on the SAT and/or ACT, and their performance on a mathematics placement exam administered by the department during Summer registration. • Majors are encouraged to take related courses in physics, mathematics, economics, and philosophy. • The laboratory and classroom components are closely integrated in Computer Science courses with a laboratory and must therefore be taken concurrently. The course grade and the laboratory grade are identical and are based on performance in both components; the relative weight of the two components is stated in each course syllabus. • Connecting Art and Computer Science, CSCI 19900, is designed specifically for students wanting a course in Computer Science to partially fulfill the College’s Learning Across the Disciplines requirements. CSCI 15100 is not recommended for these requirements. • Those students who are oriented toward the application of the computer to a specific professional objective, such as industrial or business management, medicine, engineering, computational natural science, or law, should consider a Computer Science minor or double major in consultation with the adviser for those programs. • Combined programs of liberal arts and engineering are available. (See Pre-Professional and Dual Degree Programs: Pre-Engineering.) • Only grades of C- or better are acceptable in courses for the major or minor. COMPUTER SCIENCE COURSES CSCI 11000. INTRODUCTION TO COMPUTER SCIENCE This course examines the fundamental differences between problem solving in computer science and problem solving in other disciplines. How has computing evolved since its inception? How do computers store information? How do computers communicate? What is artificial intelligence? How do computing and society interact? This course also introduces problem solving with computer programming. Annually. Not offered 2011-2012. [MNS] CSCI 15100. COMPUTER PROGRAMMING I The Java programming language is introduced in this course. Java language constructs such as variables, sequential statements, if-else, loops, classes/objects and methods are examined in relation to general problem Computer Science 79 solving strategies. Algorithmic techniques such as searching and sorting are covered. Upon completion of the course, students should be able to design, code, test, and debug at a beginning level. Prerequisite: Departmental approval, as determined by performance on mathematics placement exam. Annually. Fall and Spring. [MNS] CSCI 15200. COMPUTER PROGRAMMING II Building on the basic programming skills developed in CS 151, this course adds tools to the programmer’s repertoire to solve more complex problems using the C++ programming language. It introduces classic data structures used to store collections of data efficiently. It further develops software-engineering practices— including testing, documentation, and object-oriented programming—that aid in the construction of large programs. Prerequisite: CSCI 15100. Annually. Fall and Spring. [MNS] CSCI 19900. CONNECTING ART AND COMPUTER SCIENCE – ANIMATIONS, GAMING AND 3-D VIRTUAL WORLDS Fundamentally, Computer Science is about the application of computation to the solution of problems. Often these problems span multiple disciplines, requiring teams that bring diverse perspectives to the problem and its solution. This course explores some of the connections between two quite different disciplines, art and computer science, in the context of animation, computer games, and three-dimensional virtual worlds. In the process the student will gain insights into the basics of computing and software design, the importance of being able to communicate ideas clearly, and how to work collaboratively. Fall 2011. CSCI 25100. PRINCIPLES OF COMPUTER ORGANIZATION This course provides an overview of computer systems design and architecture, and machine language. Topics include: instruction set design, register transfers, data-path design, pipelining, controller design, memory systems, addressing techniques, microprogramming, computer arithmetic. A survey of popular computer systems and microprocessors reinforce how real computer systems are designed. Prerequisite: CSCI 15200. Annually. Fall 2011. CSCI 25200. ALGORITHMS This course covers standard and advanced algorithms for problem solving in computer science. Brute force, recursion, greedy strategies and dynamic programming techniques are applied to real world problems. Timespace analysis is performed for various algorithm and data structure pairings. The limitations of algorithms are also studied in the context of NP-completeness. Prerequisite: CSCI 15200 and MATH 12300, 21100, or 22300. Annually. Spring 2012. CSCI 25300. THEORY OF COMPUTATION The theory of abstract machines and formal languages is introduced in this course. Computability by finite automata, pushdown automata and Turing machines is examined and related to pattern matching, lexical analysis, compilation and programming for digital computer systems. Proofs by induction, construction, contradiction and reduction are used to formalize computability theory and the limitations of computing. Prerequisite: CSCI 15200 and MATH 12300 or 22300. Alternate years. Spring 2012. CSCI 27900. PROBLEM SEMINAR This course provides the opportunity for students to practice solving challenging computer science problems. Typically, this is for those students intending to prepare for the ACM programming contest in which the College participates. The ACM contest is the culmination of this course. (.25 course credit) S/NC course. May be taken more than once. Prerequisite: CSCI 15100. Annually. Fall 2011. CSCI 30900-30905. SPECIAL TOPICS The content and prerequisites of this course vary according to the topic chosen. The course is available at irregular intervals when there is a need for a special topic. Past topics include Software Quality, Parallel and Distributed Computing, and Web Programming. (Variable course credit) Prerequisite: Permission of the instructor. CSCI 35100. PROGRAMMING LANGUAGE THEORY AND COMPILER CONSTRUCTION This course examines programming languages and the use of compilers to translate from high-level languages to machine languages. We use formalisms to describe the syntax and semantics of imperative languages. We explore alternative language paradigms. We examine the algorithms and data structures used in compiler implementation. CSCI 25200 is recommended. Prerequisite: CSCI 25100. Annually. Not offered 2011-2012. CSCI 35300. OPERATING SYSTEMS An Operating System acts as an interface between the application and hardware layer of a computer system. In this course we examine how operating systems manage computing resources such as the memory hierarchy, file system, program runtime environment and peripheral devices. Several popular operating systems are examined as case studies. Prerequisite: CSCI 25100. Alternate years. Spring. Not offered 2011-2012. East Asian Studies 80 CSCI 35400. FILE AND DATABASE SYSTEMS This course provides an overview of general database topics that are relevant to any database management system. These topics include: database design (data modeling, entity-relationship modeling, relational data models, normal forms), the use of database management systems for application development (SQL query language and relational algebra), transaction processing and storage, and indexing principles. The students practice on modern database systems such as Oracle, MySQL or SQL Server. In addition, students develop a web database application. At the end of this course students will be able to design and implement a database, to query a database using SQL, and to write stored procedures to access and interact with databases. Prerequisite: CSCI 25200. Alternate years. Not offered 2011-2012. CSCI 35600. COMPUTER GRAPHICS This course explores the theory and application of computer graphics through the evolution of graphics algorithms and rendering hardware. Topics include 2-D and 3-D transformations and projections, illumination models, texture mapping, animation techniques, user interfaces, and rendering algorithms. Group projects, lab assignments and in class activities expose students to the practical problems inherent in computer graphics programming. Prerequisite: CSCI 15200 and MATH 21100. Alternate years. Fall 2011. CSCI 35700. MACHINE INTELLIGENCE This course is a hands-on introduction to machine learning and artificial intelligence. The main question addressed is: How can we design good computer algorithms that improve automatically through experience (e.g. similar to the way humans learn)? Multiple machine learning models are examined. The goal of the course is that students begin to understand some of the issues and challenges facing machine learning while being exposed to the pragmatics of implementing machine learning systems in Matlab. Prerequisites: CSCI 15200 and MATH 21100. Alternate years. Not offered 2011-2012. CSCI 40000. TUTORIAL This course is given for topics not normally covered in regular courses. Prerequisite: CSCI 25200; the approval of both the supervising faculty member and the chairperson is required prior to registration. CSCI 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: CSCI 25200. CSCI 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: CSCI 45100. EAST ASIAN STUDIES CURRICULUM COMMITTEE: Mark Graham (Religious Studies), Chair David Gedalecia (History) Setsuko Matsuzawa (Sociology) Rujie Wang (Chinese) The East Asian Studies major and minor are offered through the interdepartmental program in East Asian Studies (which also offers the Chinese Studies major and minor). Eight faculty members from five departments (six disciplinary areas) contribute to the multidisciplinary approach to East Asian Studies. The East Asian Studies major and minor focus on developing an integrated multidisciplinary understanding of the diverse but related historical and cultural traditions of East Asia, starting with a foundation in Chinese language and history, and extending that focus across the East Asian region to Japan, and across multiple disciplinary approaches to understanding China, Japan, and East Asia, broadly. This approach to the East Asian Studies major and minor recognizes the diversity of cultural and national traditions East Asian Studies 81 that exist across this region, but at the same time helps foster an understanding of the common cultural and historical concerns that make “East Asia” a coherent focus of study. Given the complexity of histories and traditions in this region of the world, the East Asian Studies major and minor requires multidisciplinary study with a core orientation in history and language, and off-campus study in an East Asian country as part of the curriculum. Our expectation is that the East Asian Studies major will be appropriate for students who seek a broad-based study of East Asia, including course work focused on China and Japan, and who are interested in off-campus study in a Wooster-endorsed program either in Japan, or in a broad-based East Asian Studies program in China. Students whose studies are focused exclusively on China, including off-campus study in China, should consult the Chinese Studies major. Major in East Asian Studies Consists of eleven courses: • CHIN 20100 • One elective Chinese course at the 200- or 300-level • HIST 20000 • HIST 20100 • One of the following courses: HIST 10100-10176 (when China-focused), 23700, SOC 21900, PHIL 23000, 23200, or RELS 21600 • HIST 20600 • RELS 22000 • ANTH 23100 • Junior Independent Study: EAST 40100 • Senior Independent Study: EAST 45100 • Senior Independent Study: EAST 45200 Minor in East Asian Studies Consists of six courses: • CHIN 20100 • Two of the following courses: a Chinese course at the 200- or 300-level, HIST 10100-10176 (when China focused), 20000, 20100, 23700, SOC 21900, RELS 21600 • Two of the following courses: ANTH 23100 (when Japan focused), HIST 20600, RELS 220 • One of the 200- or 300-level cross-listed courses accepted for EAST credit Special Notes • Off-campus Study: The major and minor in East Asian Studies requires the completion of an approved off-campus study program in an East Asian country (e.g., a program in China or Japan appropriate for this major). Up to three approved transfer credits may count toward the requirements for the major. • Students who wish to take the maximum number of courses for this major are encouraged to complete additional courses from the options offered in the major requirements. • Only grades of C- or better are accepted for the major or minor. EAST ASIAN STUDIES COURSES EAST 40100. JUNIOR INDEPENDENT STUDY A one-semester course that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. East Asian Studies 82 EAST 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: EAST 40100. EAST 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: EAST 45100. CROSS-LISTED COURSES ACCEPTED FOR EAST ASIAN STUDIES CREDIT CHINESE STUDIES CHIN 20100. INTERMEDIATE CHINESE I [C] CHIN 20200. INTERMEDIATE CHINESE II [C] CHIN 22000. REBELS, ROMANTICS, AND REFORMERS: BEING YOUNG IN CHINA [C, AH] CHIN 22200. WOMEN IN CHINESE LITERATURE [C, AH] CHIN 22300. CHINESE CINEMA AS TRANSLATION OF CULTURES [C, AH] CHIN 30100. ADVANCED CHINESE I CHIN 30200. ADVANCED CHINESE II CHIN 31100. CHINESE MODERNITY AND FILM CHIN 31200. CHINA: A CULTURAL PANORAMA CHIN 40000. TUTORIAL HISTORY HIST 10103. INTRODUCTION TO HISTORICAL INVESTIGATION: PERSONALITIES IN CHINESE HISTORY [W†, some sections count toward C, HSS] HIST 10105. INTRODUCTION TO HISTORICAL INVESTIGATION: WESTERN TRAVELERS TO CHINA [W†, some sections count toward C, HSS] HIST 20000. TRADITIONAL CHINA [C, HSS] HIST 20100. MODERN CHINA [C, HSS] HIST 20600. MODERN JAPAN [C, HSS] HIST 23700. THE UNITED STATES AND CHINA [C, HSS] PHILOSOPHY PHIL 23000. EAST/WEST COMPARATIVE PHILOSOPHY [W†, C, AH] PHIL 23200. CHINESE PHILOSOPHY [C, AH] RELIGIOUS STUDIES RELS 21600. CHINESE RELIGIONS [C, R, AH] RELS 22000. BUDDHISM [C, R, AH] SOCIOLOGY AND ANTHROPOLOGY ANTH 23101. PEOPLES AND CULTURES: JAPAN [C, HSS] SOCI 21900. GLOBALIZATION AND CONTEMPORARY CHINA [C, HSS] Economics 83 ECONOMICS James Burnell, Chair Barbara Burnell Phillip Mellizo Amyaz Moledina John Sell Lisa Verdon James Warner Jingjing Yang Affirming the mission of the college, the Economics Department enables students and faculty to collaboratively research and understand complex questions from a diversity of economic perspectives. The department uses appropriate theories and empirical methods to foster an active engagement with local and global communities. The Economics major is an academically challenging program that provides students with a foundation for understanding market-based and alternative societies and the consequences of economic policy for individual and societal behavior. The requirements are designed to provide the student with knowledge of theoretical and applied economics as well as the quantitative methods necessary for graduate study in economics or careers in business, law, or government service. Students who desire a more specifically business-oriented major should consider the major in Business Economics also offered by the Economics Department. Major in Economics Consists of fourteen courses: • ECON 10100 • ECON 11000 (see note below) • ECON 20100 • ECON 20200 • ECON 21000 (see note below) • One of the following courses: MATH 10400, 10800, or 11100 • Five elective Economics courses, one of which must be at the 300 level • Junior Independent Study: ECON 40100 • Senior Independent Study: ECON 45100 • Senior Independent Study: ECON 45200 Minor in Economics Consists of six courses: • ECON 10100 • ECON 11000 • ECON 20100 • ECON 20200 • Two 200-level or 300-level Economics courses, except ECON 21000 Special Notes • MATH 24100 can be substituted for ECON 11000 and MATH 24200 can be substituted for ECON 21000. • Students who do not place into MATH 10400 or 11100 on the Mathematics placement test should take MATH 10300 or 10700 as soon as possible in their College career to prepare them for MATH 10400 or 10800 and to provide a basis for their Economics courses. Economics 84 • ECON 10100, ECON 11000, and MATH 10400 should be completed no later than the end of the student’s fifth semester. The department recommends that students considering graduate study in Economics enroll in MATH 11100 rather than MATH 10400 and that they also take calculus through MATH 11200. • The department requires that either ECON 20100 or 20200 be taken prior to enrolling in ECON 40100. • A maximum of one Business Economics course selected from BUEC 22700, 23000, 25000, 35500, 36500, or 37000 may be counted toward an Economics major, but not a minor. • Students majoring in Economics are not permitted to take courses in the major on an S/NC basis. • A grade of C- or better is required for all courses counting toward the major, including the Mathematics course(s). Students receiving a grade below C- in ECON 10100 must retake that course before proceeding to the other Economics courses. ECONOMICS COURSES ECON 10100. PRINCIPLES OF ECONOMICS (International Relations, Urban Studies) An introductory study of the fundamental principles of the operation of the market system, the determination of national income, and the role of money in the economy. The department strongly recommends that students display a mathematics proficiency at the level of MATH 10000 or above before enrolling in ECON 10100. Annually. Fall and Spring. [Q, HSS] ECON 11000. QUANTITATIVE METHODS FOR ECONOMICS AND BUSINESS (International Relations, Urban Studies) An introduction to analytical decision-making and its role in business and economic policy. The course includes a discussion of the limitations of quantitative methods and illustrates various techniques with computer applications. Prerequisite: ECON 10100. Annually. Fall and Spring. [Q, HSS] ECON 20100. INTERMEDIATE MICROECONOMIC THEORY (International Relations) The theory of the firm and the industry; the analysis of price determination under market conditions, ranging from pure competition to monopoly; resource allocation. Prerequisite: ECON 10100, and MATH 10400 (may be taken concurrently), sophomore standing or permission of instructor. Annually. Fall and Spring. [HSS] ECON 20200. INTERMEDIATE MACROECONOMIC THEORY (International Relations) An analysis of the theory of national income determination, employment, and inflation, including a study of the determinants of aggregate demand and aggregate supply. Prerequisite: ECON 10100, sophomore standing or permission of instructor. Annually. Fall and Spring. [HSS] ECON 20500. HISTORY AND PHILOSOPHY OF ECONOMIC THOUGHT An analysis of the development of economic thought and method, with emphasis on the philosophical bases and historical context for alternative schools of thought. The course will examine the important characteristics of alternative schools of thought (e.g., Marxist, neoclassical, institutional), and will consider the implications of these alternative schools for economic research and policy. Prerequisite: ECON 10100. Alternate years. Not offered 2011-2012. [W, HSS] ECON 21000. APPLIED REGRESSION (International Relations, Urban Studies) Application of multiple regression analysis to economics. Particular attention is paid to identifying and correcting the violations of the basic model. Consideration of special topics, including time series analysis, limited dependent variables, and simultaneous models. Prerequisite: ECON 11000. Annually. Fall and Spring. ECON 23200. LABOR ECONOMICS An application of economic theory to the labor market, with particular emphasis on the U.S. labor market. Topics include: labor demand, labor supply, human capital theory, theories of labor market discrimination, unions, and inequality in earnings. Prerequisite: ECON 10100. Alternate years. Not offered 2011-2012. [HSS] ECON 24000. ENVIRONMENTAL AND NATURAL RESOURCE ECONOMICS (Environmental Studies) An examination of the economic use of natural resources in society: the economic implications of finite resource supplies, renewable resource supplies, and the use of environmental resources with consideration of policy options regarding optimal resource use. Prerequisite: ECON 10100. Alternate years. Not offered 2011-2012. [HSS] Economics 85 ECON 24500. ECONOMICS OF GENDER (Women’s, Gender, and Sexuality Studies) An investigation of the relationships between economic institutions (e.g., labor force, family, and government) and the role of women in our society, and the implications of the changing role of women for institutional change. Focus on the way traditional tools of economic analysis have been used to address issues that affect women’s economic status, and on feminist critiques of these methods. Prerequisite: ECON 10100. Alternate years. Not offered 2011-2012. [HSS] ECON 25100. INTERNATIONAL TRADE (International Relations) An examination of the basis for international trade. Evaluation of the distributional effects of trade and alternative trade policies. Analysis of free trade areas and economic integration, including the European Union and NAFTA. Prerequisite: ECON 10100. Alternate years. Not offered 2011-2012. [HSS] ECON 25400. ECONOMIC DEVELOPMENT (International Relations) This course will introduce students to the various economic schools of thought concerning the process of economic development. Traditionally economic development has been associated with increasing GDP per capita but this vision has broadened to incorporate, marxists, humanists, gender-aware economists, environmentalists, economic geographers, as well as mainstream neo-classical economists. A political economy approach that incorporates political, social, as well as economic factors affecting development will be the main focus of the course. Prerequisite: ECON 10100. Annually. Fall. [HSS] ECON 26100. URBAN ECONOMICS (Urban Studies) An analysis of economic activity in the spatial context of urban areas from the perspective of inefficient resource allocation resulting from externalities; theories of industrial location, land use, housing markets; application of models to urban problems of growth, land use, slums, ghettos, transportation, pollution, and local government, etc., with consideration of alternative policy options. Prerequisite: ECON 10100. Annually. Spring. [HSS] ECON 26300. LAW AND ECONOMICS An examination of law and legal institutions from the perspective of economics. Economics is used to explain aspects of common and statute law, and legal cases illustrate economic concepts. Prerequisite: ECON 10100. Alternate years. Not offered 2011-2012. [HSS] ECON 26800. HEALTH ECONOMICS An application of economic theory to the market for medical care and health insurance. Other topics include the role of government in these markets, health care reform, and international comparison of health care systems. Prerequisite: ECON 10100. Alternate years. Fall 2011. [HSS] ECON 29900. Special Topics in Economics A course designed to explore an application of economic analysis to a contemporary economic issue. Prerequisite: ECON 10100. ECON 31000. INTRODUCTION TO ECONOMETRICS A discussion of the mathematical and theoretical foundations of the classical linear regression model and extensions of that model. Prerequisite: ECON 21000. Alternate years. Not offered 2011-2012. ECON 31500. PUBLIC FINANCE An investigation of the economics of the public sector to determine an optimum level and structure of the revenues and expenditures of government; includes the relation between government and the private sector, the theory of public goods and collective decision-making, cost-benefit analysis, the structure and economic effects of various taxes, and inter-governmental relations among federal, state, and local governments. Prerequisite: ECON 20100. ECON 32000. INDUSTRIAL ORGANIZATION An application of microeconomic theory to firms and industries. Topics include market structure, pricing practices, advertising, antitrust, and public policy. Prerequisite: ECON 20100. Alternate years. Spring 2012. ECON 32500. AGENCY IN ECONOMICS This course surveys how economists have studied and conceptualized individual and group agency—or the capacity for human beings to make choices and to impose those choices on the world around them. Topics examining the main insights from Classical, Evolutionary, Behavioral, and Experimental Game Theory are explored. Additional topics survey the principle findings and implications of Behavioral Economics, Neuroeconomics, and Behavioral Finance for Economics and related social sciences. Prerequisite: ECON 20100. Alternate years. Fall 2011. ECON 33500. MONETARY ECONOMICS The role of money and the nature of the Federal Reserve’s management of the monetary system are examined Education 86 in the context of the U.S. financial system and economy. Topics include the term structure of interest rates, economic effects of banking regulations, formulation and execution of monetary policy, and transmission channels through which monetary policy affects employment and inflation. Prerequisite: ECON 20200. Not offered 2011-2012. ECON 35000. INTERNATIONAL FINANCE (International Relations) An analysis of the international financial system and policy issues related to world economic interdependence. Topics include exchange rate determination, balance of payments adjustments, monetary and fiscal policies in the open economy. European Monetary Union and issues of development and transition are also included. Prerequisite: ECON 20200. Alternate years. Spring 2012. ECON 39900. SPECIAL TOPICS IN ADVANCED ECONOMIC ANALYSIS A seminar designed for the advanced major. Topics will reflect new developments in the economics discipline. Prerequisite: ECON 20100 and/or ECON 20200. ECON 40100. INDEPENDENT STUDY A one-semester course that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. Prerequisite: ECON 11000, either ECON 20100 or ECON 20200, and ECON 21000 (may be taken concurrently). Annually. Spring. ECON 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: ECON 40100. ECON 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: ECON 45100. EDUCATION Megan Wereley, Chair Matthew Broda Terri Mason Alison Schmidt The Department of Education offers a teacher preparation program that leads to an Ohio, initial, four-year Resident Educator teaching license. Education is not a major at Wooster. Instead, students simultaneously pursue an academic major in a department or program while completing all of the coursework required for the Ohio teaching license. The Department of Education provides opportunities for its teacher candidates to learn and teach within a liberal arts environment that values independence, leadership, inquiry, and tradition. The teacher education program prepares reflective and competent educators for work in classrooms, schools, and a variety of educational communities. The Department recognizes that this preparation is provided in collaboration with all academic programs at Wooster. Throughout the teacher preparation program, students are immersed in hands-on, one-on-one experiences with classroom teachers, college faculty, and students. The Department emphasizes the importance of effective writing, speaking, and interpersonal skills and strives to encourage its graduates to become educational leaders in a complex and global society. Minor in Education Consists of six courses: • EDUC 10000 • One of the following courses: PSYC 11000 or 32700 Education 87 • One of the following courses: EDUC 23100 or 25100 • One of the following courses: EDUC 20000 or 30000 • Two of the following courses: AFST 10000, COMM 14500, 15200, EDUC 26500, ENGL 25000, HIST 11500, PHIL 22300, SOAN 20100, SOCI 20900 or 21400 Special Notes: • Students may earn a teaching license through the Department of Education with or without a minor. • Only grades of C- or better are accepted for the minor. DEPARTMENTAL STANDARDS FOR THE TEACHING PROFESSION The following seven standards for the teaching profession reflect a connection between the goals of Wooster’s teacher education program and the Ohio Standards for the Teaching Professions. Listed below is a brief explanation of what the Depart - ment expects teacher licensure candidates to achieve by the end of the teacher education program: • Student Learning: Candidates understand student learning and development and respect the diversity of students they will teach. • Content Knowledge: Candidates know and understand the content area for which they will have instructional responsibility. • Assessment: Candidates understand and use varied assessments to inform instruction, evaluate, and ensure student learning. • Instruction: Candidates plan and deliver effective instruction that advances the learning of each individual student. • Learning Environment: Candidates create learning environments that promote high levels of learning and achievements for all students. • Collaboration and Communication: Candidates collaborate and communicate with students, parents, families, and other educators, administrators, and the community to support student learning. • Professional Responsibility, Growth, and Reflection: Through reflection, can - didates assume responsibility for professional growth, performance, and involvement as individuals and as members of a learning community. LICENSURE AREAS: At Wooster, students can pursue a teaching license in one of the following areas: Early Childhood* Adolescent to Young Adult Multi-Age Grades preK–3 Grades 7–12 Grades preK–12 Ages 3–8 Ages 12–21 Ages 3–21 Areas of Specialty: Area of Specialty: Integrated Language Arts (English major) Music Integrated Mathematics (Mathematics major) Integrated Social Studies (usually History major) *Early Childhood licensure candidates may also seek the Grades 4/5 Endorsement after completing the additional required coursework. One hundred percent of Wooster graduates seeking the initial and/or Resident Educator Ohio teaching license passed the required Praxis II examinations during the past three academic years. Education 88 REQUIRED COURSES FOR LICENSURE: Early Childhood Education (Grades preK-3, Ages 3-8) • EDUC 10000, 11000, 14000, 20000, 21000, 23100, 26000, 26500, 31000, 49000, 49100, 49200 • COMM 14500 • PSYCH 11000 or 32700 Adolescent to Young Adult Education (Grades 7-12, Ages 12-21) • EDUC 10000, 12000, 25100, 30000, 32000, 49300, 49400, 49500 • PSYCH 11000 or 32700 For specific content courses, see Teacher Education at the College of Wooster: A Supplement to the Catalogue (which can be found at the following website: www3.wooster.edu/education/current/forms.html). Multi-Age Education: Music Education (Grades preK-12, Ages 3-21) See Music Education under MUSIC. EDUCATION COURSES EDUC 10000. INTRODUCTION TO EDUCATION This is a survey course that addresses a variety of topics that include: history of education; diversity of learners; societal changes; educational philosophy; instructional technology; school organization; family and community involvement; cultural diversity; differentiation; lesson planning; and professional development. The course includes a 50-hour supervised field placement in the appropriate content area in a local school. Enrollment in this course is typically limited to sophomores and second-semester first year students. (1.25 course credits) Annually. Fall and Spring. EDUC 11000. USING PHONICS TO TEACH READING AND DEVELOP LITERACY (Communication) In this course students explore techniques and strategies used to teach children to match, blend, and translate letters of the alphabet into sounds they represent and meaningful units. Emphasis is placed on the following topics: technology-related resources; the nature and role of word recognition; multiple literacies; methods and rationale for the instruction of phonemic awareness; fluency and vocabulary; instructional strategies for using children’s literature; diversity; differentiation; decoding; spelling; and word recognition. This class includes a series of focused observations in various early childhood classrooms. Annually. Fall. EDUC 12000. CONTENT AREA READING In this course students consider and examine the research and reading strategies used when teaching content in grades 7-12. Emphasis is placed on the following topics: diversity of learners; needs of struggling readers; developing effective strategies; reflection; ESL/ELL learners; instructional technology; differentiation; assessment; and cooperative and collaborative learning. Students observe teachers using content area reading teaching strategies. (.5 course credit) Prerequisite: EDUC 10000. Annually. Spring. EDUC 14000. INTERDISCIPLINARY FINE ARTS IN THE EARLY CHILDHOOD YEARS This course is designed to help students explore developmentally appropriate practice and curriculum design and implementation within the areas of art, music, drama, and movement. Students examine lesson planning, assessment, instructional technology, community organizations that support the arts, instructional strategies, developmentally appropriate practice, diversity, differentiation, interdisciplinary planning, teaching and learning, and program organization and classroom management available to meet the needs of all learners within the area of fine arts. Several observations and hands-on clinics sponsored by a local community arts center are required in this course. Annually. Spring. EDUC 19900. FUNDAMENTALS OF ENVIRONMENTAL EDUCATION This course is focused on preparing students with an interest in designing learning experiences within various spheres of environmental studies and education. Students will be given opportunities to think about and design environmental education experiences for a wide range of settings - classrooms, camps,nature centers, and beyond. This course will explore core components of environmental education including: its foundations, environmental literacy, planning and implementing of environmental education curricula, assessment and evaluation, and the fostering of learning in environmental education settings. EDUC 20000. TEACHING CHILDREN WITH SPECIAL NEEDS (Communication) This course is designed to explore the federal government’s exceptionalities categories and special education Education 89 models currently used in schools. Emphasis is placed on the following topics: laws governing special education; research-to-practice gap; disproportionate representation in special needs classrooms; impact of ELL/ESL; at risk students; collaborations with colleagues and students’ families; instructional differentiation; early intervention; problem-solving; writing and interpreting the I.E.P.; and cultural diversity. The course includes a 20- hour field placement within a special needs classroom. Annually. Fall and Spring. EDUC 21000. THEORY AND PRACTICE IN TEACHING READING This is a comprehensive course that introduces students to the theory and practice of acquiring literacy and developing instructional strategies for teaching reading in early childhood settings. Some course topics include: theoretical and methodological approaches; diagnostic and organizational techniques; writing; new and multiple literacies; assessment; teaching comprehension, vocabulary, phonemics awareness, writing, and working with words; content area reading; children’s literature; ESL/ELL learners; differentiation; teaching diverse populations; instructional technology; the role of family and community; and classroom environment. This course includes a 50-hour supervised field experience in a reading/literacy-related classroom. (1.25 course credits) Prerequisite: EDUC 10000. Annually. Spring. EDUC 23100. INTRODUCTION TO EARLY CHILDHOOD DEVELOPMENT AND EDUCATION This course introduces students to the theory and practice which drives current early childhood education. Designed to present an exploration of an integrated and developmentally appropriate curriculum and the implementation of that curriculum, the course provides opportunities to examine many topics related to early childhood education. A 50-hour supervised field experience in an appropriate educational setting provides exposure to a diverse student population, instructional technology in an array of social service agencies, the early childhood profession, and a variety of curriculum guidelines and expectations. (1.25 course credits) Prerequisite: EDUC 10000. Annually. Fall. EDUC 24100. INTRODUCTION TO MIDDLE CHILDHOOD EDUCATION This course introduces students to middle level education and addresses the following topics: knowledge and pedagogy in middle childhood settings; the nature of early adolescence; the needs and development of the young adolescent; assessment; middle school philosophy and organization; instructional technology; differentiation; the role of family and community; and the ways in which a young adolescent fits into the school context. A 50-hour supervised field experience in grades 4 or 5 is required. (1.25 course credits) Prerequisite: EDUC 10000. Alternate years. Fall 2011. EDUC 24200. CURRICULUM STUDIES IN THE UPPER ELEMENTARY YEARS In this course students review and use the research that informs instructional practice in curriculum and academic content standards in the upper elementary school grades. Students use these standards to design and assess instructional materials and strategies used to teach science, social studies, mathematics, language arts, fine arts, and technology in the upper elementary grades. In addition, students consider the challenges of teaching all learners, including ESL/ELL, diverse learners, and special needs students. (1.25 course credits) Prerequisite: EDUC 10000. Alternate years. Not offered 2011-2012. EDUC 25100. INTRODUCTION TO ADOLESCENT AND YOUNG ADULT EDUCATION This course is designed to introduce students to teaching at the adolescent to young adult level, grades 7-12. Emphasis is placed on the following topics: evidence-based learning; instructional technology; curriculum models; learning theories; instructional planning; assessment; motivation; the role of family and community; accountability; classroom management; and strategies for meeting the needs of all learners. A 50-hour supervised field experience in a local 7-12 classroom appropriate to the area of licensure is required. (1.25 course credits) Prerequisite: EDUC 10000. Annually. Fall. [W] EDUC 26000. CURRICULUM: MATH/SCIENCE/SOCIAL STUDIES IN THE EARLY CHILDHOOD YEARS This course is designed to help students examine curriculum and instruction in the areas of math, science, health, safety, and nutrition in the early childhood years. Topics include: developmentally appropriate practice; content area reading; content specific teaching and assessment strategies; the role of family and community; differentiation; instructional technology; ESL/ELL learners; and collaborative and cooperative learning. A 50-hour supervised field placement in a content-specific early childhood setting is required. (1.25 course credits) Prerequisite: EDUC 10000. Annually. Spring. EDUC 26500. SOCIAL AND CULTURAL ENVIRONMENTS IN EARLY CHILDHOOD EDUCATION This course examines current research that addresses the significance of the home, school, and community on the growth and development of young children. Emphasis is placed on early childhood educators establishing and maintaining collaborative, cooperative programs and activities that involve families of young children. Topics are explored through lecture, readings and discussions, student presentations, small and large group activities, community speakers, community field trips, video presentations, and 10 hours of focused, fielddirected experiences. Annually. Fall. [W, HSS] Education 90 EDUC 30000. ISSUES IN EDUCATION: TEACHING DIVERSE POPULATIONS This course examines topics relevant to teachers preparing to teach grades 7-12. Topics include: classroom management; effective professional relationships; roles and responsibilities of various school personnel; collaborative teaching and learning; differentiated instruction; teaching students with disabilities; ESL/ELL learners; content area reading; multicultural education; legal and ethical implications of teaching; school finance; educational technology; professionalism; standards and accountability; and school reform. Guest speakers from local schools and focused observations are integral to the course. Prerequisite: EDUC 10000, 25100, and MUSC 29000. Annually. Fall. EDUC 31000. ASSESSMENT AND INTERVENTION IN TEACHING READING This course is designed to provide an in-depth exploration of formal and informal assessment and intervention strategies in the early childhood years. Topics include: observation and assessment of reading skills; valueadded assessments; diagnosis and remediation of reading difficulties; use of children’s literature; multidisciplinary teaching, planning, and evaluation of instructional lessons and units; evaluation of technology tools; implementation of the I.E.P.; use of family-centered assessment; reflective practice; collegial relationships; and professionalism. An “impact on student learning” project is integral to this course and requires both pre- and post- assessments and a 12-week tutoring experience with school-aged children. Prerequisite: EDUC 10000 and 11000. Annually. Fall. EDUC 32000. CURRICULUM METHODS AND ASSESSMENT IN ADOLESCENT AND YOUNG ADULT EDUCATION: INTEGRATED LANGUAGE ARTS, INTEGRATED MATHEMATICS, OR INTEGRATED SOCIAL STUDIES This course is designed for those students who plan to teach grades 7-12 in either English/Language Arts, Mathematics or the Social Studies. Topics include: curriculum development, content area reading, implementation of Ohio Academic Content Standards and/or the Common Core State Standards; instructional models and methods; issues of diversity; integration of instructional technology and 21st century learning; assessment strategies; and research applications/best practices appropriate to the specified content area. Students will also examine and utilize appropriate professional standards (NCTE, NCTM, or NCSS). A 50-hour supervised field placement in a content-appropriate classroom setting is required. One-third of the course is taught in a public school by grades 7-12 classroom teachers licensed within the associated content area. (1.25 course credits), Prerequisite: EDUC10000 and 25100. Annually. Spring. STUDENT TEACHING Student Teaching is required in all three licensure areas. This is the culminating experience in the Teacher Education Program and consists of a full-time, twelve-week supervised teaching experience in a setting appropriate to the areas of licensure. In addition, participation in the Student Teaching Seminar, held one evening a week throughout the entire semester, is required of ALL student teachers. If completed in the Fall semester, Student Teaching begins on the first day of the public school’s academic year (usually one week before the College begins) and continues through mid-November. The remaining five weeks of the semester are dedicated to Independent Study and Student Teaching Seminar. If completed in the Spring semester, students dedicate the first four weeks of the semester to Independent Study and Student Teaching Seminar, and then begin Student Teaching in early February. If the student is completing Student Teaching as a post-graduate and the Independent Study requirement is fulfilled, the dates for Student Teaching and requirement of Student Teaching Seminar remain the same. The student teacher is responsible for providing his/her own transportation throughout the Student Teaching experience. Enrollment in this course is typically limited to seniors or recent post-graduates. Prerequisite: all professional Education courses and most-to-all content-related coursework. Annually. Fall and Spring. EDUC 49000, 49100, 49200. EARLY CHILDHOOD STUDENT TEACHING AND SEMINAR Placement consists of a full-time, 12-week supervised teaching experience in a pre-school, K, 1st, 2nd, or 3rd grade classroom. EDUC 49300, 49400, 49500. ADOLESCENT/YOUNG ADULT STUDENT TEACHING AND SEMINAR Placement consists of a full-time, 12-week supervised teaching experience in a local, approved adolescent and young adult setting (grades 7-12) within the appropriate area of licensure. EDUC 49600, 49700, 49800. MULTIAGE STUDENT TEACHING AND SEMINAR Placement consists of a full-time, 12-week supervised teaching experience in a local, approved multiage music setting (two different levels, divided among the pre-school, K-6, 7-8, and 9-12 environments). English 91 GLOBAL/URBAN STUDENT TEACHING Students may also elect to student teach in a global or urban setting through Educators Abroad- a collegeendorsed study-abroad program. .Students participating in this program must attend Student Teaching Seminar in the semester prior to their student teaching experience and complete all of the College of Wooster student teaching requirements and forms. Students interested in pursuing this placement option should inform the Field Director two-semesters prior to the semester they wish to student teach. CROSS-LISTED COURSES ACCEPTED FOR EDUCATION CREDIT AFRICANA STUDIES AFST 10000. INTRODUCTION TO AFRICANA STUDIES [C, AH, or HSS] COMMUNICATION COMM 14500. LANGUAGE DEVELOPMENT IN CHILDREN [HSS] COMM 15200. PUBLIC SPEAKING [AH] ENGLISH ENGL 25000. READERS’ RESPONSES TO TEXTS HISTORY HIST 11500. HISTORY OF BLACK AMERICA: FROM WEST AFRICAN ORIGINS TO THE PRESENT [C, HSS] PHILOSOPHY PHIL 22300. PHILOSOPHY, CULTURE, AND EDUCATION [AH] PSYCHOLOGY PSYC 11000. CHILD AND ADOLESCENT DEVELOPMENT [HSS] PSYC 32700. DEVELOPMENTAL PSYCHOLOGY: THEORY AND RESEARCH [W] SOCIOLOGY AND ANTHROPOLOGY SOAN 20100. EDUCATION IN SOCIOCULTURAL CONTEXT [C, HSS] SOCI 20900. INEQUALITY IN AMERICA [HSS] SOCI 21400. RACIAL AND ETHNIC GROUPS IN AMERICAN SOCIETY [C, HSS] ENGLISH Nancy Grace, Chair Katharine Beutner Daniel Bourne Suzanne Daly Travis Foster Jennifer Hayward Matt Hooley Mazen Naous Maria Teresa Prendergast Thomas Prendergast Debra Shostak Larry Stewart Leslie Wingard The South African writer Nadine Gordimer once said that “writing is making sense of life.” The challenge and pleasure for both writers and readers is to make sense of the writing that makes sense of life. The English Department offers the student a unique opportunity to encounter a rich variety of texts in which English, American, and Anglophone writers inscribe meaning into our world. Students dis- English 92 cover their own relationship with the world as they hone their skills in reading imaginatively, thinking analytically, and expressing their thoughts clearly, creatively and persuasively both orally and in writing. Courses in English are designed to explore texts across historical periods, cultures, geographical regions and theoretical approaches so as to invite students to ask a wide and diverse range of questions. The curriculum is organized according to those questions—whether they aim to illuminate the cultural construction of gender, sexuality, race, or ethnicity, the career of a single writer, a period in literary history, a literary genre, a reader’s response to texts, or creative writing in fictional and non-fictional forms. Major in English Consists of eleven courses: • ENGL 12000-12012 • ENGL 20000 • One elective in Literature Before 1800 • One elective in Literature Before 1900 • Four elective English courses • Junior Independent Study: ENGL 40100 • Senior Independent Study: ENGL 45100 • Senior Independent Study: ENGL 45200 Minor in English Consists of six courses: • ENGL 12000-12012 • One of the following courses: ENGL 16000, 16100, 20000, 26000, 26100, or 27000 • Four elective English courses Special Notes • ENGL 12000-12012 is strongly recommended as the first course in English for non-majors and is required for majors and minors. To enroll in English courses numbered 200 and above, first-year students must have ENGL 12000-12012 or permission of the instructor. Upperclass students who have not taken ENGL 12000-12012 may enroll in all English courses with the exception of 300-level courses. • In addition to ENGL 12000-12012 (Comparative Literature emphasis), one other Comparative Literature course from Group I may count toward the English major or minor (see Comparative Literature, Group I, in catalogue). Other crosslisted courses include SPAN 21300 (U.S. Latino Literatures and Cultures). • AP credits do not count toward the major, minor, or distribution. • Only grades of C- or better are accepted for the major or minor. ENGLISH COURSES FUNDAMENTAL ISSUES ENGL 12000-12012. INVESTIGATIONS IN LITERARY AND CULTURAL STUDIES (Comparative Literature) Inquiries into fundamental issues of literary language and textual interpretation. Each section focuses on a selected topic in literary studies to consider the ways language functions in the reading process and to explore interrelations among literature, culture, and history. Attention will be given to the following goals: 1) practicing the close reading of literary texts; 2) understanding the terminology of literary analysis as well as core concepts 3) introducing a range of genres and historical periods and discussing literature as an evolving cultural phenomenon; 4) increasing skills in writing about literature. This course is required for the major and strongly English 93 recommended as the first course in English for nonmajors; past topics have included The Gothic Imagination; Imagining America; Life as Narrative; Literatures of Conflict; Lunatics, Lovers, Poets; Modern Selves; Secrets and Lies; and Violent Modernism. Can only be taken once for credit. Annually. Fall and Spring. [AH] ENGL 20000. INVESTIGATIONS IN LITERARY THEORY AND RESEARCH METHODS A writing course designed for English majors. The course will examine reading, writing, and conducting research as interrelated processes enabling us to investigate literary texts and other cultural work. Students will: 1) become familiar with several literary theories and understand what it means to ground literary investigation in a set of theoretical principles; 2) engage with ongoing scholarly conversations and become familiar with research methods; 3) develop their own voices within the conventions of writing in the discipline. Priority given to sophomore majors. Juniors, nonmajors, and second-semester first-year students with permission of course instructor. Prerequisite: ENGL 12000-12012. Annually. Fall and Spring. [W] CULTURE A culture is a complex set of expressions and structures made up of beliefs, expectations, actions, and institutions. Among the most important expressions of a culture are the texts that are written and read within it. These texts are deeply embedded in and shaped by the beliefs and practices of the cultures in which they were first written and by the beliefs and practices of later cultures in which they are read and written about. ENGL 21000-21016. GENDER, RACE, AND ETHNICITY (Comparative Literature) Inquiries into how cultural beliefs and practices about gender, race, and ethnicity are transmitted by and sometimes transformed through texts and their readers. May be repeated for credit as offerings vary. 21002. BLACK WOMEN WRITERS (Women’s, Gender and Sexuality Studies) An examination of the writings of black women from 1746 to the present. Focusing on the major texts in the canon of African American women’s writing, we will consider the distinct cultural possibilities that enabled various forms of literary production over the course of black women’s history in America. Spring 2012. [AH] 21004. GENDER, RACE, AND THE CONSTRUCTION OF EMPIRE (Women’s, Gender and Sexuality Studies) Examines the relationship between gender and colonialism, focusing on the interaction of ideologies of sex, gender, class, and race with constructions of the British Empire. Core texts include literature, film, popular culture, and explorers’ narratives as well as colonial, postcolonial, and gender theory. [Before 1900] Not offered 2011-2012. [AH] 21008. GENDER, SEX, AND TEXTS, 350-1500 (Women’s, Gender and Sexuality Studies) In order to come to grips with what one writer has called “the image of woman” in the Middle Ages, we will explore the cultural configurations of gender and sexuality as they are represented in various kinds of writings and cultural productions (literature, philosophy, biography, legal documents, medical writings, and the visual arts). By interrogating the assumptions that colored the representations of the feminine in the medieval period, we will set the stage for exploring what women of the period (such as Marie de France and Heloise) seemed to be saying when they responded to these assumptions. [Before 1800] Not offered 2011-2012. [AH] 21009. POSTCOLONIAL LITERATURE AND FILM (Film Studies) Investigates literature, film, and theory from formerly colonized countries, with emphasis on Anglophone texts and some translated texts from South and Central Asia, Africa, the Caribbean, and the Middle East. Questions raised in the course include: How does language shape identity in the colonial and postcolonial worlds? How do factors like race, gender, or nationality affect identity? Can we identify specifically postcolonial narrative forms and techniques? Texts include literature by writers such as Salman Rushdie, Arundhati Roy, Amitav Ghosh, Caryl Phillips, J.M. Coetzee, Nadine Gordimer, Derek Walcott, Jamaica Kincaid, and Naguib Mahfouz, and films by directors such as Mira Nair, Euzhan Palcy, Vishal Bharadwaj, Ziad Doueiri and Gurinder Chadha. Spring 2012. [C, AH] 21014. RELIGION IN BLACK FILM AND LITERATURE Debates regarding religious beliefs and practices recur throughout the history of African-American film and literature. In this course, we will analyze the complicated role of religion, particularly Christianity, in black communities. Our texts were created during or about slavery, the Great Migration, the U.S. English 94 Civil Rights Movement, and the Post Civil Rights Era. We will consider such issues as ways in which religion is shown to empower and/or oppress black people; ways in which the politics of class, gender, and sexuality inflect black religious practices; and strategies by which transcendent, spiritual experiences are represented. Films to be analyzed may include: Spencer Williams’ The Blood of Jesus; Stan Lathan’s Go Tell it on the Mountain; Spike Lee’s Four Little Girls; Julie Dash’s Daughters of the Dust; and T.D. Jakes’ Woman Thou Art Loosed. Texts by Alice Walker, Melba P. Beals, Langston Hughes, James Baldwin, Zora Neale Hurston, and Ernest Gaines, as well as some visual art, may also be considered. Not offered 2011-2012. [AH] 21015. READING SEXUALITIES IN AMERICAN LITERATURE (Women’s, Gender and Sexuality Studies) This class studies depictions of queer lives in the past 150 years of American fiction, poetry, drama, and film, analyzing three prominent representational trends. We’ll begin by looking at texts that represent queerness as, to borrow from Willa Cather, “the inexplicable presence of the thing not named” (e.g. Sarah Orne Jewett’s The Country Doctor, Alfred Hitchcock’s Rope, the poetry of Walt Whitman and Elizabeth Bishop, and David Henry Hwang’s M. Butterfly). We then turn to texts that have not traditionally been defined by their queer themes (you’d never find them in the “gay lit” section of your local bookstore), but that nevertheless help us to analyze the marginalization of lesbian, gay, bisexual, and transgender people as an integral function of the U.S.’s racial, gendered, and economic histories (e.g. James Baldwin’s Another Country, Michael Chabon’s The Mysteries of Pittsburgh, Alice Walker’s The Color Purple, and David Lynch’s Mulholland Drive). And, finally, we’ll study the works of authors and artists who rely upon innovations in form in order to articulate their expressions of queer difference (e.g. Gertrude Stein’s modernist sketches, Samuel Delany’s experimental essays, and Sadie Benning’s postmodern films). Along the way, we’ll augment our own interpretations by engaging and testing the arguments of critics who have considered these same histories and texts, including Michel Foucault, Eve Kosofsky Sedgwick, Judith Butler, José Esteban Muñoz, and Christopher Nealon. Throughout these many readings, we’ll keep our attention focused on a central question about not only what it means to be queer in America today, but also about the politics of subcultural identity more broadly: What are the rewards and costs of social legitimacy? (Or, as some might prefer to ask the question, of social illegitimacy?) Not offered 2011-2012. [AH] 21016. 20th CENTURY BRITISH FICTION: WRITING FROM THE BORDERS This course will examine seminal trends in twentieth-century fiction by focusing on the theme of borders. We will examine the nature of literal and figurative borders that many of the writers face: geographical, cultural, racial, gendered, class and political borders. We will begin with the phenomenon of modernism (engaging its literary, artistic, philosophical, and historical development) and move to the second part of the century. In exploring works by Conrad, Forster, Woolf, Rhys, Coetzee, Gordimer, and Rushdie, we will see how they speak to each other in interesting and complex ways. We’ll be reading novels in English from a variety of countries and cultures in Britain, Africa, the Caribbean, and India, moving from the modern to the postmodern, the colonial to the postcolonial. As borders shift, we all have a vested interest in exploring this theme. Literary and cultural theories will guide our readings. Not offered 2011-2012. [AH] ENGL 22000-22012. WRITERS (Comparative Literature) Inquiries into how individual writers’ works are shaped in interaction with life experiences and cultural contexts. Each course will give close attention to texts by an individual writer or small group of related writers and will examine the relationship between those texts and significant issues in a writer’s life and social environment. May be repeated for credit as offerings vary. 22001. SHAKESPEARE This course follows Shakespeare’s twenty-year career as a poet and playwright by exploring the different “Shakespeares” that emerge when we read the plays and poems in light of such varied perspectives as gender, genre, race, culture, formalism, and performance. We will consider, in the process, how Shakespeare constantly develops and changes his notions of fictionality throughout his twenty-year career as a playwright and poet, and the way that these notions are at once innovative in their own right and strongly influenced by the theatrical culture of late Elizabethan and early Jacobean England. [Before 1800] Spring 2012. [AH] 22002. WILLIAM FAULKNER Explores the novels and short fiction of William Faulkner (1897-1962) within the context of the social history and literary culture of his time. Gives special attention to his innovations in form. Not offered 2011-2012. [AH] English 95 22004. CHARLOTTE BRONTË (Women’s, Gender and Sexuality Studies) This course examines the novels of Charlotte Brontë (1816-1855) in the context of her personal and family history and the social history of mid-nineteenth century England. We will give attention to gender roles and to the cultural assumptions about women, as well as to the political and social changes brought about by changes in industrial and economic conditions in early Victorian England. The course will consider her novels — and some of her sisters’ novels — in relation to subsequent texts by women and to changes in gender assumptions in the late twentieth century. [Before 1900] Not offered 2011-2012. [AH] 22008-22009. JAMES BALDWIN AND TONI MORRISON (Women’s, Gender and Sexuality Studies) James Baldwin and Toni Morrison are certainly two of the most significant authors of the 20th century. This course allows an intense study of their major works, including novels, theatre, short stories, essays, and literary critics’ responses to them all. We’ll explore answers to questions such as the following: What constitutes African American community, as well as larger U.S. and global communities? How are race, class, gender, and sexuality intersecting in our variety of selected texts? In what ways are Baldwin and Morrison using jazz and the blues, critiquing whiteness, and otherwise unraveling societal politics? And, in sum, how are Baldwin and Morrison speaking to or against one another? Texts may include Baldwin’s The Fire Next Time; If Beale Street Could Talk; Just Above My Head; Tell Me How Long the Train’s Been Gone; or, Giovanni’s Room, among others, and, Morrison’s Love; Beloved; Playing in the Dark; Tar Baby; or, Song of Solomon, among others. Interviews and documentary films will be analyzed, and some visual art may be considered. Not offered 2011-2012. [AH] ENGL 23000-23031. HISTORY (Comparative Literature) Inquiries into cultural beliefs about continuity, disruption, and change over time in the emergence, significance, and influence of texts. Special attention will be given to definitions of history and periods, the development and change of canons, and the role of authority, society, and institutions in the study of texts. May be repeated for credit as offerings vary. 23002. SURVEY OF AFRICAN AMERICAN LITERATURE (Africana Studies) A historical study of the development and change of black themes and consciousness as manifested in poetry, fiction, autobiography, and essays, and of their correspondence with the literature produced by other ethnic groups in America. Fall 2011. [AH] 23004. LITERATURE OF THE COLD WAR (Film Studies) An exploration of various English-language texts (including fiction, poetry, film, and drama) produced within the Cold War period and the ways in which the historical concerns of the era were represented in these texts. Special attention will be paid to the concept of ”the other,“ examining its function as a dramatic device as well as the numerous metaphorical representations of such a perception of dualities in conflict: east vs. west, left vs. right, patriot vs. subversive, hawk vs. dove, eagle vs. bear, and so on. Not offered 2011-2012. [AH] 23005. RESTORATION AND EIGHTEENTH-CENTURY DRAMA An examination of British plays produced between 1660 and 1800, focusing on the distinctive dramatic and theatrical conventions of the period and on the relationships of the plays to their cultural contexts. Particular emphasis on comedy, on the impact of actresses, and on the commercialization of theatre. [Before 1800] Not offered 2011-2012. [AH] 23007. NINETEENTH-CENTURY BRITISH LITERATURE This course will use three of the central preoccupations of the nineteenth century — industrialization, escalating class conflicts, and shifting views of gender — as focal points in exploring some of the major authors of the period, including Dorothy and William Wordsworth, Keats, Emily Brontë, Dickens, Barrett Browning, Collins, Eliot, and Stoker. [Before 1900] Not offered 2011-2012. [AH] 23011. LITERATURE OF THE BEAT GENERATION This course explores the historical and social contexts giving rise to that generation of writers commonly referred to as Beats: Jack Kerouac, Allen Ginsberg, William Burroughs, Neal Cassady, Gregory Corso, Gary Snyder, Diane DiPrima, and Joyce Johnson. Special attention is paid to the study of existentialism, Buddhism, and jazz, all powerful influences on Beat writing. Issues of race, gender, and sexuality are also explored. Spring 2012. [AH] English 96 23012. POETRY SINCE WORLD WAR II This course focuses on the emergence, development, and disruptions in poetic meanings and forms in American and British poetry since World War II. It also includes extensive readings in relevant critical and cultural writings. Fall 2011. [AH] 23013. TRADITION AND COUNTERTRADITION IN NINETEENTH CENTURY AMERICAN LITERATURE Examination of the cultural values that have caused works and writers to be either included in or excluded from the canon of American literature, with special attention to relationships among national concerns, national “identity,” representations of race and gender, and the rise of a distinctive literary tradition in the United States. Works by writers such as Chesnutt, Chopin, Hawthorne, James, Jewett, Melville, Stowe, Twain, and Whitman. [Before 1900] Not offered 2011-2012. [AH] 23014. THE HARLEM RENAISSANCE This course offers an examination of the literature, music, and popular culture of the period in African American cultural history that has come to be known as the Harlem Renaissance. In addition to a close examination of the major literary texts of the period, we will consider the social forces and interracial cultural dynamics that produced this unparalleled outpouring of creative activity. Not offered 2011-2012. [AH] 23016. ELIZABETHAN AND JACOBEAN DRAMA The death of Queen Elizabeth I in 1603 and the accession of James I mark a significant shift in the sociopolitical climate of Renaissance England. This course will examine this transition by comparing Elizabethan and Jacobean dramas and masques in terms of their representations of gender, race, sexuality, monarchy, and empire. Students will read authors such as Spenser, Marlowe, Shakespeare, Jonson, Carey, and Webster. We will use performance, writing, presentations, and discussion to develop an understanding of the authors and texts in relation to their historical contexts as well as to current literary theory. [Before 1800] Not offered 2011-2012. [AH] 23019. CONTEMPORARY AFRICAN AMERICAN LITERATURE In this class we will examine selected works of African American poetry and fiction published since 1970. We will pay particular attention to the impact that Hip Hop and Rap have had on African American creative culture and the degree to which film and popular music have both supplemented and displaced literature as the primary mode(s) of African American expressive culture. Among the authors whose work we will be considering will be Gil Scott-Heron, Tupac Shakur, Sister Souljah, and Carl Hancock Rux. Not offered 2011-2012. [AH] 23026. THE EARLY AMERICAN NOVEL In this class, we’ll not only survey novels written and published in the United States from the Revolutionary War to the Civil War, but also study the history of the early American novel as a widespread literary form. We’ll consider the novel’s European influences, authors, readers, commercialization, presence as a physical object, and generic subdivisions (e.g. the romance and the gothic). At the same time, we’ll examine this literary history as both an effect and an agent in the period’s social and political histories, which span state formation, the Haitian Revolution, expansion into Mexico and the West, the consolidation of U.S. capitalism, increasing tension between North and South, and the ostensible end of slavery. Readings will include multiple secondary sources along with eight early American novels: Susanna Rowson’s Charlotte Temple (London, 1792; Philadelphia, 1794); Charles Brockden Brown’s Wieland (1798); James Feminore Cooper’s The Pioneers (1823), Catharine Maria Sedgwick’s Hope Leslie (1823); Herman Melville’s Moby Dick (1851), Nathaniel Hawthorne’s The House of Seven Gables (1851); Fanny Fern’s Ruth Hall (1854), and Martin Delany’s Blake, or, the Huts of America (1861). [Before 1900] Spring 2012. [AH] 23027. RENAISSANCE FANTASIES This course explores the ways in which Renaissance authors such as Petrarch, Shakespeare, and Sor Juana de la Cruz responded to Plato’s contention that fantasy is seductive, deceitful, and subversive of established authority. The course focuses on the main conceptualizations of fantasy that the authors employ in response to Plato: fantastical characters and events, sexual fantasies, and fantastical writing styles. Discussions will also center on some of the anti-fantasy treatises of the period. [Before 1800] Not offered 2011-2012. [AH] 23029. AMERICAN LITERATURE TO 1865 This course offers a survey of American literature through the Civil War. Readings will span a full range of genres as we cover the major movements that shaped U.S. literary history: the culture of colonial set- English 97 tlers, Puritan and evangelical religiosity, Enlightenment epistemology, the Haitian and American revolutions, nationalism, reformist literature, the rise of the black public intellectual, and Transcendentalism. Authors will include Columbus, Bradford, Equiano, Franklin, Wheatley, Emerson, Thoreau, Apess, Fuller, Poe, Hawthorne, Douglass, Melville, Whitman, Dickinson, and Lincoln. [Before 1900] Not offered 2011-2012. [AH] 23030. MODERN BRITISH FICTION AND POETRY As a survey of modernist British fiction and poetry, this course will address questions such as: What is modernism? Is modernism over? If not, what applicability and possibility does modernism have in relation to current sociopolitical and cultural trends? We will investigate the phenomenon of modernism (engaging its literary, artistic, philosophical, and historical development) and pay close attention to its techniques. In exploring fiction and short stories by such authors as Joseph Conrad, E. M. Forster, Virginia Woolf, James Joyce, Vita Sackville-West, and D. H. Lawrence we will see how these works speak to each other. We’ll also be reading a collection of modern poetry by poets such as Siegfried Sassoon, Wilfred Owen, W. B. Yeats, and T. S. Eliot. Literary and cultural criticism will guide our readings. Not offered 2011-2012. [AH] 23031. FICTIONS OF BRITISH INDIA (1800-1906) In the nineteenth century, England greatly expanded its territory in and political control over India. Popular understanding of the Raj, or British rule, was shaped largely by print journalism, but a number of influential novels also purported to depict India realistically. We will read these novels alongside diaries, historical accounts, and literary criticism in order to think through the role of literature in disseminating ideas about the nature and purpose of British imperialism. We will also consider the purposes to which the novel form specifically is put: for example, what happens when imperialism is given a plot, and how values and ideologies are conveyed by novels as opposed to other kinds of writing. Novels may include Sydney Owenson, The Missionary; Walter Scott, The Surgeon’s Daughter; Philip Meadows Taylor, Confessions of a Thug; Rudyard Kipling, Kim; Sara Jeannette Duncan, Set in Authority. Fall 2011. [AH] TEXTS Texts are integral to and shaped by cultures, but as parts of culture, texts significantly shape and change cultures as well. The courses in this category inquire particularly into how the reading and writing of texts contribute to changing and defining cultures and individuals. Strategies for Reading ENGL 24000-24025. TEXTUAL FORMATION (Comparative Literature) Inquiries into changing cultural assumptions about language and its literate uses. Special attention will be given to the ways that the formulation of texts in various modes of discourse develops conventional expectations of meaning and value among writers and readers over periods of time and plays a significant role in cultural change and definition. May be repeated for credit as offerings vary. 24002. NARRATIVE AND THE REAL WORLD (Film Studies) An inquiry into narrative, both fictional and nonfictional, as a way of knowing. The course focuses on how we tell stories to make sense of our lives, our pasts, and our perceptions of the world and on how the conventions of storytelling shape our knowledge. Historical texts, fiction, and film will be used to investigate these issues. Not offered 2011-2012. [AH] 24003. THE ODYSSEY OF JAMES JOYCE’S ULYSSES This course will explore the formation of James Joyce’s Ulysses, focusing on Joyce’s composing process, identifying and analyzing historical, cultural, social, literary, and personal contexts which he used in his artistic decisions. Students will read Ulysses as well as related secondary and primary sources. Not offered 2011-2012. [AH] 24005. CONVENTIONS OF THE SHORT STORY An examination of the conventions of the English and American short story in the last two hundred years. Works will include both those within and those outside the traditional canon. Spring 2012. [AH] 24006. THE CANTERBURY TALES AND THE FORMS OF MEDIEVAL NARRATIVE A study of Chaucer’s The Canterbury Tales as a representative collection of medieval narrative forms, such as the romance, the lai, the fabliau, the saint’s life, the beast fable, the exemplum, and the moral English 98 allegory. Special attention will be given to the larger narrative framework of these tales by which Chaucer makes a critical comparison and assessment of differing cultural values on which the various narrative forms are based. [Before 1800] Not offered 2011-2012. [AH] 24017. THE AMERICAN FILM (Film Studies) The course samples the range of American film history, from the silent film to the rise of Hollywood to postmodern and independent filmmaking. The course introduces basic strategies for the interpretation of visual style, narrative, and ideological coding in the cinema and is organized around the study of such genres as comedy, the musical, populist film, the western, the historical epic, film noir, and suspense. Students should be prepared to attend evening screenings each week. Not offered 2011-2012. [AH] 24018. NINETEENTH-CENTURY BRITISH NOVEL ON FILM (Film Studies) This course will investigate 19th century novels together with their later film adaptations. We will read both 19th century fictions and their contemporary appropriations as historically and culturally embedded and debate the cultural work performed by both sets of texts: what purpose did the 19th century novel serve for its readers, what function does our fascination with the Victorian past perform for contemporary audiences, and how do discourses of nostalgia and authenticity shape filmic appropriations of 19th century fictions? Readings include novels by Austen, Emily Brontë, Dickens, Thackeray and Stoker as well as literary and film theory; students should also be prepared to attend evening screenings most weeks. [Before 1900] Not offered 2011-2012. [AH] 24019. MEDIEVAL LITERATURE: THE PLACE OF THE PREMODERN In this course we will read the imaginative literature of the later Middle Ages. In addition to experiencing the pleasures of such genres as romance, dream vision and drama, we will explore how these genres shaped medieval ideas of time and place. Along the way we will consider how the “middle age” came to be, what it was, and how it relates to modernity. Included among the works we might read are Chaucer’s Troilus and Criseyde, Langland’s Piers Plowman, Malory’s Morte Darthur, Sir Gawain and the Green Knight, the Lais of Marie de France, and The Second Shepherd’s Play. [Before 1800] Not offered 2011- 2012. [AH] 24021. BEFORE THE NOVEL This class will explore forms of writing that pre-dated and influenced the novel. Genres that we study will include the sonnet sequence as the origin of the idea of the conflicted self, Elizabethan and Jacobean theater (such as Shakespeare’s Hamlet and Much Ado About Nothing) as significant influences on the structure of the novel, the emergence of satirical works in the seventeenth century, and non-novelistic sixteenth and seventeenth-century prose fictions like Oroonoko (1688) and The Countess Pembroke’s Arcadia (1580). We will consider how these works emerge from earlier, manuscript notions of fiction as well as the developing cultures of theater and print. [Before 1800] Spring 2012. [AH] 24022. GREEN ROMANTICISM (Environmental Studies) The Romantics are thought of as nature poets first and foremost. In this course, we will interrogate the relationship between the Romantic poets and the early 19th century landscape, both “natural” and industrial. After examining the problematic notion of a unified “Romantic” ethos and establishing the divergent sub-groups within the Romantic movement (for example, the Lake poets versus the “cockney school” of London versus the Scottish Romantics), we will raise questions about the Romantics’ relationship to the environment. For example, how did the rapidly industrializing European landscape influence their works? Has Romantic poetry shaped the history of Western environmentalism? Does contemporary ecocriticism build on Romantic tropes and themes? Studying Romantic literature with an emphasis on the relationship between the Romantics’ focus on place and their emphasis on subjectivity, we will ask how the relationship between people and the landscape is imagined, and how it has been structured by institutions of class, economics, politics, gender, science, and law. [Before 1900] Not offered 2011-2012. [AH] 24024. READING RED LAND: LITERATURE & FILM ACROSS INDIAN COUNTRY This course explores 19th and 20th Native Literature as it is shaped by (and shapes) the landscapes of Native America. Course texts will be organized regionally and conceptually: opening questions about the intersection of land, politics, and literature: how, for example, ideas central to Native literary and cultural studies - religion, gender, migration, the oral tradition, law - are informed by or directed toward notions of place and geography. Fall 2011. [AH] 24025. DEMONIC ROMANCE IN 19th CENTURY BRITISH LITERATURE A wide range of 19th-century writers were compelled by the idea of attraction between humans and English 99 such not-quite human creatures as demons, vampires, goblins, and ghosts. We will explore the aesthetic and historical dimensions of these romances, considering their relationship to such literary/cultural movements as medievalism, realism, and the gothic revival as well as to contemporary political debates over science, empire, immigration, and the status of women. Poetry will include works by Coleridge, P. B. Shelley, Keats, Tennyson, and Christina Rossetti; fiction will include M. E. Braddon, Lady Audley’s Secret, Sheridan LeFanu, Carmilla, Richard Marsh, The Beetle, and Oscar Wilde, The Picture of Dorian Gray. We will also read criticism and theory by Auerbach, Burke, Dijkstra, Halberstam, Marx, Sedgwick, and others. Fall 2011. [AH] ENGL 25000-25010. READERS’ RESPONSES TO TEXTS (Communication, Comparative Literature, Education) Inquiries into the relationships among readers, texts, and experience. Attention will be given to the ways in which readers may be said to create or structure the meanings of texts; the ways in which texts may be said to govern the responses of readers; and the ways in which readers may extend these responses and meanings into the experiential world as understandings or knowledge usable in making decisions or taking actions. May be repeated for credit as offerings vary. 25003. CHILDREN AS READERS: THE TEXTS OF CHILDHOOD AND ADOLESCENCE This course introduces students to a variety of works frequently read by children and adolescents. It focuses on the responses of children and adolescents to these texts and inquires into the reasons for various individual responses. The course considers both literary and non-literary texts. Spring 2012. [AH] 25005. EIGHTEENTH-CENTURY TEXTS: READERS AND MEANINGS A study of selected novels, plays, and poems from the late seventeenth to the early nineteenth century, this course will focus on the transaction between texts and their readers. The course will inquire into the ways in which readers participate in the construction of textual meanings and the role of texts in the experience of readers. Works studied will include texts by Aphra Behn, John Dryden, Jonathan Swift, Henry Fielding, William Congreve, Laurence Sterne, Alexander Pope, Anne Finch, Samuel Johnson, and James Boswell. [Before 1800] Not offered 2011-2012. [AH] 25006. SUBVERTING FICTIONS: THE EIGHTEENTH AND NINETEENTH-CENTURY BRITISH NOVEL This course will explore the extent to which eighteenth and nineteenth century British novels may be said to subvert the ideological assumptions of their readers or, on the other hand, to reinforce those assumptions. The course will also consider whether we, as twentieth-century readers, read these novels according to our assumptions and expectations. Included will be texts by such writers as Aphra Behn, Daniel Defoe, Henry Fielding, Jane Austen, Charles Dickens, George Eliot, Thomas Hardy, and Joseph Conrad. [Before 1900] Not offered 2011-2012. [AH] Strategies for Writing ENGL 16000-16005. NON-FICTIONAL WRITING Analysis, discussion and practice of writing in a variety of non-fictional forms. Courses will explore the aims and conventions of the specified written discourse and emphasize the writing of participants. May be repeated for credit as offerings vary. Annually. Fall and Spring. [W†, AH] 16000. INTRODUCTION TO NON-FICTIONAL WRITING This class introduces students to major writers and genres of contemporary and classic non-fictional writing—particularly the genres of memoir, personal essay, literary journalism, editorial writing, critical writing, and film review. As we consider these texts, we will be answering the questions: “What is non-fiction?” “What are the boundaries between fiction and non-fiction?” and “What is the relationship between reading non-fictional writings and writing about them? “Throughout the semester, students will be writing and reading non-fiction by comparing and contrasting students’ writings in creative non-fiction, the critical essay, and the review essay with those by contemporary and classic essay writers, and with writings by other students in the class as well. Spring 2012. [W, AH] 16002. AUTOBIOGRAPHICAL WRITING (MEMOIR) Analysis, discussion and practice of autobiographical writing, with an emphasis on memoir. The course will explore the aims and conventions of the genre, and emphasize course participants’ own writing. Fall 2011. [W, AH] English 100 16003. NATURE AND ENVIRONMENTAL WRITING (Environmental Studies) Along with Henry David Thoreau, many American writers have chosen to explore their surrounding natural world and its intersections with their selves and societies. This course will be an exploration of the tradition and current practice of such writing connected with the natural world. Along with the exploration of already published works in nature and environmental writing, the course will include off-campus field trips and emphasize participants’ own writing and peer feedback workshops. Fall 2011. [W, AH] ENGL 16100. INTRODUCTION TO FICTION AND POETRY WRITING An introduction to writing in a variety of fictional forms, especially short stories and poems. Participants will analyze and discuss both published writing and their own writing. Priority given to English majors. Annually. Fall and Spring. [AH] ENGL 19900. APPRENTICESHIP IN EDITING A LITERARY-MAGAZINE By serving as an assistant editor for Artful Dodge, a nationally-distributed magazine of new American writing, graphics, and literature in translation, students will be exposed to the daily operations of editing a pro fessional literary magazine. Students will engage in a number of important activities, including design and development of the magazine’s web-site, editorial and promotional copy-writing, evaluation of manuscripts, typesetting and proofreading, and the organization of off-campus literary events. Students will be required to read histories of the American literary journal as well as explore other currently-published literary magazines. (.25 course credit) Prerequisite: Enrollment is by application to the instructor. Annually. Fall and Spring. ENGL 26000-26005. ADVANCED NON-FICTIONAL WRITING Analysis, discussion, and practice of writing in a variety of non-fictional forms. Courses will explore the aims and conventions of the specified written discourse and emphasize the writing of participants. May be repeated for credit as offerings vary. 26001. NEWS WRITING AND EDITING This course familiarizes students with the strategies and conventions of journalistic writing, specifically news stories, editorials, reviews, and feature articles. Students will participate in the publication of a news magazine featuring their own writing. Not offered 2011-2012. [W, AH] 26003. AUTOBIOGRAPHICAL WRITING (TRAVEL NARRATIVE) Analysis, discussion and practice of autobiographical writing, with an emphasis on travel narrative. The course will explore the aims and conventions of the genre, and emphasize course participants’ own writing. Not offered 2011-2012. ENGL 26100-26104. ADVANCED WRITING IN FICTIONAL FORMS Analysis, discussion, and practice of writing in one or more fictional forms, such as short stories, poems, or plays. Course will explore the aims and conventions of the specified written discourse and emphasize participants’ writing. May be repeated for credit as offerings vary. 26101. ADVANCED FICTION AND POETRY WRITING A multi-genre course that focuses on the analysis, discussion, and practice of writing in various fictional forms, such as short stories, poems, or plays. Fall 2011. 26102. ADVANCED FICTION WRITING: THE STORY CYCLE This course will focus on studying collections of linked short stories. Participants will analyze several complete story cycles and discuss the techniques authors use to connect the stories in each collection in order to create cohesive book-length narratives. The emphasis during the first half of the semester will be on studying published story collections. The class will shift into writing and workshopping during the second half of the semester. As a class, students will write and revise one complete story cycle, with each student contributing one story to the collection. Prerequisite: ENGL 16100. Spring 2012. [AH] 26103. ADVANCED POETRY WRITING This course will look at a number of different contemporary poets and approaches to poetry, including writing in various fixed as well as open forms. Students will explore (and experiment with in their own poetry) a number of traditional and contemporary techniques as well as consider prose-poetry, spoken word poetry and other artistic threads currently prominent in the poetry landscape. Along with the active reading of published works of poetry, students will also explore aspects of craft and style in their own writing as well as provide constructive feedback for their fellow writers in weekly-held workshops. Prerequisite: ENGL 16100 or permission of the instructor. Not offered 2011-2012. [AH] English 101 ENGL 27000-27007. THEORIES AND PRACTICES OF RHETORIC AND COMPOSITION Inquiries into the history of rhetoric and composition as disciplines, focusing on such topics as classical and contemporary theories of rhetoric, contemporary theories of composition and creativity, the teaching of writing, the identity of the writer, and current concerns in composition research. May be repeated for credit as offerings vary. [AH] 27003. TUTORING METHODS This course introduces students to the theory and practice of one-to-one composition instruction. Students will explore theories from psychology, sociology, and English studies. Students will also learn about the history of peer instruction and its place in a composition program. Recommended for all Writing Center peer tutors. (.5 course credit, meets first half of semester) Not offered 2011-2012. [AH] 27005. ENGLISH GRAMMAR This course examines the grammatical structures in English. It will examine the evolution of tradi tional grammar and grammatical theories. Special attention will be given to the place of grammar instruction in composition pedagogy. Recommended for all Writing Center peer tutors. (.5 course credit, meets second half of semester) Not offered 2011-2012. [AH] JUNIOR AND SENIOR SEMINAR ENGL 30000-30007. SPECIAL TOPICS IN LITERARY STUDIES (Comparative Literature) A seminar providing English majors, as well as upper-level non-majors who have completed ENGL 20000 and at least two literature courses, with the opportunity for advanced work in literature. Devoted to a specific area of investigation, the seminar will engage in close reading of primary literary and discursive texts. Topics announced in advance by the faculty member teaching the course. Prerequisite: ENGL 20000 and two literature courses. [AH] INDEPENDENT STUDY ENGL 40000. TUTORIAL Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. ENGL 40100. PERSPECTIVES AND METHODS OF INDEPENDENT STUDY Discussion and review of contemporary approaches to the study of language, texts, and culture, culminating in the student’s completion of a substantial, critically and theoretically informed essay. The course asks students to become conscious about the assumptions underlying their approaches to literary texts; conscious of the relations between their questions and some of the diverse answers that have been produced in the discipline; and aware of the kinds of evidence suitable to the arguments they wish to make. Prerequisite: ENGL 20000. Annually. Fall and Spring. ENGL 45100. SENIOR INDEPENDENT STUDY THESIS – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: ENGL 40100. ENGL 45200. SENIOR INDEPENDENT STUDY THESIS – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: ENGL 45100. Film Studies 102 ENVIRONMENTAL STUDIES CURRICULUM COMMITTEE: Richard Lehtinen (Biology), Chair Charles Kammer (Religious Studies) Matthew Mariola (Environmental Studies) Melissa Schultz (Chemistry) Issues related to the natural environment require a uniquely interdisciplinary focus to understand the way in which technological advances and human behavior affect fundamental ecological processes, what political and psychological tactics may be harnessed to address the problem, and how nature is discussed, described, and experienced. The field of environmental studies provides the opportunity to integrate multiple disciplinary perspectives in order to think about and understand environmental issues. The program at Wooster will encourage students to engage with environmental issues both inside and outside the classroom, and at both local and global levels. Environmental Studies minors will be knowledgeable about core scientific concepts that allow them to understand ecological processes and change; be able to understand different ways of assessing the value of the natural environment and be comfortable with different means of examining and communicating about the environment; and be familiar with the ways in which social institutions contribute to environmental problems and may be utilized for solutions to those problems. They should also understand their own roles as actors within the human-environment relationship. The Environmental Studies minor will complement a major in a traditional department so that students will combine a detailed understanding of the knowledge and methods within a discipline with a focus on a particular topic. Students with an Environmental Studies minor will complete their I.S. project within their major department. However, they are encouraged to include an environmental component to their I.S. when possible, and the Environmental Studies faculty will endeavor to help them to do so. Minor in Environmental Studies Consists of 6 courses: • ENVS 20000 • One cross-listed course in Natural Sciences accepted for ENVS credit • One cross-listed course in Social Science accepted for ENVS credit • One cross-listed course in Humanities accepted for ENVS credit • Two electives from Environmental Studies or cross-listed courses accepted for ENVS credit Special Notes • No more than one course within a student’s major discipline may be counted toward the Environmental Studies minor. • In departments with multiple versions of a particular course, only the sections specified in parentheses after the course listing will count toward the Environmental Studies minor. • Only grades of C- or better are accepted for the minor. ENVIRONMENTAL STUDIES COURSES ENVS 11000. SCIENCE, SOCIETY AND ENVIRONMENT Despite an ever-expanding body of scientific research about critical environmental problems facing global society, the level of knowledge evident in the public discourse remains quite low. This course will introduce stu- Environmental Studies 103 dents to a number of complex environmental issues and controversies and teach them the skills necessary to understand, interpret, and translate these issues into a form fit for consumption by a general audience. An underlying theme will be the idea that complex environmental problems have social/political causes as well as technological/biophysical ones. Fall 2011. [W] ENVS 20000. ENVIRONMENTAL ANALYSIS AND ACTION This course will present a multidisciplinary perspective on environmental topics by examining at least one issue of global significance and one of more local importance from the perspectives of the natural sciences, social sciences, and humanities. Students will apply fundamental concepts from various disciplines to understand, formulate and evaluate solutions to environmental issues. Prerequisite: at least one Natural Science course from the cross-listed courses accepted for ENVS credit, and one course from the list in either Social Sciences or Humanities. Annually. Spring 2012. ENVS 20500. ENTREPRENEURSHIP AND THE ENVIRONMENT This course will explore what we mean by the concept of “entrepreneurship”, and apply this definition to three broad areas of society that have come to define the environmental challenges of the new century: the production of food, the production of energy, and the disposal of wastes. For each of these three areas we will look at (a) the nature of the challenge; (b) what role entrepreneurial activity might play in meeting the challenge; and (c) some case studies of entrepreneurship in action to solve these social needs. As a final project, students will produce a project or business plan to address an environmental challenge of their choosing. Not offered 2011- 2012. ENVS 22000. FROM FARM TO TABLE: UNDERSTANTING THE FOOD SYSTEM Food production and consumption interfaces with disciplines from biology and chemistry to political economy, sociology, and business management. The aim of this course is to introduce students to this analysis of the food system and get them thinking critically about where our food comes from, where it goes, and how to make the entire system more sustainable. Spring 2012. [HSS] ENVS 23000. SUSTAINABLE AGRICULTURE: THEORY AND PRACTICE Agroecology is the “science of sustainable agriculture.” It serves as the scientific basis for devising more natural, less environmentally harmful farming practices that build soil fertility and plant resilience while maintaining adequate production levels. The goal of this course is to introduce students to a broad suite of sustainable agriculture principles and practices and to investigate the scientific basis for those practices. Students will learn agroecology by actually practicing it in the field during lab sessions. Includes lab. (1.25 course credits) Fall 2011. [HSS] ENVS 31000. SUSTAINABLE DEVELOPMENT: PRINCIPLES AND PRACTICES This course will explore the intersection of development and sustainability. We will begin with a historical understanding of the idea of sustainable development, then shift to a more applied and experiential focus. In the latter portion of the course we will look at sustainable development from the point of view of meeting the basic needs of a population - food, water, shelter, energy - in a sustainable manner. The experiential component will feature a number of hands-on activities and class projects. Not offered 2011-2012. ENVS 40700, 40800. INTERNSHIP In consultation with a faculty member associated with the program, students may arrange academic credit for supervised work in an applied setting that is relevant to topics in environmental studies. Placement may be on- or off-campus. Examples of on-campus internships might include work through the physical plant, exploring energy use on campus; through campus grounds, investigating aspects of campus plantings and land use; or through campus dining services, examining ways to promote local foods, reduce energy use, reduce food waste, or develop a composting program. In addition to the work, an internship will include an appropriate set of academic readings and written assignments, developed in consultation with the supervising faculty member, that will allow the student to reflect critically on his or her experience. (.5 - 1.0 credit) S/NC course. Prerequisite: prior consultation with the faculty member and permission of the chair of Environmental Studies. Annually. CROSS-LISTED COURSES ACCEPTED FOR ENVIRONMENTAL STUDIES CREDIT NATURAL SCIENCE BIOLOGY BIOL 10000. TOPICS IN BIOLOGY (Human Ecology) [MNS] BIOL 20000. FOUNDATIONS OF BIOLOGY [MNS] Film Studies 104 BIOL 20200. GATEWAY TO ECOLOGY, EVOLUTION, AND ORGANISMAL BIOLOGY [W, Q, MNS] BIOL 35000. POPULATION AND COMMUNITY ECOLOGY BIOL 35200. BEHAVIORAL ECOLOGY BIOL 35600. CONSERVATION BIOLOGY CHEMISTRY CHEM 10103 CHEMISTRY OF THE ENVIRONMENT [Q, MNS] CHEM 21600. ENVIRONMENTAL CHEMISTRY GEOLOGY GEOL 10500. GEOLOGY OF NATURAL HAZARDS [MNS] GEOL 11000. ENVIRONMENTAL GEOLOGY [MNS] GEOL 21000. CLIMATE CHANGE [Q] GEOL 22000. INTRODUCTION TO GEOGRAPHIC INFORMATION SYSTEMS (GIS) SOCIAL SCIENCE ECONOMICS ECON 24000. ENVIRONMENTAL AND NATURAL RESOURCE ECONOMICS [HSS] HISTORY HIST 30141. PROBLEMS IN HISTORY (Global Environmental History, American Environmental History) [HSS] POLITICAL SCIENCE PSCI 20200. ENVIRONMENTAL POLICY [HSS] PSYCHOLOGY PSYC 22500. ENVIRONMENTAL PSYCHOLOGY [HSS] SOCIOLOGY/ANTHROPOLOGY SOCI 20300. ENVIRONMENTAL SOCIOLOGY [HSS] HUMANITIES ENGLISH ENGL 16004. NON-FICTIONAL WRITING (Nature and Environmental Writing) [W, AH] ENGL 24022. GREEN ROMANTICISM [AH] PHILOSOPHY PHIL 21600. ENVIRONMENTAL ETHICS [AH] RELIGIOUS STUDIES RELS 26900-26929. TOPICS IN THEORIES AND ISSUES IN THE STUDY OF RELIGION (Environment) [R] THEATRE AND DANCE THTD 44304. ADVANCED SEMINAR IN THE VISUAL TEXT: GREEN THEATRE FILM STUDIES CURRICULUM COMMITTEE: Mareike Herrmann (German), Chair Brian Cope (Spanish) Carolyn Durham (French) Dale Seeds (Theatre) John Siewert (Art) The minor in Film Studies focuses on film analysis, criticism, theory, history and the cinematographic elements and techniques that translate human thought to the screen. Students learn to read, interpret, and construct complex visual and verbal images that reflect a wide range of cinematic works, styles, and movements in order to develop a critical understanding of film’s significance as an art form, a means of literary and cultural expression, and a tool for both entertainment and social change. Film Studies 105 One of the distinctive features of the interdepartmental minor is the wide range of course offerings on films from different countries and cultures. The program of study also presumes that knowledge in constructing visual and verbal imagery is integral to interpreting it. Accordingly, a component of the minor focuses on aspects of film/video production or on media studies. The Film Studies minor can play a critical supporting role to the focus of a major and the Senior Independent Study. Minor in Film Studies Consists of six courses: • CMLT 23600 • Two courses from Category I • One course from Category II • Two elective Film Studies courses, with at most one course from Category III Special Notes • Only grades of C- or better are accepted for the minor. FILM STUDIES COURSES CMLT 23600. COMPARATIVE FILM STUDIES [C, AH] Category I: FILM CRITICISM, HISTORY, AND THEORY CHIN 22300. CHINESE CINEMA AS TRANSLATION OF CULTURES (in English) [C, AH] ENGL 24000. THE AMERICAN FILM [AH] FREN 25300, 25301. TOPICS IN FRANCOPHONE LITERATURE AND SOCIETY: FRANCOPHONE FILM (in English) [C, AH] GRMN 22800-22804. TOPICS IN GERMAN SOCIETY AND CULTURE: GERMAN FILM AND SOCIETY (in English) [C] RUSS 22000. RUSSIAN CULTURE THROUGH FILM (in English) [C, AH] SPAN 28000. HISPANIC FILM (in English) [C, AH] THTD 24900. INDIGENOUS FILM [C, AH] Category II: FILM WRITING, PRODUCTION, MEDIA STUDIES ARTD 15900. INTRODUCTION TO PHOTOGRAPHY [AH] COMM 33200. VISUAL COMMUNICATION THTD 30100. TOPICS IN THE WRITTEN TEXT (as appropriate to Film Studies; see Chair) [W, AH] THTD 30200. TOPICS IN THE VISUAL TEXT (as appropriate to Film Studies; see Chair) [AH] THTD 30300. TOPICS IN THE PHYSICAL TEXT (as appropriate to Film Studies; see Chair) [AH] Category III: STUDY OF FILM AS SIGNIFICANT COURSE COMPONENT AFST 24400. CINEMA OF AFRICA AND THE AFRICAN DIASPORA [C, AH] ARTD 36000. CONTEMPORARY ART CMLT 22200. CLASSICAL TRADITION IN MODERN DRAMA, FICTION, AND FILM [W, AH] COMM 23100. RADIO, TELEVISION, AND FILM IN AMERICA ENGL 21009. POSTCOLONIAL LITERATURE AND FILM [C, AH] ENGL 23004. LITERATURE OF THE COLD WAR [AH] ENGL 24002. NARRATIVE AND THE REAL WORLD [AH] ENGL 24018. NINETEENTH-CENTURY BRITISH NOVEL ON FILM [AH] FREN 32900. STUDIES IN THE TWENTIETH-CENTURY: FICTION AND FILM (in French) [C, AH] RELS 26400. RELIGION AND FILM [C, R, AH] THTD 24800. NATIVE AMERICAN PERFORMANCE [C, AH] 106 FRENCH Carolyn Durham, Chair Marion Duval Harry Gamble John Lytle Agathe Fontaine (French Language Assistant) The French Department offers a program of courses with three broad objectives: (1) to develop high proficiency in the French language; (2) to inculcate a knowledge and appreciation of the history, literature, and cultures of French-speaking countries; and (3) to develop critical thinking and analytical skills. In recent years, graduates who have majored or minored in French have gone on to further studies or employment in a variety of areas, including teaching, international careers, publishing, translation, the travel industry, business, banking, and law. All courses in the department are taught in French, with the exception of FREN 25300 and FREN 31900. Major in French Consists of eleven courses: • FREN 21600 • FREN 21800 • FREN 22000 • FREN 22400 • Four elective French courses at the 200-level or above • Junior Independent Study: FREN 40100 • Senior Independent Study: FREN 45100 • Senior Independent Study: FREN 45200 Minor in French Consists of six courses: • FREN 21600 • FREN 22000 • FREN 22400 • Three elective French courses at the 200-level or above Special Notes • Study Abroad: To assure linguistic competence and familiarity with Fran - cophone culture, the department strongly encourages study off-campus and will provide guidance on choosing a study abroad program. All majors should plan to spend at least a semester, and preferably a full academic year, in a French-speaking country. A limited number of scholarships to assist with the travel expenses of students studying in a French-speaking country are available from the McSweeney Fund; such scholarships are awarded on the basis of both need and merit. Application information may be obtained from the department chairperson. • French House: Students with a strong interest in French are encouraged to apply to live in the French House, located in Luce Hall. The French House organizes weekly cultural and social events, which allow students to improve their French and broaden their knowledge of the Francophone world. A language assistant from a Francophone country lives in the French House and helps organize these activities. Applications can be obtained from the department chair and are normally due towards the beginning of the spring semester. French 107 • Double majors: The department supports double majors and will work closely with students to design an appropriate program of study. Students in recent years have combined their French major with majors in Art and Art History, Comparative Literature, English, Economics, History, International Relations, Political Science, Religious Studies, Sociology and Anthropology, and Spanish. • FREN 40100 will normally be taken during the junior year or, if the student plans to spend the junior year off campus, in the spring of the sophomore year. With approval, FREN 40100 can be completed off-campus. • Since the major program should provide continuity in the study of French, at least one course in French should be taken each semester of the junior year (for students on campus) and the senior year, in addition to Independent Study. • No more than one French course taught in English may count toward the major. • Students who minor in French may take up to three of the required six courses off-campus. • Students who major in French are not permitted to take courses in the department on an S/NC basis. • Only grades of C- or better are accepted for the major or minor. • Teaching Licensure: Students interested in pursuing a career in elementary or secondary school teaching must complete the requirements for Multiage Licensure in French (grades pre-kindergarten through 12, ages 3-21) as stipulated here and the general education requirements for Multiage Licensure as stipulated by the Department of Education. A minimum of 44 semester hours (11 courses) in French must be completed, beginning with FREN 20100 and including the following: FREN 21600, 21800, 22000, 22400, 31900, 40100, 45100, and 45200. In the case of students who are seeking to be certified in French as their second area of foreign language licensure, FREN 10100 and FREN 10200 will be counted toward licensure in French. • Minor in International Business Economics. Students interested in French as preparation for a career in international business or finance should consider the Interdisciplinary Minor in International Business Economics (see full description under Business Economics). • Advanced Placement: Students who receive a score of 4 or 5 on the CEEB Advanced Placement Examination may count this credit toward a major or minor in French. Students who have taken the Advanced Placement Examination are still required, regardless of the score received, to take the departmental placement exam at the College to determine the next appropriate course. • Students who wish to meet the College’s language requirement in French by taking summer school courses or by participating in an off-campus program whose courses are fewer than four semester-hours are required to consult the chairperson of the Department of French prior to such study and will be required to take the departmental placement exam to demonstrate proficiency through the FREN 10100 or 10200 level after the completion of such courses; a successful performance on the placement exam is required for transfer credit to count toward the language requirement. • If a student registers for and completes a course in French below the level at which the French placement exam placed him or her, that student will not receive credit toward graduation for that course, unless he or she has obtained the permission of the instructor of the course into which the student placed and the permission of the Department Chair. French 108 FRENCH COURSES FREN 10100. LEVEL I BEGINNING FRENCH An introduction to understanding, speaking, reading, and writing French. Acquisition of basic structure, conversational practice, short readings, and compositions. Cultural content. Extensive use of authentic video and audio materials. Annually. Fall. FREN 10200. LEVEL II BEGINNING FRENCH Continuation of FREN 10100 with increased emphasis on conversational, reading, and writing skills. Prerequisite: FREN 10100 or placement. Annually. Spring. FREN 20100. INTERMEDIATE FRENCH FOR SPOKEN COMPREHENSION AND COMMUNICATION Intensive practice in conversational French. Course includes cultural explorations in the Francophone world and work with grammar, vocabulary, and appropriate texts. Prerequisite: FREN 10200 or equivalent. Annually. Fall. [C] FREN 20300. INTERMEDIATE FRENCH FOR WRITTEN COMPREHENSION AND COMMUNICATION Intensive practice in writing and reading, with a focus on writing strategies, the writing process, and different kinds of writing. Continued study of French vocabulary and grammar. Prerequisite: FREN 10200 or equivalent. Annually. Spring. [W†] FREN 21600. ADVANCED FRENCH Practice in listening, speaking, reading, and writing at an advanced level. Review of linguistic structure focusing on questions of usage and style. Extensive use of multi-media resources. Intensive and extensive reading on multiple topics. Prerequisite: FREN 20300 or equivalent. Annually. Fall. [C, W] FREN 21800. FRENCH PHONOLOGY Introduction to phonetics and phonology of the French language. Analysis of spoken French, including phonetic transcription. Extensive use of audio materials. Oral drill to improve pronunciation and diction. Prerequisite: FREN 21600 or equivalent. Spring 2012. [AH] FREN 22000. INTRODUCTION TO FRANCOPHONE TEXTS (Comparative Literature) Introduction to textual analysis through readings in genres representative of seventeenth to twentieth centuries. Intensive study of selected passages to develop a critical approach. Practice in speaking and writing on literature. Prerequisite: FREN 21600 or permission of the instructor. Annually. Fall. [C, W, AH] FREN 22400. STUDIES IN FRANCOPHONE CULTURE Topic changes from year to year. Recent topics have included: contemporary France; Franco-American cultural encounters, women in France; the French educational system. May be repeated once for credit. Prerequisite: FREN 21600 or permission of the instructor. Annually. Spring. [C, AH] FREN 22600. THEATRE PRODUCTION A practically-oriented course focusing on the study and presentation of a play. S/NC course. Prerequisite: FREN 20100 or permission of the instructor. Not offered 2011-2012. FREN 23000-23004. TOPICS IN FRANCOPHONE LITERATURE AND SOCIETY (Comparative Literature) A special topics course. Can be taught, for example, as Francophone poetry, Quebec studies, or as a course on Francophone North Africa. Prerequisite: FREN 21600 or equivalent. Spring 2012. FREN 23500. LITERATURE AND CULTURE OF FRANCOPHONE AFRICA (Africana Studies, Comparative Literature) This course explores the fictional works of major Francophone writers such as Mariama Bâ, Mongo Beti, Fatou Diome, Ahmadou Kourouma, and Camara Laye. Considerable attention is given to the historical and cultural contexts in which these novels were produced. Students will also approach the history and culture of Francophone Africa through a selection of films. Prerequisite: FREN 22000. Not offered 2011-2012. [C] FREN 25300-25301. TOPICS IN FRANCOPHONE LITERATURE AND SOCIETY: FRANCOPHONE FILM (Comparative Literature, Film Studies) Taught in English. A study of Francophone film from its origins in the work of Méliès and the Lumière Brothers through film noir, poetic realism, and the New Wave to the contemporary period. Various cinematic techniques and theories are illustrated by texts from such major French auteurs as Renoir, Cocteau, Tavernier, Buñuel, Resnais, Carné, Truffaut, Godard, Varda, and others. Some attention to selected examples of Quebecois and/or Senegalais film. Alternate years. Not offered 2011-2012. [C, AH] French 109 FREN 31600. TRANSLATION AND STYLISTICS An advanced language course which studies linguistic and cultural differences between France and the United States through translation. Strongly recommended for majors in preparation for Independent Study. Prerequisite: FREN 21600 or equivalent. Not offered 2011-2012. [C] FREN 31900. APPLIED LINGUISTICS Taught in English. Linguistic theory and its application in the teaching of foreign languages. Offered jointly by the departments of French, German, and Spanish. Individual practice for the students of each language. Required for certification of prospective teachers of French. Prerequisite: FREN 21600. Alternate years. Not offered 2011-2012. FREN 32000. STUDIES IN THE MIDDLE AGES AND THE RENAISSANCE (Comparative Literature) An examination of works that reflect the evolution of values and institutions from the twelfth century through the sixteenth. Includes an introduction to Old French. Authors studied include Rabelais, DuBéllay, Ronsard, and Montaigne. Prerequisite: FREN 22000. Not offered 2011-2012. FREN 32200. STUDIES IN THE SEVENTEENTH CENTURY (Comparative Literature) An examination of works that reflect the crisis of values in the Age of Louis XIV. Authors studied include Mme. de Lafayette, Corneille, Racine, and Molière. Prerequisite: FREN 22000. Not offered 2011-2012. FREN 32400. STUDIES IN THE EIGHTEENTH CENTURY (Comparative Literature) An examination of works that reflect the conflict between the individual and the community in the Age of Enlightenment. Often taught with a focus on women in eighteenth-century literature and society. Prerequisite: FREN 22000. Not offered 2011-2012. [C, AH] FREN 32800. STUDIES IN THE NINETEENTH CENTURY (Comparative Literature) An examination of works which portray bourgeois society and its materialistic values and the ways in which these values alienate the developing romantic hero. Narrative and descriptive techniques also studied. Authors studied include Balzac, Flaubert, Sand, Stendhal, and Zola. Prerequisite: FREN 22000. Not offered 2011-2012. FREN 32900. STUDIES IN THE TWENTIETH CENTURY (Comparative Literature, Film Studies) An examination of works that reflect the cultural, psychological, and literary dislocation of the twentieth century. Authors studied may include Camus, Colette, Beauvoir, Sartre, Gide, Duras, and Robbe-Grillet. Prerequisite: FREN 22000. Not offered 2011-2012. [C, AH] FREN 40100. JUNIOR INDEPENDENT STUDY Includes work on skills that are useful when doing research in Francophone language, civilization, and literature. Culminates in the completion of an independent project, often a major paper on a cultural or literary topic. FREN 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor. Prerequisite: FREN 40100. FREN 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the finished thesis or an equivalent project and an oral examination. Prerequisite: FREN 45100. FRENCH STUDY OFF-CAMPUS Programs endorsed by the department include: SWEET BRIAR JUNIOR YEAR IN PARIS OR NICE Semester or academic year program offering courses in a variety of disciplines both at the Sweet Briar center, and at universities and specialized schools in Paris. A limited number of internships in government or social agencies are available. INSTITUTE FOR THE INTERNATIONAL EDUCATION OF STUDENTS (IES) IN NANTES OR PARIS Semester or academic year program offering courses in a variety of disciplines both at the Institute and at universities and specialized schools in Paris. Possibility of teaching assistantships in English. Possibility of internships at the OECD and in businesses. WOOSTER IN BESANÇON A one-semester program of intensive language study at the Centre de Linguistique Appliquée of the University Geology 110 of Besançon. A Wooster graduate who teaches at the Centre serves as the College’s agent there. Prerequisite: FREN 10100 or equivalent. DICKINSON IN TOULOUSE Semester or academic year program offering courses in a variety of disciplines both at the Dickinson Study Center and at the universities and specialized schools in Toulouse. Possibility of internships in business, education, the arts, and applied sciences. ETUDE DES LANGUES AT THE UNIVERSITE LAVAL Semester or academic year in Quebec City for intermediate or advanced students. Additional endorsed programs allow students to study in Francophone Africa. See the website of the Office of Off-Campus Studies. GEOLOGY Greg Wiles, Chair Shelley Judge Meagen Pollock Mark Wilson The Department of Geology at The College of Wooster produces liberally educated scientists who are well-versed in scientific methodology and its application, who possess a thorough knowledge of fundamental geologic concepts, who take a creative approach to problem-solving, and who are able to express themselves with clarity, both orally and in writing. Geology is an interdisciplinary science. Geologists employ principles of physics, chemistry, and biology to understand Earth history and Earth processes. Geologists should be broadly educated in the natural sciences and have diverse field and laboratory experience with rocks and fossils, which is the primary goal of the Geology major at Wooster. Major in Geology: Consists of twelve courses: • One 100-level Geology course • CHEM 11000 • GEOL 20000 • GEOL 20800 • GEOL 25000 • GEOL 26000 • GEOL 30000 • GEOL 30800 • GEOL 31300 • Junior Independent Study: GEOL 40100 • Senior Independent Study: GEOL 45100 • Senior Independent Study: GEOL 45200 Minor in Geology: Consists of six courses: • GEOL 20000 • Five elective Geology courses, with no more than two courses at the 100-level Geology 111 Special Notes • The laboratory and classroom components are closely integrated in Geology lab courses and must therefore be taken concurrently. The course and laboratory grades will be identical and are based on performance in both components; the relative weights of the two components are stated in each course syllabus. • Geology majors who intend to make a career in geology are strongly urged to supplement their curriculum with at least one additional course in chemistry, two courses in physics, and two courses in calculus (or a combination of cal culus and computer science). Other relevant courses will depend upon the student’s particular interest in Geology. • S/NC courses are not permitted in the major department and in CHEM 11000. • Only grades of C- or better are accepted for the major or minor. GEOLOGY COURSES GEOL 10000. HISTORY OF LIFE (Archaeology) Origin and evolution of life, with emphasis on biologic innovations and crises in the context of Earth history. Three hours of lecture weekly. Annually. Spring 2012. [MNS] GEOL 10300. OCEANOGRAPHY Rocks, sediments, geophysics, structure, and history of ocean basins and their margins. An interdisciplinary examination of the oceans with emphasis on physical oceanography. Three hours of lecture weekly. Fall 2011. [MNS] GEOL 10500. GEOLOGY OF NATURAL HAZARDS (Archaeology, Environmental Studies) Survey of the geologic conditions, human and environmental impacts, and regulatory consequences of natural hazards and disasters. Course focus is on earthquakes, volcanoes, flooding, landslides, and destructive coastal processes. Three hours of lecture weekly. Annually. Spring 2012. [MNS] GEOL 11000. ENVIRONMENTAL GEOLOGY (Environmental Studies) An investigation of how human activities affect and are affected by physical Earth processes. Topics include an overview of Earth’s development; minerals and rocks; internal processes such as plate tectonics, earthquakes, and volcanoes; surface processes; natural resources; waste disposal; pollution and related topics. Three hours of lecture weekly. Fieldtrips. Annually. Fall 2011. [MNS] GEOL 20000. PROCESSES AND CONCEPTS OF GEOLOGY (Archaeology) Materials, structures and surface features of the Earth; geological processes and their effects through time; origin and evolution of Earth. Three hours of lecture and three hours of laboratory weekly. One-day fieldtrips. (1.25 course credits) Prerequisite: any 100-level Geology course. Annually. Spring 2012. [MNS] GEOL 20800. MINERALOGY (Archaeology) Introduction to crystallography; detailed study of mineral structure and occurrence. Three hours of lecture and three hours of laboratory weekly. (1.25 course credits) Prerequisite: any 100-level Geology course and CHEM 11000 (which can be taken concurrently). Annually. Not offered 2011-2012. [MNS] GEOL 21000. CLIMATE CHANGE (Archaeology, Environmental Studies) Analyses of the Earth’s ocean-atmosphere system and energy balance, Quaternary dating methods and techniques of reconstructing past climates are outlined. Students will work with paleoclimate data sets from ocean cores, ice cores, tree-rings, lake cores, and corals. Labs include computer modeling, statistical analysis of time series, and various projects. Three hours of lecture and three hours of laboratory weekly. Fieldtrips. (1.25 course credits) Prerequisite: any 100-level Geology course. Annually. Fall 2011. [Q] GEOL 22000. INTRODUCTION TO GEOGRAPHIC INFORMATION SYSTEMS (GIS) (Archaeology, Environmental Studies) A lab-intensive introduction to the basic concepts in computer-based GIS. The course is designed to provide interested students a hands-on approach to spatial database design and analysis. Students will depict and evaluate spatial data to produce cartographic results in order to solve problems in a variety of disciplines, with emphasis on the natural sciences. The primary platform used will be ArcMap by ESRI and Microsoft Excel, Geology 112 but the techniques learned are applicable to other software packages. Three hours of lecture weekly. Annually. Fall 2011. GEOL 25000. INVERTEBRATE PALEONTOLOGY Identification, systematics, evolution, and paleoecologic analysis of invertebrate fossil groups. Three hours of lecture and three hours of laboratory weekly. Fieldtrips. (1.25 course credits) Prerequisite: any 100-level Geology course or BIOL 20200. Annually. Fall 2011. [W, MNS] GEOL 26000. SEDIMENTOLOGY AND STRATIGRAPHY (Archaeology) Physical and biological methods for the analysis of sedimentary environments and processes. Investigating the distribution of sedimentary rock units in space and time. Three hours of lecture and three hours of laboratory weekly. Fieldtrips. (1.25 course credits) Prerequisite: any 100-level Geology course. Annually. Spring 2012. [W, MNS] GEOL 30000. GEOMORPHOLOGY AND HYDROGEOLOGY (Archaeology) A study of the classification, genesis, and evolution of the diverse landforms which make up the surface configuration of the Earth. Relationship of soils, surficial materials and landforms to rocks, structures, climate, processes, and time. The hydrologic cycle and surface water processes, geologic settings of groundwater, groundwater flow to wells, and water quality. Three hours of lecture and three hours of laboratory weekly. Fieldtrips. (1.25 course credits) Prerequisite: GEOL 20000. Annually. Spring 2012. GEOL 30800. IGNEOUS AND METAMORPHIC PETROLOGY (Archaeology) Introduction to petrography and petrology of igneous and metamorphic rocks. Integration of theoretical petrology, geochemistry, and petrography into an understanding of the petrogenesis of rock systems. Three hours of lecture and three hours of laboratory weekly. (1.25 course credits) Prerequisite: GEOL 20800. Annually. Not offered 2011-2012. GEOL 31300. STRUCTURAL GEOLOGY Introduction to the processes of deformation and geometry of deformed rocks. Examination of rock deformation through analysis of structures at both microscopic and outcrop scales with emphasis on descriptive geometry, map interpretation, and cross-section construction methods. Three hours of lecture and three hours of laboratory weekly. Fieldtrips. (1.25 course credits) Prerequisite: GEOL 20000. Annually. Spring 2012. [Q] GEOL 35000. SPECIAL TOPICS IN GEOLOGY To allow students with significant geological background to explore interdisciplinary topics in further detail. Planetary Geology, Geochemistry, Geophysics, Desert Geology and others offered when sufficient student interest is shown. Prerequisite: GEOL 20000 and others, depending on topic offered. Spring 2012. [W] BIOL 36000. EVOLUTION GEOL 40000. TUTORIAL Advanced library, field, and laboratory research problems in geology. (.5 – 1 course credit) Prerequisite: the approval of both the supervising faculty member and the chairperson is required prior to registration. GEOL 40100. JUNIOR INDEPENDENT STUDY Concepts and techniques of geologic research culminating in a Junior I.S. thesis project. Prerequisite: GEOL 20000. Annually. Fall or Spring. GEOL 45100. INDEPENDENT STUDY THESIS – SEMESTER ONE An original geological investigation is required. An oral presentation is given to the department. Prerequisite: GEOL 40100. GEOL 45200. INDEPENDENT STUDY THESIS – SEMESTER TWO An original geological investigation is required. An oral presentation is given to the department. Projects result in a thesis and an oral defense. Prerequisite: GEOL 45100. GEOLOGY SEMINAR All senior Geology majors pursuing thesis research attend and participate in a weekly departmental seminar. Each student presents a synopsis of her or his research and leads a discussion. This presentation provides experience in oral communication and criticism in a scientific context. All junior and senior Geology majors are required to attend these seminars; other majors are encouraged to attend. Thursdays, 11:00 a.m. to 12:00 noon, Scovel 205. 113 GERMAN STUDIES Beth Ann Muellner, Chair Mareike Herrmann Anna-Lena Lock (German Language Assistant) The German Studies Department offers a program of courses that help students to attain a high level of proficiency in German, to understand and appreciate the history, literature, and cultures of German-speaking countries, and to develop critical thinking and analytical skills. Beyond the acquisition of speaking facility, language and culture courses can foster better understanding of how language both reflects and shapes consciousness of the world. In the best liberal arts tradition, language and culture study enhances our ability to deal with ambiguity and cultural pluralism. The German Studies Department offers a major and minor in German Studies. A major in German can lead to careers in teaching, research or translation work, foreign service, international business, or work in international service organizations. A minor in German can enhance one’s preparation for professions in communications, journalism, the natural and social sciences, or any work involving trans-cultural communication. In recent years, graduates who have majored in German have entered graduate programs in German Studies or embarked on careers in international business, publishing, teaching, and the sciences. Many German majors and minors have earned Fulbright teaching awards abroad in the year after graduation. In considering a major or minor in German Studies, students should consult early in the first year or sophomore year with a member of the department about how best to plan meaningful sequences of courses, ideally including at least a semester of study abroad. The curriculum as described below is intended to expose students at the intermediate level to varieties of spoken and written styles; to encourage active development of one’s written and spoken facility with German in a broad range of topic areas; to exercise skills in intercultural thinking and communication; to introduce students to the methods and questions central to the study of German literature and cultural history, and to foster critical inquiry into a number of specific areas prior to Independent Study. The German Studies major encompasses inquiries into literary, artistic, historical, and everyday cultural aspects of German-speaking areas. It includes the study of periods, genres, major themes of German culture, including film and literature. Major in German Studies Consists of eleven courses: • GRMN 20100 (see note below) • GRMN 20200 (see note below) • GRMN 25000 • GRMN 26000 (must be taken at the College of Wooster) • Two of the following courses: GRMN 30000, 32000, 33000, or 34000 • Two elective courses taken from German Studies Literature Courses (see note below) • Junior Independent Study: GRMN 40100 • Senior Independent Study: GRMN 45100 • Senior Independent Study: GRMN 45200 Minor in German Studies Consists of six courses: • GRMN 20100 (see note below) German Studies 114 • GRMN 20200 (see note below) • GRMN 25000 • GRMN 26000 (must be taken at the College of Wooster) • One 300-level course in German Studies • One of the following courses: GRMN 22700, 22800, 23000, or any 300-level course in German Studies Special Notes • Overseas Study: Majors in German Studies are required to spend a minimum of a summer, or ideally, the junior year in Germany, Austria, or German-speaking Switzerland to increase proficiency in the language and international perspective. (Consult the German Studies Department for information on the programs most suited to your interests and needs.) • If students place out of the intermediate-level courses (GRMN 20100, 20200), they have to make up the remaining courses with other classes in the German Studies Department or with transfer credits from abroad. • One of the following courses with substantial German content may be counted towards the German Studies major: ARTD 32200 The Age of the Witch-Hunts; ARTD 22200 Modern Art; CMLT 20000 Comparative Literary Theory; CMLT 24800 The Perils of Romanticism; GRMN 31900 Applied Linguistics; HIST 10121 Hitler and the Nazi State; HIST 10167 The Holocaust; HIST 20800 Europe 1890-1945; HIST 20900 Europe Since 1945; HIST 21000 Ideas that Shaped the Modern World: Intellectual History of Modern Europe; HIST 22500 Modern Germany; PHIL 26100 Continental Philosophy; RELS 25400 The Reformation. • Majors are encouraged to take additional courses in German Studies or in related disciplines as electives. • Minor in International Business Economics: Students who are interested in a fundamental preparation in international business or finance with a focus on German language and economic issues may choose a major in German Studies and a prescribed core of complementary courses. Interested students should consult with the chair of German Studies and the chair of Business Economics. • The Zertifikat Deutsch Als Fremdsprache and the Mittelstufenprüfung, administered by the Goethe Institute Centers in Germany and the U.S., are internationally recognized as certification of advanced skills in German. Students are encouraged to take the tests, usually after GRMN 25000 or equivalent, offered annually at Hiram College. • Teaching Licensure: To be certified by the State of Ohio for secondary teaching of German, a student will complete eight semester courses in German beginning at GRMN 20100 (or the equivalent as determined by placement exam). The eight courses must include GRMN 20200, 25000, 26000, 22700 or 22800, and 31900. Study abroad is highly recommended for prospective teachers. • Advanced Placement: Students who receive a score of 4 or 5 on the CEEB Advanced Placement Examination may count this credit toward a major or minor in German Studies. Students who have taken the Advanced Placement Examination are still required, regardless of the score received, to take the departmental placement exam at the College to determine the next appropriate course. • German House: Students have the opportunity to take up residence in the German House, a suite in Luce Hall that houses students along with a native Austrian assistant and serves as the focal point for most campus German language and cultural activities. Applications for residency in the German House can be obtained from the chair and are usually due early in the spring semester. German Studies 115 • One S/NC course may be included in the major. Normally the minimum grade equivalent to “Satisfactory” is C. Students considering graduate work in German are advised not to include S/NC work in the major. • Only grades of C- or better are accepted for the major or minor. GERMAN STUDIES COURSES GRMN 10100. BEGINNING GERMAN LEVEL I An introduction to understanding, speaking, reading, and writing German in a cultural context. Acquisition of basic structure, conversational practice, short readings, and compositions. Use of authentic video and audio materials. Four hours per week. Students with previous German must take the departmental placement test in order to register for GRMN 10100. See department chairperson. Annually. Fall. GRMN 10200. BEGINNING GERMAN LEVEL II Continuation of GRMN 10100 with increased emphasis on conversation, cultural material, and reading authentic texts, including two children’s books. For students who have had GRMN 10100 or equivalent training, to be determined by placement test. Four hours per week. Annually. Spring. INTERMEDIATE AND ADVANCED COURSES GRMN 20100. INTERMEDIATE GERMAN LEVEL I A skills-building course to follow GRMN 10200 or equivalent, to be determined by placement test. Emphasis on reading literary texts of moderate difficulty, improving proficiency in writing and speaking, and exposure to culture material. The German major and minor begin with GRMN 20100. Annually. Fall. [C] GRMN 20200. INTERMEDIATE GERMAN LEVEL II Current issues through the media. Advanced readings and discussion of contemporary life in the Germanspeaking countries as reflected in newspapers, magazines, television, and film. Required of majors and minors. Prerequisite: GRMN 20100 or equivalent. Annually. Spring. [C] GRMN 25000. ADVANCED GERMAN: TEXTS AND CONTEXTS Reading, discussion of, and writing about important texts (e.g. short stories, short novels, personal narratives, films) from the 20th century, presented in their socio-historical contexts. Students learn about major events of the 20th century. Special emphasis on developing students’ reading and formal conversation skills and on cultural literacy. Continued practice of complex grammar structures and systematic vocabulary building. Prerequisite: GRMN 20200. Annually. Fall. [C] GRMN 31900. APPLIED LINGUISTICS Taught in English. Linguistic theory and its application in the teaching of foreign languages. Offered jointly by the departments of French, German, and Spanish. Individual practice for the students of each language. Required for licensure of prospective teachers of German. Alternate years. Not offered 2011-2012. LITERATURE AND CULTURE COURSES (Conducted in German unless otherwise indicated) GRMN 22700. GERMAN LITERATURE IN TRANSLATION (some sections cross-listed with: Comparative Literature, Women’s, Gender, and Sexuality Studies) Taught in English. Selected readings from classical and contemporary German authors. Sample topics: German Literature East and West Since 1945; Contemporary German Literature by Women; Modern German Theater; Fairy Tales and Gender . Alternate years. Spring 2012. GRMN 22800. TOPICS IN GERMAN SOCIETY AND CULTURE (some sections cross-listed with: Comparative Literature, Film Studies, Women’s, Gender, and Sexuality Studies) Taught in English. Studies in German cultural history, varying in topic from year to year and often interdisciplinary in approach. Not offered 2011-2012. [C] GRMN 23000. THEATERPRAKTIKUM Dramatic readings and play production, in German. Ideal for students wishing to maintain and build speaking proficiency and self-confidence. No acting experience required. May be taken more than once, but only one of these may count toward the minimum eleven courses for the major or minor. Prerequisite: GRMN 20100 or permission of instructor. Alternate years. Not offered 2011-2012. German Studies 116 GRMN 26000. KULTURKUNDE: INTRODUCTION TO GERMAN STUDIES (Comparative Literature) A survey of the cultural history of the German-speaking world, with particular attention to the social matrix in which German cultural institutions function. An introduction to the methods and resources of German Studies as an interdisciplinary area of study. Must be taken at the College of Wooster. Prerequisite: GRMN 25000. Annually. Spring. [W, C, AH] CMLT 24800. THE PERILS OF ROMANTICISM: NINETEENTH CENTURY EUROPEAN LITERATURE (Taught in English) [C, AH] GRMN 30000. MAJOR EPOCHS OF GERMAN LITERATURE AND CULTURE (Comparative Literature) Each of five subcourses deals with a distinct period of German literature and culture marked by watershed events and characterized by certain concerns and issues which find significant expression in the literature of the period. Each course will focus on major literary works in a broad cultural context. Prerequisite: GRMN 26000. Not offered 2011-2012. GRMN 30000A. Faith, Love, and Reason: The Middle Ages to the Enlightenment GRMN 30000B. The Coming of Age of German Culture (1770-1830) GRMN 30000C. Poetry and Politics: Literature, Revolution and Nationalism (1830-1918) GRMN 30000D. The Weimar Republic and the Third Reich (1918-1945) GRMN 30000E. After the Holocaust (Post-1945) GRMN 32000. MAJOR AUTHORS IN GERMAN LITERATURE AND CULTURE (Comparative Literature) A seminar concentrating on one or more authors of the German-speaking world. Close readings of shorter and longer works in all genres: consideration of methods of criticism and interpretation, the authors’ reception and influence in various periods and across national boundaries; thematic comparisons among authors of different periods. Course topic varies from year to year. Examples: Kleist and Kafka, Büchner and Brecht; Goethe and Schiller; Christa Wolf and Sarah Kirsch; Keller and Fontane. May be taken more than once for credit in the major. Prerequisite: GRMN 26000, or permission of the instructor. Not offered 2011-2012. GRMN 33000. GENRES OF GERMAN LITERATURE AND CULTURE (Comparative Literature) A survey of literature of important genres (Novelle, ballad, lyric poetry, Bildungsroman, drama, short story, autobiography, etc.). While focusing attention on representative works, the course considers genres as cultural conventions, asking how the history of a culture is reflected in the directions taken by such cultural forms, and why particular genres have flourished at a specific time in history. May be taken more than once for credit in the major. Prerequisite: GRMN 26000 or permission of instructor. Not offered 2011-2012. [C] GRMN 34000. MAJOR THEMES IN GERMAN LITERATURE AND CULTURE (Comparative Literature) A study of dominant recurring themes that cross period and genre lines and are important to the German cultural tradition. Topics will vary from year to year — e.g., Travel and Migration; Images of Women; The Artist and Society; The German Middle Ages; Fiction, History, and Memory; Nature, Space, and Place. May be taken more than for credit in the major. Prerequisite: GRMN 26000 or permission of instructor. Fall 2011. [Depending on the topic, C, AH] GRMN 40000. TUTORIAL Individually supervised readings on a special topic. By prior arrangement with the department only. Prerequisite: GRMN 25000 or equivalent; the approval of both the supervising faculty member and the chairperson is required prior to registration. GRMN 40100. INTRODUCTION TO INDEPENDENT STUDY Bibliography and research methods in German, including the preparation of two shorter papers or one longer research paper. Normally taken Semester II of the junior year. If a Junior Year Abroad is planned, GRMN 40100 should be taken Semester II of the sophomore year. If a one-semester program abroad is planned, it should be Semester I so that GRMN 40100 can be taken Semester II. GRMN 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, a two-semester course in thesis preparation taken in the senior year, supervised by a departmental adviser and approved by oral examination by the department in the second semester. Prerequisite: GRMN 40100. GRMN 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: GRMN 45100. German Studies 117 GERMAN STUDY OFF-CAMPUS One option for fulfilling the one-year language requirement is to participate successfully in one of the programs described below. Students continuing beyond the 102- level are encouraged to ask a member of the department (at least three semesters in advance of scheduled study abroad) about summer, semester, and year-long programs available to advanced students. INSTITUTE FOR THE INTERNATIONAL EDUCATION OF STUDENTS (IES) IN FREIBURG AND BERLIN A one-semester or one-year program for juniors in good standing at the College. Students will take intensive language courses taught by IES instructors and a combination of IES tutorials and German-university-taught courses in a variety of disciplines as well as in German literature and history. Courses at Freiburg and at the Humboldt University in Berlin are conducted entirely in German and require a minimum proficiency of GRMN 250 or equivalent. Courses at Vienna are conducted mostly in English, and students with beginning German proficiency are usually eligible. WAYNE STATE UNIVERSITY JUNIOR YEAR IN MUNICH A year-long (or one-semester option) program for juniors in good standing at the college. Students will take an intensive language course offered by JYM staff and enroll directly at the prestigious Ludwig-MaximiliansUniversität München, which offers a vast selection of courses in 150 degree-granting areas. The oldest intercollegiate study abroad program in Germany, the JYM is especially recommended for German majors. It offers a special independent study tutorial course, which can be counted for Junior I.S. credit in German. MACALESTER SEMESTER IN BERLIN-VIENNA A spring semester program for juniors in good standing at the college. The Macalester German Study Abroad program is a unique six-month program based in Berlin and Vienna that provides students with the opportunity to gain high proficiency in German and to immerse themselves academically, culturally, and socially in both Germany and Austria. Students will spend two months studying intensive German at the Goethe Institut in Berlin, before heading for Vienna, where they spend four months taking two program-specific courses, and two courses at the University of Vienna. STUDY-TRAVEL SEMINAR A summer or one-semester program in German language and culture with a practical focus determined by the particular groups and institutions visited (theater, social organizations, hospitals, farms, etc.). Prerequisite: GRMN 20200 or equivalent. HEBREW (see RELIGIOUS STUDIES) 118 HISTORY Gregory Shaya, Chair (Fall 2011) Madonna Hettinger, Chair (Spring 2012) Kabria Baumgartner Monika Flaschka Joan Friedman David Gedalecia Katherine Holt Shannon King Sarah Mirza Peter Pozefsky Jeffrey Roche Hayden Schilling Ibra Sene History is one of the oldest fields of knowledge, but it has never been more relevant than in the fast-changing, interconnected world in which we live. The study of history is the foundation for a complex understanding of the world. It offers a rich view upon the developments that have shaped the society we live in; it helps us understand distant cultures; it provides a set of rigorous tools for understanding changes and continuities over time; it offers a high perspective to make sense of the tumult of current events. The study of history cultivates skills and habits of mind that are essential to a liberal arts education. Students of history will develop the ability to research complex topics, to analyze evidence, to assess conflicting interpretations, to convey ideas with clarity and persuasion, and to build strong arguments. History encourages a subtle understanding of difference. What is more, the study of history provides a set of deep pleasures. Vastly enlarging our experience, the study of the past is a profound source of personal meaning and collective identity. We believe the best way to study history is to do history. In their coursework, students will develop a wide knowledge of the past and a practical understanding of the skills of the historian, culminating in the year-long Senior Independent Study. The major in history is flexible, allowing students to design a course of study that fits their interests and builds upon work in other disciplines. Students are required to complete ten courses, including four courses at the 200-level or above, Junior Independent Study (HIST 40100) and the two-semester Senior Independent Study (HIST 45100-45200). Major in History Consists of ten courses: • Four courses at the 200-level or above • Three elective History courses • Junior Independent Study: HIST 40100 • Senior Independent Study: HIST 45100 • Senior Independent Study: HIST 45200 Minor in History Consists of six courses: • Four courses at the 200-level or above • Two elective History courses History 119 Special Notes • Majors and minors in history are strongly encouraged to complete The Craft of History (HIST 20100) in their sophomore year, after they’ve taken a first course in history at the College but before Junior I.S. Majors and minors are also strongly encouraged to complete one course at the 200-level or above in the history of a non-U.S. and non-European field and one course at the 200-level or above in the history of a society before 1800. • Advanced Placement: A student obtaining a score of 5 on one of the CEEB Advanced Placement Examination in history will receive two course credits in history; a student scoring 4 will receive one course credit. A student will receive a maximum of three course credits in history for any combination of Advanced Placement Examinations. Students may count these credits toward a major or minor in history. Students receiving Advanced Placement credit should consult with the department before registering for 100-level survey courses in the department. The Advanced Placement policy of the College is explained in the section on Admission. • Only grades of C- or better are accepted for the major or minor. HISTORY COURSES INTRODUCTORY TOPICS COURSES HIST 10100-10176. INTRODUCTION TO HISTORICAL STUDY (some sections cross-listed with: Africana Studies, Chinese Studies, East Asian Studies, International Relations, Latin American Studies, Russian Studies, Women’s, Gender, and Sexuality Studies) An introduction to the study of history through the examination of a specific historical theme. Topics frequently offered include: America in the Sixties, Crime & Punishment, The Civil Rights Movement, Hitler & the Nazi State, and Western Travelers to China. Class format includes lecture and discussion. May be repeated for credit as offerings vary. Scheduled for 2011–2012 HIST 10103. PERSONALITIES IN CHINESE HISTORY Fall 2011. [C, HSS] HIST 10121. HITLER & THE NAZI STATE Fall 2011. [HSS] HIST 10165. WEST AFRICA & THE AFRICAN-AMERICAN CONNECTION Fall 2011. [C, HSS] HIST 10173. THE SOCIAL HISTORY OF HIP HOP: RACE, CRIME, GENDER, AND POPULAR CULTURE Fall 2011. [C, HSS] HIST 10143. AMERICA IN THE SIXTIES Spring 2012. [HSS] HIST 10161. RUSSIA’S WORLD WAR II Spring 2012. [HSS] HIST 10176. THE HISTORY OF ISLAM Spring 2012. [C, HSS] INTRODUCTORY SURVEY COURSES HIST 10600. WESTERN CIVILIZATION TO 1600 A survey of the rise of western civilization to 1600. European history. Not offered 2011-2012. [HSS] HIST 10700. WESTERN CIVILIZATION SINCE 1600 (International Relations) The development of western civilization from 1600 to the present. European history. Spring 2012. [C, HSS] HIST 10800. AN INTRODUCTION TO GLOBAL HISTORY (International Relations) Global history examines the interactions between different cultures across the globe from ancient times to the present. These interactions range from trade, to warfare, to the exchange of ideas, technology and disease. More specifically, global history explores the ways that those interactions have changed over time, and the impact they have had on economics, society, culture, politics and the environment at the local level. The course will introduce students to Global history through readings in the historiography of the field and in selected topics. The course will also critique the phenomenon of globalization from a historical perspective. Fall 2011. [C, HSS] HIST 11000. THE UNITED STATES EXPERIENCE TO 1877 A survey of the development of United States society to 1877. Fall 2011. [HSS] History 120 HIST 11100. THE UNITED STATES EXPERIENCE SINCE 1877 A survey of United States history from 1877 to the present. Spring 2012. [HSS] HIST 11500. HISTORY OF BLACK AMERICA: FROM WEST AFRICAN ORIGINS TO THE PRESENT (Africana Studies, Education) This course covers the history of black Americans from their origins in West Africa to the present. Although this course is a survey, it will have a topical approach. Topics will include the following: West African origins, the southern slavery experience, Black Reconstruction, the Great Migration, the Civil Rights Movement, and the Black Power Movement. The current situation of black people is the result of this heroic and yet sometimes tragic history. This course will view the development of America from the black perspective, displaying a history that is not the traditional view of the United States. Spring 2012. [C, HSS] THE METHODS OF HISTORY HIST 20100-20105. THE CRAFT OF HISTORY (some sections cross-listed with International Relations) An introduction to the critical skills of the historian—including the analysis of primary sources, historiography, historical research and writing, and historical argument—through the study of a specific historical theme. Topics frequently offered include: The History of the News, Slavery in the Americas, Plagues in History, and the Harlem Renaissance. A writing-intensive course, the class is taught as a seminar. The course is strongly recommended for majors, but it is open to students from all departments and programs. It is normally taken in the sophomore year and before HIST 40100—Junior I.S. Prerequisite: one course in History or permission of instructor. May be repeated for credit with permission of the Department Chair. Scheduled for 2011–2012 HIST 20101. HISTORY OF THE NEWS Fall 2011. [W, HSS] HIST 20102. HISTORY OF SCHOOL IN AMERICA Fall 2011 and Spring 2012. [W, HSS] HIST 20103. PLAGUES IN HISTORY Spring 2012. [W, HSS] HIST 20104. LATIN AMERICA & THE UNITED STATES Spring 2012. [W, C, HSS] HIST 20201. HISTORY WORKSHOP (.25 credit) This course will provide a theoretical foundation and practical training in a historical methodology. Topics offered may include: Public History, Oral History, Documentary History, Cultural History, and Digital History. Prerequisite: one course in History or permission of instructor. May be repeated for credit as offerings vary. Scheduled for 2011–2012 HIST 20201. THE HISTORICAL DOCUMENTARY Fall 2011. HIST 29800. MAKING HISTORY (International Relations) Explores both the theoretical debates that shape current historical thinking and the methodological challenges of working with original historical materials. Topics include philosophies of history, the use of interdisciplinary methods in history, the influence of technological developments on historical research and writing, archival methods, and research design. Fall 2011. [HSS] THE UNITED STATES HIST 23700. THE UNITED STATES AND CHINA (Chinese Studies, East Asian Studies, International Relations) The historical development of relations between the United States and China from the late eighteenth century to the present day, as seen through diplomatic, economic, political, and intellectual contacts. Spring 2012. [C, HSS] HIST 23800. THE AMERICAN WEST This course examines the development of the American West as a recognized region over the past 500 years. It focuses on several primary themes: ideologies of expansion, ethnic conflict, environmental change, technology, politics, and myth. Moreover, the course will examine how shifting historical interpretations of the West (including those of novelists and filmmakers) have reflected contemporary society. Fall 2011. [HSS] HIST 23900. RECENT AMERICA: THE UNITED STATES SINCE 1945 An examination of selected themes and topics of importance in recent American history, such as the Cold War, the Vietnam War, political coalitions, Presidential leadership, the 1960s as a decade, and contemporary cultural and economic concerns. Not offered 2011–2012. [HSS] History 121 HIST 24300. MODERN AMERICAN THOUGHT Major ideas, intellectual movements, and cultural institutions with emphasis on the influence of economic change, science, and world upheaval in the formation of contemporary thought. Not offered 2011–2012. [HSS] HIST 24400. ISSUES IN EARLY AMERICAN SOCIAL HISTORY The development of American societies through the early nineteenth century, focusing on the family, national character, and economic and cultural institutions. Not offered 2011–2012. [HSS] HIST 24500. THE AMERICAN REVOLUTION AND THE CONSTITUTION Focuses on the ways in which the European, African American, and Native American cultures interacted to form both the context for and content of democracy in America, particularly in the ways men and women of all three races and all classes understood and participated in the Revolution and the shaping of the Constitution. Not offered 2011–2012. [HSS] HIST 24600. UNITED STATES URBAN HISTORY (Africana Studies) A study of the urbanization process from colonial settlements through the development of the modern metropolis. The course will focus on those forces that have shaped the modern American city. Not offered 2011–2012. [HSS] HIST 24700. WOMEN’S HISTORY IN THE UNITED STATES (Women’s, Gender, and Sexuality Studies) An exploration of women’s experience as it was limited by their roles as daughter, wife, and mother; how women used their roles to participate in the construction of American society and change the course of American history, emphasizing race, class, and gender. Not offered 2011–2012. [HSS] HIST 24900. INTELLECTUAL HISTORY OF BLACK AMERICA A basic survey of some of the leading black thinkers in American history. Spring 2012. [W, HSS] EUROPEAN HISTORY HIST 20400. GREEK CIVILIZATION (Archaeology, Classical Studies) A survey of the civilization of ancient Greece from the Bronze Age to the Hellenistic period, with concentration on the Classical period (490-340 B.C.). Readings in primary sources, especially the Greek historians, with particular attention to the problems of recording and interpreting historical data. Not offered 2011-2012. [HSS] HIST 20500. ROMAN CIVILIZATION (Archaeology, Classical Studies) A survey of the civilization of ancient Rome from the Iron Age to the age of Constantine, with concentration on the late Republic and early Empire (133 B.C. - A.D. 180). Readings in primary sources, especially the Roman historians, with particular attention to the problems of recording and interpreting historical data. Not offered 2011-2012. [HSS] HIST 20600. MEDIEVAL EUROPE, 500-1350 (Archaeology) Organized thematically, the course examines the political and economic development of Europe in the Middle Ages, including feudalism and manorialism, and their social and cultural underpinnings. Special attention will be given to the problem of the “invisible” people of the Middle Ages: peasants, women, and Jews. Fall 2011. [HSS] HIST 20700. RENAISSANCE AND REFORMATION EUROPE, 1350-1650 Examines the great intellectual and religious events of the fourteenth to the seventeenth centuries within their political and social contexts. In particular, the course will examine how the “new thought” of these centuries provided Europe with a new intellectual language for describing and evaluating the growth of absolutism and the conquest of the Americas. Spring 2012. [HSS] HIST 20800. EUROPE, 1890 TO 1945 (International Relations) An investigation into European politics, society, and culture from 1890 to 1945. Topics include: mass politics and their discontents, modernism in the arts, new theories of society and personality, European imperialism, the second industrial revolution and the rise of socialist parties, feminism, the First World War, the Russian Revolution, the Versailles Treaty, the rise of fascism, Stalin’s Russia, the Depression, the Spanish Civil War, the Nazi threat to Europe, the Second World War, and the Holocaust. Not offered 2011–2012. [C, HSS] HIST 20900. EUROPE SINCE 1945: FILM AND HISTORY (International Relations) This course examines politics, society and culture in Europe from the immediate aftermath of the devastation of the Second World War to the present. Topics include: the reconstruction of Europe, the Cold War, the dilem- History 122 ma of Americanization, the expansion of the social welfare state, decolonization and immigration, student protest, the radical right, (the challenges of) European integration, and more. A large part of our studies will be devoted to a consideration of how the larger political and social struggles of Europe have been refracted and interpreted in the art of cinema. Fall 2011. [C, HSS] HIST 21000. IDEAS THAT SHAPED THE MODERN WORLD: INTELLECTUAL HISTORY OF MODERN EUROPE An investigation of the central trends in European thought through readings and discussions of primary texts. Topics include the enlightenment, romanticism, liberalism, socialism, fascism, existentialism, and post-modernism. Not offered 2011–2012. [W, C, HSS] HIST 21100. EUROPE OF THE REVOLUTIONS (1789-1914) Europe from the French Revolution through the eve of the Russian Revolution. Topics include the French Revolution, Napoleon, the Industrial Revolution, the establishment of liberal regimes, the rise of revolutionary movements on the right and left (liberalism, socialism, nationalism), the Revolution of 1848, and imperialism. Not offered 2011–2012. [HSS] HIST 21200. PLAGUE IN THE TOWNS OF TUSCANY When the Black Death arrived in Europe in the middle of the fourteenth century, Tuscany’s advanced urban centers were hit first and hardest. Within the first two years of bubonic plague in Western Europe, such thriving commercial cities as Siena, Florence and Pisa, saw their populations cut in half. While these cities eventually recovered the experience of epidemic disease left its mark on the survivors. This course will explore the impact of the Black Death on the social, religious, and economic lives of these cities. By mapping the spread of the plague on location, we will consider how these cities responded with new public health measures and new interventions into the private and public lives of citizens. Not offered 2011–2012. [C, HSS] HIST 21400. MYSTICS, POPES AND PILGRIMS From the late twelfth to the late fourteenth century, western Christendom grew simultaneously in two very different directions. While the papacy became increasingly involved in temporal concerns, often competing with kings and emperors for earthly power, ordinary believers sought more personal means of engaging with their faith. In the cases of more extra-ordinary believers, mystics and pilgrims, extreme physical hardship and the sacrifice of worldly possessions was seen as an avenue toward salvation. This course will explore the nature of these alternative expressions of faith and examine how the popularity and influence of such famous mystics as Francis of Assisi and Catherine of Siena challenged the worldly aspirations of the hierarchy of the Church. Field trips to the Vatican, Assisi, the pilgrim route to Rome, and a working monastery will emphasize the role landscape and location played in the experience of popular religion. Not offered 2011–2012. [C, R, HSS] HIST 22000. TUDOR-STUART ENGLAND, 1485-1688 The emergence of the Tudor state, the English Renaissance and the Reformation: the Age of Elizabeth and overseas expansion, the early Stuarts and the struggle over the constitution, parliamentary politics and the Civil War, Cromwell and the Interregnum, Restoration politics and culture, the Glorious Revolution. Spring 2012. [HSS] HIST 22100. MODERN BRITAIN The Hanoverian Succession, rise of cabinet and party politics, the structure of oligarchy, the Trans-Atlantic Revolutions, the Industrial Revolution, the reform movements, Victorian prosperity, the rise of Labor, the World Wars, the rise of the Welfare State, decolonization, and the crisis of Europe. Fall 2011. [HSS] HIST 22200. THE MAKING OF INDUSTRIAL SOCIETY: BRITAIN AND EUROPE, 1760-1900 A comparative study of Britain and Europe from the mid-eighteenth through the end of the nineteenth centuries. Topics covered include the origins of the Industrial Revolution in England and its expansion in Britain and Western Europe, technological expansion, the transformation of rural and urban communities, workplace organizations, the division of labor, popular protest and trade unionism. Not offered 2011–2012. [HSS] HIST 22300. MODERN FRANCE A survey of French politics, society, and culture from the mid-nineteenth century to the present. Topics include: the revolutionary tradition and the revolutions of 1848, Napoleon III and the Second Empire, consumer culture, the Franco-Prussian War, the Paris Commune, peasants and workers, the belle époque and the Dreyfus Affair, the First World War, avant-garde culture, the crises of the interwar era, Vichy France, the wars of decolonization, May 1968, Immigration. Not offered 2011-2012. [C, HSS] History 123 HIST 22500. MODERN GERMANY An examination of continuity and change in German political culture from the mid-nineteenth century to the present. Topics emphasized are imperial Germany, the two World Wars, Weimar and the rise of Hitler, Nazi culture, post-war trends, and reunification. Not offered 2011-2012. [HSS] HIST 23000. RUSSIA TO 1900 (Russian Studies) The rise and fall of the Kiev State, the origins and expansion of Muscovy, and the Tsarist empire. Emphasis on nineteenth century intellectual history. Fall 2011. [C, HSS] HIST 23300. RUSSIA SINCE 1900 (Russian Studies) Modern Russia, focusing on the Bolshevik Revolution, the Stalin era, World War II, the fall of the USSR and the rise of the new Russia under Boris Yeltsin and Vladimir Putin. Not offered 2011–2012. [C, HSS] AFRICAN, ASIAN, JEWISH, LATIN AMERICAN & MIDDLE EASTERN HISTORY HIST 21500. COLONIAL LATIN AMERICA (Latin American Studies) Latin American history from the pre-Columbian period to the 1830s. The course will emphasize the clash between European colonizers and indigenous populations, the development of Spanish and Portuguese colonial institutions and culture in America, and the overthrow of colonial rule in the early years of the nineteenth century. Fall 2011. [C, HSS] HIST 21600. MODERN LATIN AMERICA (International Relations, Latin American Studies) Latin American history from the 1830s to the present. The course will emphasize the difficult problems encountered by Latin American nations forced to face the demands of the modern world with political, economic, and social institutions developed in a colonial past. Not offered 2011–2012. [C, HSS] HIST 22700. THE MODERN MIDDLE EAST (International Relations) Emphasis on the heritage of religious unity, the political tradition of universal empire, the contrast between cultural unity and ethnic division, the special role of cities, the ecological constants, and the heritage of imperialism. Fall 2011. [C, HSS] HIST 22800. ISRAEL/PALESTINE: HISTORIES IN CONFLICT (International Relations) The history of the current conflict from the late 19th century down to the immediate present. Emphasis will be on understanding Israeli and Palestinian national identities; the parties’ incompatible interpretations of history and their role in perpetuating the conflict; and the specific terms of a possible solution to the conflict. Students will be registered for this course upon being accepted to the 12-day spring break “Wooster in Israel/Palestine” study trip. Spring 2012. [C, HSS] HIST 23100. AFRICA BEFORE 1800 (Africana Studies) From early antiquity to the late 16th century, Africa and Africans have been key players in world affairs. Ancient Egypt, Kush, Aksum, Ancient Zimbabwe, the west African empires of Ghana, Mali, Songhai, and Asante, as well as the state of Kongo in central Africa, the various Muslim dynasties in North Africa, and the Swahili citystates on the Indian Ocean coast, to name but a few examples, were the centers of this fascinating historical development. From the 16th century, the Atlantic slave trade, which lasted for at least three hundred years, destroyed African political, social, and economic institutions that sustained the continent on the world scene up to that time. As a consequence of that, this trade paved the way to the colonization of almost every single corner of Africa by European powers, beginning in the nineteenth century. In this course we will be exploring the various ways in which these developments have been shaping African societies, politics, and cultures over this long period of time. Fall 2011. [C, HSS] HIST 23200. AFRICA SINCE 1800 (Africana Studies, International Relations) With the official abolition of the Atlantic Slave Trade in the early 19th century, the encounter between Africa and Europe took a new and dramatic turn, with the beginning of the “legitimate trade.” This course will investigate how this change paved the way to the conquest and colonization of most of the continent by countries such as Belgium, France, Germany, Great Britain, and Portugal. We are also going to examine the important role played by Africans during the two World Wars, the severe impact of the Great Depression on them, and the origins of the nationalist movement that led to the end of colonialism in the 1960s. We will then turn to the ways in which the combined effects of the Cold War, neocolonialism, and the failure of many of the first postcolonial leaders created a deep sentiment of disillusionment among millions of Africans and ushered into a tumultuous period that literally engulfed the continent from the early 1970s to the late 1980s. Starting in the 1990s, strong civil soci- History 124 ety groups began to emerge and, against all odds in Africa and beyond, pushed forcefully for Africans to define their own place in the world. Spring 2012. [C, HSS] HIST 23400. TRADITIONAL CHINA (Archaeology, Chinese Studies, East Asian Studies) Chinese civilization, thought, and institutions from earliest times to 1644: the development of the imperial system, the Buddhist influx, the rise of gentry society, foreign invasions, and late empire. Not offered 2011–2012. [C, HSS] HIST 23500. MODERN CHINA (Chinese Studies, East Asian Studies) Chinese history from 1644 to the present: the modernization of traditional institutions in response to the foreign challenge in the nineteenth and twentieth centuries; rebellion, reform, nationalism, and communism as components of a Chinese revolution in process. Fall 2011. [C, HSS] HIST 23600. MODERN JAPAN (East Asian Studies) Japanese history from the nineteenth century to the present: the decline of feudal society and the Western impact, Meiji transformation and growth as a world power, militaristic expansion and the Second World War, post-war recovery, and industrial development in the contemporary world. Not offered 2011–2012. [C, HSS] HIST 24000. HISTORY OF THE JEWS This course spans three millennia, from antiquity to 1948. It breaks the broad outline of Jewish civilization into these areas: the origins and early history of the nation and religion of Israel; the transformation of the Jews into a diaspora people and the emergence of classical/rabbinic Judaism; Jewish existence as a tolerated minority under Christian and Muslim rule and the salient cultural characteristics of Jewish life in each domain; the redefinition of the geographical, communal, and religious parameters of Jewish life as the result of expulsions and persecutions in the early modern period; the fragmentation of Jewish identity in the modern period; and the enormous upheavals in Jewish life of the twentieth century: mass migrations, the Holocaust, and the establishment of the State of Israel. Two themes provide the threads of continuity throughout this chronological narrative: Jewish culture and forms of group life, and political, social, and cultural interaction with others. Not offered 2011–2012. [C, HSS] UPPER-LEVEL TOPICS COURSE HIST 27500-27502. STUDIES IN HISTORY (some sections cross-listed with International Relations) An advanced course devoted to a specific historical topic. Topics regularly offered include: Environmental History, Black Social Movements, The City in History, The Plague, Islam & Africa, Women in Latin America. Format includes lecture and discussion. Prerequisite: one course in History or permission of instructor. May be repeated for credit as offerings vary. Scheduled for 2011–2012 HIST 27502. HISTORY OF BRAZIL Spring 2012. [C, HSS] JUNIOR-SENIOR SEMINAR HIST 30100-30142. HISTORY COLLOQUIUM (some sections cross-listed with: Environmental Studies, International Relations, Latin American Studies, Russian Studies, Women’s, Gender, and Sexuality Studies) A reading-intensive seminar, focusing on a particular historical problem or field. Normally, this course is only open to Juniors and Seniors. Prerequisite: one course in History or permission of instructor. May be repeated for credit as offerings vary. Scheduled for 2011–2012 HIST 30142. AFRICAN-AMERICAN WOMEN’S HISTORY Spring 2012. [C, HSS] INDEPENDENT STUDY & TUTORIAL HIST 40000. TUTORIAL A one-semester tutorial that explores a specialized field of study. Specific readings and assignments are worked out by the student and the supervising faculty member together. Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. HIST 40100. JUNIOR INDEPENDENT STUDY A one-semester tutorial that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. Fall and Spring. Interdepartmental Courses 125 HIST 45100. SENIOR INDEPENDENT STUDY ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and culminating in a thesis and an oral examination in the second semester. Prerequisite: HIST 40100. Fall and Spring. HIST 45200. SENIOR INDEPENDENT STUDY TWO The second semester of the Senior Independent Study project, culminating in the thesis and an oral examination. Prerequisite: HIST 45100. Fall and Spring. INTERDEPARTMENTAL COURSES FIRST-YEAR SEMINAR First-Year Seminar provides a unique intellectual opportunity for faculty and firstyear students to participate in a small, discussion-oriented, multidisciplinary course. The course introduces students to critical thinking and other academic skills that will be needed in subsequent courses, including Junior and Senior Independent Study. Students are expected to develop their abilities in writing, interpreting complex texts, constructing an argument, supporting the argument with evidence, and defending the argument orally. The course also requires students to appreciate and critique multiple perspectives, including their own. Students normally will complete the First-Year Seminar in Critical Inquiry in their first semester. Each year approximately 32 sections of First-Year Seminar are offered by faculty from departments and zprograms across the College. For information on the First-Year Seminar Program, contact the Dean for Curriculum and Academic Engagement. IDPT 10100. FIRST-YEAR SEMINAR IN CRITICAL INQUIRY Required of all first-year students, the First-Year Seminar in Critical Inquiry focuses on the processes of critical inquiry in a writing-intensive, small seminar. Each seminar invites students to engage a set of issues, questions, or ideas that can be illuminated by the disciplinary and interdisciplinary perspectives of the liberal arts. Seminars are designed to enhance the intellectual skills essential for liberal learning and for successful participation in the College’s academic program. First-Year Seminar may not be taken S/NC. Annually. Fall. COLLEGE WRITING In conjunction with the Program in Writing (see The Academic Program), the College Writing course provides students with individualized instruction and opportunity to share their written work with the course instructor and their peers. The course seeks to improve the student’s ability to competently employ the grammar and syntax of the English language, to improve the student’s understanding of and ability to construct an expository essay as well as other genres used in academic communication, and to improve the student’s ability to critique and edit his/her own writing. For information on the College Writing course, contact the Dean for Curriculum and Academic Engagement. IDPT 11000. COLLEGE WRITING The course is designed specifically for students who have been determined to need intensive instruction in grammar, syntax, and basic essay format in order to fulfill the College’s Writing Proficiency Requirement. Students will compose essays and other texts appropriate for academic writing. Drafting, revising, and peer editing will be emphasized throughout the course, and reading skills will be integrated with writing instruction. Students required to take the course should enroll in their first year, and completion of the course is a prerequisite for enrollment in the sophomore-level “W” course. Other students may register for the course if space permits or upon the recommendation of their academic adviser. College Writing may not be taken S/NC. Annually. Fall 2011 and Spring 2012. Interdepartmental Courses 126 INTERDISCIPLINARY COURSES IDPT 19905. INTRODUCTION TO ENTREPRENEURSHIP This course will introduce students to the multidisciplinary world of entrepreneurship. The course will survey and explore the fundamental components of entrepreneurship and its connectedness to a liberal arts education. The course will take students through various entrepreneurial phases including pre-launch, launch, growth, and maturity of an entrepreneurial endeavor. Students will be introduced to the basic elements of entrepreneurship and highlight both entrepreneurial success and failure. Students will be challenged to think differently by being innovative, creative, and forward thinking. Not offered 2011-2012. IDPT 20001. THE PRESOCRATICS This course will focus on the emergence of philosophical and scientific thinking out of the archaic Greek culture informed by the literature of Homer, Hesiod, the lyric poets, and Aeschylus. The Presocratic philosophers from Thales to the Sophists contain the roots of our concept formation in philosophy, science, and theology. Archaic Greek authors, those ancestral to and contemporary with these thinkers, create the foundational literature that grounds these concerns in artistic, religious, mythological, and ethical contexts. In addition to gaining an appreciation of these roots of modern thought, a close study of these emergent systems of thought will sharpen students’ concept formation and critical faculties. Not offered 2011-2012. [AH] IDPT 20010. THE COLD WAR ON FILM This course examines the history of the Cold War through film and literature. Students will be exposed to major theories in History and Political Science regarding the origins of the Cold War, struggles between the superpowers, foreign policy decision making, and linkages between politics and popular culture. This course employs film, original government documents, academic works, and popular fiction to contextualize the theoretical principles that scholars use to explain the past. This exploration will yield a deeper understanding of the political and cultural climate of the United States between 1945 and 1990 and convey what the Cold War really meant to Americans in political, military, economic, and cultural terms. Not offered 2011-2012. [W, HSS] IDPT 20011. NEUROSCIENCE OF LEARNING AND MEMORY (Neuroscience) The focus of this course is learning and memory, from the molecular events responsible for the memory trace to whole organism behavior. A survey of relevant empirical research articles is combined with laboratory experiences to demonstrate major techniques and findings in the field. Laboratory experiences expose students to techniques used to study animal behavior and biological mechanisms that underlie learning and memory. This course is an upper-level course for Psychology, Biology, or Neuroscience majors. Prerequisite: C- or better in BIOL 20100 and PSYC 23000, or permission of instructor. Not offered 2011-2012. [MNS] IDPT 20012. THE ETHICS OF ACHILLES This interdisciplinary course will focus on the emergence of ethical thinking out of the archaic culture informed by Homeric poetry, primarily the Iliad. The study of Aristotle’s Nichomachean Ethics applied to Achilles in battle will provide a framework for understanding not only for the foundation of ethical concepts, but for a contemporary reading of Achilles. Not offered 2011-2012. IDPT 20013. INTRODUCTION TO BIOINFORMATICS (Biochemistry and Molecular Biology) Bioinformatics applies the tools of computer science to the research questions of molecular biology and biological chemistry. In this class, students are first introduced to the basic concepts of molecular biology and computer programming. Subsequently, students work collaboratively to develop and explore the analytical tools of bioinformatics, as applied to the analysis of genomes, the prediction of RNA and protein structure, and the analysis of evolutionary relationships. Prerequisite: C- or higher in CSCI 15200 or BIOL 20100 or permission of instructor. Not offered 2011-2012. IDPT 20014. GLOBALIZING HEALTH The twenty-first century will present numerous public health challenges, such as the AIDS crisis, the rise of multidrug-resistant tuberculosis, and trafficking in human organs and tissues. Such problems can only be addressed by a combination of local and global responses. This course applies contemporary globalization theories to such public health challenges, and critically examines the ways in which Western medical techniques and attitudes toward health are disseminated throughout the world, and the tensions generated in local cultures by this globalization of health. Not offered 2011-2012. [C] IDPT 21000. THE COFFEE COURSE An examination of an important global commodity, coffee, from a multi-disciplinary perspective. The goal is to Interdepartmental Courses 127 achieve a rich and complex understanding of coffee and its role in human society through a broad implementation of the liberal arts perspective. Not offered 2011-2012. IDPT 24000. GREEK ARCHAEOLOGY AND ART (Archaeology, Art and Art History, Classical Studies) A study of the major archaeological sites and monuments in Greece from the prehistoric, archaic, classical, and Hellenistic periods. Emphasis on the interrelationship between artistic creativity, material culture, and their social, historical, and intellectual context. Recommended: ARTD 12000. Alternate years. Not offered 2011-2012. [AH] IDPT 24100. ROMAN ARCHAEOLOGY AND ART (Archaeology, Art and Art History, Classical Studies) A study of Roman art, architecture, and archaeology, from the Early Empire through Constantine. Emphasis on the interrelationship between artistic creativity, material culture, and their social, historical, and intellectual context. Recommended: ARTD 12000. Alternate years. Spring 2012. [AH] IDPT 25000. THE THEORY AND PRACTICE OF LAW This course examines basic principles of the Western legal tradition and their incorporation into the U.S. Constitutional framework from an interdisciplinary perspective. Incorporating historical, philosophical, ethical, rhetorical, and political perspectives, the course will analyze how the theory and practice of law are connected to fundamental issues of individual freedom, social order, justice, fairness, scarcity, and human rights. In addition, students will investigate the historical underpinnings of the American legal system, contemporary legal debates, and ethical and political problems that arise within the U.S. Constitutional system. Readings and assignments are designed to develop the critical reading, writing, research, and reasoning skills that are crucial to the law. Not offered 2011-2012. [W, AH, HSS] LEADERSHIP AND LIBERAL LEARNING As an institution committed to distinction in the liberal arts and to the education of persons who will assume significant leadership roles, the College believes that a liberal arts curriculum can address the understanding and practice of leadership in both its theoretical and practical aspects. In the Leadership and Liberal Learning Program, the concept of leadership is studied in an interdisciplinary seminar that examines the complexity of leadership from diverse points of view. In addition to the Leadership Seminar, students will have an opportunity to participate in a summer apprenticeship program to learn leadership and entrepreneurial skills in a real world environment. IDPT 39000. LEADERSHIP IN THE PUBLIC AND PRIVATE SECTORS A study of theories of leadership and their practical application with focus on accounts of leadership and entrepreneurship, past and present. Special emphasis on diverse cultural contexts, global interdependence, and consequences of leadership. A summer apprenticeship experience will be available for successful participants. Not offered 2011-2012. TEACHING APPRENTICESHIP Students often serve as teaching apprentices in departmental academic courses across the College as well as in the First-Year Seminar program. Students benefit from the experience of working in a different way with familiar material, from the relationship with the faculty teaching mentor, and from the opportunity to share their enthusiasm for a subject with other students. Student peers, faculty members, and teaching apprentices themselves come to recognize the importance of the teaching apprentice’s roles as a mentor, a model of academic participation, and a tutor in the course. IDPT 39800. TEACHING APPRENTICESHIP An apprenticeship in teaching in which a student, under the supervision of a faculty member, examines critically a specific process of education and learns through practice to impart the basic concepts of a course. May be taken only twice toward graduation and only by invitation of the instructor with the approval of the faculty adviser and the Dean for Curriculum and Academic Engagement. Annually. Fall and Spring. International Relations 128 INTERNSHIPS For more information on internships, see Academic Policies – Internships. IDPT 40600. GLOBAL SOCIAL ENTREPRENEURSHIP SEMINAR A problems-centered, preparatory seminar that seeks to understand solutions to the social and economic challenges faced by people living in poverty. Students refine their understanding of social entrepreneurship and economic development, explore the ethics and philosophy of global engagement, build cultural sensitivity skills that enable them to work in the developing world, and research a social problem from a multidisciplinary perspective with an eye to innovation. The problems studied within the course are tailored to fit the summer experience IDPT 40700. Students are also asked to attend a fundraising/social venture capital clinic. Prerequisite: course registration requires an application. IDPT 40700, 40800. INTERNSHIP A multidisciplinary or interdisciplinary structured normally off-campus experience (which is not part of a study-abroad program) in which a student extends classroom knowledge through experience in a responsible position within a community, business, or government organization. Student interns work and learn under the joint oversight of a site supervisor and a faculty adviser. The student must arrange the internship in advance through the appropriate department or program. Internships must be unpaid. No more than two internships, and a maximum of four Wooster course credits, will count toward graduation. The form for registering for an internship is available in the office of the Registrar. (.25-4 course credits) S/NC course. Prerequisite: The approval of both the faculty adviser and the Dean for Curriculum and Academic Engagement is required. Annually. INTERNATIONAL RELATIONS CURRICULUM COMMITTEE: James Warner (Economics), Chair Katherine Holt (History) Kent Kille (Political Science) (Spring 2012) Matthew Krain (Political Science) Jeffrey Lantis (Political Science) Michelle Liebby (Political Science) Amyaz Moledina (Economics) Peter Pozefsky (History) (Spring 2012) Ibra Sene (History) Gregory Shaya (History) (Fall 2011) The International Relations major is administered by a committee consisting of faculty who teach in the program. A major in International Relations (IR) provides a body of knowledge, perspectives, and critical skills for understanding global politics. The major combines work in political science, economics, and history to focus on such specific areas of study as the dynamics of international politics, diplomacy, and conflict; the nature of the global economic system; and the structure and function of diverse political and economic systems. The major in International Relations consists of twelve to fourteen courses in political science, economics, and history; one foreign language course beyond the first four courses in a foreign language; and an overseas term. Courses must include ECON 10100, ECON 20100 or 20200, PSCI 12000, PSCI 22700, two history courses and a research sequence including the relevant social science methods Independent Study courses (see below). At the time of declaring the major each student will select Political Science, Economics, or History as his or her home department. The home department will International Relations 129 have the responsibility for supervising the student’s research training, including methodology and Independent Study in International Relations. Students should also present a plan for the completion of the major requirements, including the timing of the overseas term, social science methods courses, and Independent Study. For more information about these courses, see listings in the appropriate home department. Major in International Relations—Home Department: Economics Consists of fifteen courses: • ECON 10100 • One of the following courses: HIST 10100 (when focused on global phenomena or IR themes), 10700, or 10800 • PSCI 12000 • One upper-level foreign language course (see note below) • One of the following courses: ECON 20100 or 20200 • One of the following courses: ECON 25100, 25200, or 25400 • One of the following courses: HIST 20100*, 20800, 20900, 21600, 22700, 22800, 23200, 23700, 27500*, or 30100* (*when focused on global phenomena or IR themes) • One of the following courses: PSCI 22100, 22200, 22300, 22400, 22500, 22600, 22800, or 22900 • PSCI 22700 • One of the following courses: PSCI 24200, 24400, 24600, 24700, or 24900 • ECON 11000 • ECON 21000 • Junior Independent Study: ECON 40100 • Senior Independent Study: ECON 45100 • Senior Independent Study: ECON 45200 Major in International Relations—Home Department: History Consists of fourteen courses: • ECON 10100 • One of the following courses: HIST 10100 (when focused on global phenomena or IR themes), 10700, or 10800 • PSCI 12000 • One upper-level foreign language course (see note below) • One of the following courses: ECON 20100 or 20200 • One of the following courses: ECON 25100, 25200, or 25400 • One of the following courses: HIST 20100, 20800, 20900, 21600, 22700, 22800, 23200, 23700, 27500*, or 30100* (*when focused on global phenomena or IR themes) • One of the following courses: PSCI 22100, 22200, 22300, 22400, 22500, 22600, 22800, or 22900 • PSCI 22700 • One of the following courses: PSCI 24200, 24400, 24600, 24700, or 24900 • One of the following courses: HIST 20100 or 29800 • Junior Independent Study: HIST 40100 • Senior Independent Study: HIST 45100 • Senior Independent Study: HIST 45200 Latin American Studies 130 Major in International Relations—Home Department: Political Science Consists of thirteen courses: • ECON 10100 • One of the following courses: HIST 10100 (when focused on global phenomena or IR themes), 10700, or 10800 • PSCI 12000 • One upper-level foreign language course (see note below) • One of the following courses: ECON 20100 or 20200 • One of the following courses: ECON 25100, 25200, or 25400 • One of the following courses: HIST 20100, 20800, 20900, 21600, 22700, 22800, 23200, 23700, 27500*, or 30100* (*when focused on global phenomena or IR themes) • One of the following courses: PSCI 22100, 22200, 22300, 22400, 22500, 22600, 22800, or 22900 • PSCI 22700 • One of the following courses: PSCI 24200, 24400, 24600, 24700, or 24900 • Junior Independent Study Equivalent: PSCI 35000 • Senior Independent Study: PSCI 45100 • Senior Independent Study: PSCI 45200 Special Notes • Overseas Study: Credit for the overseas term will be given typically for participation in a Wooster-endorsed program. Normally the overseas term will be at least one academic semester in length. Summer programs must be a minimum of eight weeks in length. Programs other than Wooster-endorsed programs will count only toward the fulfillment of the requirement by special permission, obtained in advance through written petition. • The International Relations major must have one foreign language course beyond the first four courses in a foreign language (i.e., three semesters more than the existing College graduation requirement in a single language). • HIST 10100, 20100, 27500 and 30100 are accepted for International Relations credit when the courses focus on global phenomena or underlying political themes that characterize the international system. See Chair of International Relations (IR) for approval. • Majors in the home department of History cannot count the same HIST 20100 course twice. • Only grades of C- or better are accepted for the major or minor. LATIN AMERICAN STUDIES CURRICULUM COMMITTEE: Katherine Holt (History), Chair Cynthia Palmer (Spanish) Latin American Studies combines a multidisciplinary approach to Latin America and the Hispanic Caribbean, Spanish language study, and off-campus study to deepen participating students’ knowledge of the area. Contributing courses are not restricted to the region’s geographic limits but also include the experiences of dias- Latin American Studies 131 poric communities as well as courses that provide a broader theoretical perspective to help students understand Latin Americans’ diverse lived experiences. This firm grounding in the history, cultures, and languages of Latin America will allow students from any major to bring a wider global perspective to their disciplinary projects. Minor in Latin American Studies Consists of six courses: • One of the following courses: HIST 21500 or 21600 • SPAN 22400 • One elective taken from Latin American Studies courses in a department other than History or Spanish • Three electives taken from Latin American Studies courses Special Notes • Overseas Study: Students must study abroad in an endorsed program in the region. This may be a summer, semester, or year-long program. • Students may take either HIST 21500 or HIST 21600 as a foundational course, although students are encouraged to take both to further their knowledge of regional history. • In general, courses from LAST endorsed programs analyzing regional issues will automatically count as elective credit towards the minor. • No more than three off-campus courses can be counted toward the minor. • Supervised internships, experiential learning opportunities, or research projects awarded credit during the off-campus study term may also be counted towards the LAST minor with approval. • Students may count no more than one Spanish elective in English towards the minor. • Only grades of C- or better are accepted for the minor. LATIN AMERICAN STUDIES COURSES HISTORY HIST 10100-10176 INTRODUCTION TO HISTORICAL INVESTIGATION (depending on topic) [HSS, some sections count toward C and/or W] HIST 20100-20105. THE CRAFT OF HISTORY (depending on topic) [W, HSS, some sections count toward C] HIST 21500. COLONIAL LATIN AMERICA [C, HSS] HIST 21600. MODERN LATIN AMERICA [C, HSS] HIST 27500-27502. STUDIES IN HISTORY (depending on topic) [HSS, some sections count toward C] HIST 30100-30142 HISTORY COLLOQUIUM (depending on topic) [HSS, some sections count toward C and/or R] POLITICAL SCIENCE PSCI 24700. SPECIAL TOPICS IN COMPARATIVE POLITICS: LATIN AMERICAN POLITICS [HSS] RELIGIOUS STUDIES RELS 25100. MODERN RELIGIOUS THINKERS [W, R] SOCIOLOGY AND ANTHROPOLGY ANTH 23100-23112. PEOPLES AND CULTURES (Latin American focus only) [C, HSS] SPANISH SPAN 21200. LITERATURE AND CULTURE OF THE HISPANIC CARIBBEAN (in English) [C, AH] SPAN 21300. US LATINO LITERATURES AND CULTURES (in English) [C, AH] SPAN 24800. TWENTIETH AND TWENTY-FIRST CENTURY SPANISH AMERICAN WRITERS [C, AH] Mathematics 132 SPAN 25000. COMMERCIAL LANGUAGE AND CULTURE IN THE HISPANIC WORLD [C] SPAN 27000. SPANISH PHONOLOGY [AH] SPAN 28000. HISPANIC FILM (in English) [C, AH] SPAN 30500. THE CONTEMPORARY LATIN AMERICAN NOVEL [C, AH] SPAN 30900. TRENDS IN SPANISH AMERICAN LITERATURE [C, AH] SPAN 31000. THE STRUCTURE OF MODERN SPANISH [AH] SPAN 31100-31101. ADVANCED SEMINAR: SPECIAL TOPICS IN HISPANIC LANGUAGE, LITERATURE, AND CULTURE (depending on topic) [AH, some sections count toward C] THEATER AND DANCE THTD 24100. LATINA/O DRAMA AND PERFORMANCE [AH, C] WOMEN’S, GENDER, AND SEXUALITY STUDIES WGSS 20400. GLOBAL FEMINISMS [C, HSS] While the amount of Latin American content in this course will vary depending on the instructor, it will provide a broader theoretical perspective to help students understand Latin Americans’ diverse lived experiences. MATHEMATICS Pamela Pierce, Chair Jennifer Bowen James Daehn James Hartman Ronda Kirsch Mary Joan Kreuzman R. Drew Pasteur John Ramsay The study of mathematics develops the ability to think carefully – it sharpens analytical and problem-solving skills and trains the mind to reason logically and with precision. The program in Mathematics serves students from many majors, with a variety of academic goals. For the benefit of both majors and non-majors, the course offerings include an array of topics from both pure and applied mathematics. Some courses are theoretical, stressing the development of rigorous, well-written mathematical proof and communication, while others are computational, using appropriate software as an aid. In preparation for Senior Independent Study, there is an emphasis on clear and precise written and oral communication of mathematical concepts. Most upper-level courses culminate in a final paper, project, or presentation. First-year or transfer students are given a recommended placement in mathematics based upon their previous records, their scores on the SAT and/or ACT, and their performance on a placement exam administered by the department during Summer registration. In some cases, incoming students have multiple options from which to choose their first mathematics course at Wooster. Major in Mathematics Consists of twelve courses: • One of the following courses: MATH 11100 or 10800 • MATH 11200 • MATH 21100 • MATH 21200 • CSCI 15100 Mathematics 133 • Two of the following courses: MATH 21900-21901 (when topic is applied mathematics, full-credit), 22100, 22300, 22500, 22700, 23500, 24100, or 24200 • Two of the following courses: MATH 30000, 30200, 30300, 30400, 30500, 30600, or 31900-31901 (when topic is theoretical, full-credit) • One elective full-credit Mathematics course numbered above 21200 • Junior Independent Study: See note below • Senior Independent Study: MATH 45100 • Senior Independent Study: MATH 45200 Minor in Mathematics Consists of six courses: • One of the following courses: MATH 11100 or 10800 • MATH 11200 • MATH 21100 • Three elective full-credit Mathematics courses numbered above 21100 Special Notes • Junior Independent Study: In lieu of a MATH 40100 course, the College requirement of a third unit of Independent Study is satisfied through the independent work done as part of the courses numbered above 20000 which are taken to fulfill the requirements of the major. • Advanced Placement: At most two courses of advanced placement may be counted toward a major or minor. Advanced Placement of one or two courses in Mathematics is available to students who have taken the Advanced Placement Examination or an equivalent furnished by the Department of Mathematics and Computer Science. Students are urged to take the AP Examination for this purpose when possible. A minimum score of 3 on the AP Calculus AB examination is required to receive credit for MATH 11100; a minimum score of 4 on the AP Calculus BC examination is required to receive credit for both MATH 11100 and 11200. A student placed in MATH 11200 will receive one course credit; two course credits will be granted if the student is placed in a course above the level of MATH 11200. In cases not involving AP examinations, the decision about grant ing such placement will be made by the Department of Mathematics and Computer Science. The advanced placement policy of the College is explained in the section on Admission. • Majors are encouraged to pursue a minor and/or second major in related fields. Double majors often write an interdisciplinary Independent Study thesis, typically using mathematics as a tool to better understand a problem in the other field. Students considering a Mathematics major should discuss their plans with a member of the department, ideally during their first year as a student. • Although MATH 21500 is not required, majors are strongly encouraged to take this course prior to the 300-level Mathematics courses, to help develop the proof-writing skills necessary in theoretical mathematics. • Majors are encouraged to complete the core requirements of the major (Math 11100, 11200, 21100, and 21200) and at least one additional course in mathematics by the end of their junior year. • Minors should contact a member of the department to determine which Mathematics electives would be most applicable to their major. • Mathematics Study Abroad: The College has direct connections with the overseas program Budapest Semesters in Mathematics in Budapest, Hungary. This program is designed for Amer ican and Canadian undergraduate mathematics Mathematics 134 students interested in a one-semester overseas study experience in which they continue their study of mathematics. The program is primarily for junior mathematics students with a strong mathematics background. All courses are taught in English by Hungarian mathematicians, most of whom have spent some time teaching in the U.S. or Canada. Courses taken in Budapest appear on the student’s transcript, but grades do not count toward the student’s grade point average. Only courses receiving a grade of C or above will receive Wooster credit. Most financial aid is applicable to the program, but students with financial aid should consult directly with the Director of Financial Aid. • Teaching Licensure (Early Childhood): Students who are planning to receive Ohio licensure in early childhood education are required to take EDUC 26000 Curriculum: Math/Science/Social Studies in the Early Childhood Years. No mathematics beyond this course is required to fulfill the State requirement; however, MATH 10000 would be an excellent choice to help meet Wooster’s Learning Across the Disciplines requirements. Any student wishing to pursue licensure in early childhood education should plan a program carefully with the Depart - ment of Education. • Teaching Licensure (Middle School or Adolescent to Young Adult/ Secondary): For Ohio licensure in middle school or adolescent to young adult/secondary teaching of mathematics, State requirements call for at least a minor in Mathematics. Because specific courses in Education and Mathematics are required for licensure, Mathematics majors seeking licensure for teaching middle school or adolescent to young adult/secondary mathematics should plan their program early, in consultation with the Department of Education. These students may choose to write a Senior Independent Study Thesis on a topic related to the teaching of middle school or adolescent to young adult/ secondary mathematics. • Combined programs of liberal arts and engineering are available. (See PreProfessional and Dual Degree Programs: Pre-Engineering.) • Only grades of C- or better are acceptable in courses for the major or minor. MATHEMATICS COURSES MATH 10000. MATHEMATICS IN CONTEMPORARY SOCIETY This course is designed for students wanting to partially satisfy the Learning Across the Disciplines requirements. This is a survey course that explores a broad spectrum of mathematical topics; examples include the search for good voting systems, the development of efficient routes for providing urban services, and the search for fair procedures to resolve conflict. The emphasis is on observing the many practical uses of mathematics in our modern society and not on mastering advanced mathematical techniques. This course does not satisfy the prerequisites for further Mathematics courses, nor does it count toward a major or minor. Annually. Fall and Spring. [Q, MNS] MATH 10200. BASIC STATISTICS This course covers an introduction to basic statistical methods and concepts - the basic elements of descriptive and inferential statistics. Topics include exploratory data analysis, experimental design, sampling, inference for means and proportions, regression, and categorical data. This course does not satisfy the prerequisites for further Mathematics courses, nor does it count toward a major or minor. Annually. Spring. [Q, MNS] MATH 10300. MATRIX ALGEBRA AND PROBABILITY FOR SOCIAL SCIENCE This course is designed primarily for students in the social sciences. Topics include probability, math of finance, matrix algebra, and linear programming. This course does not count toward a major or minor. Not offered 2011- 2012. [Q, MNS] MATH 10400. CALCULUS FOR SOCIAL SCIENCE This course is designed primarily for students in the social sciences. The course covers the basic concepts of single variable calculus and, to a lesser extent, multivariable calculus. This includes the topics of limits, differenti- Mathematics 135 ation, integration, and applications of these topics. The emphasis is on fundamental themes, computational skills, and problem solving, rather than on mathematical theory. This course does not count toward a major or minor. Credit cannot be given for both MATH 10400 and either 10800 or 11100. Prerequisite: Departmental approval, as determined by performance on placement exam. Annually. Spring. [Q, MNS] MATH 10700. CALCULUS WITH ALGEBRA A This course is the first in a two-course sequence that integrates precalculus and first-semester calculus topics. This course will examine the algebraic, geometric, and analytic properties of polynomial and rational functions. Limits, continuity, differentiation, and integration in connection with these functions will be studied, along with applications. This course does not count toward a major or minor and may not be taken by anyone with credit for MATH 10400 or 11100. Prerequisite: Departmental approval, as determined by performance on placement exam. Annually. Fall. [Q, MNS] MATH 10800. CALCULUS WITH ALGEBRA B This course is a continuation of MATH 10700 and will further cover topics in differential and integral calculus. It will examine algebraic, geometric, and analytic properties of trigonometric, exponential, and logarithmic functions. Limits, continuity, differentiation, and integration in connection with these functions will be studied, along with applications. This course counts toward a major or minor and may not be taken by anyone with credit for MATH 10400 or 11100, nor can a student receive credit for both this course and MATH 10400 or 11100. Prerequisite: MATH 10700. Annually. Spring. [Q, MNS] MATH 11100. CALCULUS AND ANALYTIC GEOMETRY I This course and MATH 11200 cover the calculus of functions of one variable. Topics include limits, continuity, differentiation and integration, applications of the calculus, elements of analytic geometry, and the Fundamental Theorem of Calculus. Prerequisite: Departmental approval, as determined by performance on placement exam. Annually. Fall and Spring. [Q, MNS] MATH 11200. CALCULUS AND ANALYTIC GEOMETRY II This course is a continuation of MATH 11100. Topics include calculus of transcendental functions, integration techniques, infinite series, polar and parametric representations and/or first-order differential equations. Prerequisite: MATH 11100 or MATH 10800, or AP/equivalent credit. Annually. Fall and Spring. [Q, MNS] MATH 12300. DISCRETE MATHEMATICS This course covers logic, proofs, sets, relations, functions, algorithms, counting methods, recurrence relations, graph theory, trees, Boolean Algebras, automata and grammars. Alternate years. Fall 2011. MATH 21100. LINEAR ALGEBRA This course covers systems of linear equations, matrix theory, vector spaces and linear trans formations, determinants, eigenvalues and eigenvectors, and inner product spaces. Prerequisite: MATH 11200 or permission of the instructor. Annually. Fall. [W†, Q, MNS] MATH 21200. MULTIVARIATE CALCULUS This course covers analytic geometry of functions of several variables, limits and partial derivatives, multiple and iterated integrals, non-rectangular coordinates, change of variables, line and surface integrals and the theorems of Green and Stokes. Prerequisite: MATH 11200. Annually. Spring. [Q, MNS] MATH 21500. TRANSITION TO ADVANCED MATHEMATICS This is a transition course from the primarily computational and algorithmic mathematics found in calculus to the more theoretical and abstract mathematics in the 300-level Math courses. The emphasis is on developing the skills and tools needed to read and write proofs, and to understand their importance in mathematics. The course examines topics such as set theory and logic, mathematical induction, and a number of other proof techniques. Prerequisite: MATH 21100 (may be taken concurrently). Annually. Fall. [W] MATH 21900-21901. SPECIAL TOPICS The content and prerequisites of this course will vary according to the needs of students. It will be given at irregular intervals when there is need for some special topic. (Variable course credit) MATH 22100. DIFFERENTIAL EQUATIONS This course covers the classification of equations, forms of solution (algebraic, numeric, qualitative, geometric), solution and application of first-order and constant-coefficient second-order equations, systems of linear differential equations, phase plane analysis, applications to modeling, and computational methods (including the use of appropriate software). Prerequisite: MATH 11200. Alternate years. Not offered 2011-2012. Mathematics 136 MATH 22300. COMBINATORICS AND GRAPH THEORY This course introduces the basic techniques and modes of reasoning of combinatorial problem-solving in the same spirit that calculus introduces continuous problem-solving. It will include topics in graph theory, combinatorics, inclusion/exclusion principle, recurrence relations, and generating functions. Prerequisite: MATH 12300 or 21100. Alternate years. Spring 2012. MATH 22500. MATHEMATICAL MODELING This course considers a variety of mathematical models in the physical, life, and social sciences. In addition to analyzing models, a major component of the course is using computational tools to construct mathematical models and test their validity against empirical data. Prerequisite: MATH 11200. Alternate years. Spring 2012. MATH 22700. OPERATIONS RESEARCH This course begins with an introduction to the general methodology of operations research supported by examples and a brief history. A fairly extensive coverage of the theory and applications of linear programming leads to both discrete and continuous models used in economics and the management sciences. Among those models are nonlinear programming, continuous and discrete probability models, dynamic programming, and transportation and network flow models. Prerequisite: MATH 21100 and MATH 21200 (may be taken concurrently) or permission of instructor. Alternate years. Not offered 2011-2012. MATH 23500. NUMERICAL ANALYSIS This course covers error analysis, interpolation theory, solution of nonlinear equations and systems of linear and nonlinear equations, numerical differentiation and integration, and solution of ordinary differential equations. While theoretical results are discussed, there is also an emphasis on implementing algorithms and analyzing computed results. Prerequisite: CSCI 15100, MATH 11200, and MATH 21100, or permission of instructor. Alternate years. Not offered 2011-2012. MATH 24100. PROBABILITY AND STATISTICS I This course is an introduction to probability and statistics. Topics include permutations and combinations, sample spaces, probability, random variables, discrete probability distributions, continuous probability distributions, multivariate distributions, transformations of random variables, and moment generating function techniques. Prerequisite: MATH 11200. Alternate years. Fall 2012. MATH 24200. PROBABILITY AND STATISTICS II This course is a continuation of MATH 24100. Topics include random vectors and random sampling, estimation and hypothesis testing, analysis of variance, regression, and nonparametric statistics. Prerequisite: MATH 21100 and 24100. Alternate years. Spring 2012. MATH 27900. PROBLEM SEMINAR This course is a seminar in problem solving. In the Fall semester, the seminar focuses on analysis and solution of advanced contest-type problems, concluding with the taking of the Putnam Examination. In the Spring semester, the seminar may include the International Mathematical Contest in Modeling, in addition to introduction to problem solving. (.25 course credit) S/NC course. May be repeated for credit. Annually. Fall and Spring. MATH 30000. TOPOLOGY This course covers sets and functions, metric spaces, topological spaces, compactness, separation, and connectedness. Prerequisite: MATH 21100 and 21200 or permission of instructor. Every third semester. Spring 2012. MATH 30200. REAL ANALYSIS I This course develops the theoretical background for many Calculus concepts. The course begins with a study of sets, mathematical induction, and proof techniques. We then focus on the properties of the real numbers, sequences, convergence, and the Bolzano-Weierstrass Theorem. The course finishes with a study of functions defined on the real numbers, limits, continuity, and differentiation. Prerequisite: MATH 21100 and 21200 or permission of instructor. Every third semester. Fall 2011. MATH 30300. REAL ANALYSIS II This course is a continuation of MATH 30200, covering uniform continuity, uniform convergence, and further topics in differentiation and integration. Some discussion of metric spaces, introductory measure theory, and the Lebesgue integral will be included. Prerequisite: MATH 30200. Offered as needed. Not offered 2011-2012. MATH 30400. ABSTRACT ALGEBRA I This course is an introduction to abstract algebraic structures. This course and MATH 30500 include an axiomatic approach to familiar number systems, equivalence, polynomials, rings, isomorphism, and fields. Emphasis is on understanding and writing mathematics proofs. Prerequisite: MATH 21100. Annually. Fall 2011. Music 137 MATH 30500. ABSTRACT ALGEBRA II This course is a continuation of MATH 30400. Topics include groups, subgroups, symmetric groups, congruence, Lagrange’s Theorem, and further topics in ring and field theory. Prerequisite: MATH 30400. Offered as needed. Not offered 2011-2012. MATH 30600. FUNCTIONS OF A COMPLEX VARIABLE This course covers complex numbers, elementary functions, Cauchy’s theorem and formula, infinite series, elements of conformal mapping, and residues. Prerequisite: MATH 21200 and permission of instructor. Every third semester. Not offered 2011-2012. MATH 31900-31901. SPECIAL TOPICS The content and prerequisites of this course will vary according to the needs of students. It will be given at irregular intervals when there is need for some special topic. (Variable course credit) MATH 40000. TUTORIAL This course will be given for topics not normally covered in regular courses. Prerequisite: The approval of both the supervising faculty member and the chairperson are required prior to registration. MATH 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE Senior Independent Study is a two-semester project culminating in the I.S. Thesis and an oral presentation. In the first semester, the student will produce a project abstract, an annotated bibliography, and a substantial written portion of the thesis. The semester concludes with a short oral presentation on the project and progress in the first semester. MATH 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO In the second semester of Senior Independent Study the student completes the I.S. Thesis and an oral presentation. Prerequisite: MATH 45100. MUSIC Nancy Ditmer, Chair Carrie Culver Theodor Duda Jack Gallagher Jeffrey Lindberg Peter Mowrey Jake Rundall Thomas Wood Josephine Wright Lisa Yozviak The College of Wooster has been an accredited institutional member of the National Association of Schools of Music since 1947. The requirements for entrance and for graduation as set forth in this catalogue are in accordance with the published standards of the National Association of Schools of Music. The Department of Music provides students with comprehensive training in performance, composition, music theory, music education, music therapy, and music history and literature. The successful Wooster Music major will graduate with greatly enhanced musicality and technique, a deeper understanding of musical structure and style, and thorough preparation for a lifetime of musicianship. Depending on the degree, the Music major will be well prepared to seek a career as a professional musician; to teach music in public and private schools or in private studios; to utilize music as a therapeutic tool; and/or to continue study at the graduate level. The Department of Music has the following learning goals. By the completion of their studies, Wooster’s music graduates should be able: Music 138 • as performers with secure techniques, to communicate effectively a wide range of expressive content in ways appropriate to music of diverse historical periods; • to practice and learn music effectively independent of a teacher; • to interact effectively in music ensembles of various sizes and musical styles; • to speak and write effectively about music; • to understand the common elements and organizational patterns of music and how they contribute to the style and design of any particular musical work; • to understand the stylistic evolution of music of various cultures over at least the past four centuries, and to possess some knowledge of the lives and works of major composers • to possess a working knowledge of electronic music technology applications; • with the B.M. degree in Performance and the B.M.E. degree in public school teaching, to teach effectively their principal instrument or voice to students of at least elementary and intermediate levels; • with the B.M.E. Degree in public school teaching, to possess the knowledge and teaching skills to design and implement effectively a comprehensive music program in a public or private school, grades K–12; • with the B.M.E. degree in music therapy to possess the knowledge and skills to design and implement effectively a comprehensive music therapy program for a variety of populations. Major in Music The Department of Music offers the following six degree programs in Music: The liberal arts degree: Bachelor of Arts in Music Pre-professional degrees: Bachelor of Music in Music History and Literature Bachelor of Music in Performance* Bachelor of Music in Theory/Composition * Bachelor of Music Education in Music Therapy* Bachelor of Music Education in Public School Teaching* Degrees marked with an asterisk (*) require a successful audition for entrance into the program. Please see the Handbook for Music Students and Faculty for further details. Copies of the Handbook for Music Students and Faculty are available at the Administrative Coordinator’s office, Scheide Music Center, Room 112. They are also distributed to all students taking MUSC 10100 during the first few days of classes. The Handbook contains thorough information on the following topics: • Music Department mission statement and learning goals • Music facilities and policies for their use • Descriptions and requirements of the six degree programs in Music • Selecting and declaring the appropriate Music major • Entrance auditions for the pre-professional degree programs in Music • Student recitals • Recital and concert attendance requirements for Music majors • Staff accompanists • Applied music study (private lessons) • Independent Study in Music • Piano Proficiency Exam, required of all Music majors • Student employment in Music • College-owned instruments • Music Department faculty members Music 139 The descriptions below provide only an “at-a-glance” summary of requirements for the six degree programs in Music. Please see the Degree Requirements section of this Catalogue and the Handbook for Music Students and Faculty for complete information. Required of all Music majors, regardless of degree: • Recital attendance requirement: 10 events per semester (see Handbook for details) • Successful performance on the Piano Proficiency Examination (see Handbook for details) • Specific course requirements (see individual degree listings in Degree Requirements section of this Catalog) BACHELOR OF ARTS IN MUSIC Consists of twelve to fifteen course credits: • MUSC 10100, 10200, 20100, 20200, and 30100 • MUSC 21000, 21200, and 21300 • 1 credit in applied music lessons (MUSC 12000-14000, 22000-24000) • 0–3 credits in music electives • Junior Independent Study: MUSC 40100 • Senior Independent Study: MUSC 45100 • Senior Independent Study: MUSC 45200 BACHELOR OF MUSIC IN MUSIC HISTORY AND LITERATURE Consists of twenty-four course credits: • MUSC 10100, 10200, 20100, 20200, 30100, 30200, 30300, 30400, and 30500 • MUSC 21000, 21100, 21200, and 21300 • Three of the following courses: MUSC 21400, 21500, 21600, 21700, 21800, 21900, 31100, or AFST 21200 • MUSC 28000 • 2 credits in applied music lessons (MUSC 12000–14000, 22000–24000) • 1.25 credits in music ensembles (see Degree Requirements for details) • 1.25 credits in music electives • Junior Independent Study: MUSC 40100 • Senior Independent Study: MUSC 45100 • Senior Independent Study: MUSC 45200 Special Note: Required courses outside the major differ from those of the B.A. degree; see Degree Requirements for details. BACHELOR OF MUSIC IN PERFORMANCE Consists of twenty-four course credits: • MUSC 10100, 10200, 20100, 20200, 30100, 30200, 30300, and 30400 • MUSC 21000, 21100, 21200, and 21300 • MUSC 28000 • One of the following courses: MUSC 37000 or 37100 • 3–5 credits in applied music lessons (MUSC 12000–14000, 22000–24000) • 1.25 credits in music ensembles (see Degree Requirements for details) • 1.75–3.75 credits in music electives • Junior Independent Study: MUSC 40100 • Senior Independent Study: MUSC 45100 • Senior Independent Study: MUSC 45200 Special Note: Required courses outside the major differ from those of the B.A. degree; see Degree Requirements for details. Music 140 BACHELOR OF MUSIC IN THEORY/COMPOSITION Consists of twenty-four course credits: • MUSC 10100, 10200, 20100, 20200, 30100, 30200, 30300, 30400, and 30500 • MUSC 21000, 21100, 21200, 21300, and 31100 • MUSC 28000 • 2 credits in applied music lessons (MUSC 12000–14000, 22000–24000) • 1 credit in composition (MUSC 20800, 20900, 30800, 30900) • 1.25 credits in music ensembles (see Degree Requirements for details) • 2.25 credits in music electives • Junior Independent Study: MUSC 40100 • Senior Independent Study: MUSC 45100 • Senior Independent Study: MUSC 45200 Special Note: Required courses outside the major differ from those of the B.A. degree; see Degree Requirements for details. BACHELOR OF MUSIC EDUCATION IN MUSIC THERAPY Consists of 24.5 course credits: • MUSC 10100, 10200, 20100, 20200, 30100, 30300, and 30500 • MUSC 17000, 17100, 17300, 17400, 17500, and 17700 • MUSC 19000, 19100, 29000, 29100, 29200, 29300, 29400, and 29500 • MUSC 21000 • One of the following courses: MUSC 21200 or 21300 • MUSC 34200 • One of the following courses: MUSC 34300 or 34400 • MUSC 37000 and 37200 • MUSC 39200, 39300, and 39400 • 3.5 credits in applied music lessons (MUSC 12000–14000, 22000–24000) • Half-recital on major instrument (see Handbook for details) • 1 credit in music ensembles (see Degree Requirements for details) • MUSC 40700–40800 (Music Therapy Internship) Special Note: Required courses outside the major differ from those of the B.A. degree (including EDUC 20000; PSYC 10000, 21200, and 25000; SOCI 10000 and either SOCI 20400 or SOCI 21300); see Degree Requirements for details. BACHELOR OF MUSIC EDUCATION IN PUBLIC SCHOOL TEACHING Consists of 22.25 course credits: • MUSC 10100, 10200, 20100, 20200, 30100, 30300, 30500, and 30600 • MUSC 21000, 21200, and 21300 • MUSC 17000, 17100, 17200, 17300, 17400, 17500, 17600, and 17700 • MUSC 28000 • MUSC 29000, 34200, 34300, and 34400 • MUSC 37000 and 37200 • MUSC 39500 • 4 credits in applied music lessons (MUSC 12000–14000, 22000–24000) • Half-recital on major instrument (see Handbook for details) • 1.25 credits in music ensembles (see Degree Requirements for details) • EDUC 49600, 49700, and 49800 (Multiage Student Teaching and Seminar) Special Note: Required courses outside the major differ from those of the B.A. degree (including EDUC 10000, 12000, and 30000; and PSYC 11000); see Degree Requirements for details. Music 141 Minor in Music Consists of six course credits: • Two courses in music theory • Two courses in music history and literature (may include MUSC 11100 and other courses without prerequisite) • 2 credits in music electives (may include music performance, music ensemble, music theory, and/or music history) Special Notes for all Music Students • Only grades of C- or better are accepted for the major or minor. • Music minors may count a total of two full credits in music performance courses or groups towards graduation. • Advanced Placement: The advanced placement policy of the College is explained in the section on Admission. • Gateway Courses/Non Majors Courses: Many students have found music courses to be a valuable supplement to their major in the natural and social sciences and other humanities departments. Any student may take these courses, regardless of prior musical background. The 200-level courses below may also be taken as Music electives by Music majors. Students who wish to take upperlevel music history courses and advanced music theory courses are strongly encouraged, given appropriate background, to take MUSC 10100 (Music Theory I) as a first course in music. All courses below earn one course credit. • MUSC 10000. Fundamentals of Music • MUSC 10100. Music Theory I, with the demonstrated ability to read music • THTD 10000. Arts and Entrepreneurship • MUSC 11100. Introduction to Music • AFST 21200. Survey of African-American Folklore: The Creative and Performing Arts • MUSC 21400. History of African American Music • MUSC 21500. Music of the United States • MUSC 21600. The Art of Rock Music • MUSC 21700. Survey of Jazz • MUSC 21800. Masterpieces of Musical Theatre • MUSC 21900. Women in Music • MUSC 29000. Foundations of Music Education • Course Sequence Suggestions: Courses in the department are systematically related; skills and knowledge developed in some courses are presupposed and/or integrated into other courses. Thus, there is a timeline or schedule that helps students most effectively progress through the various majors. In general, we expect students to follow this schedule. • -First Year All Music majors • MUSC 10100, Fall semester • MUSC 10200, Spring semester • MUSC 12000–14000 or 22000–24000, both semesters • MUSC 16000, 16100, 16200, 16300, or 16400, both semesters (as required by degree program) • MUSC 21100, Spring semester (B.M. candidates only) Music 142 Public School Teaching majors • MUSC 29000, Fall semester Music Therapy majors • MUSC 19000, Spring semester • -Sophomore Year All Music majors • MUSC 20100, Fall semester • MUSC 20200, Spring semester • MUSC 21000, Spring semester • MUSC 21200, Fall semester; and/or MUSC 21300, Spring semester (as required by degree program) • MUSC 12000–14000, 22000–24000, 20800, or 30800, both semesters • MUSC 16000, 16100, 16200, 16300, or 16400, both semesters (as required by degree program) Public School Teaching majors • MUSC 17000 • MUSC 34200, Spring semester Music Therapy majors • MUSC 17000, Spring semester • MUSC 19100, Fall semester • MUSC 29000, Fall semester • MUSC 29200, Spring semester • MUSC 29300, Spring semester • -Junior Year All Music majors • MUSC 30100, Fall semester • MUSC 30300, Spring semester (except B.A. majors) • MUSC 40100, usually Spring semester (only B.A. and B.M. majors) • MUSC 12000–14000, 22000–24000, 20800, or 30800, both semesters • MUSC 16000, 16100, 16200, 16300, or 16400, both semesters (as required by degree program) Public School Teaching majors • MUSC 34300, Spring semester • MUSC 34400, Spring semester • MUSC 37000, Fall semester Music Therapy majors • MUSC 29100, Fall semester • MUSC 29400, Fall semester • MUSC 37000, Fall semester • MUSC 29300, Spring semester • MUSC 29500, Spring semester • MUSC 34300 or 34400, Spring semester • -Senior Year All Music majors • MUSC 45100–45200, both semesters (only B.A. and B.M. majors) • MUSC 12000–14000 or 22000–24000 (as required by degree program) • MUSC 16000, 16100, 16200, 16300, or 16400, both semesters (as required by degree program) Public School Teaching majors • MUSC 30500, Fall semester Music 143 • MUSC 30600, Fall semester • MUSC 39500, Spring semester • EDUC 39600, 39700, and 39800, Spring semester • MUSC 12000–14000 or 22000–24000, Fall semester • Half Recital Music Therapy majors • MUSC 30500, Fall semester • MUSC 39200, Fall semester • MUSC 39400, Fall semester • MUSC 37000, Fall semester • MUSC 39300, Spring semester • MUSC 12000–14000 or 22000–24000, both semesters • Half Recital MUSIC COURSE DESCRIPTIONS COURSES OPEN TO ALL STUDENTS, WITHOUT PREREQUISITES Any student may take these courses, regardless of prior musical background. The 200-level courses may also be taken as Music electives by Music majors. One credit per course. MUSC 10000. FUNDAMENTALS OF MUSIC Reading and aural recognition of single pitches, intervals, scales, triads, time values, key signatures, and other basic elements of music. Recommended for students with little or no musical background. Does not count toward either the major or minor in Music. Spring 2012. [AH] MUSC 11100. INTRODUCTION TO MUSIC An introduction to the appreciation of Western art music with an emphasis on hearing, recognizing, and relating the elements of music in an increasingly informed context. Topics will focus on major composers from the Middle Ages to modern times and will explore the range of meaning and value that their works have had and continue to have, by drawing connections between music and other humanities as well as the social and natural sciences. The course might include some jazz, popular music, and non-Western music. In addition to listening and reading, students will attend concerts and prepare written assignments. No previous musical background necessary. Does not count toward a major in Music. Fall. Not offered 2011-2012. [AH] THTD 10000. ARTS AND ENTREPRENEURSHIP [AH] THTD 10400. THE IMPULSE TO CREATE [AH] MUSC 21400. HISTORY OF AFRICAN AMERICAN MUSIC (Africana Studies) Study of the history of African American music from 1619 through the present day. Focuses on the socio-historical context in which popular music, folk music, classical music, and religious music evolved. Topics include spiritual, blues, gospel, jazz, rhythm and blues, and contemporary music as well as women in music. Open to non-Music majors. No technical knowledge required. Fall 2011. [C, AH] MUSC 21500. MUSIC OF THE UNITED STATES A survey of music created within the multi-cultural mosaic of this country over the past four centuries. Topics may include Native American music; Anglo-American folk song; popular song to the twentieth century; bluegrass and country music; band music; instrumental and vocal concert music; and the role of composers, performers, and listeners in American life. No previous musical background necessary. Spring. Not offered 2011- 2012. [AH] MUSC 21600. THE ART OF ROCK MUSIC The study of the artistic and aesthetic potential of rock music. Areas of emphasis may include the history and analysis of rock music; rock music aesthetics and their relationship to the aesthetics of other music and art forms; the evolution of rock musical styles; the connections between rock, poetry, and literature; “covering,” quotation, and stylistic borrowing in rock music; the impact of the electronic music revolution; and the live performance of rock. Offered every two to three years. Spring. Not offered 2011-2012. [AH] MUSC 21700. SURVEY OF JAZZ (AFRICANA STUDIES) A study of jazz from its inception to the present, including the New Orleans, swing, bebop, cool, hard bop, free Music 144 jazz, and jazz-rock fusion styles, as well as major individual musicians such as Louis Armstrong, Duke Ellington, and Charlie Parker. Special assignments for Music majors and minors. Annually. Spring. [C, AH] MUSC 21800. MASTERPIECES OF MUSICAL THEATRE A study of approximately twelve classic operas, operettas, and musicals from the eighteenth century to the present, with attention to general characteristics of the three genres. The music and its relationship to plot are emphasized; occasional guest lectures on other aspects of the works. Attendance at performances when appropriate. Works studied have included Don Giovanni, The Magic Flute, The Barber of Seville, La Traviata, Otello, Carmen, La Bohème, Tosca, Madame Butterfly, Treemonisha, Porgy and Bess, Die Fledermaus, The Mikado, The Pirates of Penzance, Oklahoma!, Guys and Dolls, My Fair Lady, Candide, Fiddler on the Roof, Sweeney Todd, and Into the Woods. No previous musical background necessary. A few special assignments for Music majors. Not offered 2011-2012. [AH] MUSC 21900. WOMEN IN MUSIC (WOMEN’S, GENDER AND SEXUALITY STUDIES) Examination of the history of women in Western music, focusing upon black and white women in classical music, jazz, gospel, popular music, and the blues. Topics will include the status of women as professional musicians, the economics of mainstreaming women in the music industry, and the collaborative efforts of women to achieve parity with men in the creative and performing arts. No prior musical knowledge required. Spring. Not offered 2011-2012. [C, AH] AFST 21200. SURVEY OF AFRICAN-AMERICAN FOLKLORE: THE CREATIVE AND PERFORMING ARTS [C, AH] MUSIC THEORY-COMPOSITION One credit per course unless otherwise specified. MUSC 10100. THEORY I Fundamentals review, diatonic triads in root position, three- and four-part writing, principles of harmonic progression. Elementary dictation, sightsinging, and keyboard skills. Required of all majors and minors in Music. Strongly recommended for all: concurrent enrollment in MUSC 13200 unless the piano proficiency requirement for Music majors has already been completed. Prerequisite: MUSC 10000; or fluent ability as measured by timed testing in the first ewek of class to read pitches in treble and bass clefs through the second ledger lines above and below those staves. Annually. Fall. [AH] MUSC 10200. THEORY II First and second inversions, cadences, elementary form, non-chord tones, common seventh chors. Related dictation, sightsinging, and keyboard skills. Required of all majors and minors in Music. Strongly recommended for all: concurrent enrollment in MUSC 13200 unless the piano proficiency requirement for Music majors has already been completed. Prerequisite: C- or better in MUSC 10100. Annually. Spring. [AH] MUSC 20100. THEORY III Modulation, less common seventh chords, binary and ternary forms. Related dictation, sightsinging, and keyboard skills. Required of all majors in Music. Prerequisites: C- or better in MUSC 10200, and concurrent enrollment in MUSC 13200 or completion of the piano proficiency requirement. Annually. Fall. [AH] MUSC 20200. THEORY IV Advanced chromatic techniques in tonal music prior to the 20th century. Related dictation, sightsinging, and keyboard skills. Required of all majors in Music. Prerequisites: C- or better in MUSC 20100, and concurrent enrollment in MUSC 13200 or completion of the piano proficiency requirement. Annually. Spring. [AH] MUSC 20800. ACOUSTIC COMPOSITION Original writing for various instrumental and vocal media in small and large forms. Emphasis will be placed on acquiring a foundation in the basic compositional techniques and developing an ability to organize musical ideas into logical and homogeneous forms. One half-hour private lesson per week. (.5 course credit) Prerequisite: MUSC 10200. Annually. Fall and Spring. MUSC 20900. ELECTRONIC COMPOSITION Original writing for electronic media. Emphasis will be placed on acquiring a foundation in the basic compositional techniques and developing an ability to organize musical ideas into logical and organic forms. One halfhour private lesson per week. (.5 course credit) Prerequisite: MUSC 10200 and either MUSC 18000 or 28000. Annually. Fall and Spring. Music 145 MUSC 30100. THEORY V Twentieth century techniques and related sightsinging/keyboard skills. Required of all majors in Music. Prerequisite: C- or better in MUSC 20200, and successful completion of the piano proficiency examination. Annually. Fall. [AH] MUSC 30200. FORM AND ANALYSIS Advanced harmonic, contrapuntal, and structural analysis of all types of musical composition. Required of all B.M. majors. Prerequisite: MUSC 20200. Alternate years. Spring 2012. MUSC 30300. BASIC CONDUCTING A course designed to introduce the fundamental skills of conducting, including basic symmetric and asymmetric patterns, expressive gestures, cues, fermatas, and the development of independence of the right and left hands. Attention is also given to transposition, instrumental score reading, score preparation, and ensemble rehearsal techniques. Required of all B.M. and B.M.E. majors. Prerequisite: MUSC 10200. Annually. Spring. MUSC 30400. COUNTERPOINT Study of the basic polyphonic principles of the sixteenth and eighteenth centuries, including species counterpoint, imitation, canon, invertible counterpoint, two- and three-part inventions, and fugue. Required of all B.M. majors. Prerequisite: MUSC 20200. Alternate years. Spring 2012. MUSC 30500. ORCHESTRATION A theoretical and practical study of instrumentation and scoring music for various instrumental combinations. Required of B.M. in Composition, B.M. in Music History/Literature, and B.M.E. majors. Composition majors should take the course as early as possible. Prerequisite: MUSC 20200. Annually. Fall. MUSC 30600. CHORAL CONDUCTING A course devoted to the specific skills and techniques required for choral conductors. Score preparation, gestures, text analysis, diction, and general aspects of good singing are among the several foci of this course. Two class hours per week. (.5 course credits) Prerequisite: MUSC 30300. Fall 2011. MUSC 30800. ACOUSTIC COMPOSITION Original writing for various instrumental and vocal media in small and large forms. Emphasis will be placed on acquiring a foundation in the basic compositional techniques and developing an ability to organize musical ideas into logical and homogeneous forms. One hour private lesson per week. Prerequisite: MUSC 10200. Annually. Fall and Spring. MUSC 30900. ELECTRONIC COMPOSITION Original writing for electronic media. Emphasis will be placed on acquiring a foundation in the basic compositional techniques and developing an ability to organize musical ideas into logical and organic forms. One hour private lesson per week. Prerequisite: MUSC 10200 and either MUSC 18000 or 28000. Annually. Fall and Spring. MUSIC HISTORY AND LITERATURE One credit per course unless otherwise specified. MUSC 21000. BASIC REPERTOIRE Guided listening to standard works of the Western classical repertoire. The list of works is determined by the entire Music faculty and is revised periodically. Required of all Music majors; others admitted by permission of the instructor. Prerequisite: MUSC 10100 or permission of the instructor. Annually. Spring. MUSC 21100. MUSIC HISTORY I Early music. The development of major musical styles from antiquity through the early baroque. Required of all B.M. majors. Prerequisite: MUSC 10100 or permission of the instructor. Not offered 2011-2012. [W, AH] MUSC 21200. MUSIC HISTORY II Monteverdi to Mozart. The development of major musical styles in the baroque and classical periods. Required of all B.A., B.M., and B.M.E. (Public School Teaching) majors. Prerequisite: MUSC 10100 or permission of the instructor. Spring 2012. [W†, AH] MUSC 21300. MUSIC HISTORY III Beethoven to the present. The development of major musical styles in the nineteenth and twentieth centuries. Required of all B.A., B.M., and B.M.E. (Public School Teaching) majors. Prerequisite: MUSC 10200 and 21200 or permission of the instructor. Annually. Fall. [AH] Music 146 MUSC 31100. SEMINAR IN MUSIC LITERATURE Selected historical studies. Topics have included The Song Cycle, Music of Living Composers, Bach, Haydn, Brahms, Piano Literature, and Romantic Concerto. Required of B.M. (Composition) majors. Prerequisite: MUSC 10200 or permission of the instructor. Not offered 2011-2012. [AH] PERFORMANCE MUSC 12000-14000, 22000-24000. PERFORMANCE Please see the “Applied Music Study” section of the Handbook for Music Students and Faculty for details about performance study, special requirements for Performance majors on different instruments, applied music requirements for Music Education majors, required recitals for all Music majors, and private lessons for nonMusic majors. For non-Music majors, no more than one credit in music performance courses or groups may count toward the minimum of 32 courses required for graduation, unless the student is a minor in Music, in which case two credits of such courses may count towards graduation. For non-majors, private performance lessons are normally taken at the 100-level for one-half (.5) course credit. Full-credit (200-level) lessons are reserved for Music majors; non-majors may take full-credit lessons only with the approval of the Music Department chair. Please see the Handbook for further information, and please see the Expenses section of this Catalogue for information about lesson fees. 12000/22000. BAGPIPE 12700/22700. FRENCH HORN 13400/23400. STRING BASS 12100/22100. BASSOON 12800/22800. GUITAR 13500/23500. TROMBONE 12200/22200. CELLO 12900/22900. OBOE 13600/23600. TRUMPET 12300/22300. CLARINET 13000/23000. ORGAN 13700/23700. TUBA 12400/22400. ELECTRIC BASS 13100/23100. PERCUSSION 13800/23800. VIOLA 12500/22500. EUPHONIUM 13200/23200. PIANO 13900/23900. VIOLIN 12600/22600. FLUTE 13300/23300. SAXOPHONE 14000/24000. VOICE CLASS INSTRUCTION IN MUSIC MUSC 15000-15700. ENSEMBLE In addition to the larger performing groups (Band, Orchestra, etc.), smaller groups such as string, woodwind, percussion, and brass ensembles function as there is a demand or requirement. One to one and one-half hours per week. (.125 course credit) S/NC course. Prerequisite: permission of instructor. Annually. Fall and Spring. 15000. ACCOMPANYING 15400. KEYBOARD ENSEMBLE 15100. BRASS ENSEMBLE 15500. PERCUSSION ENSEMBLE 15200. GUITAR ENSEMBLE 15600. STRING ENSEMBLE 15300. JAZZ COMBO 15700. WOODWIND ENSEMBLE Students are expected to practice 30-45 minutes per day for courses MUSC 17000-17800 and 37200. MUSC 17000. CLASS VOICE Study and development of basic individual vocal technique. Instruction in the International Phonetic Alphabet and its application to singing. Designed for Music Education and Music Therapy students. Two class hours per week. Required of all B.M.E. majors whose primary performance area is instrumental. (.25 course credit) Prerequisite: MUSC 10200 or permission of the instructor. Alternate years. Spring 2012. MUSC 17100, 17200. CLASS BRASS INSTRUMENTS Study of the mechanics of playing and instructional procedures and materials relative to brass instruments of the orchestra and band. MUSC 17100 covers trumpet and french horn; MUSC 17200 covers trombone, euphonium, and tuba. One class hour per week for each of two semesters. Both required of B.M.E. (Public School Teaching) majors. B.M.E. (Music Therapy) majors must complete MUSC 17100. (.25 course credit) Alternate years. Not offered 2011-2012. MUSC 17300, 17400. CLASS STRING INSTRUMENTS MUSC 17300 covers violin and viola; MUSC 17400 covers cello and string bass. Limit of six in a class. One class hour per week for each of two semesters. Required of all B.M.E. majors. (.25 course credit) Alternate years. MUSC 17300 in Fall 2011; MUSC 17400 in Spring 2012. MUSC 17500, 17600. CLASS WOODWIND INSTRUMENTS Study of the mechanics of playing and instructional materials and procedures relative to woodwind instruments of the orchestra and band. MUSC 17500 covers flute and clarinet; MUSC 17600 covers saxophone, oboe, Music 147 and bassoon. One class hour per week for each of two semesters. Both required of B.M.E. (Public School Teaching) majors. B.M.E. (Music Therapy) majors must complete MUSC 17500. (.25 course credit) Alternate years. MUSC 17500 in Fall 2011; MUSC 17600 in Spring 2012. MUSC 17700. CLASS PERCUSSION INSTRUMENTS Study of the mechanics of playing and instructional materials and procedures relative to percussion instruments of the orchestra and band. One class hour per week. Required of all B.M.E. majors. (.25 course credit) Alternate years. Fall 2011. MUSC 17800. FUNCTIONAL GUITAR A course designed for teaching Music Education and Therapy students how to use the guitar in their work. Basic strumming and finger-picking styles for song-leading and accompaniment, transposition of song material, and chording in several major and minor keys. One or two class hours per week. (.25 course credit) Spring 2012. MUSC 18000. INTRODUCTION TO THE ELECTRONIC STUDIO Hands-on experience with keyboard synthesizers, a sampler, a drum machine, sequencers, a multitrack recorder, a digital effects processor, and other electronic instruments, culminating in a creative musical project. No musical background necessary. (.25 course credit) Not offered 2011-2012. MUSC 26400. INTRODUCTION TO JAZZ IMPROVISATION Notation, standard forms and chord progressions, transcribing jazz solos from recordings, study of recordings, and other activities. (.25 course credit) Prerequisite: permission of instructor required. Spring 2012. MUSC 28000. INTRODUCTION TO MUSIC TECHNOLOGY Topics may include the MIDI electronic studio; computer applications in music including music notation, music education and music theory software, and musicological research; recording technology; and other appropriate technological developments. Assignments will be tailored insofar as possible to individual students’ needs and interests. Required of all B.M. and B.M.E. majors. (.5 course credit) Annually. Spring. MUSC 37000. VOCAL PEDAGOGY Study of the anatomy and physiology of all singing voices. Examination of instructional materials and pedagogy texts relative to the vocal instrument used singly and collectively. Two class hours per week. Required of all B.M.E. and B.M. Vocal Performance majors. (.5 course credit) Prerequisite: MUSC 17000 or two semesters of MUSC 14000. Alternate years. Fall 2011. MUSC 37100. INSTRUMENTAL PEDAGOGY Study of the literature, instructional materials and procedures relative to the teaching of the major instrument. (.5 course credit) Annually. Fall and Spring. MUSC 37200. FUNCTIONAL PIANO A course designed to give practical experience in sight-reading, transposition, accompanying, improvisation, and aural dictation, as required for certification to teach in Ohio public schools. Two hours per week. Required of all B.M.E. majors. (.5 course credit) Prerequisite: completion of all parts of the Piano Proficiency Examination. Annually. Fall and Spring. MUSIC EDUCATION One credit per course unless otherwise specified. MUSC 29000. FOUNDATIONS OF MUSIC EDUCATION This is an introductory course for all students planning to pursue teacher licensure in music. Emphasis is on historical, cultural, and social contexts for music education as well as the role of personal expression, arts criticism, and the nature and meaning of the arts in the education of children and adolescents, ages 3-21. Topics include philosophical foundations for music teaching and learning, curriculum planning and development, goals and objectives of music programs, materials, technology, and assessment strategies. Students will also examine the music education profession, its history, and the qualities, competencies, and skills required of music teachers. Clinical experiences in the classroom and field experiences in the schools are a major component of the course. Required of all B.M.E. majors. Annually. Fall. MUSC 34200. METHODS AND MATERIALS FOR TEACHING PRE-K AND ELEMENTARY GENERAL MUSIC This course provides a study of specific methods of delivering standards-based instruction to children, ages 3- 12, in pre-school and general music classroom settings. Included is significant use of the National Standards for Music 148 Arts Education and the Ohio Academic Content Standards in Music. Emphasis is on specific teaching techniques in the implementation of curriculum, classroom procedures and materials, integration of technology, instructional strategies for special needs students, and the use of various assessment strategies. Field experiences in elementary general music and preschool settings are a major component of the course. Required of all music education and music therapy majors. (.5 course credit) Prerequisite: MUSC 29000. Spring 2012. MUSC 34300. METHODS AND MATERIALS FOR TEACHING SECONDARY CHORAL AND GENERAL MUSIC This course addresses the role of choral and general music instruction in secondary public school education, techniques of teaching choral music, and the study of music from various cultures appropriate to students in choral ensembles. Included is significant use of the National Standards for Arts Education and the Ohio Academic Content Standards in Music. Emphasis is on literature selection, specific teaching techniques in the implementation of curriculum, classroom procedures and materials, integration of technology, instructional strategies for special needs students, and the use of various assessment plans. Field experiences in middle and high school choral and general music settings are a major component of the course. Required of all music education majors. (.5 course credit) Prerequisite: MUSC 29000 and 34200. Spring 2012. MUSC 34400. METHODS AND MATERIALS FOR TEACHING INSTRUMENTAL MUSIC This course provides a study of specific methods of delivering instruction in instrumental music, covering band and orchestra instruments. Emphasis is on recruitment and retention of instrumental music students, appropriate teaching techniques for musical and technical concepts for instrumentalists from the beginning years through high school, integration of technology into the instrumental classroom, and differentiation of instruction for all students and especially for those with special needs. Administrative and organizational aspects are also addressed. Field experiences in grades 5-12 instrumental music settings are a major component of the course. Required of all music education majors. (.5 course credit) Prerequisite: MUSC 29000 and 34200. Spring 2012. MUSC 39500. SPECIAL TOPICS IN MUSIC EDUCATION This course provides a study of the administrative responsibilities of music educators with a focus on projects that address the specific needs of students enrolled in the course. Topics include but are not limited to contemporary issues in education and music education; educational technology; budget and finance; facilities and equipment; music library and instrument inventory management; travel; design and purchase of uniforms; music support groups; professional development for teachers; philosophical foundations and advocacy; and relationships with parents, administrators, music dealers, and private teachers. Field experience in the student teaching setting is a strong component of the course. Prerequisite: MUSC 29000, 34200, 34300, and 34400. Spring 2012. MUSIC THERAPY All courses listed below, with the possible exception of MUSC 40700-40800, will normally be taught at Baldwin-Wallace College in Berea by the Music Therapist who is also the Director of the Music Therapy Consortium. One credit per course is standard unless otherwise specified. Please see the Handbook for Music Students and Faculty for further information about the Music Therapy major, including acceptance requirements and the entrance exam. MUSC 19000. INTRODUCTION TO MUSIC THERAPY Provides an overview of the profession including current terminology, history, and practical application of Music Therapy for several client populations. Assessment of personal qualities necessary to become a music therapist is an on-going process of the class. Observation of music and related-area therapists is required in addition to classwork. Required of all B.M.E. (Music Therapy) majors. (.5 course credit) Annually. Spring. MUSC 19100. RECREATIONAL MUSIC - PROGRAMMING AND LEADERSHIP The main focus of this course is students’ development of a repertoire of activities which will provide a foundation for their initial fieldwork experiences. Adaptation of activities and instruments, basic assessment of client interests and needs, and evaluation by observation are addressed as part of the fieldwork that is required as part of this course. Group leadership skills, time management, and musical skills are also emphasized through student-led activities and class demonstrations. Required of all B.M.E. (Music Therapy) majors. (.5 course credit) Prerequisite: MUSC 19000. Annually. Fall. MUSC 29100. MUSIC THERAPY IN PSYCHIATRY AND REHABILITATION Clinical methods as they relate to working with psychiatric, elderly, medical, head-injured, corrections, and addiction clients. Includes a review of behavioral characteristics, treatment adaptations, current therapeutic intervention models, goals and objectives, and applicable resources. Required of all B.M.E. (Music Therapy) majors. Prerequisite: MUSC 19100. Fall 2011. Music 149 MUSC 29200. MUSIC THERAPY WITH THE DEVELOPMENTALLY DISABLED Clinical practice as it relates to working with mentally retarded, autistic, sensory impaired, physically challenged, and learning-disabled clients. Includes review of behavioral characteristics, treatment considerations, current therapeutic intervention models, goals and objectives, and current literature. Required of all B.M.E. (Music Therapy) majors. Prerequisite: MUSC 19100. Spring 2012. MUSC 29300. PRACTICUM I IN MUSIC THERAPY Practical experience with clients in approved institutions, including a musical and behavioral assessment of the group or individual, the development and implementation of ongoing treatment procedures, and evaluation. To be taken in conjunction with MUSC 29100, 29200, and 39400. Required of all B.M.E. (Music Therapy) majors. (.25 course credit) Prerequisite: MUSC 19100. Annually. Spring. MUSC 29400. PRACTICUM II IN MUSIC THERAPY Practical experience with clients in approved institutions. Continuation of MUSC 29300. Required of all B.M.E. (Music Therapy) majors. (.25 course credit) Prerequisite: MUSC 29300. Annually. Fall. MUSC 29500. ADVANCED PRACTICUM IN MUSIC THERAPY Practical experience with clients in approved institutions. Continuation of MUSC 29400. Required of all B.M.E. (Music Therapy) majors. (.25 course credit) Prerequisite: MUSC 29400. Annually. Spring. MUSC 39200. PSYCHOLOGY OF MUSIC Study of the basic principles of musical acoustics and the relationship between the human apparatus of hearing and actual perception of music. Research literature is reviewed for the psychology of musical abilities, emotion and meaning in music, development of musical preference, and behavior of music listeners. Required of all B.M.E. (Music Therapy) majors. (.5 course credit) Prerequisite: MUSC 19100. Annually. Fall. MUSC 39300. RESEARCH SEMINAR IN MUSIC THERAPY This course provides students with practical exposure to research methods. Students will pursue independent research projects using the critical review of literature completed by them in the preceding course and augmented by instruction in test design and the most common methods of data analysis: correlation, analysis of variance, non-parametric and parametric statistics. Also includes critique of several consumer-oriented periodicals and the benefit of these publications to public education about Music Therapy. Required of all B.M.E. (Music Therapy) majors. (.5 course credit) Prerequisite: MUSC 39200. Annually. Spring. MUSC 39400. PROGRAM DEVELOPMENT AND ADMINISTRATION IN MUSIC THERAPY Program planning, scheduling, budgeting, and public relations strategies are main topics. Documentation procedures, including current standards for various types of agencies, and legislative issues relating to Music Therapy practice are also covered. Music Therapy in the milieu approach and the Music Therapist as a member of the treatment team. Structure and function of local, state, and national Music Therapy organizations, including Standards of Practice and Code of Ethics. Required of all B.M.E. (Music Therapy) majors. Prerequisite: MUSC 19100. Alternate years. Spring 2012. MUSC 40700, 40800. INTERNSHIP A six-month, full-time (1,040 clock hours) clinical experience in an American Music Therapy Association (AMTA)-approved facility. Involves general orientation to the institution, observation of the therapist, and personal involvement in observing, describing, and providing music therapy to clients. Documentation and special research projects are included according to the clinical internship training plan. Application for internship is generally initiated late in the junior year; the internship must be completed within two years of completing coursework. Required of all B.M.E. (Music Therapy) majors. (.25 course credit) S/NC course. Annually. GENERAL COURSES IN MUSIC One credit per course unless otherwise specified. MUSC 40000. TUTORIAL Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. MUSC 40100. JUNIOR INDEPENDENT STUDY A one-semester, creative, individual program of study in music performance, music history and literature, or music theory-composition, corresponding to the student’s degree track. The Junior I.S. in music performance leads to the presentation of a public recital 25-30 minutes in length. The Junior I.S. in music history and literature emphasizes bibliographical and research methods, major library resources, and writing style, and results in a major paper. In music composition the Junior I.S normally consists of at least two pieces in small forms Music 150 planned for public performance by performers or ensembles available at the College. Junior I.S. projects in music theory yield written analyses of music. MUSC 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study, in which the student engages in a creative, individual program of study in music performance, music history and literature, or music theory-composition, corresponding to the student’s degree track, which will be ultimately completed in the second semester of Senior Independent Study. The Senior I.S. in music performance leads to the presentation of a public recital 50-60 minutes in length, with a supporting document of ten pages length in the case of B.A. majors. The Senior I.S. in music history and literature emphasizes bibliographical and research methods, major library resources, and writing style, and results in a major paper at least 60 pages in length. In music composition the Senior I.S normally consists of one composition on a larger scale planned for public performance by performers or ensembles available at the College. Senior I.S. projects in music theory yield written analyses of music at least 60 pages in length. Prerequisite: MUSC 40100. MUSC 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, in which the student engages in and completes a creative, individual program of study in music performance, music history and literature, or music theorycomposition, corresponding to the student’s degree track. Prerequisite: MUSC 45100. GLCA ARTS PROGRAM IN NEW YORK See Off-Campus Study. MUSIC PERFORMANCE GROUPS Non-majors may receive up to a total of one course credit for participation in the following music performance groups, all of which are graded S/NC. MUSC 16000. WOOSTER SINGERS A choir open to all without audition. This ensemble explores choral music of a wide range of styles and historic periods and develops sightsinging skills. Performances will be scheduled depending on the size and preparation of the ensemble. Two hours per week. Two semesters of enrollment required of all B.M. and B.M.E. majors (except B.M. Voice majors, who may substitute MUSC 16100 instead); these semesters must be Fall and Spring of the same year, except by permission of the instructor. (.125 course credit) Annually. Fall and Spring. MUSC 16100. WOOSTER CHORUS A performing choir dedicated to the performance of the finest sacred and secular choral works of the past five centuries. In addition to presenting several programs on campus each year, the Wooster Chorus makes a concert tour during spring vacation. Admission is by audition. Four hours per week. (.125 course credit) Annually. Fall and Spring. MUSC 16200. WOOSTER SYMPHONY ORCHESTRA A performing organization comprised of students and members of the community devoted to the study and performance of the standard orchestral repertoire from the baroque to the contemporary. Admission is by audition. Four hours per week. Four regular concerts per year. (.125 course credit) Annually. Fall and Spring. MUSC 16300. SCOT BAND A performing organization whose emphasis during the fall season is on marching, with the latest techniques and best quality of appropriate music being prepared and performed. In winter and spring the band studies and performs the best in band literature from all periods for performance on tour and for home audiences. Admission to the Symphonic Band is by audition. Membership in the Marching Band is open to any student without audition. Selected music education majors are given the opportunity to prepare and conduct compositions. Four hours per week. (.125 course credit) Annually. Fall and Spring. MUSC 16400. WOOSTER JAZZ ENSEMBLE A performing organization which prepares and performs suitable literature in the jazz idiom for large ensemble. Opportunity is given for composing, arranging, and improvisation. Three hours per week. (.125 course credit) Annually. Fall and Spring. MUSC 16500. GOSPEL CHOIR (AFRICANA STUDIES) A performing organization, open to any student, faculty, or staff person at the College and to members of the community, offering live performance in a secular context of serious African American choral music. Two hours per week. (.125 course credit) Annually. Fall and Spring. Neuroscience 151 MUSC 16600. OPERA WORKSHOP Study of basic stage movement through the analysis and staging of scenes and arias from the standard and contemporary repertoire. (.25 course credit) May be taken more than once. Prerequisite: permission of instructor required. Annually. Spring. PREPARATORY DEPARTMENT The Music Department accepts each year for private instruction in a performance area a limited number of non-matriculated students. For such students, the College provides no housing or meals. Information regarding teachers and entrance requirements may be obtained from the Administrative Coordinator of the Department of Music. NEUROSCIENCE CURRICULUM COMMITTEE: Amy Jo Stavnezer (Psychology), Chair Dean Fraga (Biology) Gary Gillund (Psychology) Sharon Lynn (Biology) Neuroscience is an exceptionally diverse and interdisciplinary field that incorporates aspects of biology, psychology, chemistry, philosophy, computer science, and other disciplines in the study of the nervous system. Neuroscientists seek to understand the function of the brain, spinal cord and peripheral nervous system at multiple levels, from the complex processes that occur in single neurons to the expansive cellular networks that ultimately give rise to perception, emotion, cognition, and even social behavior. The Neuroscience Program is thus a multidisciplinary program with the curriculum consisting of a combination of nine required foundational courses currently required for majors in Chemistry, Biology, and Psychology. Neuroscience continues to draw from, inform and expand the disciplines of Biology and Psychology in a variety of ways, and therefore emphasizes these areas in its curriculum. Students can choose from a variety of upper level electives according to their personal interests and career goals. The goals of the Neuroscience Program are to provide students with the essential foundational knowledge, skills, confidence and research experiences that will allow them to identify and meet their intellectual and professional goals. In addition, it will produce liberally educated scientists who are well-versed in scientific methodology and its application, who possess a thorough knowledge of fundamental neuroscientific concepts, and who are able to express themselves with clarity, both orally and in writing. Major in Neuroscience Consists of fifteen courses: • PSYC 10000 • CHEM 12000 • BIOL 20000 • BIOL 20100 • PSYC 23000 • One of the following courses: PSYC 25000 or MATH 10200 Neuroscience 152 • NEUR 32300 • NEUR 38000 • Four elective courses, from two or more departments, from cross-listed courses accepted for NEUR credit • Junior Independent Study: NEUR 40100 • Senior Independent Study: NEUR 45100 • Senior Independent Study: NEUR 45200 Special Notes • See Chemistry Department information on placement exams for CHEM 11000/12000. • First year students are advised to complete all 100-level courses and at least one 200-level course by the end of the first year. • The Core courses (PSYC 10000, 23000, 25000, BIOL 20000, 20100, CHEM 12000, NEUR 32300 and 38000) and at least two electives should be completed by the end of the Junior year. • The electives BIOL 34400, 35200, and 37700 require BIOL 20200 as a prerequisite or special permission of instructor. • The laboratory and classroom components are closely integrated in the upper-level Biology and Psychology courses and must therefore be taken concurrently. The course and laboratory grades will be identical and are based on performance in both components; the relative weights of the two components are stated in each course syllabus. • For I.S., students can work with faculty advisers that are on the Neuroscience curriculum committee or other faculty members in Psychology, Biology, Chemistry or Biochemistry and Molecular Biology, with their permission. • Students are also encouraged to take the following courses, which are requirements for many graduate, medical and other pre-professional programs: CHEM 21100 and 21200 (Organic Chemistry sequence), CHEM 33100 and 33200 (Biochemistry sequence), PHYS 20300, and MATH 11100 (OR both MATH 10700 and 10800). • A double-major with Biochemistry and Molecular Biology, Biology, Chemistry or Psychology is not an option. • If a student majors in Neuroscience, a minor in Biochemistry and Molecular Biology, Biology, Chemistry or Psychology must consist of six courses that do not double-count with the Neuroscience major. • No minor in Neuroscience is offered. • Only grades of C- or better are accepted for the major. NEUROSCIENCE COURSES PSYC 10000. INTRODUCTION TO PSYCHOLOGY [HSS] CHEM 12000. PRINCIPLES OF CHEMISTRY [Q, MNS] BIOL 20000. FOUNDATIONS OF BIOLOGY [MNS] BIOL 20100. GATEWAY TO MOLECULAR AND CELLULAR BIOLOGY [Q, MNS] PSYC 23000. HUMAN NEUROPSYCHOLOGY [HSS] PSYC 25000. INTRODUCTION TO STATISTICS AND EXPERIMENTAL DESIGN [Q] NEUR 32300. BEHAVIORAL NEUROSCIENCE (Communication, Psychology) An introduction to the anatomical and physiological basis of animal and human behavior. Content areas Neuroscience 153 include basic neuronal physiology and brain anatomy, neural/endocrine interactions, methods in neuroscience, control of movement, sexual development and behavior, sleep, learning and memory, and physiological correlates of psychopathology. Includes a 3-hour laboratory in addition to class. The laboratory and classroom components are closely integrated and must be taken concurrently. (1.25 course credits) Prerequisite: PSYC 25000. Annually. Fall. [W] NEUR 38000. CELLULAR NEUROSCIENCE (Biochemistry and Molecular Biology, Biology) This course focuses on the cellular and molecular aspects of the nervous system. Topics include nerve cell physiology, synapse structure and formation, axon guidance, simple pattern generators, and the cellular basis of learning and memory. Three lecture periods and one laboratory period weekly. Recommended: one upper-level Biology course or NEUR 32300. Prerequisite: C- or better in BIOL 20100, CHEM 12000 or permission of instructor. Annually. Spring. NEUR 40100. INTRODUCTION TO INDEPENDENT STUDY Students will attend weekly classroom meetings which focus on science writing, accessing and evaluating primary literature, and experimental design. The major paper will include a literature review and a detailed research proposal related to their I.S. thesis research. Students will also participate in the peer review process as well as present an oral research proposal presentation at the end of the semester. Annually. Spring. NEUR 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: NEUR 40100. NEUR 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: NEUR 45100. CROSS-LISTED COURSES ACCEPTED FOR NEUROSCIENCE CREDIT BIOLOGY BIOL 30400. HUMAN PHYSIOLOGY BIOL 30500. CELL PHYSIOLOGY [W†] BIOL 30600. GENES AND GENOMES BIOL 30700. DEVELOPMENT BIOL 34400. COMPARATIVE ANIMAL PHYSIOLOGY BIOL 35200. BEHAVIORAL ECOLOGY BIOL 37700. BEHAVIORAL ENDOCRINOLOGY INTERDEPARTMENTAL IDPT 20011. NEUROSCIENCE OF LEARNING AND MEMORY [MNS] PHILOSOPHY PHIL 21500. BIOMEDICAL ETHICS [AH] PHIL 30400. PHILOSOPHY OF MIND AND COGNITIVE SCIENCE [AH] PSYCHOLOGY PSYC 21200. ABNORMAL PSYCHOLOGY [HSS] PSYC 32100. LEARNING AND BEHAVIOR [W] PSYC 32200. MEMORY AND COGNITION [W] PSYC 33500. PERCEPTION AND ACTION [W] 154 PHILOSOPHY Elizabeth Schiltz, Chair Grant Cornwell Ronald E. Hustwit Henry B. Kreuzman Lee McBride Evan Riley John Rudisill Garrett Thomson The Philosophy Department has as its fundamental mission the cultivation of skills, dispositions, and knowledge in its students that contribute to their development as autonomous persons and as responsible and engaged members of society. These skills and dispositions are acquired and honed through studying and doing philosophy. They facilitate a student’s development by enabling the critical, systematic, and philosophically informed examination of beliefs, values, and conceptions of the world. Such an individual has an independent mind: one that is open, flexible, creative, critical, and capable of making well-reasoned decisions. Philosophy is the critical search for understanding through argumentation and the analysis of concepts. Philosophical issues arise in all areas of human inquiry, and consequently the types of questions that philosophy examines are surprisingly diverse. Does the world consist only of matter? What does it mean to be rational? What is the relationship between law and morality? Do computers think? What obligations do we have to the environment? In answering such questions, one acquires skills in critical reading, writing, and discussion, conceptual analysis, argumentation, and identification of presuppositions. Thus, philosophy helps to enrich, expand, and develop one’s liberal arts education. Many students have found a minor in philosophy to be a valuable supplement to other majors in the natural and social sciences and other humanities departments. Major in Philosophy Consists of ten courses: • PHIL 22000 • PHIL 25000 • PHIL 25100 • One of the following 300-level courses: PHIL 30100, 30200, 30300, or 30400 • PHIL 31100 • Two elective Philosophy courses • Junior Independent Study: PHIL 40100 • Senior Independent Study: PHIL 45100 • Senior Independent Study: PHIL 45200 Minor in Philosophy Consists of six courses: • One of the following 200-level courses: PHIL 25000 or 25100 • One 300-level course: PHIL 30100, 30200, 30300, 30400, 31000, 31100, or 31200 • Four elective Philosophy courses Special Notes • Students are strongly encouraged to take PHIL 10000 as a first course in Philosophy. • Majors and minors are not permitted to take any courses within the department for S/NC credit. • Only grades of C- or better are accepted for the major or minor. Philosophy 155 PHILOSOPHY COURSES ETHICS, JUSTICE, AND SOCIETY PHIL 10000. ETHICS, JUSTICE AND SOCIETY Philosophy aims to understand and solve fundamental conceptual problems in all areas of human inquiry. Philosophical reasoning deals with such problems in a systematic and rigorous way. The aim of this course is to introduce the practice of doing philosophy. This course will focus upon questions relating to ethics and political philosophy, and will address methods of argumentation and critical reasoning. Annually. Fall and Spring. [AH] PHIL 21000. JURISPRUDENCE: LAW AND SOCIETY This course examines the nature of law, its relation to coercive power and to morality. How should one define law? In what way should precedent determine the decisions of judges? As well as investigating these classical questions of jurisprudence, it will also study contemporary criticism of legal theory, the relationship of the law to justice, and important legal cases. Spring 2012. [AH] PHIL 21200. RACE, GENDER AND JUSTICE (Women’s, Gender, and Sexuality Studies) This course examines various historical and contemporary attempts to theorize race and gender and answer the questions ‘what is race?’ and ‘what is gender?’ Further, we will look at the ways in which “race” and “gender” pose problems for traditional conceptions of justice and inquire into the degree to which these problems warrant substantive revision of our favored theories of justice. Authors discussed include W.E.B. DuBois, Alain Locke, Franz Fanon, Anthony Appiah, Iris Marion Young, and Nancy Fraser. Alternate years. Not offered 2011- 2012. [C, AH] PHIL 21500. BIOMEDICAL ETHICS (Neuroscience) This course examines the ethical problems that arise within medicine and health care. Ethical questions relating to the physician-patient relationship, reproductive rights, abortion, AIDS, physician-assisted suicide, patient autonomy, and the allocation of resources will be addressed. Alternate years. Fall 2011. [AH] PHIL 21600. ENVIRONMENTAL ETHICS (Environmental Studies) This course is an examination of the ethical obligations that humans have toward the environment. What is the nature and source of our obligations to animals, plants, and the environment as a whole? Can non-human entities have rights? We will evaluate various approaches to these questions including anthropocentrism, biocentrism, and eco-feminism. Alternate years. Fall 2011. [AH] PHILOSOPHY AND THE LIBERAL ARTS PHIL 22000. LOGIC AND PHILOSOPHY This course examines the development of formal logic from categorical logic to sentential and predicate logic. In addition, the course evaluates the nature of formal logical systems and the philosophical issues related to them. Such issues include puzzles about sets, conditional statements, induction, contradiction, and the nature of truth and meaning. Annually. Fall and Spring. [AH] PHIL 22100. PHILOSOPHY AND THE RELIGIOUS LIFE (Religious Studies) In one part of this course we will look at traditional issues in the philosophy of religion: the nature of religious experience, classical proofs for the existence of God, and the problem of evil. In the second part of the course we will focus on issues in religious language, “seeing God,” the place of ceremony and liturgy in religious life, and religious pluralism. Alternate years. Spring 2012. [R, AH] PHIL 22200. SCIENTIFIC REVOLUTIONS AND METHODOLOGY The traditional view of scientific method, with its emphasis on observation, prediction, falsification, and hypothesis forming, is often thought to be a model of rationality. Yet there have been several conceptual revolutions in science that seem to challenge this view. The course will critically evaluate the scientific method, including empiricist, post-modern, and feminist critiques. Alternate years. Spring 2012. [AH] PHIL 22300. PHILOSOPHY, CULTURE, AND EDUCATION (Education) The philosophical study of education includes such issues as the formation of knowledge, curriculum rationale, conceptions of human nature, the requirements of citizenship, and the cultivation of intellectual and moral virtues. Alternate years. Not offered 2011-2012. [AH] Philosophy 156 PHIL 22400. ART, LOVE, AND BEAUTY What is the relationship between the artist, the work of art, and the audience? In this course, we will learn to say something meaningful about different forms of art, such as dance, music, architecture, and visual arts. What is it to appreciate them? What do we see, hear, feel? What is art’s relationship to culture, to perception, to judgment? How do classical theories of aesthetics interface with modern and post-modern views? Alternate years. Spring 2012. [AH] COMPARATIVE PHILOSOPHY PHIL 23000. EAST/WEST COMPARATIVE PHILOSOPHY (East Asian Studies, South Asian Studies) This course is an examination of fundamental issues in philosophy, focusing on the work of philosophers in the Indian, Chinese, and Western traditions. Special attention will also be given to critical reflection on the project of comparative philosophy. Alternate years. Fall 2011. [W†, C, AH] PHIL 23100. INDIAN PHILOSOPHY AND ITS ROOTS (South Asian Studies) This course is an examination of the unique Indian tradition of philosophy, including careful study and analysis of the Vedic and Upanishadic inheritance, ”Heterodox” developments, such as the Buddhist and Jaina systems, and the “Orthodox” schools of Hindu philosophy, as well as later developments in Indian thought. Each offering of this course will focus on a distinct philosophical theme. Alternate years. Spring 2012. [W†, C, AH] PHIL 23200. CHINESE PHILOSOPHY (Chinese Studies, East Asian Studies) An examination of traditional Chinese thought, in translation, with emphasis on philosophical problems. The topics to be covered in lectures and discussions will include Confucianism, Taoism, Buddhism, Neo-Confu - cianism, and Ch’ing empiricism. Alternate years. Spring 2012. [C, AH] PHIL 23400. AFRICAN PHILOSOPHY (Africana Studies) An examination of the African tradition of philosophy, including the epistemology and metaphysics, ethics, and political philosophy. The primary focus will be the various concepts in ethics and political philosophy, particularly, as these issues arise within the political and social structures in post-colonial Africa. Alternate years. Not offered 2011-2012. [C, AH] HISTORICAL FOUNDATIONS PHIL 25000. ANCIENT PHILOSOPHY: PLATO AND ARISTOTLE (Classical Studies) This course examines the major philosophical texts of Ancient Greece and the presocratic writings out of which they grew. The writings of these philosophers have implications for contemporary politics, education, morality, and knowledge. Annually. Fall and Spring. [AH] PHIL 25100. RATIONALISM AND EMPIRICISM During the period from about 1600 to 1800, modern science emerged, and the Medieval worldview receded. These deep changes led to a re-evaluation of our understanding of knowledge, God, and the human mind. This course focuses on the Empiricist philosophies of Locke, Berkeley, and Hume, and the Rationalism of Descartes, Spinoza, Leibniz, and Kant. Their work will be used to introduce some crucial debates in philosophy today. Annually. Fall 2011. [AH] PHIL 26100. THEMES IN CONTINENTAL PHILOSOPHY This course is meant to give an introduction to the major figures and schools of thought of phenomenology, hermeneutics, post-structuralism, and critical theory, paying particular interest to continental conceptions of subjectivity, rationality, and ethics. We will become well acquainted with the theoretical frameworks of four challenging and provocative philosophers, namely: Martin Heidegger, Hans-Georg Gadamer, Michel Foucault, and Jürgen Habermas. This will entail the careful reading, interpretation, and discussion of difficult texts as well as the exposition, critique, and construction of arguments. Alternate years. Not offered 2011-2012. [AH] PHIL 26400. EXISTENTIALISM What are the philosophies by which people live? Can abstract systems of philosophy be a guide to life? Existentialism claims that existence is an enigma and that abstract systems of philosophy have failed to explain it. What philosophy will stand in the place of these systems? Readings will be taken from such writers as Camus, Sartre, Dostoevsky, Heidegger, and Kafka. Alternate years. Spring 2012. [AH] PHIL 26600. AMERICAN PHILOSOPHY This course offers a detailed examination of the central doctrines of two or more of the following American philosophies: transcendentalism, American idealism, pragmatism, and neo-pragmatism. General topics Philosophy 157 include: (i) the effects of evolutionary theories to our conceptions of reality and truth, (ii) the motivations behind individualism and collectivism, and (iii) melioristic faith in moral and religious ideals. Readings will be drawn from such writers as Ralph Waldo Emerson, Margaret Fuller, Charles Sanders Peirce, William James, Josiah Royce, George Herbert Mead, John Dewey, Jane Addams, Alain Locke, Cornel West, and Richard Rorty. Alternate years. Not offered 2011-2012. [AH] ADVANCED SEMINARS IN PHILOSOPHY PHIL 30100. ONTOLOGICAL COMMITMENTS Ontology, as part of metaphysics, investigates the general features of what there is, and takes up questions about topics as diverse and central as universals, particulars, space, time, causation, and persistence. This class undertakes a rigorous investigation of the ontological commitments we have – and works toward an understanding of which ones we should have. At the same time, it develops students’ skills in critical interpretation, analysis, argumentation, and expression. Prerequisite: a minimum of two Philosophy courses. Alternate years. Spring 2012. [AH] PHIL 30200. EPISTEMOLOGY: RATIONALITY AND OBJECTIVITY This course examines the nature and scope of human knowledge. What does it mean to be rational? What is objectivity? Can humans obtain knowledge and truth? We will critically examine answers presented by foundationalism, coherentism, reliabilism, and naturalized epistemology. Prerequisite: a minimum of two Philosophy courses. Alternate years. Not offered 2011-2012. [AH] PHIL 30300. UNDERSTANDING LANGUAGE What is meaning? How do we understand each other? To what do words refer? Formal theories of meaning and syntax offer one kind of answer to these questions. Other answers focus on communicative behavior and speech acts. Still others focus on the metaphorical use of language and context. We will critically evaluate these different approaches. Prerequisite: a minimum of two Philosophy courses. Alternate years. Fall 2011. [AH] PHIL 30400. PHILOSOPHY OF MIND AND COGNITIVE SCIENCE (Neuroscience) What is the relation between the mind and the brain? Is consciousness a neurological function? What are the limits of artificial intelligence? During this century, there has been a dramatic revolution in our understanding of these and other issues. We will follow and critically evaluate some of these changes. Prerequisite: a minimum of two Philosophy courses. Alternate years. Not offered 2011-2012. [AH] PHIL 31000-31009. SEMINAR IN PHILOSOPHY A topical seminar which focuses upon a special issue or the work of a particular philosopher. Prerequisite: a minimum of two Philosophy courses. Alternate years. Fall 2011. [W†, AH] PHIL 31100. ETHICAL THEORY In this course, we will examine and compare the main theories of ethics: utilitarianism, Kant’s Ethics, virtue theory, feminist ethics, and moral cognitivism. The focus of this course will be on the foundations of moral principles. Prerequisite: a minimum of two Philosophy courses. Annually. Fall and Spring. [AH] PHIL 31200. POLITICAL PHILOSOPHY This course explores themes in political philosophy from the 19th century to the present. It addresses fundamental questions about the conditions for a political state’s legitimacy, citizens’ obligations, the nature of justice and rights, and the concept of fairness in respect to the distribution of resources. We will also examine questions about pluralism, the good life, and the relationship between conceptions of the good life and public/political institutions. Can and should our political institutions be neutral with respect to conceptions of the good life? Prerequisite: a minimum of two Philosophy courses. Alternate years. Fall 2011. [AH] PHIL 40000. TUTORIAL A tutorial course on a special topic offered to an individual student under the supervision of a faculty member. (.25 – 1 course credit) Prerequisite: The approval of both the supervising faculty member and the chairperson are required prior to registration. INDEPENDENT STUDY PHIL 40100. JUNIOR INDEPENDENT STUDY A seminar designed to help students further develop their ability to do independent research in philosophy and to write a philosophical thesis. In order to achieve this goal, the course will require students to examine questions about the nature and methodology of philosophy, engage in research using philosophical journals and Physical Education 158 electronic data bases, deliver oral presentations, participate in peer review of others’ writing, and plan and write a philosophical paper. PHIL 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: PHIL 40100. PHIL 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: PHIL 45100. PHYSICAL EDUCATION Keith Beckett, Chair Lisa Campanell-Komara Brenda Meese Steve Moore Tim Pettorini Mike Schmitz The Department of Physical Education, Athletics, and Recreation supports the belief that participation in physical activity and sports are integral components of the culture in which we live. The values and concepts inherent in sports are parallel to those developed within the framework of a liberal arts education. Skills learned through physical activity and sport participation are valuable personal, social, and recreational tools which may be used to enrich the lives of men and women within society. The department is committed to create and develop a unique program of health, fitness, and leisure education dedicated to improving the quality of life and promoting longevity. The discipline of Physical Education challenges us to: • acquire and maintain a level of fitness and wellness necessary to enhance the quality of life; • develop a coordinated body and efficient movement patterns that will be understood and utilized by us during activity; • become more proficient in one or more activities which give personal satisfaction, enjoyment, and leisure time resources during and beyond college; • develop through sport experiences and physical activity the values and standards of conduct inherent in participation in sport and recreational activity. Minor in Physical Education Consists of six courses: • Six Physical Education courses at the 200-level or beyond Special Notes • Only grades of C- or better are accepted for the minor. LIFETIME SPORT AND PERSONAL CONDITIONING COURSES (.25 course credit) The Department of Physical Education offers courses in a variety of lifetime sports and personal conditioning activities. These courses meet for one-half semester. The focus of these courses is for students to acquire and further develop the fundamental Physical Education 159 skills/knowledge that would allow them to participate in a selected sport or activity. Students may earn one-quarter credit for each lifetime sport course, and no more than four of these courses may count for degree completion credit. Students who participate on intercollegiate teams may earn .25 course credit (one time) for their participation by registering for PHED 13001-13002. PHED 10001-10002. ARCHERY PHED 10101-10102. BADMINTON, BEGINNING PHED 10301-10302. BASIC SELF DEFENSE, BEGINNING PHED 10401-10402. BASIC SELF DEFENSE, INTERMEDIATE PHED 10801-10802. BOWLING, BEGINNING PHED 10901-10902. BOWLING, INTERMEDIATE PHED 11101-11102. GOLF, BEGINNING PHED 11201-11202. GOLF, INTERMEDIATE PHED 11501-11502 KARATE, BEGINNING PHED 11601-11602. KARATE, INTERMEDIATE PHED 11801-11802. PERSONAL CONDITIONING PHED 11901-11902. PERSONAL CONDITIONING, ADVANCED PHED 12001-12002. PLYOMETRICS PHED 12201-12202. SCUBA, BEGINNING PHED 12301-12302. SCUBA, ADVANCED PHED 12601-12602. TABLE TENNIS PHED 12701-12702. TENNIS, BEGINNING PHED 12801-12802. TENNIS, INTERMEDIATE PHED 13001-13002. VARSITY SPORTS (S/NC course) PHED 13201-13202. YOGA PHYSICAL EDUCATION COURSES PHED 20000. WOMEN IN SPORT (Women’s, Gender, and Sexuality Studies) Psychological, sociological, and physiological factors that contribute to an interest and ability to participate in sports, with special reference to those factors particularly significant to women. This course also reviews relevant historical and current events. Spring 2012. PHED 20100. COACHING OF INDIVIDUAL AND TEAM SPORTS The philosophies, methods, and strategies involved in the coaching of individual and team sports. Not offered 2011-2012. PHED 20200. ELEMENTARY PHYSICAL EDUCATION Development of a fundamental movement foundation along with skills and knowledge necessary for sequencing educational games, rhythms, and gymnastics. Spring 2012. PHED 20300. KINESIOLOGY An examination of the structure and function of the human muscular and skeletal systems. Emphasis will be placed on the mechanical analysis of human movement. Not offered 2011-2012. PHED 20400. EXERCISE, NUTRITION, AND STRESS MANAGEMENT Study of the basic concepts of nutrition, the elementary principles of exercise physiology, and the physiological principles of stress as well as the relationship of these subject areas to one another. Not offered 2011-2012. PHED 20500. SPORT IN AMERICAN LIFE A study of the social phenomena, economic roles, and the psychological and cultural consequences of sport in American life, with particular reference to social and psychological factors. Topics such as the interaction of sport and other social institutions and the competitive process will be examined. Not offered 2011-2012. PHED 20600. PREVENTION AND CARE OF ATHLETIC INJURIES Personal and team conditioning methods, standard first aid techniques, methods and materials for prevention and care of injuries common in athletic activities and their appropriate rehabilitation techniques. Spring 2012. Physics 160 PHED 20700. ADMINISTRATION OF PHYSICAL EDUCATION Professional planning of physical education programs with special reference to curriculum development, facilities, equipment, legal liability, and public relations. Spring 2012. PHED 20800. EXERCISE PHYSIOLOGY A study of the effects of various activities and environmental factors on the system of the body and an investigation of the capacity of individuals to meet the demands imposed on them to determine how this capacity can be influenced by training and acclimatization. Not offered 2011-2012. PHED 30800. PRACTICUM IN COACHING/ATHLETIC TRAINING AND PHYSICAL THERAPY Prerequisite: Approval of the department chairperson. Fall 2011 and Spring 2012. PHED 40000. TUTORIAL A tutorial course on special topics offered to an individual student under the supervision of a faculty member. Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. Fall and Spring. PHYSICS John Lindner, Chair Shila Garg Donald Jacobs Cody Leary Susan Lehman Karen Lewis Nicole Moore Why is the sky blue? Why is water wet? In seeking to understand natural phenomena as simply as possible, physicists have made a remarkable discovery: whatever questions they ask the answers ultimately involve the same elegant principles of energy and momentum, mass and charge. Physicists seek and study rhythms and patterns among natural phenomena, including those that are readily apparent (like the orbits of planets) and those that are apparent only to deep analysis and careful observation (like the quantum fluctuations of atoms). Abetted by the power of mathematics, they ultimately comprehend and express the fundamental regularities of the physical universe in uniquely human metaphors. In this way, the universe comes to know itself in human terms. A Physics major provides a rigorous grounding in the scientific process and a firm scientific understanding of the world. It fosters critical thinking and provides broad practical training in science and technology. It can lead to graduate study and basic research (in a variety of disciplines), to stimulating jobs in industry, or to challenging and rewarding careers in teaching. Our faculty is engaged in original research, and our students are drawn early into collaborative research projects with faculty. Major in Physics Consists of fifteen courses: • MATH 11100 • MATH 11200 • MATH 21200 • PHYS 20300 • PHYS 20400 • PHYS 20500 • One of the following courses: PHYS 22000 or 23000 Physics 161 • PHYS 20800 • PHYS 30100 • PHYS 30200 • PHYS 30400 • One of the following courses: PHYS 30300, 30500, 32000, 35000, or 37700 • Junior Independent Study: PHYS 40100 • Senior Independent Study: PHYS 45100 • Senior Independent Study: PHYS 45200 Minor in Physics Consists of six courses: • PHYS 20300 • PHYS 20400 • PHYS 20500 • Three elective Physics courses, only one of which can be PHYS 11000, 12100, or 12200 Special Notes • The Foundations sequence PHYS 20300, 20400 is a prerequisite for the selection of Physics as a major and is best taken the first year, although one can still complete the major if the sequence is taken the second year. • The Calculus sequence MATH 11100, 11200 must be taken at least concurrently with the Foundations sequence, although MATH 10700, 10800 may substitute for MATH 11100. • Those students considering graduate study in physics should also take PHYS 35000, MATH 21100, CHEM 11000, 12000, and as many advanced Physics courses as can be scheduled. • Those students considering astronomy or astrophysics as a career should major in Physics and take PHYS 12100, 12200, and 32000. • For students interested in engineering, Physics is a natural basis for 3-2 engineering programs, which are described under Pre-Professional and Dual Degree Programs. However, such students must complete enough physics in three years to complete the major in the fourth year, if necessary. • PHYS 10100, 10200, 11000, 12100, and 12200 do not count toward a Physics major (except by special permission of the department). • Advanced Placement: A student may receive credit if a score of 4 or 5 is obtained on any of the following AP examinations: Physics B Physics C: Mechanics Physics C: Electricity and Magnetism. •Students need to check with the chairperson of the department to determine whether they will receive one or two credits toward graduation and at what level they should begin their college Physics courses. The advanced placement policy of the College is explained in the section on Admission. Students who have taken a college level physics course (other than Advanced Level or AP Exam) and would like to place beyond the first Physics course need to take a placement exam that the chairperson administers. • No student may receive credit for both PHYS 10100 and 20300 or PHYS 10200 and 20400. • The laboratory and classroom components are closely integrated in Physics courses with a laboratory and must therefore be taken concurrently. The course Physics 162 grade and the laboratory grade will be identical and are based on performance in both components; the relative weight of the two components will be stated in each course syllabus. • Physics majors cannot use S/NC grading option for the required courses, and the department recommends they not use it for any course in Physics, Mathematics, or Chemistry. • Physics minors can use the S/NC grading option for no more than two of the required courses. • Only grades of C- or better are accepted for the major or minor. PHYSICS COURSES PHYS 10100. GENERAL PHYSICS (Communication) Mechanics, heat, wave motion and sound. For students who do not intend to major in physics. Students who have completed one semester of calculus with a grade of C+ or better should take PHYS 20300. Three hours per week plus laboratory. Knowledge of algebra and trigonometry is expected. (1.25 course credits) Annually. Fall. [Q, MNS] PHYS 10200. GENERAL PHYSICS Optics, electricity and magnetism, and atomic and nuclear physics. Three hours per week plus laboratory. (1.25 course credits) Prerequisite: PHYS 10100. Annually. Spring. [Q, MNS] PHYS 11000. PHYSICS REVOLUTIONS Designed for non-science majors, this course explores how physics has revolutionized our understanding of the natural world. Revolutions include the unification of the terrestrial and the celestial in Newton’s Mechanics; of electricity, magnetism and light in Maxwell’s Electromagnetism; of space and time in Einstein’s Theory of Relativity; of particles and waves in Quantum Mechanics. No mathematics beyond high school algebra is assumed. Three hours per week. Fall 2011. [Q, MNS] PHYS 12100. ASTRONOMY OF STARS AND GALAXIES The brilliant and sometimes fuzzy objects in the night sky are dynamic, volatile stars and gigantic galaxies. We will study the general properties of stars as well as how they evolve from birth to death. We will also study the shape and composition of galaxies and the ultimate fate of our universe. Knowledge of high school algebra and trigonometry is expected. Three hours per week. Spring 2012. [Q, MNS] PHYS 12200. ASTRONOMY OF THE SOLAR SYSTEM In just one generation, space exploration has revolutionized our understanding of the solar system. Planets, moons, asteroids and comets have been transformed from obscure and remote objects with mythical names to remarkable and detailed real worlds. In this course, we will study the surprising new solar system that the Space Age continues to reveal. Knowledge of high school algebra and trigonometry is expected. Three hours per week. Not offered 2011-2012. [MNS] PHYS 20300. FOUNDATIONS OF PHYSICS Quantitative development of classical mechanics and thermodynamics. For students who intend to major in physics or chemistry or attend a professional school. Three hours per week plus laboratory. (1.25 course credits.) Prerequisite: MATH 11100 (may be taken concurrently; MATH 10700-10800 may substitute for MATH 11100, but taking MATH 10700 concurrently with PHYS 20300 will defer PHYS 20400 to the next academic year). Annually. Fall. [Q, MNS] PHYS 20400. FOUNDATIONS OF PHYSICS Quantitative development of classical electromagnetism and optics. Three hours per week plus laboratory. (1.25 course credits) Prerequisite: PHYS 20300, and MATH 11200 must be taken at least concurrently. Annually. Spring. [Q, MNS] PHYS 20500. MODERN PHYSICS Space-time physics (relativity, gravitation) and quantum physics (the microworld). Three hours per week plus laboratory. (1.25 course credits) Prerequisite: PHYS 20400 or PHYS 10200 with permission of the instructor. Annually. Fall. [W, Q, MNS] Physics 163 PHYS 20800. MATHEMATICAL METHODS FOR THE PHYSICAL SCIENCES Introduces skills of differential equations, linear algebra, and Fourier analysis essential to the physical sciences and engineering. Three hours per week. Prerequisite: MATH 11200 and PHYS 20400 or permission of the instructor. Annually. Spring. PHYS 22000. ELECTRONICS FOR SCIENTISTS An introduction to the principles and applications of circuit components, operational amplifiers, oscillators, digital logic, analog-to-digital and digital-to-analog, and an introduction to LabVIEW. Three hours per week plus laboratory. (1.25 course credits) Prerequisite: PHYS 10200 or 20400 or permission of the instructor. Fall 2011. [Q, MNS] PHYS 23000. COMPUTATIONAL PHYSICS A project-based introduction to computer simulation that develops increasingly sophisticated numerical models of physical systems in parallel with proficiency in either a modern computer language like C++ or in computational software like Mathematica. Three hours per week plus laboratory. (1.25 course credits) Prerequisite: PHYS 20500 (may be taken concurrently) or permission of the instructor. Alternate years. Not offered 2011-2012. PHYS 30100. MECHANICS Viscous forces, harmonic motion, rigid bodies, gravitation and small oscillations in Newtonian mechanics, Lagrange and Hamilton formulations, computer simulation and numerical methods. Three hours per week. Prerequisite: PHYS 20300 and MATH 21200, PHYS 20800 or permission of the instructor. Fall 2011. PHYS 30200. THERMAL PHYSICS Classical and quantum treatment of problems in thermodynamics and statistical mechanics. Three hours per week. Prerequisite: PHYS 20500. Alternate years. Not offered 2011-2012. PHYS 30300. MODERN OPTICS An introductory course in the basic concepts, principles, and theories of modern optics, including lasers. Topics include wave optics, light and matter interactions, basic laser principles, holography, and specific optical systems. Three hours per week. Prerequisite: PHYS 20500. Spring 2012. PHYS 30400. ELECTRICITY AND MAGNETISM Introduction to classical field theory and Maxwell’s equations of electromagnetism. Three hours per week. Prerequisite: PHYS 20400, 20800, MATH 21200, or permission of the instructor. Alternate years. Not offered 2011-2012. PHYS 30500. PARTICLE PHYSICS An introduction to the concepts and techniques of nuclear and elementary particle physics. Three hours per week. Prerequisite: PHYS 20500. Every three years. Not offered 2011-2012. PHYS 32000. ASTROPHYSICS A quantitative introduction to astronomy and astrophysics. Topics include classical astronomy; stellar structure, stellar atmospheres, and stellar evolution; galactic structure, cosmology, and cosmogony. Emphasis will be on quantitative application of physical theory to astronomical phenomena. Three hours per week. Prerequisite: PHYS 20500. Every three years. Not offered 2011-2012. PHYS 35000. QUANTUM MECHANICS A rigorous introduction to the formalism and interpretation of microworld physics. Probability amplitudes, interference and superposition, identical particles and spin, 2-state systems, Schrodinger evolution, applications. Three hours per week. Prerequisite: PHYS 20500 and 20800, MATH 21200, or permission of the instructor. Alternate years. Spring 2012. PHYS 37700. SELECTED TOPICS Condensed Matter, Nonlinear Dynamics, General Relativity, Introduction to Quantum Field Theory, and others offered when sufficient student interest is shown. For Fall 2011 the topic is Nonlinear Dynamics. PHYS 40000. TUTORIAL Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. PHYS 40100. INDEPENDENT STUDY Laboratory investigations in Mechanics, Thermal Physics, Optics, Quantum, Electricity and Magnetism. Techniques of statistics and data analysis, library utilization, computer interfacing and simulation are explored. One hour per week plus two laboratories. Prerequisite: PHYS 20800 and one of the following: PHYS 30100, 30200, or 30400. Annually. Spring. Political Science 164 PHYS 45100. INDEPENDENT STUDY THESIS – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: PHYS 40100. PHYS 45200. INDEPENDENT STUDY THESIS – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: PHYS 45100. POLITICAL SCIENCE Eric Moskowitz, Chair Elissa Alzate Angela Bos Kent Kille Matthew Krain Jeffrey Lantis Michele Leiby Boubacar N’Diaye Bas van Doorn Mark Weaver Political Science is concerned with the study of power, government, and the state. Power relationships among individuals, groups, nations, and their governmental and policy results are examined using a variety of political science methods, including case studies, textual analysis, field research, interviews, and statistical analysis of quantitative data. The discipline is divided into four major fields, listed below. Students of United States politics examine the interactions among citizens, political parties, interest groups, social movements, and government institutions in the United States. Comparative politics provides students with a broader view of their own society by putting their experience into the context of how other societies in different parts of the world have attempted to solve problems of governance, justice, economic development, and political stability. International relations is concerned with patterns of conflict and cooperation among nations, countries, international organizations, and non-governmental actors such as human rights organizations, terrorist groups, and multinational corporations. Political theorists question the philosophical underpinnings of our understanding of the political world and implications for justice and the common good. A major in Political Science provides the diverse analytical and critical skills appropriate to a liberal arts education at The College of Wooster. Political Science majors often continue their education by attending graduate school or law school. Many of our majors are employed by interest groups, government officials, research organizations, campaigns, and law and business firms. Major in Political Science, Field I: Government and Politics in the United States Consists of eleven courses: • Two 100-level courses: PSCI 11000, 12000, 13000, or 14000 • Three courses in Field I, one of which is PSCI 11000 • Three electives, one from each of the other fields • Two elective Political Science courses Political Science 165 • Junior Independent Study Equivalent: PSCI 35000 • Senior Independent Study: PSCI 45100 • Senior Independent Study: PSCI 45200 Major in Political Science, Field II: International Relations Consists of eleven courses: • Two 100-level courses: PSCI 11000, 12000, 13000, or 14000 • Three courses in Field II, one of which is PSCI 12000 • Three electives, one from each of the other fields • Two elective Political Science courses • Junior Independent Study Equivalent: PSCI 35000 • Senior Independent Study: PSCI 45100 • Senior Independent Study: PSCI 45200 Major in Political Science, Field III: Political Theory Consists of eleven courses: • Two 100-level courses: PSCI 11000, 12000, 13000, or 14000 • Three courses in Field III, one of which is PSCI 13000 • Three electives, one from each of the other fields • Two elective Political Science courses • Junior Independent Study Equivalent: PSCI 33000 • Senior Independent Study: PSCI 45100 • Senior Independent Study: PSCI 45200 Major in Political Science, Field IV: Comparative Politics Consists of eleven courses: • Two 100-level courses: PSCI 11000, 12000, 13000, or 14000 • Three courses in Field IV, one of which is PSCI 14000 • Three electives, one from each of the other fields • Two elective Political Science courses • Junior Independent Study Equivalent: PSCI 35000 • Senior Independent Study: PSCI 45100 • Senior Independent Study: PSCI 45200 Minor in Political Science Consists of six courses: • One 100-level Course: PSCI 11000, 12000, 13000, or 14000 • Five elective Political Science courses, with at least one course in each of two additional fields Special Notes • The two 100-level courses should be completed by the end of the sophomore year. • Students will be asked to confirm their concentration field when they declare their major. • Students who declare a concentration in Field I, II, or IV are required to take PSCI 35000, usually in the junior year. Students who declare a concentration in Field III are required to take PSCI 33000, usually in the junior year. • Students should consult their advisor or the chair of the department concerning which courses might best complement their chosen concentration and interests. • Senior Independent Study is completed in the field of concentration. Political Science 166 • Students may count towards graduation as many as three additional elective courses in Political Science. Indeed, students are strongly encouraged to take additional upper-division political science courses in order to acquire depth of understanding in preparation for internships and Senior Independent Study. • Teaching Licensure: The requirements for the Teacher Education Licensure Program can be found in Teacher Education at the College of Wooster: A Supplement to the Catalogue (which can be found at the following website: www3.wooster.edu/education/current/forms.html). Interested students should consult with the chairs of Political Science and Education during their first year of study. • Advanced Placement: A student may receive advanced placement credit in Political Science if a score of 4 or 5 is obtained on the following AP tests: United States Government and Politics Test: credit for PSCI 11000 Comparative Government and Politics Test: credit for PSCI 14000 •• Qualifying students must see the chair of Political Science. The advanced placement policy of the College is explained in the section on Admission. • Only grades of C- or better are accepted for the major or minor. POLITICAL SCIENCE COURSES Field I: GOVERNMENT AND POLITICS IN THE UNITED STATES PSCI 11000. INTRODUCTION TO UNITED STATES NATIONAL POLITICS An introduction to the major governmental institutions and processes in the United States, and the political forces that continue to shape them. Annually. Fall and Spring. [HSS] PSCI 20100. STATE POLITICS AND POLICY A comparative analysis of state behavior and public policy. The course examines the function of the most significant state institutions (governor, legislature, and courts) as well as the role of state political parties and interest groups. It also focuses on the impact of federalism on state politics and on the causes and consequences of diversity in state politics and public policy. Not offered 2011-2012. [HSS] PSCI 20200. ENVIRONMENTAL POLICY (Environmental Studies) Examines the theories and politics of the U.S. environmental movement and analyzes the process through which environmental policy is made. The first part of the course focuses on the contemporary environmental movement, the environmental critique of present policies, and their proposals for changing the way we think about and interact with the environment. The second part of the course focuses on the political process through which environmental policy is made and on the policy alternatives regarding such topics as air pollution and hazardous waste. Alternate years. Fall 2011. [HSS] PSCI 20300. THE POLITICS OF PUBLIC POLICY Analyzes the nature of the policy-making process with an emphasis on the interactions among the various individual and institutional actors involved at all levels in the U.S. federal system. It examines the processes through which public policies are made in the United States and the various factors that influence their content. Both case studies of policy making and general models of the determinants of public policies are discussed. Alternate years. Fall 2011. [HSS] PSCI 20400. PUBLIC POLICY ANALYSIS An inquiry into the sources and consequences of public policy in the United States. The emphasis is on evaluation and impact rather than process; the approach is by case study in selected areas of contemporary policy. Alternate years. Not offered 2011-2012. [HSS] PSCI 20500. URBAN POLITICS (Urban Studies) An exploration of urban political processes in the context of a federalist governmental structure and a private economic system. Special emphasis is given to the distribution of community power, racial and ethnic conflict, community development, and the economic development of cities. Annually. Spring 2012. [C, HSS] PSCI 20600. POLITICAL PARTIES AND ELECTIONS A systematic examination of elections and political parties focused on how well elections perform their representative function in the United States. Alternate years. Not Offered 2011-2012. [HSS] Political Science 167 PSCI 20700-20712. SPECIAL TOPICS IN UNITED STATES POLITICS A seminar focusing on a selected topic concerning U.S. politics. May be taken more than once. Spring 2012. [HSS] PSCI 20800. RACE AND POLITICS (Africana Studies) The course will explore the role of race in the development of the American political system. The course will evaluate a number of competing theoretical explanations for racial dynamics of contemporary American politics and public policy. While primarily focusing on the United States, there will also be a comparative dimension to the course. Not offered 2011-2012. [C, HSS] PSCI 20900. POLITICAL MOVEMENTS AND COLLECTIVE ACTION Examines collective political action and participation outside of electoral politics. The course focuses on the variety of ways that citizens participate in politics in the United States and Europe, and it examines the conditions under which citizens identify common concerns and join together in political movements to bring about change. Not offered 2011-2012. [C, HSS] PSCI 21000. WOMEN, POWER, AND POLITICS (Womens, Gender, and Sexuality Studies) A comprehensive examination of women as political actors, as candidates for political office, and as elected or appointed governmental officials in the United States. Alternate years. Fall 2011. [C, HSS] PSCI 21100. CONGRESS Examines the U.S. Congress as a representative and policy-making institution. Among topics included are the recruitment and selection process, the organization of Congress, Congressional procedures, the interaction of Congress with other American political institutions, and the impact of these aspects of Congress on public policies. Alternate years. Not Offered 2011-2012. [HSS] PSCI 21200. THE PRESIDENCY The course considers the question of whether the power of the contemporary presidency is appropriate for both effective national policy-making and constitutional democratic accountability. Examines the various political factors that influence the quality of the decision-making process within the modern presidency. Alternate years. Not offered 2011-2012. [HSS] PSCI 21300. THE CONSTITUTIONAL LAW OF CIVIL RIGHTS (Africana Studies) Examines the devel opment and institutionalization of civil rights for racial, ethnic, religious, gender, and class groups in American society. The issue of the Court as an agent for social change will also be explored. Not offered 2011-2012. [C, HSS] PSCI 21400. CONSTITUTIONAL INTERPRETATION AND CIVIL LIBERTIES Examines important political and theoretical questions regarding the rule of law, the nature of constitutional law, and the role of the Supreme Court in the U.S. system of government. The course focuses on these issues in the context of the interpretation and development of civil liberties, such as freedom of expression, freedom of religion, and the right to privacy. Alternate years. Not offered 2011-2012. [HSS] PSCI 21500. TOPICS IN CONSTITUTIONAL LAW AND APPELLATE ADVOCACY Each year this course will focus on detailed analysis of two related constitutional questions that are presented in a hypothetical case problem. The selected constitutional questions will reflect important public policy issues that are currently being litigated in the lower courts, but have not yet reached the Supreme Court. Students will research the relevant authorities cited in the case problem, argue the case before a moot court, and learn to write analytical briefs, legal memoranda, and persuasive briefs. Annually. Fall 2011. [W] PSCI 21600. THE IMPERIAL PRESIDENCY AND THE CONSTITUTION Examines the historical growth of presidential authority in the U.S. through an investigation of presidential prerogative powers and emergency presidential powers delegated by Congress. The course seeks to answer the question of whether the contemporary U.S. constitutional system (including the courts, Congress, and the public) is capable of limiting the powers of the presidency. Among the issues to be considered are: the use of executive orders, presidential war making authority, executive detention of enemies of the state, warrantless wiretapping within the U.S., and the use of executive privilege and the classification of documents. Alternate years. Spring 2012. [HSS] PSCI 21700. MEDIA AND POLITICS A comprehensive analysis of the ways in which the mass media influence politics in the United States. Special attention is paid to the interaction between the media, citizens, and political campaigns. Alternate years. Spring 2012. [W, HSS] Political Science 168 PSCI 21800. POLITICAL PSYCHOLOGY OF MASS BEHAVIOR An introduction to the field of political psychology, an interdisciplinary field that employs cognitive and social psychological theories to examine mass political behavior. The course focuses on United States politics and, specifically, on how ordinary citizens makes sense of their political world. Alternate years. Fall 2011. [HSS] Field II: INTERNATIONAL RELATIONS PSCI 12000. INTRODUCTION TO INTERNATIONAL RELATIONS (International Relations) An introductory level course that focuses on key actors, issues, theories, and political dynamics that shape world politics. The course explores opposing trends toward integration (globalization) and disintegration (conflict) in international politics. Theories are tested in case studies of particular regions, problems, and historical moments. Annually. Fall and Spring. [C, HSS] PSCI 22100. INTERNATIONAL SECURITY (International Relations) An examination of the changing realities of security in the 21st century. Topics include the defense policies of various states and their implications for international stability; the proliferation of nuclear weapons; international terrorism; theories of war; and the prospects for security through negotiation, cooperation, and international organization. Alternate years. Not offered 2011-2012. [HSS] PSCI 22200. PROBLEMS OF THE GLOBAL COMMUNITY (International Relations) A critical analysis of problems confronting the global community — such as population expansion, economic development, environmental degradation, and anarchy — and individual and collective efforts to cope with them. Not offered 2011-2012. [C, HSS] PSCI 22300. UNITED STATES FOREIGN POLICY (International Relations) A critical assessment of the development of United States foreign policy from World War II to the present; examines the key actors and institutions involved in the foreign policy-making process (the President, Congress, interest groups, bureaucracy, public opinion, etc.); and surveys contemporary foreign policy challenges. Annually. Spring 2012. [HSS] PSCI 22400. COMPARATIVE FOREIGN POLICY (International Relations) This course analyzes foreign policy development in comparative perspective. It examines prominent theoretical perspectives and explores the behavior of different countries in Asia, Europe, Latin America, Africa, and the Middle East; and in different issue areas, including national security policy, foreign economic policy, and environmental policy. Alternate years. Fall 2011. [C, HSS] PSCI 22500. THE UNITED NATIONS SYSTEM (International Relations) An in-depth examination of the United Nations System, including historical background, organizational structure, procedures, and global problems handled. An extended Model United Nations simulation provides a detailed feel for the decision-making process involved in addressing issues through the United Nations. Alternate years. Spring 2012. [C, HSS] PSCI 22600. INTERNATIONAL POLITICAL ECONOMY (International Relations) This course explores mutual relationships between politics and economics in the relations of states; political effects of economic disparities; foreign economic policies of states in trade, aid, investment, and debt management; the roles of international institutions in the global economy; policy implications. Annually. Spring 2012. [C, HSS] PSCI 22700. THEORIES OF INTERNATIONAL RELATIONS (International Relations) This course examines the assumptions and implications of the major theories of international relations. Students will explore, compare, and debate the merits of contending theoretical explanations of international interactions, and explore how they might be applied to research and policymaking. Recommended for juniors. Annually. Fall. [HSS] PSCI 22800. NATIONALISM AND INTERDEPENDENCE (International Relations) This course explores the contrasting trends of fragmentation and integration in international relations by examining challenges to the predominance of sovereign states; including nations, regional and universal governmental organizations, nongovernmental organizations, and economic and cultural interdependence. Alternate years. Not offered 2011-2012. [HSS] PSCI 22900-22909. SPECIAL TOPICS IN INTERNATIONAL RELATIONS (International Relations) A seminar focusing on a selected topic concerning International Relations. May be taken more than once. Spring 2012. [C, HSS] Political Science 169 Field III: POLITICAL THEORY PSCI 13000. INTRODUCTION TO CONTEMPORARY POLITICAL IDEOLOGIES An introductory level course that focuses on the comparative analysis of competing ideologies that have dominated Western politics in the twentieth century: liberalism, libertarianism, conservatism, democratic socialism, communism, anarchism, and fascism. The second part of the course analyzes several of the newer ideologies that are transforming politics in the twenty-first century: minority liberation, liberation theology, gay liberation, feminism, environmentalism, animal liberation, and religious fundamentalism. Annually. [HSS] PSCI 23100. MODERN WESTERN POLITICAL THEORY A critical examination of the works of selected major theorists in the “modern” period which begins with Machiavelli and includes Hobbes, Locke, Rousseau, Burke, Hegel, John Stuart Mill, Harriet Taylor Mill, and Marx, among others. Alternate years. Fall 2011. [HSS] PSCI 23200. KNOWLEDGE AND POWER A critical analysis of the philosophical underpinnings of the study of politics and of the complex connections between knowledge and power in contemporary political life. Alternate years. Not offered 2011-2012. [HSS] PSCI 23400. CONTEMPORARY WESTERN POLITICAL THEORY A survey of major political and social theorists who have shaped twentieth century Western thought, such as Nietzsche, Weber, Freud, Woolf, Gadamer, Habermas, and Foucault, among others. Alternate years. Spring 2012. [HSS] PSCI 23500. CONTEMPORARY FEMINIST POLITICAL THEORY (Women’s, Gender, and Sexuality Studies) A critical analysis of selected contemporary feminist political theorists, including Davis, Eisenstein, Elshtain, Flax, Haraway, Hartstock, MacKinnon, O’Brien, and Watkins, among others. Alternate years. Fall 2011. [HSS] PSCI 23900-23906. SPECIAL TOPICS IN POLITICAL THEORY A seminar focusing on a selected topic concerning Political Theory. May be taken more than once. Spring 2012. [HSS] Field IV: COMPARATIVE POLITICS PSCI 14000. INTRODUCTION TO COMPARATIVE POLITICS This course introduces students to the basic concepts, tools, and theories of comparative politics. The main focus is on the emergence and development of major types of political systems and political institutions. Different political systems and institutions are systematically compared and analyzed in terms of how they respond to developmental tasks at different stages in the historical process. Annually. Fall and Spring. [C, HSS] PSCI 24200. THE POLITICS OF WEST EUROPE (International Relations) A comparative analysis of the economic and political development of major countries in West Europe, with special consideration of parliamentary representation, political mobilization, governability, the rise of new social movements, and European unification. Not offered 2011-2012. [HSS] PSCI 24400. POLITICS IN DEVELOPING COUNTRIES (International Relations) This course examines the main problems confronting developing countries, the political tools and strategies used for addressing them, and their relative success and failure given the constraints of the international economic and political order. The problems of developing countries are examined in the light of modernization, dependency, world system, political-cultural, and institutional theories and approaches, and cases from all the main parts of the developing world. Annually. Fall 2011. [C, HSS] PSCI 24600. PEACE STUDIES (International Relations) An exploration of the numerous dimensions of violence present in the world and the variety of peace tools available to address this violence. Understanding of ways to build both negative and positive peace are bolstered through review of cases of violence. Annually. Spring 2012. [C, HSS] PSCI 24700-24725. SPECIAL TOPICS IN COMPARATIVE POLITICS (International Relations) A seminar focusing on a selected topic concerning Comparative Politics. Multiple Sections. May be taken more than once. Fall 2011 and Spring 2012 [W†, C, HSS] PSCI 24900. THE GOVERNMENT AND POLITICS OF AFRICA (Africana Studies, International Relations) A general overview of Africa’s encounter with Europe and its after-effects. The course will also be concerned Psychology 170 with the various ways in which African countries have attempted to build viable political and economic systems. Alternate years. Not offered in 2011-2012. [C, HSS] RESEARCH AND METHODS COURSES PSCI 33000. RESEARCH IN POLITICAL THEORY This tutorial surveys the major contemporary approaches to political theory, including textual analysis, hermeneutics, critical theory and conceptual analysis, and focuses on research design and writing in political theory. Course requirements include the design and completion of a substantial research paper in political theory. This course is a prerequisite to enrolling in PSCI 45100 in Field III, Political Theory. By arrangement with the instructor and the chair of the department. Annually. Fall and Spring. PSCI 35000. RESEARCH METHODS AND DESIGN (International Relations) This course is a survey of various methodologies employed in the study of political science as a foundation for Senior Independent Study. It emphasizes research design, hypothesis construction, data collection, and a variety of forms of empirical political analysis. PSCI 35000 is a prerequisite for enrolling in PSCI 45100. Political science majors normally take PSCI 350 in their junior year. In the rare case of a student spending their entire junior year off-campus, they must notify the Chair of the Department of Political Science no later than fall semester of their sophomore year so arrangements can be made for the student to take the course in spring semester of their sophomore year. The department recommends that students have at least one introductory course and one 200-level course in their concentration field prior to enrolling in PSCI 35000. Students with a field specialization in Political Theory are exempt from this requirement but are required to take PSCI 33000 instead. Annually. Fall and Spring. PSCI 39100, 39200, 40200, 40700, 40800. INTERNSHIPS For a detailed discussion of the various internships available through the Washington Semester Program, see the description under Off-Campus Study and Internships. For seminars and internships with a focus in Political Science, the Washington Semester’s part-time internship is accredited as PSCI 40700; the two credit seminar is accredited as PSCI 39100 and 39200; and the research project can be accredited as PSCI 40200. Since the prerequisites differ for the different seminars and internships, you should consult the Washington Seminar adviser within the Political Science Department. S/NC course. PSCI 40000. TUTORIAL A tutorial course on a special topic may be offered to an individual student under the supervision of a faculty member. Prerequisite: The approval of both the supervising faculty member and the chairperson are required prior to registration. PSCI 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: PSCI 35000 or 33000 (depending on concentration field). PSCI 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: PSCI 45100. PSYCHOLOGY Michael Casey, Chair Susan Clayton Amber Garcia Gary Gillund John Jewell Brian Karazsia John Neuhoff Amy Jo Stavnezer Claudia Thompson Psychology 171 Psychology combines perspectives from both the natural and social sciences to gain an understanding of the processes underlying human and animal behavior, by examining influences ranging from the neurological to the sociocultural. The Psychology curriculum prepares students for diverse professional and career experiences. Approximately two thirds of its graduates enter professional programs at either the M.S. or Ph.D. level in psychology or related areas (e.g., education, law, social work, medicine). Other students enter the job market in a variety of settings immediately after graduation (e.g., technology, sales, finance, real estate, and social services). The Psychology major stresses an empirical approach to the broad range of psychological and behavioral issues and problems. As such, the curriculum is intended to expose students to both scientific and applied aspects of the discipline. As part of its facilities, the Department of Psychology maintains a statistical/computer facility and well-equipped animal, developmental, cognition, sensory/perception, and social/personality laboratories. Students also have access to the College’s nursery school for observational studies. Major in Psychology Consists of ten courses: • PSYC 10000 • PSYC 25000 • One elective 200-level Psychology course • One of the following 300-level courses: PSYC 32100, 32200, 33500, or NEUR 32300 • One of the following 300-level courses: PSYC 32500, 32700, or 33000 • Two elective 300-level Psychology courses • Junior Independent Study: PSYC 40100 • Senior Independent Study: PSYC 45100 • Senior Independent Study: PSYC 45200 Minor in Psychology Consists of six courses: • PSYC 10000 • PSYC 25000 • One elective 200-level Psychology course • One of the following 300-level courses: PSYC 32100, 32200, 33500, or NEUR 32300 • One of the following 300-level courses: PSYC 32500, 32700, or 33000 • One elective 200- or 300-level Psychology course Special Notes • Majors are encouraged to take a two-semester sequence of a laboratory course in either Biology or Chemistry and at least one course in Mathematics and Computer Science. • Advanced Placement: A student who has received a 4 or 5 on the Advanced Placement examination in Psychology may receive credit for PSYC 10000 and does not need to take that course as a prerequisite for advanced courses. The advanced placement policy of the College is explained in the section on Admission. • A student who has earned a D or F in the same course two times may not repeat that course or count it within the major or minor. Psychology 172 • A minimum grade of C is required in PSYC 25000 to advance in the major or minor. • Majors and minors are not permitted to take any courses within the department for S/NC credit, except for internships. • A student must earn a grade of C- or higher for a course to count toward the major or minor, or to count as a prerequisite for any Psychology course. PSYCHOLOGY COURSES PSYC 10000. INTRODUCTION TO PSYCHOLOGY (Neuroscience) An introduction to psychological theory, research, and methods. Coverage includes basic neurological pro cesses, principles of learning and cognition, individual differences in personality, developmental processes, sensation and perception, mental health, and social influences on behavior. Students may take the course only once for course credit. Annually. Fall and Spring. [HSS] PSYC 11000. CHILD AND ADOLESCENT DEVELOPMENT (Communication, Education) A study of the processes that contribute to the development of the individual as a person. The emphasis is typically on the child from conception to early adolescence. This course is intended primarily for students seeking licensure in Education. Psychology majors and minors are strongly encouraged to enroll in PSYC 10000. A 2-hour per week field placement at the College of Wooster Nursery School is required of all students. The field placement satisfies licensure requirements for Education minor students. Precludes enrollment in PSYC 32700. Annually. Fall 2011. [HSS] PSYC 21100. MATURITY AND OLD AGE (Communication) A course exploring the individual’s needs and developmental tasks to be accomplished by people as they progress from young adulthood to retirement and beyond. The impact of biological, sociological, and psychological factors on the aging process will be examined in an attempt to separate myth from reality about aging. The emphasis will be on middle aged people to senior citizens. Prerequisite: PSYC 10000. Alternate years. Not offered 2011-2012. [HSS] PSYC 21200. ABNORMAL PSYCHOLOGY (Neuroscience) Examines the origin, development, and classification of abnormal behavior and human psychopathology. Topics will include mood and anxiety disorders, psychosis, substance-related disorders, and disorders usually diagnosed in childhood. Prerequisite: PSYC 10000. Alternate years. Not offered 2011-2012. [HSS] PSYC 21500. PSYCHOLOGY OF WOMEN AND GENDER (Women’s, Gender, and Sexuality Studies) This course focuses on the societal construction and significance of gender, as well as the psychological implications of events unique to women. We will engage in a critical examination of theories and evidence concerning differences between women and men. Prerequisite: PSYC 10000. Alternate years. Not offered 2011-2012. [HSS] PSYC 22500. ENVIRONMENTAL PSYCHOLOGY (Environmental Studies) The field of environmental psychology explores the interrelationships between people and their physical environments, including both built and natural environments. This course covers the major areas of research in environmental psychology, including effects of the environment on humans, human perception of the environment, the relationship between humans and the natural world, and psychological factors affecting human care for the natural environment. We will also consider how this information can be applied to promote a healthier relationship between humans and their environment. Prerequisite: PSYC 10000, or permission of the instructor. Alternate years. Fall 2011. [HSS] PSYC 23000. HUMAN NEUROPSYCHOLOGY (Communication, Neuroscience) This course will explore the functioning of the fascinating human brain by discussing how we make decisions, how we rationalize choices, how we consider emotions and how we learn, to name a few. The course emphasizes the various methodologies used to assess the functions of brain regions and behavior through case studies as well as empirical research. Prerequisite: PSYC 10000. Alternate years. Spring 2012. [HSS] PSYC 23500. EVOLUTIONARY PSYCHOLOGY The course provides an integrated approach to studying human behavior based on an evolutionary model. Using Darwin’s theory of natural and sexual selection we will investigate adaptive problems such as predator avoidance, inter-group aggression, mate selection, child rearing, and negotiating social relationships. Other topics include: “human nature,” the origins and functions of various behavioral sex differences, the evolutionary Psychology 173 basis of nepotism, gene-behavior relations, reproductive behavior, and how culture and social learning interface with Darwinian evolution. Prerequisite: PSYC 10000. Alternate years. Not offered 2011-2012. [HSS] PSYC 24000. TOPICS IN APPLIED PSYCHOLOGY A course in which traditional concepts, methods, and theories in psychology are applied to a practical issue. Topics selected yearly and announced in advance by the faculty member responsible for the course. Prerequisite: PSYC 10000. Annually. Fall and Spring. [HSS] PSYC 24500. HUMAN SEXUALITY A survey course examining the evolutionary, comparative, biological, developmental, social, and historical cultural aspects of human reproductive behavior. Additional topics include: sexually transmitted disease, sex in the context of human relationships, and issues of sexual orientation. Prerequisite: PSYC 10000. Alternate years. Not offered 2011-2012. [HSS] PSYC 25000. INTRODUCTION TO STATISTICS AND EXPERIMENTAL DESIGN (Neuroscience) Introduction to the basic principles of descriptive statistics, inferential statistics, and experimental design. Includes SPSS instruction and a one-hour laboratory. Minimum grade of C is required to advance in the major or minor. Prerequisite: PSYC 10000. Annually. Fall and Spring. [Q] PSYC 32100. LEARNING AND BEHAVIOR (Neuroscience) Detailed critical examination of theory, research and applications of learning processes, from simple associative processes (classical and operant conditioning) to complex processes (conceptual abstraction and reasoning). Scientific writing is emphasized. Three-hour weekly laboratory with additional outside hours for animal testing. Class and laboratory components are closely integrated and must be taken concurrently. (1.25 course credits) Prerequisite: PSYC 25000. Annually. Spring 2012. [W] PSYC 32200. MEMORY AND COGNITION (Communication, Neuroscience) Analysis of complex human behavior, including learning, memory, perception, and cognition. Scientific writing is emphasized in this course. Includes a 3-hour laboratory in addition to class. The laboratory and classroom components are closely integrated and must be taken concurrently. (1.25 course credits) Prerequisite: PSYC 25000. Annually. Not offered 2011-2012. [W] PSYC 32500. PERSONALITY: THEORY AND RESEARCH A basic course emphasizing theories of human personality and research generated from the theories. Scientific writing is emphasized in this course, which includes a 3-hour laboratory in addition to class. The laboratory and classroom components are closely integrated and must be taken concurrently. (1.25 course credits) Prerequisite: PSYC 25000. Annually. Fall 2011. [W] PSYC 32700. DEVELOPMENTAL PSYCHOLOGY: THEORY AND RESEARCH (Education) A survey of methods, research topics, and theory in developmental psychology. Scientific writing is emphasized in this course. Includes a 3-hour laboratory in addition to class. The laboratory and classroom components are closely integrated and must be taken concurrently. A 2-hour per week field placement at The College of Wooster Nursery School is required of all students. The field placement satisfies licensure requirements for Education minor students. (1.25 course credits) Prerequisite: PSYC 25000. Annually. Spring 2012. [W] PSYC 33000. SOCIAL PSYCHOLOGY: THEORY AND RESEARCH This course surveys theory and research on human social cognition and behavior, addressing the ways in which human beings are affected by others and covering topics such as social influence, prosocial and antisocial interactions, and relationships. Scientific writing is emphasized in this course. Includes a 3-hour laboratory in addition to class. The laboratory and classroom components are closely integrated and must be taken concurrently. (1.25 course credits) Prerequisite: PSYC 25000. Annually. Fall 2011. [W] PSYC 33100. CLINICAL METHODS Primarily for majors, the course includes an introduction to current methods of psychotherapy, counseling, behavioral modification, and other selected topics concerning treatment and evaluation. Prerequisite: PSYC 21200 and 25000. Alternate years. Not offered 2011-2012. PSYC 33200. PSYCHOLOGICAL TESTING An introduction to basic principles of psychological testing (reliability, validity, and normative data) and types of psychological tests. Assignments are intended to familiarize the students with administration, interpretation, and evaluation of psychological tools. Prerequisite: PSYC 25000. Alternate years. Not offered 2011-2012. Psychology 174 PSYC 33500. PERCEPTION AND ACTION (Communication, Neuroscience) A basic introduction to sensations, sensory processes, and their organization into perceptions. Both psycho - physical and physiological perspectives are emphasized. Scientific writing is emphasized in this course. Includes a 3-hour laboratory to addition to class. The laboratory and classroom components are closely integrated and must be taken concurrently. (1.25 course credits) Prerequisite: PSYC 25000. Annually. Spring 2012. [W] PSYC 34000-34016. ADVANCED TOPICS IN PSYCHOLOGY A seminar for junior and senior majors and minors that explores current theory and research in selected topics in psychology. Topics selected yearly and announced in advance by the faculty member responsible for the seminar. Prerequisite: PSYC 25000, junior or senior standing with advanced background in Psychology. Annually. Fall and Spring. PSYC 39500. HISTORY AND SYSTEMS OF PSYCHOLOGY A study of changing views of psychology from Aristotle to the present, with emphasis on the influences of ideas and methodologies of the evolution of systems and theories of psychological thought over the past hundred years. The course offers an integrative perspective on the varied courses of the Psychology major. Prerequisite: Psychology major, a 300-level Psychology lab course. Annually. Fall 2011. PSYC 39900. SPECIAL PROBLEMS Special courses on selected topics offered for a single time only to groups of students. Prerequisite: as specified by the instructor. NEUR 32300. BEHAVIORAL NEUROSCIENCE [W] PSYC 40000. TUTORIAL A tutorial course on special topics offered to an individual student under the supervision of a faculty member. Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. Annually. Fall and Spring. PSYC 40100. RESEARCH METHODS FOR INDEPENDENT STUDY A one-semester course, with instruction in research methods and statistical analysis. The course emphasizes active engagement in all stages of research, including experimental design and methodology, data collection and analysis, and scientific writing. Requires literature searches on three psychological topics, design of empirical research projects, and preparation of materials related to ethical considerations pertaining to research with humans and other animals. Prerequisite: Psychology major, any of the 300-level writing intensive laboratory courses in Psychology. Annually. Fall and Spring. PSYC 40700, 40800. INTERNSHIP An academically-oriented, applied experience that provides off-campus placement in an approved clinic, agency, institution, or research center. Focuses on the practical application and implications of theory and research under supervision and within the limits of the student’s competence. Number of credits allowed for the experience (1-2 units) will be determined in advance by the department. Advanced planning and permission of the department chairperson are required. Credits cannot be substituted for major requirements. (1 – 2 course credits) S/NC course. Prerequisite: advanced standing and permission of the department chairperson. Annually. Fall and Spring. PSYC 45100. SENIOR INDEPENDENT STUDY - SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. The Independent Study Thesis must be a data-gathering research project. Students are encouraged to base their projects on a study from the experimental, comparative, personality, developmental, social, clinical, or neuroscience literature. Prerequisite: senior standing and PSYC 40100. PSYC 45200. SENIOR INDEPENDENT STUDY - SEMESTER ONE The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: PSYC 45100. 175 RELIGIOUS STUDIES Mark Graham, Chair Mary Bader Lisa Crothers Joan Friedman Jennifer Graber Charles Kammer Jeremy Rapport Religious Studies is an interdisciplinary approach to the academic study of religion. The department provides for students a broad, yet nuanced, understanding of the place of religion in human experience. Although Religious Studies is a mode of intellectual inquiry, for many students the study of religion involves a personal journey as academic study and religious faith intersect and challenge one another. Religious Studies does not endorse a particular creed or religious position, but creates the context for discussion and study that allows students to explore academic and personal questions about religion and society within the framework of their growing knowledge. The natural connection of Religious Studies to other liberal arts disciplines is reflected in the range of courses offered. Courses in the department examine religion from the dual perspectives of methodology and content. Courses in the department are divided into two general areas: Area I focuses on religious traditions and histories; Area II focuses on issues and theories in the study of religions. Major in Religious Studies Consists of ten courses: • Three courses in Area I • Three courses in Area II • One elective Religious Studies course • Junior Independent Study: RELS 40100 • Senior Independent Study: RELS 45100 • Senior Independent Study: RELS 45200 Minor in Religious Studies Consists of six courses: • Two courses in Area I • Two courses in Area II • Two elective Religious Studies courses Special Notes • No more than two 100-level courses may count toward the major or minor. • Only grades of C- or better are accepted for the major or minor. BIBLICAL HEBREW COURSES The Religious Studies Department also offers courses in Biblical Hebrew. As with other introductory (10100-10200) foreign language courses, Biblical Hebrew I and II may be taken by any student to fulfill the College’s foreign language requirement, or may be taken as elective credits by students who have already fulfilled the language requirement. Students with some prior knowledge of Hebrew language who have questions about placement in HEBR 10200 should contact the Religious Studies Department. Religious Studies 176 HEBR 10100. BIBLICAL HEBREW I (Classical Studies) Introduction to the grammar and vocabulary of Biblical Hebrew, beginning with the alphabet. Students will master basic grammatical forms and will read simple prose passages from the textbook and selected Biblical verses and phrases. No prior knowledge of Biblical Hebrew is expected. Alternate years. Not offered 2011-2012. HEBR 10200. BIBLICAL HEBREW II (Classical Studies) Continued study of Biblical Hebrew grammar and vocabulary, reading selected prose passages from the Hebrew Bible, and discussion of the cultural and religious context. Prerequisite: successful completion of HEBR 10100 or placement/instructor permission. Alternate years. Not offered 2011-2012. RELIGIOUS STUDIES COURSES Area I: RELIGIOUS TRADITIONS AND HISTORIES RELS 11000. RELIGIONS EAST AND WEST An examination of basic issues in religious studies and an overview of the beliefs and practices of some of the major religions of the world, such as Islam, Judaism, Hinduism, Buddhism, and Christianity. Fall and Spring. [C, R, AH] RELS 12000. INTRODUCTION TO BIBLICAL STUDIES: INTERPRETATION AND CULTURE (Classical Studies) Introduces the examination of basic issues of reading the Bible in an academic setting. Special attention will be given to the biblical texts as resources for understanding political, social, and religious discourses in the ancient world. The student will encounter introductions to historical, literary and feminist methodologies. Not offered 2011-2012. [C, R, AH] RELS 13000. AMERICAN RELIGIOUS COMMUNITIES An examination of the tension between religious power and religious pluralism in American history. Fall and Spring. [C, R, AH] RELS 21600. CHINESE RELIGIONS (Chinese, East Asian Studies) This course primarily examines Chinese “popular religions,” and the three formalized traditions of Confucianism, Daoism, and Buddhism, as practiced both historically and in contemporary life in China, Taiwan, and Chinese Diaspora communities in Asia and the West. This course also examines the presence of other non-indigenous Chinese religions (e.g., Islam and Christianity) in China. Annually. Not offered 2011-2012. [C, R, AH] RELS 21700. AFRICAN RELIGIONS This course explores African religious thought and practice. While the focus is on traditional African religions, it also investigates the impact of African thought and culture on Christianity and Islam on the African continent. The course includes the study of the role of religion in contemporary African culture and politics. Alternate years. Not offered 2011-2012. [C, R, AH] RELS 21800. HINDUISM (South Asian Studies) Hindu concepts and practices as reflected in texts such as the Vedas, Upanishads, and Bhagavad-Gita and in religious practice in Indian cultures through the centuries, with attention to sects and modern reform movements. Alternate years. Fall 2011. [C, R, AH] RELS 22000. BUDDHISM (East Asian Studies, South Asian Studies) Buddhist concepts and practices, including karma, rebirth, and devotion, as found in religious writings and as practiced through history, across Asian cultures. Alternate years. Spring 2012. [C, R, AH] RELS 22200. ISLAM (South Asian Studies) The foundations of Islam as set forth in the Qur’an, the life of the prophet Mohammad, Muslim philosophers and mystics as reflected in Middle Eastern and South Asian cultures, with attention to central concepts of revelation, community, law, and worship. Alternate years. Spring 2012. [C, R, AH] RELS 22400. HEBREW PROPHECY AS RELIGIOUS IMAGINATION (Classical Studies) An exploration into the historical, political, and religious traditions of the Hebrew prophets within both Jewish and Christian scholarship. The prophetic books of the canon will be examined from historical, literary and feminist viewpoints. Prerequisite: RELS 12000 or permission of the instructor. Alternate years. Not offered 2011-2012. [R, AH] Religious Studies 177 RELS 22500. THE LIFE AND TEACHINGS OF JESUS (Classical Studies) An examination of the views, problems, and hypotheses about the identities of Jesus in the first few centuries C.E. Historical issues and religious-cultural implications of the “afterlife” of Jesus will be investigated. Attention will be given to Gnostic and Rabbinic references to Jesus. The course encourages students to develop a critical awareness about the complexities involved when we talk about Jesus in today’s world. Prerequisite: RELS 12000 or permission of instructor. Alternate years. Not offered 2011-2012. [W, C, R, AH] RELS 23000. THE LIFE AND THOUGHT OF MAHATMA GANDHI (South Asian Studies) A study of the life and thought of Mahatma Gandhi and his philosophy of non-violence as reflected in his ideas about religion, politics, economics, social work, etc., in comparison with such movements as Sarvodaya, Civil Rights, Liberation Theology, and the Women’s Movement. Also includes discussion of contemporary Gandhians and their application of Gandhian thought to their personal lives and social movements. Prerequisite: RELS 10000 or 11000 or permission of the instructor. Alternate years. Not offered 2011-2012. [W†, C] RELS 23300. JUDAISM Presents the Jewish religious tradition and its historical evolution, its sacred texts, practices and beliefs, and modern movements within Judaism, with particular attention to central concepts of covenant, divine authority, and the interdependence of religion and people hood. Alternate years. Spring 2012. [C, R, AH] RELS 23900. GLOBAL CHRISTIANITY Focuses on the history, theology and practice of Christianity as an international religion, especially the global zones of Asia, Africa, Europe, and Latin America. Alternate years. Fall 2011. [R] RELS 24700. NATIVE AMERICAN RELIGIONS AND CULTURES A study of tradition and change within the historical and modern religions of various regional Native American tribal groups, including Pan-Indian activism and revitalization. Alternate years. Fall 2011. [C, R, AH] RELS 25200. THIRD WORLD FEMINIST THEOLOGY (Women’s, Gender, and Sexuality Studies) To be human is to think about questions of meaning, transcendence, purpose and relationship. This course on Third World Feminist Theology will explore ways that women of Asian heritage are asking these questions. The course will begin with an experience of dislocation through an immersion experience in Thailand. This will provide the students with a taste of the context and experience from which Asian and Asian-American feminist theology emerges. Students’ direct engagement with the lived realities of several Thai communities forms the core of a course that will use a case-study approach. The second part of the course will take place at The College of Wooster during the regular semester. We will use the lens of sacred writing, theological discussion, and the immersion experience to better comprehend and appreciate the voice and perspectives that Asian and AsianAmerican feminists bring to the larger liberation conversation. Not offered 2011-2012. RELS 25400. THE REFORMATION: PAST AND PRESENT TRADITIONS A study of the theological, cultural, and political issues that prompted a variety of 16th Century Protestant movements. The course connects these new traditions to their modern-day instantiations around the globe. Alternate years. Not offered 2011-2012. [R] RELS 26100. BLACK RELIGIOUS EXPERIENCE IN AMERICA (Africana Studies) An interdisciplinary study of Black religious experience, institutions, leadership, thought, and social movements in American society, with emphasis on the work of King, Malcolm X, and the Womanist tradition. Alternate years. Fall 2011. [C, R] RELS 26700-26722. TOPICS IN RELIGIOUS TRADITIONS AND HISTORIES (some sections cross-listed with: Classical Studies, South Asian Studies, Women’s, Gender, and Sexuality Studies) An in-depth study of central issues in the history of religious traditions, such as Global Catholicism in America, Asian Religions in America, Modern Jewish Identities. Fall 2011and Spring 2012. [R] Area II: ISSUES AND THEORIES IN THE STUDY OF RELIGION RELS 10000. RELIGIOUS THOUGHT AND ACTION Approaches to selected religious ideas, themes, and problems in the thought of diverse traditions, religious issues, or major thinkers of the past and the present. Annually. Fall and Spring. [C, R, AH] RELS 20600. SOCIOLOGY OF RELIGION An analysis of the nature of religion, religious movements and institutions, belief and ethics in religion today Religious Studies 178 from the dual perspectives of sociology and religious studies. A focus on the interaction of religion, politics, and culture. Alternate years. Not offered 2011-2012. [R] RELS 21900. ETHICS IN A SOCIAL PERSPECTIVE A comparison of the ethical insights of a variety of Eastern and Western religious traditions as they relate to current social problems, such as war and peace, social justice, death and dying, and bioethics. Alternate years. Fall 2011. [C, R] PHIL 22100. PHILOSOPHY AND THE RELIGIOUS LIFE [R, AH] In one part of this course we will look at traditional issues in the philosophy of religion, the nature of religious experience, classical proofs for the existence of God and the problem of evil. In the second part of the course we will focus on issues in religious language, seeing God, the place of ceremony and liturgy in religious life and religious pluralism. Alternate years. Not offered 2011-2012. [R, AH] RELS 22900. WOMEN AND RELIGION (Women’s, Gender, and Sexuality Studies) An investigation into the roles of women as depicted in sacred texts of the Jewish, Christian, Muslim, and Goddess traditions. This course will use feminist narrative skills as the primary methodology. Using leading feminists’ work from all four traditions, students will investigate what texts may have to say about women’s roles in both ancient and modern religious traditions, in world religions, the lives and thoughts of prominent women in religious history, and central issues in feminist theology. Alternate years. Not offered 2011-2012. [C, R] RELS 24100. NEW RELIGIOUS MOVEMENTS An examination of America’s marginal but influential religious movements. 19th Century groups include Mormons, Spiritualism, Jehovah’s Witnesses, Christian Scientists, 20th Century practices and traditions include Hare Krishnas, the Unification Church, New Age spiritualities, Scientology, Branch Davidians, and Wicca. Alternate years. Spring 2012. [C, R, AH] RELS 24300. RELIGIOUS AND SPIRITUAL AUTOBIOGRAPHY This course studies the many religious purposes (e.g. the understanding of religious experience, formation of religious identity, presentation of a moral or religious ideal, social criticism) that religious autobiographies serve. Such writings also provide readers a window into individual religious lives, experiences, and cultures. Writings selected may include classic Western religious autobiographies such as Augustine’s Confessions as well as other writings, both historical and contemporary, from a variety of religious traditions. Recommended: one 100-level Religious Studies course. Alternate years. Fall 2011. [W†, R, AH] RELS 24500. CHRISTIAN ETHICS Historical overview of the structure of Christian ethics with the focus on its biblical and theological foundations and its application to important personal and social issues. Alternate years. Not offered 2011-2012. [R] RELS 25100. MODERN RELIGIOUS THINKERS (Latin American Studies) An introduction to selected religious thinkers of the 20th Century. Attention will be given to figures representative of major movements, such as neo-orthodoxy, existentialism, process theology, and third world theologies. Not offered 2011-2012. [W, R] RELS 26300. RELIGION AND LITERATURE This course examines the ways in which modern and contemporary writers represent religious traditions and experiences, make use of religious narratives and themes and confront religious questions in their novels, short stories, and poetry. Recommended: one 100-level Religious Studies course. Alternate years. Spring 2012. [R, AH] RELS 26400. RELIGION AND FILM (Film Studies) This course examines the interactions of religions and religious life with the electronic media technologies of film and video. Through such a course, students can arrive at better understandings of the place of religions in contemporary cultures, the aesthetics of film and video, and the place of these media as communicators of cultural phenomena such as religion. Alternate years. Spring 2012. [C, R, AH] RELS 26900-26929. TOPICS IN THEORIES AND ISSUES IN THE STUDY OF RELIGION (some sections cross- listed with: Classical Studies, Environmental Studies, South Asian Studies, Women’s, Gender, and Sexuality Studies) An examination of one major issue involving the interface of sociological, ethical, and theoretical factors, such as Religion, Violence and Peacemaking; Interfaith Dialogue; Religion and the Environment; and Third World Feminist Theology. Annually. Fall and Spring. [R] Russian Studies 179 RELS 40700, 40800. ETHICS AND SOCIETY INTERN PROGRAM Students will be placed for one semester in an agency, organization, or other context where the academic study of religion can be joined with a practical experience in dealing with ethical and religious issues in American society. Three credits, with the possibility of a fourth. (1-3 course credits) S/NC course. Prerequisite: Permission of instructor is required, and previously taken appropriate courses in the department are desirable. Annually. OTHER COURSES RELS 40000. TUTORIAL Individual readings and reports may be required by the instructor. The course may be given an Area I or II designation with departmental approval. (.5 - 1 course credit) Prerequisite: The approval of both the supervising faculty member and the chairperson is required prior to registration. RELS 40100. JUNIOR INDEPENDENT STUDY A one-semester course that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. Combines tutorial-seminar format. Spring (unless the student is studying off-campus Spring semester). RELS 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: RELS 40100. RELS 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: RELS 45100. RUSSIAN STUDIES Beth Ann Muellner (German), Chair Yuri Corrigan (Russian Studies) Engeniya Gerdt (Russian Language Assistant) Russian Studies is an interdisciplinary area focused on Russia and Eastern Europe in cultural, literary, historical, philosophical, and political contexts. It is one of several programs at the College that provides students with the opportunity to develop proficiency in a foreign language in connection with their other academic and professional interests. The department offers courses in three areas: 1) Russian language, 2) Russian culture, and 3) Russian literature. These, together with courses in history and comparative literature, give students a strong background in Russia and Eastern Europe. Recent graduates in Russian Studies have embarked on employment in government service, non-governmental organizations, and private companies, both in the United States and abroad. Some have gone on to graduate school, pursuing careers in such areas as law, education, library science, and academia. Recent graduates have also served in the Peace Corps in the former Soviet Union. Major in Russian Studies Consists of twelve courses: • RUSS 20100 • RUSS 20200 • RUSS 21000 • RUSS 22000 Russian Studies 180 • One of the following courses: HIST 10100-10176 (when topic focuses on Russian history), 23000, 23300, or 30100-30142 (when topic focuses on Russian history) • Three of the following courses: RUSS 23000, 25000, 26000, CMLT 24800, HIST 10100-10176 (when topic focuses on Russian history), 23000, 23300, or 30100- 30142 (when topic focuses on Russian history) • RUSS 40000 • Junior Independent Study: RUSS 40100 • Senior Independent Study: RUSS 45100 • Senior Independent Study: RUSS 45200 Minor in Russian Studies Consists of six courses: • RUSS 20100 • Five of the following courses: RUSS 20200, 21000, 22000, 23000, 25000, 26000, CMLT 24800, HIST 10100-10176, 23000, 23300, or 30100-30142 Special Notes • The College language requirement may be satisfied in Russian by completing a 102-level course or receiving a score equivalent to the 102-level on the placement examination administered during registration week. If a student registers for and completes a course in language below the level at which the language department’s placement exam placed him or her, that student will not receive credit toward graduation for that course, unless he or she has obtained the permission of the instructor of the course into which the student placed and permission of the department chair. • Study Abroad: Students will be encouraged to enhance their educational experience (or fulfill requirements for a major in International Relations) by studying in Russia or East Central Europe, and are advised to consult with the chairperson of the department in the first term of their first year of study at the College. Early planning is essential for the CIEE program in St. Petersburg, as well as for other programs such as SIT in St. Petersburg and Irkutsk, and IPSL in Moscow. Approved courses taken in an off-campus semester will provide additional variety in the courses offered for the major. • Russian House: Students have the opportunity to take up residence in Russian House, a suite in Luce Hall that houses students along with a native Russian assistant and serves as the focal point for most campus Russian language and cultural activities. • Related Interdepartmental Programs: Students interested in Russian and East Central European literature and culture should be aware of several interdepartmental programs in which the Department of Russian Studies cooperates: Comparative Literature, Film Studies, and International Relations. • S/NC courses are not permitted in the major or minor. • Only grades of C- or better are accepted for the major or minor. RUSSIAN STUDIES COURSES RUSS 10100. BEGINNING RUSSIAN (LEVEL I) An introduction to understanding, speaking, reading, and writing Russian; acquisition of basic grammar; conversational practice and short readings. Cultural content. Five hours per week. Annually. Fall. RUSS 10200. BEGINNING RUSSIAN (LEVEL II) Continuation of RUSS 10100, with increased emphasis on conversational, reading, and writing skills. Cultural content. Prerequisite: RUSS 10100 or placement. Annually. Spring. Russian Studies 181 HIST 10100-10136. INTRODUCTION TO HISTORICAL INVESTIGATION (when topic focuses on Russian history) [W, some sections count toward C, HSS] RUSS 20100. INTERMEDIATE RUSSIAN (LEVEL III) Review and enhancement of basic grammar; practice through speaking, listening, reading, and writing. Attention to reading strategies. Exposure to cultural material. Four hours per week. Prerequisite: RUSS 10200 or placement. Annually. Fall. RUSS 20200. INTERMEDIATE RUSSIAN (LEVEL IV) Continuation of RUSS 20100, with still greater emphasis on speaking, reading, and writing. Cultural content. Prerequisite: RUSS 20100. Annually. Spring. RUSS 21000. RUSSIAN CIVILIZATION: FROM FOLKLORE TO PHILOSOPHY (Comparative Literature) An introductory and interdisciplinary study of fundamental aspects of Russian culture from medieval Russia through the post-Soviet era, with emphasis on the changing and evolving concept of Russian identity over the centuries. A broad range of texts will include folktales, memoirs, fiction, painting, poetry, philosophy, music and film. Every three years. Spring 2012. [W, C, AH] RUSS 22000. RUSSIAN CULTURE THROUGH FILM (Comparative Literature, Film Studies) An introduction to twentieth-century Russian society and culture through the medium of cinema, covering the immediate pre- and post-revolutionary periods, Stalinism, the post-Stalin “thaw,” stagnation under Brezhnev, Gorbachev’s “perestroika” and “glasnost,” and the post-communist era. Weekly screenings of films will be supplemented with readings in Russian film theory and criticism. Every three years. Not offered 2011-2012. [C, AH] RUSS 23000. RUSSIAN DRAMA PRACTICUM (Comparative Literature) This course has two components. The first is an in-depth study of the works of one major Russian playwright. The course will address figures such as Nikolai Gogol, Anton Chekhov, and Mikhail Bulgakov. Since these artists were prose writers to the same extent as they were playwrights, we will read a wide selection of both their prose and their dramatic works in order to understand the significance of their artistic innovations. The second part of the course will be to produce one of our author’s major plays as a class and to present it to the public at the end of the semester. No acting experience required. Every three years. Not offered 2011-2012. [C, AH] RUSS 25000. RUSSIAN LITERATURE IN THE AGE OF DOSTOEVSKY AND TOLSTOY (Comparative Literature) In the nineteenth century, Russia witnessed an unprecedented explosion of literary and intellectual activity, a renaissance which yielded some of the greatest masterpieces world literature has seen. Our course will examine the seven most prominent authors of this period, with special emphasis on Russia’s unique handling of the sudden influx of European philosophy and culture (Rationalism, Idealism, Romanticism, Atheism, Socialism). Through its literary canon, we will explore how Russia envisioned the problems of modern individualism in a culture divided between European and Slavic roots. Every three years. Not offered 2011-2012. [C, AH] RUSS 26000. THE ARTIST AND THE TYRANT: TWENTIETH-CENTURY RUSSIAN LITERATURE (Comparative Literature) Russian literature developed side by side with the myths and horrors of a cataclysmic twentieth century. In this course, we will read some of the most powerful artistic meditations on the collapse of imperial Russia, on the dream and nightmare of the Soviet experiment, and on the search for dignity and meaning in the post-Soviet contemporary world. Authors include Nobel laureates Pasternak, Bunin, Solzhenitsyn and Brodsky. We will also read novels by Bulgakov and Nabokov, short stories from a host of writers from Babel to Petrushevksaya, and some of the major poetry of the era in translation. Every three years. Not offered 2011-2012. [C, AH] CMLT 24800: THE PERILS OF ROMANTICISM: NINETEENTH-CENTURY EUROPEAN LITERATURE [C, AH] HIST 23000. RUSSIA TO 1900 [C, HSS] HIST 23300. RUSSIA SINCE 1900 [C, HSS] HIST 30100-30142. PROBLEMS IN HISTORY (when topic focuses on Russian history) [C, R, HSS] RUSS 40000. TUTORIAL Individually supervised advanced language learning. By prior arrangement with the department only. Prerequisite: RUSS 20200 or equivalent; the approval of both the supervising faculty member and the chairperson is required prior to registration. Sociology and Anthropology 182 RUSS 40100. INDEPENDENT STUDY Bibliographical and research methods in Russian Studies, including the preparation of one longer research paper. Normally taken Semester II of the junior year. RUSS 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research of a specific topic in Russian Studies guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. Prerequisite: RUSS 40100. RUSS 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: RUSS 45100. SOCIOLOGY AND ANTHROPOLOGY Thomas Tierney (Sociology), Chair Abigail Adams (Anthropology) J. Heath Anderson (Anthropology) Christa Craven (Anthropology) Heather Fitz Gibbon (Sociology) Pamela Frese (Anthropology) Raymond Gunn (Sociology) P. Nick Kardulias (Anthropology, Archaeology) Setsuko Matsuzawa (Sociology) David McConnell (Anthropology) Anne Nurse (Sociology) Craig Willse (Sociology) The Department of Sociology and Anthropology offers a diverse curriculum exploring the institutions and processes that maintain and change human societies. Our program places special emphasis on the development of students’ abililty to analyze contemporary social and cultural issues, and their problem-solving and research skills. Students choose a major in either Sociology or Anthropology, but all majors are introduced to the concepts, methods and theories appropriate to research in both disciplines. SOCIOLOGY The basic challenge in sociology is to understand ourselves and others more fully. The discipline asks us to probe beneath the surface and to question why people behave as they do, especially in group situations. The sociological perspective asks us to question what we often take for granted, why our society operates as it does, and how our social arrangements could be different. Major in Sociology Consists of twelve courses: • SOCI 10000 • ANTH 11000 • One of the following courses: SOCI 20700, 20900, 21400, 21500, or 21700 • SOAN 24000 • One of the following courses: SOCI 34200 or ANTH 34100 • SOCI 35000 • SOCI 35100 Sociology and Anthropology 183 • Two elective Sociology courses (see note below) • One elective Sociology, Anthropology, or Sociology/Anthropology course (see note below) • Junior Independent Study: (see note below) • Senior Independent Study: SOCI 45100 • Senior Independent Study: SOCI 45200 Minor in Sociology Consists of six courses: • SOCI 10000 • One of the following courses: SOCI 20700, 20900, 21400, 21500, or 21700 • SOAN 24000 • One of the following courses: SOCI 35000 or 35100 • Two elective Sociology courses Special Notes • A second or third course from SOCI 20700, 20900, 21400, 21500, or 21700 or a second 300-level methods course (SOCI 34200 or ANTH 34100) may count as electives for the requirements for the major. • The Junior Independent Study requirement is met by completing SOCI 35000 or 35100 prior to Senior Independent Study. • Sociology majors who plan to attend graduate school are strongly encouraged to take SOCI 34200 (Social Statistics). • Sociology majors who elect to participate in the 3-2 program in Social Work at Case Western Reserve University (see Pre-Professional and Dual Degree Programs) must complete all requirements in the major except Senior Independent Study. Students should see the department chairperson for more details about this arrangement. • Teaching Licensure: The requirements for teacher licensure can be found in Teacher Education at The College of Wooster: A Supplement to the Catalogue (which can be found at the following website: www3.wooster.edu/education/ current/forms.html). Students should consult with the chairpersons of Sociology and Anthropology and of Education. • Only grades of C- or better are accepted for the major or minor. ANTHROPOLOGY Anthropology explores the variety of human groups and cultures that have developed across the globe and throughout time. Anthropologists hope that by seeing ourselves in the mirror of alternative cultural and historical possibilities, we can come to a better understanding of our own assumptions, values and patterns of behavior. Major in Anthropology Consists of twelve courses: • ANTH 11000 • SOCI 10000 • One of the following courses: ANTH 21000, 22000, or ARCH 10300 • ANTH 23100 • SOAN 24000 • One of the following courses: ANTH 34100 or SOCI 34200 • Two elective Anthropology courses (see note below) • One elective Anthropology, Sociology, or Sociology/Anthropology course (see note below) Sociology and Anthropology 184 • Junior Independent Study Equivalent: ANTH 35200 • Senior Independent Study: ANTH 45100 • Senior Independent Study: ANTH 45200 Minor in Anthropology Consists of six courses: • ANTH 11000 • One of the following courses: ANTH 21000, 22000, or ARCH 10300 • SOAN 24000 • ANTH 23100 • Two elective Anthropology courses Special Notes • A second or third course from ANTH 21000, 22000, or ARCH 10300, or a second 300-level methods course (SOCI 34200 or ANTH 34100) may count as electives for the requirements for the major. • Anthropology majors who plan to attend graduate school are strongly encouraged to take ANTH 34100 (Ethnographic Methods). • Anthropology majors who elect to participate in the 3-2 program in Social Work at Case Western Reserve University (see Pre-Professional and Dual Degree Programs) must complete all requirements in the major except Senior Independent Study. Students should see the department chairperson for more details about this arrangement. • Only grades of C- or better are accepted for the major or minor. SOCIOLOGY COURSES SOCI 10000. INTRODUCTION TO SOCIOLOGY An examination of ociological principles and methods of investigation, and their relationship to the major issues in society, such as social change, social class, urbanization, and intergroup relations. Attention will also focus upon the major social institutions and the relationship between the individual and society. Class sessions will utilize lectures, seminar discussions, data analysis, and audio-visuals. Annually. Fall and Spring. [HSS] SOCI 11100. TOPICS IN SOCIOLOGY A seminar focused on a special topic in sociology. Topics are chosen by the instructor and announced in advance. Annually. Fall. [HSS] SOAN 20100. EDUCATION IN SOCIOCULTURAL CONTEXT (Education) An acquaintance with selected anthropological and sociological approaches to the study of education. It seeks to communicate a cross-cultural perspective on the educative process through case studies of education and socialization in diverse societies. Theories and research on the social effects of schooling will also be covered. Special attention will be given to the situation of minorities in the schooling process and to understanding educational policy debates in American society. Alternate years. Not offered 2011-2012. [C, HSS] SOAN 20200. GLOBALIZING HEALTH An examination of public health issues from a global perspective. The twenty-first century has presented numerous public health challenges, such as the AIDS crisis, the rise of multidrug-resistant tuberculosis, and trafficking in human organs and tissues. Such problems can only be addressed by a combination of local and global responses. This course applies contemporary globalization theories to such public health challenges, and critically examines the ways in which Western medical techniques and attitudes toward health are disseminated throughout the world, and the tensions generated in local cultures by this globalization of health. Every third year. Not offered 2011-2012. [C, HSS] SOCI 20300. ENVIRONMENTAL SOCIOLOGY (Environmental Studies) An investigation of the dynamic relation between society and the environment. Sociology points us beyond mere technical and scientific problems to the social roots of contemporary ecological issues, as well as the justice issues these circumstances entail. We explore the many ways in which environmental issues are, in fact, Sociology and Anthropology 185 social issues. The topics we cover include: causes of environmental degradation, environmental movements, environmental activism and organizations, corporate social responsibility, social construction of the environment, collective behavior, Genetically Modified Organisms (GMO), and locavorism. Prerequisite: SOCI 10000 or permission of instructor. Alternate years. Not offered 2011-2012. [HSS] SOCI 20400. SELF AND SOCIETY An examination of social psychological perspectives on the inter relationships among the individual, the small group, and the larger culture. Topics emphasized include socialization, the development of self, deviance, the individual and social change, and attitude formation. Prerequisite: SOCI 10000 or ANTH 11000 or permission of instructor. Alternate years. Spring 2012. [W, HSS] SOCI 20500. SOCIOLOGY OF LAW In this course we will read some influential legal cases, but our task will not be the technical application of the law (such as in a law school course). The social science field of law and society is designed to show both the impacts of the broader social context on law-making and judicial decision-making and the impacts of the law and the courts on society. Students will also be introduced to some classic law and society research. The topics we cover include: courts and social science, courts and economic interests, courts and social expectations, law and citizenship, the death penalty, law and culture, the limits of justice, litigation crisis, and legal globalization. Prerequisite: SOCI 10000 or permission of instructor. Alternate years. Spring 2012. [HSS] SOCI 20600. URBAN SOCIOLOGY (Urban Studies) An analysis of contemporary urban problems with an emphasis on race, class and gender. The course examines the historical roots of urban areas; global urban development; and present spatial, economic and political trends in cities. Prerequisite: SOCI 10000 or permission of instructor. Fall 2011. [HSS] SOCI 20700. SOCIOLOGY OF GENDER (Women’s, Gender, and Sexuality Studies) An examination of the role of gender in society, exploring how gender intersects with race, ethnicity, social class, sexuality, and nationality. The course examines biological, psychological, and social structural explanations of gender roles, with emphasis on the experiences of women and men within social institutions such as family, work, and education. Prerequisite: SOCI 10000 or permission of instructor. Alternate years. Not offered 2011-2012. [HSS] SOCI 20900. INEQUALITY IN AMERICA (Africana Studies, Education) An examination of the structure and process of inequality in the United States. Included will be an analysis and explanation of the extent of lifestyle as well as economic, occupational, and political inequality among groups, including gender and race as dimensions of inequality. Policies aimed at dealing with inequality will also be addressed. Prerequisite: SOCI 10000 or permission of instructor. Alternate years. Spring 2012. [HSS] SOCI 21100-21105. ADVANCED TOPICS IN SOCIOLOGY (some sections cross-listed with: Chinese Studies, Women’s, Gender, and Sexuality Studies) A seminar focusing on a specialized area of sociology. Topics are chosen by the instructor and announced in advance. Prerequisite: SOCI 10000. Annually. Not offered in 2011-2012. [HSS] SOCI 21300. DEVIANCE AND CRIMINOLOGY An analysis of deviant and criminal behavior. The focus is on definitions and measurement of deviant and criminal behavior. The major types of criminal behavior that occur in the United States are discussed, followed by a review of several sociological theories that explain criminal behavior. The course concludes with a general overview and assessment of major agencies that comprise the Criminal Justice System. Prerequisite: SOCI 10000 or permission of instructor. Annually. Not offered 2011-2012. [W†, HSS] SOCI 21400. RACIAL AND ETHNIC GROUPS IN AMERICAN SOCIETY (Africana Studies, Education) An analysis of racial and ethnic groups in the United States. Emphasis is placed on investigating discrimination based on race, gender, and culture; how discrimination develops; and the solutions proposed for solving the problems associated with it. Prerequisite: SOCI 10000 or permission of instructor. Annually. Fall. [C, HSS] SOCI 21500. AMERICAN MASCULINITIES An introduction to the sociological study of masculinity in its various guises in the contemporary United States. The theoretical perspective of the course is based on three fundamental premises: there is no single masculinity, but rather multiple masculinities; individuals in society are best understood as doing gender rather than as being gender; and masculinities are not static identities, but are fluid, fragile, negotiated, and always subject to contestation. The course explores the complex world of American masculinities through a series of overlapping Sociology and Anthropology 186 themes that students will reflect on and analyze as the class progresses through a variety of writing assignments. The course material is presented through readings and visual images. Prerequisite: SOCI 10000 or permission of instructor. Alternate years. Not offered 2011-2012. [W, C, HSS] SOCI 21700. BLACKS IN CONTEMPORARY AMERICAN SOCIETY (Africana Studies) A sociological study of the life experiences of African Americans, including a focus upon a critical analysis of race relations as it impacts intra- and intergroup dynamics. The primary focus of the course may vary (i.e., family, community, development, leadership). Prerequisite: SOCI 10000 or permission of instructor. Alternate years. Spring 2012. [C, HSS] SOCI 21900: GLOBALIZATION AND CONTEMPORARY CHINA (East Asian Studies) An exploration of the social causes, including globalization, and consequences of the economic, cultural, and social changes that China is undergoing today. Following a roughly chronological order, we will focus mainly on events and trends of the past twenty years: from the social movements of 1989 and the economic expansion of the early 1990s to the consequent changes in a consumer-driven popular culture, as well as renewed quests for moral and religious meaning and emerging social activism (e.g., the environment, women’s rights, etc.). Prerequisite: SOCI 10000 or permission of instructor. Alternate years. Not offered 2011-2012. [C, HSS] SOAN 24000. RESEARCH METHODS (Archaeology, Urban Studies) A general introduction to research methods and the analysis of social science data. Students will learn about the process of doing research—from forming a research question, to collecting data, to analyzing the data. Basic qualitative and quantitative data collection techniques taught will include surveys, interviewing, content analysis, and participation observation. The course will cover elementary statistical analysis and qualitative data coding. Prerequisite: SOCI 10000 or ANTH 11000 or permission of instructor. Annually. Fall and Spring. [HSS] SOCI 34200. SOCIAL STATISTICS (Urban Studies) An examination of the statistical analysis of social science data. Students will be trained to use statistical techniques, including chi square, t-testing, and regression. Emphasis will be placed on understanding the logic behind the numbers. The course will enable students to think critically about statistics in social research and in the popular media. Prerequisite: SOCI 24000 or permission of instructor. Annually. Fall. [Q, HSS] SOCI 35000. CLASSICAL SOCIAL THEORY (Archaeology) An examination of classical social theories of the nature of society and of human behavior. Included are the works of Marx, Durkheim, Weber, and Simmel. Emphasis will be placed on understanding these theories and their relevance in contemporary society. Prerequisite: SOCI 10000 or permission of instructor. Annually. Fall. [HSS] SOCI 35100. CONTEMPORARY SOCIAL THEORY An examination of the wide range of contemporary social theories that developed out of the classical tradition. Among the theories examined in this course are: functionalism, conflict theory, feminist theory, critical race theory, queer theory, globalization theory, and various forms of late- or post-modern theory. Students will be expected to develop an understanding of the relevance of these theories for the critical analysis of contemporary social issues and structures. Over the course of the semester all students will use one or more of these contemporary social theories to develop a theoretical perspective on a research question or topic that the students will examine in their Senior Independent Study thesis. This course, or SOCI 35000, is a prerequisite for enrolling in SOCI 45100. Prerequisite: SOCI 10000, or permission of instructor. Annually. Spring. SOCI 40000. TUTORIAL A tutorial course on a special topic(s) offered to an individual student under the supervision of a faculty member. Prerequisite: SOCI 10000 or permission of instructor; the approval of both the supervising faculty member and the chairperson is required prior to registration. SOCI 40700, 40800. INTERNSHIPS In close consultation with a faculty member in the department, students may arrange for credit for a supervised work situation that relates to their major course of study. It is expected that in addition to the work experience itself, this course will include both regular discussion of a set of readings chosen by the faculty member and written assignments that allow the students to reflect critically on their work experiences. Internship credit will be approved by the chairperson of the department on a case-by-case basis. S/NC course. Prerequisite: SOCI 10000, ANTH 11000, or permission of instructor. Sociology and Anthropology 187 SOCI 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the second semester. The student will normally do the thesis during the Fall and Spring semesters of the senior year. The suggested fields include papers or projects in any of the standard subcategories of sociology, such as family, community, race, urban, mental health, or social work. The student is assigned to an appropriate adviser by the chairperson following submission of a proposal. Prerequisite: SOCI 35000 or 35100. SOCI 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: SOCI 45100. ANTHROPOLOGY COURSES ANTH 11000. INTRODUCTION TO ANTHROPOLOGY (Archaeology) An introduction to the five fields used by anthropologists to explore broadly the variety of human groups that have developed across the globe and throughout time. The five fields include biological, cultural, linguistic, applied anthropology, and archaeology. The course will prepare students to take a holistic perspective on contemporary human cultures. It will also foster an appreciation of cultural relativity in the sense of understanding other cultures in their own terms as coherent and meaningful designs for living. Annually. Fall and Spring. [C, HSS] ANTH 11100. TOPICS IN ANTHROPOLOGY A seminar focused on a special topic in anthropology. Topics are chosen by the instructor and announced in advance. Annually. Not offered 2011-2012. [HSS] SOAN 20100. EDUCATION IN SOCIOCULTURAL CONTEXT (Education) An acquaintance with selected anthropological and sociological approaches to the study of education. It seeks to communicate a cross-cultural perspective on the educative process through case studies of education and socialization in diverse societies. Theories and research on the social effects of schooling will also be covered. Special attention will be given to the situation of minorities in the schooling process and to understanding educational policy debates in American society. Alternate years. Not offered 2011-2012. [C, HSS] SOAN 20200. GLOBALIZING HEALTH The twenty-first century has presented numerous public health challenges, such as the AIDS crisis, the rise of multidrug-resistant tuberculosis, and trafficking in human organs and tissues. Such problems can only be addressed by a combination of local and global responses. This course applies contemporary globalization theories to such public health challenges, and critically examines the ways in which Western medical techniques and attitudes toward health are disseminated throughout the world, and the tensions generated in local cultures by this globalization of health. Every third year. Not offered 2011-2012. [C, HSS] ANTH 20500. POLITICAL ANTHROPOLOGY (Archaeology) A comparative analysis of politics as the cultural process through which people make binding decisions for groups. The course examines this process in western and non-western cultures at all stages of complexity from bands to stages within an evolutionary model. Prerequisite: ANTH 11000 or permission of instructor. Alternate years. Fall 2011. [W†, C, HSS] ANTH 21000. PHYSICAL ANTHROPOLOGY (Archaeology) An introduction to the role of physical anthropology in defining humans as biological and cultural entities. This course examines a variety of topics, including the genetic basis for evolution, primate behavior, the process of primate and human development, and contemporary variation among human populations. Prerequisite: ANTH 11000 or permission of instructor. Alternate years. Fall 2011. [C, HSS, MNS] ANTH 21100. ADVANCED TOPICS IN ANTHROPOLOGY A seminar focusing on a specialized area of anthropology. Topics are chosen by the instructor and announced in advance. Prerequisite: ANTH 11000. Annually. Fall and Spring. [HSS] ANTH 22000. LINGUISTIC ANTHROPOLOGY (Archaeology, Communication) A critical analysis of language and all other forms of human communication within the context of culture and society, human thought, and behavior. Special attention is paid to the relationships between culture and language, the social uses of language, language as a model for interpreting culture, language and all forms of nonverbal communication within speech interactions. Prerequisite: ANTH 11000 or permission of instructor. Alternate years. Spring 2012. [C, HSS] Sociology and Anthropology 188 ANTH 22500. GENDER IN WORLD CULTURES (Women’s, Gender, and Sexuality Studies) An examination of the ways in which the boundaries of gender construct, reflect, and influence cultural ideology and social interaction from a cross-cultural perspective. This course also examines the development of gender studies within the discipline of anthropology. Prerequisite: ANTH 11000 or permission of instructor. Every three years. Fall 2011. [C, HSS] ANTH 23000. MAGIC, WITCHCRAFT, AND RELIGION Focuses on anthropological approaches to the study of cultural beliefs in the sacred: analysis of what is “religious” in many cultures; covers a variety of anthropological topics related to these practices, including myth, ritual, totemism, magic, and shamanism. Examination of the role that the study of religion, magic, and witchcraft has played in the theoretical development of anthropology. Prerequisite: ANTH 11000 or permission of instructor. Alternate years. Not offered 2011-2012. [C, R, HSS] ANTH 23100-23112. PEOPLES AND CULTURES (some sections cross-listed with: Archaeology, East Asian Studies, Latin American Studies) An exploration of the richness and diversity of a particular world culture. Readings and lectures provide the historical background for each culture area and an examination of the contemporary cultures. Generally focused on religious beliefs, economics, politics, kinship relationships, gender roles, and medical practices. Consideration of this culture area in the world economic system. Prerequisite: ANTH 11000 or permission of instructor. Annually. Fall and Spring. [C, HSS] SOAN 24000. RESEARCH METHODS (Archaeology, Urban Studies) A general introduction to research methods and the analysis of social science data. Students will learn about the process of doing research—from forming a research question, to collecting data, to analyzing the data. Basic qualitative and quantitative data collection techniques taught will include surveys, interviewing, content analysis, and participation observation. The course will cover elementary statistical analysis and qualitative data coding. Prerequisite: SOCI 10000 or ANTH 11000 or permission of instructor. Annually. Fall and Spring. [HSS] ANTH 34100. ETHNOGRAPHIC METHODS A course that is designed to build on the required Research Methods course (SOAN 240) and explore a variety of methods that are essential components to qualitative ethnographic research. The readings for this course include a selection of ethnographies as products of qualitative research methods. These ethnographies also illustrate the many ways in which the ethnography as final product can be constructed. Students learn how to design their own qualitative research projects; how to conduct the actual research; and, how to present a series of brief “ethnographic” descriptions based on the data they collect. Students will need to purchase a digital or tape recorder, as well as batteries and tape cassettes for this equipment. Annually. Spring. [HSS] ANTH 35200. CONTEMPORARY ANTHROPOLOGICAL THEORY An examination of key theoretical perspectives in anthropology from the mid-1900s to the present day. Among the perspectives examined in this course are: evolutionary theory, historical particularism, functionalism, culture and personality, cultural and ecological materialism, ethnoscience, symbolic anthropology, feminist anthropology, practice theory, and postmodernism. Students will be expected to develop an understanding of the relevance of these theories for the critical analysis of contemporary social and cultural issues. Over the course of the semester all students will use relevant concepts and theorists to develop a theoretical perspective on a research question or topic that they will examine in their Senior Independent Study thesis. This course is a prerequisite for enrolling in ANTH 45100. Prerequisite: ANTH 11000 or permission of instructor. Annually. Spring. ANTH 40000. TUTORIAL A tutorial course on a special topic(s) offered to an individual student under the supervision of a faculty member. Prerequisite: ANTH 11000 or permission of instructor; the approval of both the supervising faculty member and the chairperson is required prior to registration. ANTH 40700, 40800. INTERNSHIPS In close consultation with a faculty member in the department, students may arrange for credit for a supervised work situation that relates to their major course of study. It is expected that in addition to the work experience itself, this course will include both regular discussion of a set of readings chosen by the faculty member and written assignments that allow the students to reflect critically on their work experiences. Internship credit will be approved by the chairperson of the department on a case-by-case basis. S/NC course. Prerequisite: SOCI 10000, ANTH 11000, or permission of instructor. ANTH 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research guided by a faculty mentor and which culminates in a thesis and an oral examination in the South Asian Studies 189 second semester. The student will normally do the thesis during the Fall and Spring semesters of the senior year. Suggested fields include papers or projects in any of the standard subcategories of anthropology, such as kinship, politics, economics, religion, education, media, gender, or ethnicity. The student is assigned to an appropriate adviser by the chairperson following submission of a proposal. Prerequisite: ANTH 35200. ANTH 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: ANTH 45100. SOUTH ASIAN STUDIES CURRICULUM COMMITTEE: Mark Graham (Religious Studies), Chair Lisa Crothers (Religious Studies) Shirley Huston-Findley (Theatre and Dance) Elizabeth Schiltz (Philosophy) The interdepartmental minor in South Asian Studies focuses on developing an understanding of the diverse but related historical and cultural traditions of South Asia (a region that is comprised primarily of, but not necessarily limited to, the nations of India, Pakistan, Bangladesh, Nepal, Bhutan, Sri Lanka, Afghanistan, and Myanmar), both in their “home” locations and in their global and diasporic situations. The South Asian Studies minor recognizes the diversity of cultural and national traditions that exist across this region, but at the same time helps foster an understanding of the common cultural and historical concerns that make “South Asia” a coherent focus of study. Given the complexity of histories and traditions in this region of the world, the minor requires multidisciplinary study, and the integration of off-campus study in a South Asian country as part of the curriculum. Minor in South Asian Studies Consists of six courses: • Six elective South Asian Studies courses from at least two departments Special Notes • Off-campus Study: The minor in South Asian Studies requires the completion of an approved off-campus study program in a South Asian country. Acceptable programs can be either Wooster-endorsed semester-long programs in a South Asian country (e.g., India or Nepal), or South Asia-focused “Wooster In” programs led by Wooster faculty members (e.g., Exploring India at Home and Abroad Through the Arts; Global Social Entrepreneurship, focused on India). • A maximum of three courses completed for transfer credit during an approved off-campus study program in South Asia may, with the South Asian Studies curriculum committee’s approval, be counted toward completion of the minor. • College of Wooster courses not listed below (e.g., new interdepartmental or special topics courses) may also be approved for the minor, if such courses are focused on South Asia. For example, in 2009-2010, the following courses were offered: Global Social Entrepreneurship Seminar/Internship (IDPT 40600, 40700), which was focused on India; History of Indian Politics Since 1948 (IDPT 19902). Though not listed below as part of the regular course offerings, each of these could be applied to the minor. • Only grades of C- or better are accepted for the minor. Spanish 190 SOUTH ASIAN STUDIES COURSES PHILOSOPHY PHIL 23000. EAST/WEST COMPARATIVE PHILOSOPHY [W†, C, AH] PHIL 23100. INDIAN PHILOSOPHY AND ITS ROOTS [W†, C, AH] RELIGIOUS STUDIES RELS 21800. HINDUISM [C, R, AH] RELS 22000. BUDDHISM [C, R, AH] RELS 22200. ISLAM [C, R, AH] RELS 23000. THE LIFE AND THOUGHT OF MAHATMA GANDHI [W, C, R] RELS 26700-26722. TOPICS IN RELIGIOUS TRADITIONS AND HISTORIES (approved when topic is South Asia-related) [R] RELS 26900-26929. TOPICS IN THEORIES AND ISSUES IN THE STUDY OF RELIGION (approved when topic is South Asia-related) [R] THEATRE AND DANCE THTD 24300. EXPLORING INDIA AT HOME AND ABROAD THROUGH THE ARTS [C, AH] SPANISH Brian Cope, Chair Mary Addis John Gabriele Cynthia Palmer J. Morgan Robison Diane Uber Cecilia Fraga (Spanish Language Assistant) The curriculum of the Department of Spanish is designed to develop a critical understanding of cultural difference in a variety of contexts that correspond to the three general areas in the study of Hispanic languages, literatures, and cultures: Peninsular Spanish Literature and Culture, Latin American Literature and Culture (including the Caribbean and U.S. Latino), and Hispanic Linguistics. The department strongly recommends that students take courses in all three areas. The department’s curriculum seeks to develop skills in spoken and written language, linguistic and literary analysis, and cultural knowledge — all of which are considered inseparable and complementary. Spanish is the language of instruction at all levels. The Spanish curriculum may be utilized for specialization leading to public school or university teaching; research in Peninsular Spanish literature and culture, Latin American literature and culture or Hispanic Linguistics; business and government work; work with the Peace Corps and a wide variety of professional, service, and voluntary agencies in Spanish-speaking areas of the world and urban and rural concentrations of Hispanic peoples in the USA. Students interested in Spanish as preparation for a career in international business or finance should consider the Interdisciplinary Minor in International Business Economics. Students interested in the history, cultures, and languages of Latin America may consider the multidisciplinary Minor in Latin American Studies. Major in Spanish Consists of eleven courses: • SPAN 20100 • SPAN 20200 Spanish 191 • SPAN 22300 or 22400 • SPAN 27000 or 31000 • One of the following courses: SPAN 30100, 30200, 30500, or 30900 • Three elective Spanish courses at the 200-level or above • Junior Independent Study: SPAN 40100 • Senior Independent Study: SPAN 45100 • Senior Independent Study: SPAN 45200 Minor in Spanish Consists of six courses: • SPAN 20100 • SPAN 20200 • SPAN 22300 or 22400 • SPAN 27000 or 31000 • Two elective Spanish courses at the 200-level or above Special Notes • Language requirement and Placement Exam: Successful completion of the first two courses of a foreign language satisfies the College’s Foreign Language Graduation Requirement. In Spanish, this corresponds to SPAN 10100 and 10200. The Spanish Placement Exam is administered each year to incoming students during first-year registration to determine the proficiency level of students who have previously studied Spanish and to determine whether they have met the graduation requirement in foreign language, and to determine course selection for those students who wish to continue to study Spanish. If a student registers for and completes a course in a language below the level at which the language department’s placement exam placed him or her, that student will not receive credit toward graduation for that course, without prior permission of the instructor of the course into which the student placed, and of the Department Chair. • Transfer Credit for the College’s Foreign Language Requirement: In order to receive transfer credit toward satisfying The College of Wooster’s language requirement for taking the equivalent of SPAN 10100 (Beginning Spanish I) or SPAN 10200 (Beginning Spanish II) from another institution, the following requirements must be met: (1) The course must be taken at an accredited institution. Consult with the Office of the Registrar for this information; (2) A minimum of sixty contact hours is required for the transfer of credit; (3) If the institution is on a semester system, the course must be worth at least four semesterhours credit; (4) If the institution is on a quarter system, the course must be worth at least six quarter-hours credit; (5) The student must receive a grade of C or higher in the course. Students who wish to meet the College’s foreign language requirement in Spanish through transfer credit for courses that do not meet the minimum requirements above must consult with the chairperson prior to such study, and will be required to take the departmental placement exam to demonstrate proficiency through the SPAN 10200 level. The Department of Spanish does not accept transfer credit from dual enrollment programs to fulfill the graduation requirement or requirements in the major or the minor. A student who seeks to fulfill the College’s foreign language requirement for graduation based on work completed through a dual enrollment program must take the Spanish Placement Exam. The College does not accept transfer credit for online or distance learning courses. • SPAN 20100, 20200, either 22300 or 22400, either 27000 or 31000, one 300-level Spanish 192 literature course, the Junior Seminar (SPAN 40100), SPAN 45100 and 45200 are required of all majors. Students who place above SPAN 20100 may take another upper-level course to complete the major. Whenever possible, the department strongly encourages students to take more than the required minimum of eleven courses. • The Junior Seminar is to be completed before Senior Independent Study. A student may fulfill the Junior Seminar requirement by completing SPAN 31000, if not already taken to fulfill the department’s requirements for the major, or an additional 300-level literature course. SPAN 31900 (Applied Linguistics) does not fulfill the Junior Seminar requirement. • Students who place above SPAN 20100 may take another upper-level course to complete the minor. It is strongly recommended that one 300-level literature course be one of the six required courses for the minor. • Regarding the Major and the Minor: SPAN 27000, 31000, the required 300-level literature course, and Junior Seminar may not be completed through transfer credit. A student may take both SPAN 22300 and 22400 for credit toward the major or the minor only with the permission of the Department Chair. A single 300-level literature course may not count as both the required literature course and Junior Seminar. SPAN 31000 may not count as both the required linguistics course and Junior Seminar. No more than one Spanish course taught in English may count toward the major. No Spanish courses taught in English may count toward the minor. Courses taken S/NC are not permitted in either the major or the minor. • Teaching Licensure: Students interested in pursuing a career in elementary or secondary school teaching must complete the requirements for Multiage Licensure in Spanish as listed under Teacher Licensure. Complete information about the requirements may be found in Teacher Education at the College of Wooster: A Supplement to the Catalogue (which can be found at the following website: www3.wooster.edu/education/current/forms.html). Students should consult with the chairpersons of the departments of Spanish and Education. • Please see the Degree Requirements section of this Catalogue, and the Department of Spanish Majors’ Handbook, available on the Spanish Department webpage, for more complete information. In case of questions about the requirements for the major or the minor in Spanish, students should consult with the chairperson of the department. • Only grades of C- or better are accepted for the major or minor. SPANISH COURSES SPAN 10100. BEGINNING SPANISH LEVEL I Oral-aural instruction and practice with grammar, reading, and some writing. Emphasis on practical everyday language for direct communication. Instruction focuses on the cultural meaning of language. Annually. Fall. SPAN 10200. BEGINNING SPANISH LEVEL II Additional oral-aural instruction and continued practice with grammar, reading, and writing. Further emphasis on practical everyday language for communication. Instruction focuses on the cultural meaning of language. Prerequisite: SPAN 10100 or placement. Annually. Spring. SPAN 20100. INTERMEDIATE SPANISH FOR GRAMMAR, CONVERSATION, AND COMPOSITION I Extensive practice in conversation and composition with comprehensive grammar review. Reading and discussion of short texts. Structured to improve oral and written proficiency and to develop reading ability by way of vocabulary building, recognition of grammatical structures, and determining meaning from context. Prerequisite: SPAN 10200, equivalent, or permission of the instructor. Annually. Fall and Spring. Spanish 193 SPAN 20200. INTERMEDIATE SPANISH FOR GRAMMAR, CONVERSATION, AND COMPOSITION II A continuation of Spanish 201. Extensive practice in conversation and composition with comprehensive grammar review. Reading and discussion of short texts. Structured to improve oral and written proficiency and to develop reading ability by way of vocabulary building, recognition of grammatical structures, and determining meaning from context. Prerequisite: SPAN 20100, equivalent, or permission of the instructor. Annually. Fall and Spring. SPAN 21200. LITERATURE AND CULTURE OF THE HISPANIC CARIBBEAN (Africana Studies, Compar - ative Literature, Latin American Studies) Taught in English. A study of Caribbean culture, literature, and film with special emphasis on the African heritage and the cultural politics of race. Topics include colonization and transculturation, slavery and plantation culture, maroon resistance, negrismo, race, and nationalism. Primary texts include readings in social and cultural history, film, autobiography, historical fiction, and poetry. Works by Juan Francisco Manzano, Miguel Barnet, Alejo Carpentier, Nicolás Guillén, Nancy Morejón, Rosario Ferré, and Ana Lydia Cabrera. Alternate years. Not offered 2011-2012. [C, AH] SPAN 21300. U. S. LATINO LITERATURES AND CULTURES (Comparative Literature, English, Latin Amer ican Studies) Taught in English. A study of U. S. Latino literature, culture, and film that focuses on questions and issues of ethnic identity as presented in works by Puerto Rican, Chicano, Mexican, Cuban, and Dominican authors who live and write in the United States. Topics will include self-representation, “ethnic” autobiographical discourse, the concept of language literacy and legacy, border theory, and the notions of (be)longing and displacement. Alternate years. Not offered 2011-2012. [C, AH] SPAN 22300. READINGS IN SPANISH PENINSULAR CULTURES The study of selected, key issues in the cultures of Spain through the close reading and analysis of appropriate texts. The focus is on the nature of cultural values, political and gender ideologies, social norms, institutions, and cultural practices as manifested in the literature and the visual arts of Spain. Prerequisite: SPAN 20200, equivalent, or permission of the instructor. Annually. Fall. [W, C] SPAN 22400. READINGS IN LATIN AMERICAN CULTURES The study of selected, key issues in the cultures of Latin America through the close reading and analysis of appropriate texts. The focus is on the nature of cultural values, political and gender ideologies, social norms, institutions, and cultural practices as manifested in the literature and the visual arts of Latin America. Prerequisite: SPAN 20200, equivalent, or permission of the instructor. Annually. Spring. [W, C] SPAN 24700. TWENTIETH AND TWENTY-FIRST CENTURY SPANISH PENINSULAR WRITERS (Compar ative Literature) Introduction to Spanish Peninsular literature and textual analysis through readings of representative genres of the twentieth and twenty-first centuries. Intensive study and discussion of selected passages to develop a critical approach to the literary currents that have most clearly contributed to the development of Spanish Peninsular literature of the period. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Every three years. Not offered 2011-2012. [C, AH] SPAN 24800. TWENTIETH AND TWENTY-FIRST CENTURY SPANISH AMERICAN WRITERS (Compar - ative Literature, Latin American Studies) Introduction to Spanish American literature and textual analysis through readings of representative genres of the twentieth and twenty-first centuries. Intensive study and discussion of selected passages to develop a critical approach to the literary currents that have most clearly contributed to the development of Spanish American literature of the period. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Every three years. Not offered 2011-2012. [C, AH] SPAN 25000. COMMERCIAL LANGUAGE AND CULTURE IN THE HISPANIC WORLD (Latin American Studies) The study of the general linguistic, geographic, and cultural proficiency essential to conducting business in Spanish successfully, both in the United States and abroad. The focus is on vocabulary building, written and spoken business communications, and role plays. Recommended: ECON 10100. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Alternate years. Fall 2011. [C] SPAN 27000. SPANISH PHONOLOGY (Latin American Studies) Introduction to Spanish Phonology and its historical development from Latin. The focus is on the principles of Spanish 194 phonetics and diction. Attention is given to speech characteristics and to dialectal differences in Peninsular and Spanish American phonology. Oral drill to improve pronunciation and diction. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Alternate years. Spring 2012. [AH] SPAN 27500. INTERMEDIATE SEMINAR: SPECIAL TOPICS IN HISPANIC LANGUAGE, LITERATURE, AND CULTURE (Comparative Literature) Studies in Hispanic language, literature, and culture varying in topic from year to year. Topics will be chosen for their significance and impact on the Hispanic cultures and may include, but are not limited to, religion, politics, philosophy, feminism, minority groups, linguistics. May be taken more than once. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Not offered 2011-2012. [Depending on the topic, C, AH] SPAN 28000. HISPANIC FILM (Comparative Literature, Film Studies, Latin American Studies) Taught in English. A study of the history and evolution of cinema in Spain and Latin America with special attention paid to the documentaries and avant-garde films of the silent era, the neo-realist trends of the 40s, 50s, and 60s, the national cinemas of the 70s, 80s, and 90s, and the new directions of the contemporary period. The course focuses on the continuity of the auteur tradition in an industry dominated by Hollywood. Topics for discussion include: film as a means of exposing or confronting social injustice, nation-building, (de)constructing identity, problematizing modernity, subverting social codification/codifying subversion. Requirements: two evening film screenings per week and pre-assigned readings on film criticism, history and theory. Every three years. Not offered 2011-2012. [C, AH] SPAN 30200. GOLDEN AGE LITERATURE (Comparative Literature) A study of the principal trends and themes in Golden Age literature. Reading, analysis, and discussion of selected literary works of the Renaissance and Baroque periods that most clearly reflect the cultural, social, and psychological temperament of sixteenth and seventeenth century Spain. Readings include the poetry of Góngora and Quevedo, and the plays of Alarcón, Lope de Vega, Tirso de Molina, Calderón de la Barca, and Rojas Zorrilla. Introduction to the research methods for the study of Hispanic literature and culture. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Every three years. Not offered 2011-2012. [C, AH] SPAN 30500. THE CONTEMPORARY LATIN AMERICAN NOVEL (Comparative Literature, Latin American Studies) The study of selected Latin American novels of the Boom and post-Boom. Consideration of technical innovation, gender difference, literature and history. Novelists studied include Rulfo, Vargas Llosa, García Márquez, Traba, Puig, and Skármeta. Introduction to the research methods for the study of Hispanic literature and culture. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Every three years. Fall 2011. [C, AH] SPAN 30900. TRENDS IN SPANISH AMERICAN LITERATURE (some sections cross-listed with: Comparative Literature, Latin American Studies, Women’s, Gender, and Sexuality Studies) The study of major literary currents of Spanish America from the nineteenth century to the present through the readings, discussion, and criticism of key literary works that have most clearly contributed to the development of Spanish American literature. Emphasis on the realist and regionalist novel, the essay, and late nineteenth century and twentieth century theater. Introduction to the research methods for the study of Hispanic literature and culture. May be taken more than once. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Every three years. Not offered 2011-2012. [C, AH] SPAN 31000. THE STRUCTURE OF MODERN SPANISH (Latin American Studies) A contrastive study of morphological, syntactic, and semantic structures of Spanish and English. This course is designed to help advanced students and prospective teachers of either language to gain knowledge of the particular areas of difficulty and correct problems. Introduction to the research methods for the study of Hispanic language and linguistics. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Alternate years. Not offered 2011-2012. [AH] SPAN 31100. ADVANCED SEMINAR: SPECIAL TOPICS IN HISPANIC LANGUAGE, LITERATURE, AND CULTURE (some sections cross-listed with Latin American Studies) An advanced seminar exploring a specific author or a limited number of authors, a literary period or genre, or a specific linguistic, literary, cultural topics or methodological approach. Topics will be chosen for their significance in Hispanic language, linguistics, literature, or culture. Introduction to the research methods for Hispanic Studies. May be taken more than once. Prerequisite: SPAN 20200 and either 22300 or 22400, or permission of the instructor. Not offered 2011-2012. [Depending on the topic, C, AH] Theatre and Dance 195 SPAN 31900. APPLIED LINGUISTICS Taught in English. Linguistic theory and its application in the teaching of foreign languages. Offered jointly by the departments of French, German, and Spanish. Individual practice for the students of each language. Required for licensure of prospective teachers of Spanish. This course does not fulfill the Junior Seminar requirement. Prerequisite: SPAN 27000, 31000, or permission of the instructor. Alternate years. Spring 2012. SPAN 39900. DON QUIXOTE: METAFICTION AND THE DAWNING OF THE MODERN NOVEL (COMPARATIVE LITERATURE) This course is offered in English. An in-depth study of Don Quixote as the beginning of the modern novel in the western world. Discussion of the inherent national values of Cervantes’s masterpiece and its intrinsic universal appeal. Study of the structure, motives, and motifs of the novel; of Cervantes’s narrative technique and narrative theory; of point of view in the novel; of the the themes of self-conscious literature and metafiction; of Don Quixote’s heroism and folly; and of the “quixotic principle” and its impact on the evolution of western narrative tradition. Prerequisite: prior study of literature or film recommended. Spring 2012. {C, AH} SPAN 40000. TUTORIAL Individual study of a topic developed in consultation with the faculty member of the department supervising the project. Prerequisite: The approval of both the supervising faculty member and the chairperson are required prior to registration. SPAN 40100. JUNIOR SEMINAR A student may fulfill the Junior Seminar requirement by completing SPAN 31000, if not already taken to fulfill the department’s requirements for the major, or an additional 300-level literature course. All 300-level courses provide an introduction to the research methods for the study of Hispanic language and linguistics, literature, and culture in preparation for Senior Independent Study. SPAN 31900 (Applied Linguistics) does not fulfill the Junior Seminar requirement. To be completed before Senior Independent Study. Fall and Spring. SPAN 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student engages in creative and independent research of a specific topic in Spanish language, culture, or literature under the direction of a faculty member of the department, and which culminates in a thesis and an oral examination in the second semester. Prerequisite: SPAN 40100, Spanish major of senior standing. SPAN 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: SPAN 45100. THEATRE AND DANCE Kim Tritt, Chair Shirley Huston-Findley James Levin Jimmy Noriega Dale Seeds Theatre and Dance, as studied at The College of Wooster, emphasizes the relationship between scholarship and artistry, investigating both the range and depth of the human experience. The Theatre and Dance Major and Minor curricula offer a broad range of knowledge designed to examine acting, directing, dance, design and technology, history, literature, playwriting, and theory, focusing in each area on the importance of analyzing texts in their various modes: the written text, the visual text, and the physical text. While the Theatre and Dance student may choose to specialize in one of these particular areas of the discipline for their Senior Independent Study, the departmental philosophy remains dedicated to the liberal arts belief in developing, through its interdisciplinary curricular structure, a combination of historical and Theatre and Dance 196 critical analysis in relationship to the study of various performance texts, resulting in the creation of the artist/scholar. The Theatre and Dance major consists of a minimum of 12 course credits: three 100-level foundational courses focusing on the understanding of text from a variety of perspectives, three 200-level history/literature/theory/criticism courses, two 300- level Topics courses, one 400-level Advanced Seminar, Junior Independent Study, and two semesters of Senior Independent Study. Major in Theatre and Dance Consists of twelve courses: • THTD 10100 • THTD 10200 • THTD 10300 • Two of the following courses: THTD 20100, 20200, 24400, or 24600 • One of the following courses: THTD 24100, 24200, 24300, 24500, or 24800 • Two of the following 300-level courses: THTD 30100, 30200, or 30300 • One of the following 400-level courses: THTD 44100, 44200, or 44300 • Junior Independent Study: THTD 40100 • Senior Independent Study: THTD 45100 (see note below) • Senior Independent Study: THTD 45200 Minor in Theatre and Dance Consists of six courses: • THTD 10100 • THTD 10200 • THTD 10300 • One of the following courses: THTD 20100, 20200, 24100, 24200, 24300, 24400, 24500, 24600, or 24800 • One of the following 300-level courses: THTD 30100, 30200, or 30300 • One of the following 400-level courses: THTD 44100, 44200, or 44300 Special Notes • Students choosing to include a production/performance component in their Senior IS must also take three sections of either THTD 12101 Performance Practicum or THTD 12102 Production Practicum, and one section of THTD 12103 Stage Management Practicum (.25 credit each), for a total of one credit. • Only grades of C- or better are accepted for the major or minor. THEATRE AND DANCE COURSES FOUNDATION COURSES THTD 10000. ARTS AND ENTREPRENEURSHIP (Art and Art History, Music) This course will provide an introduction to and overview of the philosophy of entrepreneurship as well as the operational, management and “real life” aspects of launching and maintaining a non-profit arts organization in contemporary America. The course will explore general principles and theories of the following aspects: legal, organizational, mission, board development, branding and marketing, fiscal and budgetary, fund-raising, programming and strategic planning. It will also examine the issues that modern artistic and managerial leadership confront including intellectual property, first amendment issues, successors, capital campaigns, and institutional survival. Classes will focus on the student’s conceptualization, formation, launch and management of a faux non-profit cultural organization. Fall 2011. [AH] THTD 10100. THE WRITTEN TEXT (W) The Theatre and Dance program at Wooster emphasizes the importance of analyzing texts in their various Theatre and Dance 197 modes: the written text, the visual text and the physical text of the performer’s body. These modes interact simultaneously with each other in the process of performance. This course specifically poses fundamental questions about the nature of written texts, and how they become transformed in the performance process. This understanding of texts is fundamental to both the enlightened theatre and dance audience member and to the work we do as actors, dancers, directors, choreographers, dramaturges, designers, technical personnel, and support staff. Annually. Spring. [AH] THTD 10200. THE VISUAL TEXT The visual text, which includes the images created by the body, scenery, lighting, costumes, properties, film, and digital imagery, forms the ways in which the written text is performed. Students will focus on the following: how visual elements narrate the story; the basic tools and principles of design and the visual arts which communicate space, meaning, mood and emotion; and how visual communication in a performance context is culturally based and informed by historical and stylistic insight. The student is expected to develop a visual literacy and to apply this knowledge to both the understanding of how these elements create meaning and the development of creative visual representations of a text. Annually. Fall 2011 and Spring 2012. [AH] THTD 10300. THE PHYSICAL TEXT An introductory level course intended to engage students in the study of movement as a primary text necessary for developing the art and craft of performance. Students will be introduced to the diversity of physical tools that shape movement of the performative body and how to analyze physical text with critical literary and cultural theory of Western and non-western performance systems. Annually. Fall. [AH] THTD 10400. THE IMPULSE TO CREATE (Art and Art History, Music) Creativity has been defined as “the process involving the generation of new ideas or concepts or associations.” The impulse to create is at the core of entrepreneurialism, which can be defined as the transformation of incoherent elements into a tangible “something,” to create art or a product or an event. Class discussion will range over diverse topics such as theories of creativity, student research into the sources of creativity for artists, innovators, inventors and creators, and experiential, in-class exercises exploring the students’ own creative impulses. Spring 2012. [AH] THTD 12101. PERFORMANCE PRACTICUM Performing in a faculty-directed theatre or dance production. Rehearsal and performance time must total a minimum of 40 hours. Only those students who are cast in faculty-directed productions should register for the Performance Practicum. Students cast in non-faculty directed productions may receive credit pending faculty approval through a student petition. (.25 course credit) Annually. Fall and Spring. THTD 12102. PRODUCTION PRACTICUM Practical experience in the production of a faculty directed play, musical or dance concert, including scene, costume or props design or construction; lighting design or execution; or serving on a stage or wardrobe crew. Non-faculty directed productions may receive credit pending faculty approval through a student petition. A minimum of 40 hours during the semester is required. Permission and arrangements are made through the instructor and the Department’s Technical Director. (.25 course credit) Annually. Fall and Spring. THTD 12103. STAGE MANAGEMENT PRACTICUM Practical experience in stage management of a faculty directed play, musical or dance concert, including serving as an assistant stage manger or assistant director. Non-faculty directed productions may receive credit pending faculty approval through a student petition. A minimum of 40 hours during the semester is required. Prerequisite: Permission and arrangements are made through the instructor and the Department’s Technical Director. (.25 credit). Annually. Fall and Spring. HISTORY, LITERATURE, THEORY & CRITICISM THTD 20100. CONTEMPORARY DANCE HISTORY This course explores the development of contemporary dance as an art form. Rich in diversity, the modern dance is world-conscious, concerned with social, cultural, and personal issues. Beginning with an introduction to late-nineteenth-century theatrical dance, this class will examine twentieth-century concert dance choreographers and their work as evidence of identity and change through dance literature, critical essays, and film. Alternate years. Not offered 2011-2012. [C, AH] Theatre and Dance 198 THTD 20200. DANCE IN WORLD CULTURES An introductory overview to selected dance traditions of the world. The course will examine such issues as the role of the physical text in dance, influences from other cultures, and culture-specific choices of the physical body. Students will gain understanding of how dance is embedded in the belief systems of the people who created it, how dance forms have changed and why, and develop skills in communicating about dance orally and in written form. Alternate years. Not offered 2011-2012. [C, AH] THTD 24100. Latina/o Drama and Performance (Latin American Studies) This course is an introduction to the history of Latina/o theatre and performance in the United States. By dismantling borders and opening up the public space of performance, students will explore topics related to identity and representation within the various Latina/o communities in the U.S. Analyzing a variety of performance genres and styles, the course examines how creative forms challenge dominant ideology and culture. Topics of emphasis include: immigration and diaspora, family and heritage, gender and sexuality, assimilation and resistance, violence, politics, and class struggle. Students will engage in historical, social, political, and cultural analyses of the theatre being created by Latina/os and the ways that their works bridge the gap between Latin America and the United States. Fall 2011. [C, AH] THTD 24200. AFRICAN AMERICAN THEATRE HISTORY (Africana Studies) An overview of the history and literature of African Americans in theatre from the pre-Civil War era to the emergence of contemporary theatre. Students will compare images of blacks as created by both black and white playwrights and the effect of those images on social attitudes, through the reading and analyses of various plays. In addition, the lives and contributions of noted African American artists will be researched. Not offered 2011-2012. {C, AH] THTD 24300. EXPLORING INDIA AT HOME AND ABROAD THROUGH THE ARTS (South Asian Studies) This interdisciplinary course provides students an opportunity to examine the rich history of the arts and culture of India both at home during the fall semester and abroad in a three-week field study experience during winter break. Through readings, discussions and guest lectures, the fall semester course, meeting one day per week, focuses on developing a foundational knowledge regarding the geography, religions, history, and cultural practices of India, as well as a more in depth awareness of the richness of the arts in their various forms. The three-weeks abroad provides students with a field experience where they will attend traditional January festivals in Chennai, engage in folk arts in the village of Dakshinachitra, interact with Indian artists and scholars in Kerala Kalamandalam University of Arts and Culture, and participate in a service-project in Wooster Nagar. Fall 2011. [C, AH] THTD 24400. ORIGINS OF DRAMA This course introduces students to the origins of eastern and western dramas, focusing primarily on Europe, the U.S, and India, emphasizing the relationships between history, dramatic literature, and theory. Alternate years. Spring 2012. [AH] THTD 24500. FEMINISM AND THEATRE (Women’s, Gender, and Sexuality Studies) This course is designed to explore theories of feminism and gender issues in relation to dramatic literature from a wide range of time periods and perspectives. Emphasis will be placed on developing student appreciation of and critical responses to traditional and non-traditional forms of drama as they relate to women as bodies in performance; the relationship of the male gaze (in film and on stage) to both canonical and non-canonical works; and marginalized voices (e.g.; women of color). Every third year. Not offered 2011-2012. [C, AH] THTD 24600. REALISM AND BEYOND This course traces the various theoretical movements found in the development of world theatre from the introduction of Realism to the present, emphasizing the relationships between history, theory, criticism, and dramatic literature. Alternate years. Not offered 2011-2012. [AH] THTD 24800. NATIVE AMERICAN PERFORMANCE (Film Studies) The performance traditions within Native American cultures are extremely rich and diverse, embracing ritual, myth, spirituality, oral literature, art, music, dance, film, and, more recently, improvised and written scripts. A survey of this tremendous diversity would be impossible; accordingly, the course intends to indicate and suggest the diversity of recent Native performance in two ways: first, by focusing specifically on the range of recent performance practices of specific Native Alaskan and Native American peoples, and second, by the study of recent texts and performances by Native theatre groups, and performance artists such as Tomson Highway, William S. Yellow Robe, Drew Hayden Taylor, Chris Eyre, Marie Clements, and Hanay Geiogamah. Every third year. Not offered 2011-2012. [C, AH] Theatre and Dance 199 THTD 24900. INDIGENOUS FILM (Film Studies) The course explores how indigenous cultures throughout the world have combined ritual, myth, oral literature, art, music, and dance with contemporary film. It will focus primarily on the films that have recently emerged from indigenous cultures of North American, Northern Europe Australia Africa, Asia and Polynesia. We will examine traditional culture, stories and performance practices as a means to gain awareness as to how a culturally specific indigenous film genre, free from colonial domination, develops its own voice and unique visual language. Alternate years. Spring 2012. [C, AH] TOPICS Intended to create a natural extension from 100-level foundation courses, THTD 30100-30103 Topics in the Written Text, THTD 30200-30209 Topics in the Visual Text, and THTD 30300-30308 Topics in the Physical Text educate students in a variety of areas pertaining to the many possible foci available in the performing arts: acting, dance, directing, design, writing, and/or history, as well as practical application to Film Studies when possible. Four 300-level Topics courses will be provided each year, two per semester, rotating emphasis upon the Written Text, the Visual Text, and the Physical Text as appropriate. THTD 30100-30103. TOPICS IN THE WRITTEN TEXT (some sections cross-listed with Film Studies) Prerequisite: THTD 10100 or permission of the instructor. [W, AH] THTD 30200-30209. TOPICS IN THE VISUAL TEXT (some sections cross-listed with Film Studies) Prerequisite: THTD 10200 or permission of the instructor. [AH] THTD 30300-30308. TOPICS IN THE PHYSICAL TEXT (some sections cross-listed with Film Studies) Prerequisite: THTD 10300 or permission of the instructor. [AH] ADVANCED SEMINAR An Advanced Seminar course intended to engage students in theatre and dance through the written text, the visual text or the physical text as they connect to and reinforce the production program. Students will have an opportunity to experience the relationship between the classroom and the stage by being challenged with advanced theoretical and critical thinking that mingles with the skills inherent in production. THTD 44100-44102. ADVANCED SEMINAR IN THE WRITTEN TEXT Prerequisite: THTD 10100 and 30000. At the discretion of the instructor, prerequisites may be waived for students whose major has prepared them for the specific topic of a particular Advanced Seminar. Not offered 2011-2012. THTD 44200-44201. ADVANCED SEMINAR IN THE VISUAL TEXT Prerequisite: THTD 10200 and 30000. At the discretion of the instructor, prerequisites may be waived for students whose major has prepared them for the specific topic of a particular Advanced Seminar. Not offered 2011-2012. THTD 44300-44304. ADVANCED SEMINAR IN THE PHYSICAL TEXT Prerequisite: THTD 10300 and 30000. At the discretion of the instructor, prerequisites may be waived for students whose major has prepared them for the specific topic of a particular Advanced Seminar. OFF-CAMPUS STUDY THTD 39100, 39200. INDIVIDUAL SUMMER STUDY This course is intended to provide the advanced theatre student an opportunity to develop professionally by accepting a Summer Internship or Apprenticeship with a recognized theatre or dance company. Students will submit a detailed course proposal to the departmental faculty at the beginning of the second semester prior to commencing summer study. The reputation and operational procedures of each theatre organization will be closely scrutinized by the faculty in order to assure a significant experience for the student. Special attention will be paid to the supervision and evaluation of the summer experience by a Theatre and Dance faculty member. Students must turn in a journal to the supervising faculty member at the conclusion of the course. (1 – 2 course credits) Prerequisite: permission of the department. THTD 40700, 40800. PROFESSIONAL THEATRE INTERNSHIPS Internships with established professional theatres are included under this classification. Interns are assigned responsibilities by the host theatre, which they are expected to fulfill, and the theatre will make available other Urban Studies 200 opportunities for observation and participation. The student’s choice of theatre and its intern program must be approved by the department. Students choosing to study off-campus for a full semester are strongly encouraged to do so in either the Spring of their sophomore year or the Fall of their junior year to avoid conflict with the Junior Independent Study offered in the Spring only. S/NC course. GLCA NEW YORK ARTS PROGRAM A semester of study and work in New York with professionals in various aspects of theatre and dance according to individual interest. Students live in a dormitory-type environment where they also attend a number of seminars. The major portion of time is spent on-the-job as an intern with a well-known artist or artists and companies. Prerequisite: recommendations by the department chairperson and adviser, and acceptance by the administrators of the program in New York. (4 credits) INDEPENDENT STUDY THTD 40100. JUNIOR INDEPENDENT STUDY An application of methods of research (historical, theoretical, analytical) pertinent to the performing arts, with emphasis on developing a research agenda and writing process, including the formation of a critical question, sentence outline, and annotated bibliography. Requirements include a 25-30-page research paper. Prerequisites: THTD 101, 102, 103, the completion of a writing-intensive (W) course, at least one 300-level Topics course, and at least one of the three required history courses. Annually. Spring. THTD 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student produces a thesis and/or a project. The project can be in stage management, directing, acting, play writing, design, dance, or a devised production and must include a companion research paper that articulates and explores a critical question posed by the project. Prerequisite: THTD 40100. THTD 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and/or project. Prerequisite: THTD 45100. URBAN STUDIES CURRICULUM COMMITTEE: James Burnell (Economics), Chair Heather Fitz Gibbon (Sociology) Eric Moskowitz (Political Science) The Urban Studies program provides both an interdisciplinary major and an off-campus urban experience for non-majors. Urban Studies is sponsored by the departments of Economics, Political Science, and Sociology and administered by a faculty committee. The Urban Studies major allows students to gain a perspective on the different urban phenomena that crucially affect the quality of life of most Americans, and to relate their liberal education to specific and real human concerns. The Urban Studies major provides a social-scientific core from which students may elect to branch out into various curricular tracks that either broaden the disciplinary bases of urban understanding or deepen competence within a particular discipline. Major in Urban Studies Consists of fourteen courses: • URBN 10100 • ECON 10100 • One of the following courses: ECON 11000 or SOAN 24000 • URBN 20100 Urban Studies 201 • PSCI 20500 • SOCI 20600 • One of the following courses: ECON 21000 or SOCI 34200 • ECON 26100 • URBN 29100 • URBN 29200 • URBN 29300 • Junior Independent Study: URBN 40100 • Senior Independent Study: URBN 45100 • Senior Independent Study: URBN 45200 Minor in Urban Studies Consists of six courses: • URBN 10100 • ECON 10100 • One of the following course: ECON 11000, SOCI 24000, or Urban Semester • PSCI 20500 • SOCI 20600 • ECON 26100 Special Notes • Off-Campus Study: The Urban Studies Program requires off-campus study – the Urban Semester. The off-campus program should consist of a city seminar and an urban related internship. Contact the chairperson of Urban Studies about the opportunities and arrangements for the Urban Semester. • To be eligible for the Urban Semester, the major must complete either URBN 10100 or one of the following: ECON 26100, PSCI 20500, or SOCI 20600. • For the Urban Semester, the students enroll in URBN 29100-29200 Urban Field Study. This is the field placement for which the students receive two course credits. In addition, they enroll in URBN 29300 Urban Field Seminar, a course designed to familiarize the student with the particular problems of the host city. URBN 29100, 29200, and 29300 are graded S/NC. • Participation in the Urban Semester is also available to non-majors. The prerequisites for Urban Semester for the non-major are either URBN 10100 or two of the following courses: ECON 10100, 26100, PSCI 20500, SOCI 20600, or URBN 20100. The Urban Semester for the non-major consists of URBN 29100-29200 Urban Field Study and the additional options as provided for majors. • S/NC evaluation is not permitted for courses in the major, except for URBN 29100, 29200, and 29300. • Only grades of C- or better are accepted for the major or minor. URBAN STUDIES COURSES URBN 10100. CONTEMPORARY URBAN ISSUES An interdisciplinary approach to issues and institutions present in American cities. Contemporary urban problems related to growth, housing, poverty, race, social relations, etc., and public policies designed to alleviate them are analyzed from a social science perspective. Alternative ideological perspectives are presented. Annually. Fall. [HSS] URBN 20100-20103. SPECIAL TOPICS IN URBAN STUDIES A seminar exploring the current theories and research regarding selected issues facing urban areas. Topics will be announced in advance by the faculty member teaching the course. Prerequisite: URBN 10100 or any course in Economics, Political Science, or Sociology. Annually. Spring. [HSS] Women’s, Gender, and Sexuality Studies 202 URBN 29100, 29200. URBAN FIELD STUDY The city itself is the laboratory in which this learning experience takes place. The student becomes engaged in the activity of that “laboratory” through thirty or more hours a week of intern-type service in any one of a variety of public or private agencies. Placements are designed to meet the student‘s particular curricular and pre-professional interests. The field experience is supervised by a mature employee of the agency. On location in various cities. (2 course credits) URBN 29300. URBAN FIELD SEMINAR The cross-disciplinary analysis of the city as a political, social, and economic entity will draw upon and help interpret the student’s experience in urban field study. Utilizing various resources, including local citizens and leaders, attention will focus on acquisition and analysis of information about the host city. Seminar directed by staff on location in the city. URBN 40100. JUNIOR INDEPENDENT STUDY This course will introduce Urban Studies majors to the process of conducting social scientific research in an urban context. Students will be exposed to the practical techniques for accomplishing an urban research project. This includes providing the appropriate theoretical framework and specification of methodology that will be used to test hypotheses on urban phenomena. URBN 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which each student undertakes a significant, independent, interdisciplinary analysis of an urban-related topic, and which culminates in a thesis and an oral examination in the second semester. Prerequisite: URBN 40100. URBN 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: URBN 45100. WOMEN’S, GENDER, AND SEXUALITY STUDIES CURRICULUM COMMITTEE: Nancy Grace (English), Chair Travis Foster (English) Amber Garcia (Psychology) Raymond Gunn (Sociology) Katherine Holt (History) Monika Flaschka Stacia Kock The WGSS curriculum is based in feminist scholarship—both within traditional disciplines across the academic divisions and in response to questions that cannot be answered within the framework of a single discipline. To foster this interdisciplinary inquiry, the Women’s Studies Program was established in 1978 and has been built upon the feminist teaching, scholarship, and activism of faculty and students with a wide variety of disciplinary and cross-disciplinary perspectives. In the past few decades, the program has grown and evolved, changing its name to the Women’s, Gender, and Sexuality Studies Program in 2008 to recognize important changes within feminist scholarship. Acknowledging this important history, WGSS courses retain Women’s Studies’ focus on examining previously unavailable information about the lives and contributions of women and analyzing the effects of cultural attitudes, power and inequal- Women’s, Gender, and Sexuality Studies 203 ity, and social structures on the experiences of women. The courses also incorporate the important and transformative scholarship and questions that have been revealed by Men’s and Queer Studies, about the relationships between patriarchy, men’s experiences, and the dynamic mutability of gender. By valuing the relationship between theoretical and experiential knowledge, and privileging historically marginalized voices, WGSS encourages scholarship and teaching that is committed to the feminist principle of creating a more just world for all. Major in Women’s, Gender, and Sexuality Studies Consists of eleven courses: • WGSS 12000 • Two of the following courses: WGSS 20200, 20400, or 20600 • WGSS 31000 • Four electives from cross-listed courses accepted for WGSS credit. Two of these must be courses accepted for WGSS credit from one division (AH, HSS, or MNS). One of these must be a course accepted for WGSS credit from a second division (AH, HSS, or MNS). One elective must be a course accepted for WGSS credit and focused on race, class, or culture in an area other than the United States. • Junior Independent Study: WGSS 40100 • Senior Independent Study: WGSS 45100 • Senior Independent Study: WGSS 45200 Minor in Women’s, Gender, and Sexuality Studies Consists of six courses: • WGSS 12000 • One of the following courses: WGSS 20200, 20400, or 20600 • WGSS 31000 • Two electives from cross-listed courses accepted for WGSS credit from two different divisions (AH, HSS, or MNS). One of these must be a course accepted for WGSS credit and focused on race, class, or culture in an area other than the United States. • WGSS 33000 Special Notes • Majors and minors may substitute one WGSS 200-level course, WGSS 40000, or WGSS 40700 for one of the cross-listed courses. • WGSS 40700 is strongly recommended. • Only grades of C- or better are accepted for the major or minor. WOMEN’S, GENDER, AND SEXUALITY STUDIES COURSES WGSS 12000. INTRODUCTION TO WOMEN’S, GENDER, AND SEXUALITY STUDIES This course is an interdisciplinary overview of the discipline of Women’s Studies and the issues, theories, and feminist approaches to the study of gender and sexuality. It explores the meaning of the terms “gender” and “sexuality” in a historical and transnational context—-particularly through their complex intersections with race, class, and nationality—-and the political movements mobilized around these terms. Students will gain the critical tools necessary to examine how gender and sexual orientation are constructed in society and how these constructions might be shaped by individual experience. Annually. Fall and Spring. [W†, AH or HSS] WGSS 20200. HISTORY OF WESTERN FEMINIST THOUGHT A broad introduction to the history and literature of Western feminist thought, beginning with precedents in the middle ages and the early modern eras and focusing on the major thinkers of the suffrage movement (First Women’s, Gender, and Sexuality Studies 204 Wave) through the Second and Third Waves of the women’s movement. A consideration of the impact of feminists of color and of feminist postcolonial critiques, queer studies, and queer theory on contemporary feminist thought concludes the course. Prerequisite: WGSS 12000 or permission of instructor. Every three years; alternates with WGSS 20400 and 20600. Not offered 2011-2012. [W†, AH or HSS] WGSS 20400. GLOBAL FEMINISMS (Latin American Studies) This course will explore how feminism is understood throughout the world and examine struggles for women’s equality in both a historical and transnational perspective. It examines the relationship between Western and so-called “Third World” feminisms, especially as efforts to empower women are impacted by nationalism, race, class and caste, religion, sexuality, and immigration. It will also interrogate the complex process of globalization to understand why it is experienced differently based on gender, as well a geographical location. Theoretical developments in feminist and postcolonial theory and case studies of transnational feminist activism will allow students to critically explore political movements to address a variety of human rights issues throughout the world. Prerequisite: WGSS 12000. Every three years; alternates with WGSS 20200 and 20600. Spring 2012. [C, HSS] WGSS 20600. QUEER LIVES This course addresses a broad range of “queer” issues and the lived experiences of sexual minorities throughout the world. It explores major events in the history of lesbian, gay, bisexual, transgender, transsexual, and queer political movements in the United States and transnationally to understand the social construction of identities and movements and how they have changed in different times and places—-often as a result of race, class-, and gender-based inequities. This course also considers the categories we use to describe same-sex desire. How do Western terms used above help (or hinder) our understandings of the experiences of Indian hijras, Thai “Toms” & “Dees,” Native American two-spirit people, drag queens & kings, and others who don’t fit “neatly” within single categories of gender, sex, and sexuality? Prerequisite: WGSS 12000. Every three years; alternates with WGSS 20200 and 20400. Not offered 2011-2012. [C, HSS] WGSS 31000. SEMINAR IN FEMINIST LEARNING AND TEACHING A rethinking of students’ previous work in Women’s, Gender, and Sexuality Studies through an in-depth immersion in advanced theoretical readings, literature, and personal writings pertaining to women, gender, and sexuality. The course is taught through feminist pedagogy and collaborative learning. The seminar is required of majors and minors but open to other interested students. S/NC course. Prerequisite: WGSS 12000 and one 200-level WGSS course, or permission of instructor. Annually. Fall 2011. WGSS 32000-32003. SPECIAL TOPICS IN WGSS An advanced seminar exploring current theory and research on selected interdisciplinary issues in Women’s, Gender, and Sexuality Studies. Topics will be announced in advance by the faculty member teaching the course. Prerequisite: WGSS 12000 or permission of instructor. Annually. Not offered 2011-2012. [W†] WGSS 33000. FEMINIST METHODOLOGIES What makes a research methodology feminist? Through advanced interdisciplinary readings and short writing assignments, students will be introduced to feminist research methods as well as distinctive feminist critical approaches to issues in the social sciences, physical sciences, and the humanities. This course is required of Women’s, Gender, and Sexuality Studies majors and minors, but it is designed for other students planning to incorporate feminist perspectives into their senior research. Prerequisite: WGSS 12000 and/or permission of instructor. Annually. Spring 2012. WGSS 40000. TUTORIAL IN WGSS Independent research on a topic in consultation with a supervising faculty member. Prerequisite: WGSS 12000 and at least one other course from Women’s, Gender, and Sexuality Studies courses or cross-listings; the approval of both the supervising faculty member and the chairperson is required prior to registration. WGSS 40100. FEMINIST METHODOLOGIES What makes a research methodology feminist? Through advanced interdisciplinary readings and short writing assignments, students will be introduced to feminist research methods as well as distinctive feminist critical approaches to issues in the social sciences, physical sciences, and the humanities. This course is required of Women’s, Gender, and Sexuality Studies majors and minors, but it is designed for other students planning to incorporate feminist perspectives into their senior research. Prerequisite: WGSS 12000 and one 200-level WGSS course, or permission of instructor. Annually. Spring 2012. Women’s, Gender, and Sexuality Studies 205 WGSS 40700. WGSS PRACTICUM Supervised participation in practical efforts toward understanding and/or working for gender equity; to be undertaken through approved placement in an organization in the community or a student-defined project addressing these goals. The work will culminate in written analysis of the practicum experience in relation to coursework in WGSS. Students interested in a practicum experience are also urged to explore the Antioch Women’s Students Semester in Europe (Fall), the GLCA Philadelphia Center Urban Program, and make prior arrangements with a Women’s, Gender, and Sexuality Studies faculty member to count their off-campus work as a practicum upon submission of a reflective paper or journal entries. Prerequisite: WGSS 12000 and at least one other WGSS course; permission of the chairperson is required before registration. Annually. Fall and Spring. WGSS 45100. SENIOR INDEPENDENT STUDY – SEMESTER ONE The first semester of the Senior Independent Study project, in which students use the methods and perspectives of feminist interdisciplinary scholarship to pursue questions of their own design, developed within the context of their prior course work and their interests within the major, and which culminates in a thesis and an oral examination in the second semester. Prerequisite: WGSS 40100. WGSS 45200. SENIOR INDEPENDENT STUDY – SEMESTER TWO The second semester of the Senior Independent Study project, which culminates in the thesis and an oral examination. Prerequisite: WGSS 45100. CROSS-LISTED COURSES ACCEPTED FOR WOMEN’S, GENDER, AND SEXUALITY STUDIES CREDIT AFRICANA STUDIES AFST 24000. AFRICANA WOMEN IN NORTH AMERICA: EARLIEST TIMES THROUGH THE CIVIL RIGHTS MOVEMENT [C, HSS] AFST 24100. AFRICANA WOMEN IN CONTEMPORARY SOCIETY [C, HSS] ART AND ART HISTORY ARTD 21600. GENDER IN TWENTIETH-CENTURY ART [AH] ARTD 31018. THE VISUAL CULTURE OF CHILDHOOD IN EUROPE, 1300-1800 [AH] CHINESE STUDIES CHIN 22200. WOMEN IN CHINESE LITERATURE [C, AH] CLASSICAL STUDIES AMST 22300. GENDER AND SEXUALITY IN CLASSICAL ANTIQUITY [AH] COMPARATIVE LITERATURE CMLT 23000. COMPARATIVE SEXUAL POETICS [W, AH] ECONOMICS ECON 24500. ECONOMICS OF GENDER [HSS] ENGLISH ENGL 21002. BLACK WOMEN WRITERS [AH] ENGL 21004. GENDER, RACE, AND THE CONSTRUCTION OF EMPIRE [AH] ENGL 21008. GENDER, SEX, AND TEXTS [AH] ENGL 21017. FEMINIST/QUEER THEORY [AH] ENGL 22003. VIRGINIA WOOLF [AH] ENGL 22011. JAMES BALDWIN AND TONI MORRISON [AH] ENGL 22015. REPRESENTING SEXUALITIES IN AMERICAN LITERATURE [AH] ENGL 30000. QUEENS [AH] GERMAN STUDIES GRMN 22700. GERMAN LITERATURE IN TRANSLATION: GENDER, POWER, TEXT GRMN 22800. TOPICS IN GERMAN SOCIETY AND CULTURE: WOMEN IN GERMAN SOCIETY AND CULTURE [C] HISTORY HIST 10100-10176. INTRODUCTION TO HISTORICAL INVESTIGATION (depending on topic) [W, some sections count toward C, HSS] Pre-Professional and Dual Degree Programs 206 HIST 10101. HISTORY OF MEN IN AMERICA [W†, C†, HSS] HIST 20100. CRAFT OF HISTORY: TOPIC IN GENDER HISTORY [W, AH] HIST 24700. WOMEN IN U.S. HISTORY [HSS] HIST 27502. STUDIES IN HISTORY: HISTORY OF BRAZIL [HSS] HIST 30126. WOMEN’S LIVES IN LATIN AMERICA [C, R, HSS] HIST 30142. SOCIAL HISTORY OF AFRICAN AMERICAN WOMEN [HSS] MUSIC MUSC 21900. WOMEN IN MUSIC [C, AH] PHILOSOPHY PHIL 21200. RACE, GENDER, AND JUSTICE [C, AH] PHYSICAL EDUCATION PHED 20000. WOMEN IN SPORT POLITICAL SCIENCE PSCI 21000. WOMEN, POWER, AND POLITICS [C, HSS] PSCI 23500. CONTEMPORARY FEMINIST POLITICAL THEORY [HSS] PSYCHOLOGY PSYC 21500. PSYCHOLOGY OF WOMEN AND GENDER [HSS] RELIGIOUS STUDIES RELS 22900. WOMEN AND RELIGION [C, R] RELS 25200. THIRD WORLD FEMINIST THEOLOGY [R] RELS 26719. GLOBAL CATHOLICISM IN AMERICA [R] SOCIOLOGY AND ANTHROPOLOGY ANTH 22500. GENDER IN WORLD CULTURES [C, HSS] SOCI 20700. SOCIOLOGY OF GENDER [HSS] SOCI 21102. AMERICAN MASCULINITIES [HSS] SPANISH SPAN 30900. TRENDS IN SPANISH-AMERICAN LITERATURE: SPANISH-AMERICAN WOMEN WRITERS AND THE FEMALE LITERARY TRADITION [C, AH] THEATRE THTD 24500. FEMINISM AND THEATRE [C, AH] PRE-PROFESSIONAL AND DUAL DEGREE PROGRAMS The College of Wooster provides pre-professional advising programs to support and assist students who want to combine the study of the liberal arts with preparation for a specific profession. Professional schools in a variety of fields, from architecture and business to law and medicine, recognize the value of the range of skills that a liberal arts background provides. Wooster students have excellent success in pursuing advanced degrees in architecture, business, engineering, forestry and environmental studies, health care (e.g., dentistry, medicine, nursing, and veterinary medicine), law, seminary studies, and social work. The pre-professional advising programs at Wooster provide students with advice on the development of an appropriate academic program, co-curricular and volunteer experiences that expand a student’s understanding of a given profession, guidance on summer research opportunities, lectures by leaders in the various profes- Pre-Professional and Dual Degree Programs 207 sions, and information regarding the process of applying to graduate/professional schools. In addition to the pre-professional advising programs, the College also has established formal cooperative relationships with a number of leading universities to offer dual degree programs. DUAL DEGREE PROGRAMS The College provides students with the opportunity to pursue a liberal arts degree from Wooster in conjunction with a graduate/professional degree from a number of leading universities. Graduate or professional programs in medicine, dentistry, nursing, physical therapy, engineering, polymer engineering, architecture, law, and for - estry and environmental studies are examples of eligible programs that may be approved. Specific requirements for some of these programs are provided below. The Dean for Curriculum and Academic Engage ment will exercise judgment as to which graduate and professional programs are consistent with a baccalaureate degree and will set conditions for awarding the degree. Students who intend to pursue dual degrees may complete the senior year in absentia and upon the completion of a specified portion of the graduate/professional program receive the baccalaureate degree. A candidate for the in absentia privilege should apply in writing to the Dean for Curriculum and Academic Engagement by the end of the second semester of the sophomore year and must receive the recommendation of the major department. Wooster does not offer financial aid for the senior year in absentia. The general conditions under which approval of participation in a combined program is granted are as follows: • The student must have completed at least 24 semester courses of which not fewer than 16 courses have been completed at Wooster. No more than two transfer courses may be offered, if approved, in fulfillment of degree requirements for participation in a Combined Professional Program. • All other requirements of the College for the degree of Bachelor of Arts must have been met, except in the major and in Independent Study. In the major, the student must have completed a minimum of six courses, excluding the Senior Independent Study, and in Independent Study, a student must have completed one course, which if taken in the major may be included among the six courses required in the major. Students enrolled in the 3-2, 3-3, or 3-4 programs may declare a major in only one department. PRE-ARCHITECTURE An undergraduate B.A. degree from a liberal arts college such as Wooster can provide an excellent foundation for graduate training as a professional architect. Since the built environment both shapes and is shaped by society, an architect needs not only the technical training in design and engineering that would be provided by an advanced degree in architecture but also the broader understanding of history and culture that is best attained within the context of a liberal arts education. Moreover, an architect must think and write critically and be able to articulate his or her vision—another reason why a liberal arts B.A. is good career preparation. Two options are available to students interested in graduate study in architecture. Pre-Architecture Program Students considering a career in architecture can major in any discipline while completing a four-year B.A. at The College of Wooster. While fulfilling their major and general education requirements, they should plan to take the following recommended courses as preparation for graduate study: Pre-Professional and Dual Degree Programs 208 • one semester of Calculus • one or two semesters of Physics • introduction to Psychology • two semesters of History of Western Civilization • four semesters of Studio Art (drawing, design, photography, painting, sculpture, and ceramics particularly recommended—either four introductory classes in different studio areas, or three introductory studio classes and one upperlevel class) • one or both of the Architectural History courses (ARTD 22300 and 22400) are highly recommended. While this pre-architecture curriculum can be undertaken in conjunction with the requirements of any major, many students interested in architecture major in Studio Art, given the emphasis on that area in this recommended program. Cooperative Program in Architecture (also referred to as 3+4 Cooperative Program) Under agreement with Washington University’s School of Architecture in St. Louis, students may complete three years at The College of Wooster before applying to transfer to Washington University for a senior year of accelerated architectural study, leading to a B.A. from Wooster. Upon acceptance into the graduate program at Washington University, three additional years of study then lead to a Master of Architecture degree. Washington University recommends that students include the coursework outlined above in the program of their first three undergraduate years, although it does not include introduction to psychology and art history courses in its suggested preparation. The co-advisers for the pre-architecture program are John Siewert and Walter Zurko, Department of Art and Art History. PRE-BUSINESS The liberal arts provide excellent preparation for a career in business and for graduate study in business-related areas. The communication, decision-making, and analytical skills required at higher levels of corporate management and in small businesses are well served by Wooster’s emphasis on a broad education from a variety of areas. In addition to the specific business courses offered as part of the Business Economics major (Finance, Accounting, Marketing, Management), students are encouraged to consider courses in the languages, mathematics, English, computer science, speech, psychology, and sociology. For students interested in International Business, there is an integrated course of study that includes language, culture, and business economics components. Students should consult with the pre-business adviser or with the chairpersons of French, German, or Spanish for additional information. Students who are interested in graduate study in business (M.B.A., D.B.A., or Ph.D.) can select any undergraduate major but are encouraged to include courses in mathematics (calculus or above), statistics (ECON 11000), accounting, and several Business Economics courses at the 200-level in their plan of study. The pre-business adviser is John Sell, Department of Economics. PRE-ENGINEERING Bachelor Degrees The College of Wooster has established formal 3-2 cooperative engineering programs with two universities: Case Western Reserve University in Cleveland, Pre-Professional and Dual Degree Programs 209 Ohio, and Washington University in St. Louis, Missouri. Under these cooperative programs, the student is eligible to apply for admission to the engineering school upon satisfactory completion of a specific set of courses taken in the first three years at Wooster. The student transfers to the engineering school after the junior year to complete the last two years of the undergraduate engineering program. Upon completing the program, the student has earned a B.A. from Wooster and a B.S. from the engineering school. The bachelor degree programs in engineering available at one or more of the cooperating universities include aerospace, biomedical, chemical, civil, computer science, electrical, environmental science, materials science, mechanical, polymer, and systems engineering. Students who are considering this program should consult with the pre-engineering advisor and the chairpersons of the departments of Biology, Chemistry, Mathematics, or Physics before arranging their schedules. Bachelor/Master of Science B.A. in Physics/Chemical Physics and M.S. in Polymer Engineering (with the University of Akron) A special 3-2 program has been approved for strong science students who can complete the required set of courses in three years at The College of Wooster and be admitted by special arrangement to the M.S. program in polymer engineering at the University of Akron. The College of Wooster will award a B.A. degree to these students after successful completion of the fourth year of coursework at the University of Akron. In the fifth year at Akron, students will complete a master’s thesis and may have the opportunity to be co-advised by a Wooster faculty member. Students who are considering this program should consult with the pre-engineering advisor at the earliest opportunity to arrange their course schedules. The adviser for all the pre-engineering programs is John Lindner, Department of Physics. FORESTRY AND ENVIRONMENTAL STUDIES Qualified students may participate in a joint program with the Nicholas School of the Environment and Earth Sciences at Duke University. The program leads to a B.A. from The College of Wooster and either a Master’s of Environmental Man - agement (M.E.M.) or a Master’s of Forestry (M.F.) from Duke University. The Wooster degree will be awarded upon the successful completion of the first year of the twoyear professional curriculum. The purpose of the program is to educate students to apply knowledge from the natural, social, and management sciences in the analysis of problems in natural resources and environment. Students may major in any area at The College of Wooster, and may enroll in one of nine specialty areas at the Nicholas School. In addition to the Master’s of Forestry, the Master’s of Environmental Man agement programs are: Coastal Environmental Management; Environmental Toxicology, Chemistry, and Risk Assessment; Conservation Science and Policy; Ecosystem Science and Management; Water and Air Resources; Global Environmental Change; Environmental Health and Security; and Forest Resource Management. In addition to satisfying the requirements for a combined professional training program at Wooster, students should have taken at least one semester of college calculus, a statistics course, and some courses in the natural or social sciences related to their area of specialty. Graduate Record Examination scores (verbal, quantitative, and analytical) must accompany the application to the program in the third year. The adviser for this program is Richard Lehtinen, Department of Biology. Students Pre-Professional and Dual Degree Programs 210 aspiring to this program are encouraged to discuss their goals with Dr. Lehtinen early in their first year. HEALTH PROFESSIONS (DENTISTRY, MEDICINE, NURSING, AND VETERINARY MEDICINE) A liberal arts education is designed to address the complex scientific, societal, and practical challenges facing modern health care practitioners. Correspondingly, The College of Wooster offers its pre-health students a range of opportunities including: 1) a strong curriculum in the sciences that emphasizes undergraduate research; 2) courses in the social sciences and humanities that address ethical, economic, and social issues in health care; 3) practical programs that focus on themes such as how to plan for a career in health care and how to apply to medical and other professional schools; 4) job shadowing at local facilities; and 5) lectures by physicians and other health care professionals. The Pre-Health Advising Committee is composed of faculty from the natural sciences, social sciences, and humanities in addition to staff from the Longbrake Wellness Center and Office of Career Services. This integrated approach to advising reflects the multifaceted nature of health care as well as the recommendation of the American Association of Medical Colleges that undergraduates take a balanced distribution of courses across many different disciplines. Wooster offers several courses in the social sciences and humanities that address current issues in health care. Research and clinical experience are strongly encouraged for students pursuing a career in medicine, and Wooster students are provided with excellent opportunities for undergraduate research through the College’s Independent Study and summer research programs. The Pre-Health Advising Committee assists students in choosing the most appropriate courses, informs them about the range of health career options, and conducts workshops on preparing for the application process. Students can also gain firsthand experience through the Medic Aide program at Wooster Community Hospital in addition to several other volunteer and job shadowing opportunities. A few students have even done Senior Independent Study projects with co-advisors from the Cleveland Clinic-Wooster. While most students enter professional school after completing four years of undergraduate education, some have taken advantage of Wooster’s dual degree programs. In medicine and dentistry, the dual degree option applies to any accredited medical or dental school that admits students with three years of pre-medical preparation. With approval of the in absentia privilege, the Bachelor of Arts degree is granted upon the successful completion of the first year of the professional program. The College has established a Seven-Year Pre-Dental/Dental Program in which students spend three years at Wooster followed by four years at Case Western Reserve University School of Dental Medicine. Students who have been accepted to Wooster but have not yet started their first year may apply to this program if they notify the Office of Admissions of their intent. Provided that they meet certain guidelines, participants in this program will have guaranteed placement in the School of Dental Medicine upon completion of their junior year. The College also has a cooperative 3- 4 program with the Frances Payne Bolton School of Nursing at Case Western Reserve University, which requires the completion of three years at Wooster and four years at Case Western Reserve. Students in this program follow a prescribed set of courses in the physical sciences, social sciences, and humanities at Wooster. The graduate entry program at Case progresses from licensure as a Registered Nurse (RN) to a Master of Nursing (MN) degree and ultimately to the Doctor of Nursing Practice (DNP) degree. Students have the option of entering the workforce or continuing their training at any of these stages. Pre-Professional and Dual Degree Programs 211 Further information is available from Paul Bonvallet, Chairperson of the Pre-Health Advising Committee. PRE-LAW The College of Wooster has a network of Pre-Law advisers committed to assisting students in constructing an appropriate academic program, selecting and applying to law schools, and examining career opportunities in law. The Pre-Law Advising Committee includes both faculty members with interest and experience in law and attorneys in private practice, as well as community and staff members with legal backgrounds. In addition to a network of advisers, Wooster offers a diversified, demanding, and traditional liberal arts curriculum which is most effective in developing the necessary skills to be successful in law school and the legal profession. Law schools and the American Bar Association point out that there is no correlation between academic major and success in law school. Extremely successful legal careers have been launched by Wooster graduates from a wide variety of academic majors, ranging from history, philosophy, and political science to communication, biology, and economics. Wooster graduates have pursued law degrees at a wide variety of institutions, including Harvard University, Columbia University, Georgetown University, the University of Virginia, Ohio State University, Case Western Reserve University, University of Michigan, University of Chicago, Northwestern University, and Stanford University. Wooster graduates have gone on to distinguished and successful careers in the judiciary, private practice, corporate counsel, government, academic administration, and public interest work. The Pre-Law Advising Program provides various sessions that focus upon practical advice for students preparing for law school and a legal career, including such themes as “Considering Law School,” “Applying to Law School,” and “Choosing a Legal Career.” The Program also aids students in their preparation for the LSAT by offering a Mock LSAT on campus. In addition, the Pre-Law Advising Program sponsors co-curricular programs that expand students’ understanding of law and provide exposure to the legal profession. Wooster has an active Moot Court Program. One of the unique features of the College’s Moot Court Program is that students are guided in their preparation not only by Wooster faculty but also by local attorneys and judges. The Pre-Law Advising Program sponsors “The Bell Lectureship in Law,” an annual lectureship endowed by Jennie M. Bell and Federal Judge Samuel H. Bell (’47). The purpose of the Bell Lectureship is to engage students, faculty, members of the legal profession, and members of the community in a legal issue that has broad implications for society. The College of Wooster participates in the Accelerated Interdisciplinary Legal Education (AILE) Program with Columbia University, whereby two Wooster students may be admitted to Columbia School of Law after their junior year. The students are selected jointly by the College and Columbia School of Law. Applications are made through the Pre-Law Committee chairperson at the College. Students accepted into this 3-3 program receive their B.A. from Wooster after completing their first year at Columbia. In addition, this program requires that students incorporate twelve hours of interdisciplinary study into their law school program after the first year. For further information, contact John Rudisill, Chairperson of the Pre-Law Advising Committee or Mark Weaver, Coach of the Moot Court Team. PRE-SEMINARY STUDIES The curricular program at The College of Wooster provides for a course of study that serves the educational needs of those students interested in seminary or gradu- Off-Campus Study 212 ate study in religion as preparation for religious vocations or other service-oriented professions related to religion and religious vocations. The Association of Theological Schools recommends a broad liberal arts program that includes courses across the liberal arts, with a particular focus in the humanities and social sciences. Courses in religious studies, which may result in a major or minor, are strongly urged for those interested in exposure to religious studies prior to seminary or graduate school. The Department of Religious Studies and the Department of Classical Studies, in addition to regular offerings, provide courses in the languages (Classical Greek, Latin, Biblical Hebrew) crucial for seminary education. Off-campus credit programs wherein students can gain experience in religious and religion-related fields are among the offerings of the Department of Religious Studies. Representatives from seminary and graduate schools of religion visit the campus frequently. Those interested in structuring a course of study that will lead to advanced study and vocational alternatives in the field of religious studies are urged to consult Mark Graham, Department of Religious Studies. PRE-SOCIAL WORK Students who wish to combine a liberal arts education with a social work profes - sional degree have an opportunity to participate in a 3-2 program that leads to a graduate degree in an accredited school of social work. The College of Wooster is one of a select group of schools cooperating with the Mandel School of Applied Social Sciences at Case Western Reserve University. Under this program, a student would complete three years of liberal arts education at Wooster and then transfer into a twoyear social work program at Case Western Reserve University. The Bachelor’s Degree will be granted by The College of Wooster when the student has earned 30 semester hours of credit through the Mandel School of Applied Social Sciences. Admission to the social work phase of the program is determined by the admissions office of the Mandel School of Applied Social Sciences of Case Western Reserve University. Students are not recommended to apply for the 3-2 program with Case Western Reserve University if their cumulative grade point average is below 3.2. Students interested in the details of the program and the specific course requirements for the 3-2 option should discuss their program with Thomas Tierney, Department of Sociology and Anthropology. OFF-CAMPUS STUDY Off-campus study is integral to global education at the College. It offers students the opportunity to study and live in another cultural setting, whether domestic or foreign, and to pursue academic work that is not available on campus but that complements and supports Wooster’s curriculum. Off-campus study is coordinated by the Director of Off-Campus Studies through the office of Off-Campus Studies (OCS). The OCS office promotes global events on campus, advises students on off-campus opportunities, and facilitates both domestic and international off-campus study. Please consult the OCS website (www.wooster.edu/Academics/Off-CampusStudy). For further information, contact Jessica DuPlaga, Director, Off-Campus Studies. The off-campus study application deadlines to study off-campus for the 2012-2013 academic year (fall or spring semester, or full year) are: December 1, 2011 — Declaration of Intent to Study Off-Campus March 1, 2012 — Off-Campus Study Application. Off-Campus Study 213 Application forms may be downloaded from the OCS website. Application to OCS is a request for permission from the College to participate in off-campus study. To be eligible for off-campus study, students must meet the following prerequisites: • sophomore or junior status at the time of the program; • good standing under the College’s Codes of Academic Integrity and Social Responsibility; • a minimum cumulative GPA of 2.500 (some programs require a higher GPA); • completion of the Off-Campus Study Application forms by the College deadline; • approval by the student’s academic adviser. Following OCS approval of their application to study off-campus, students must also apply for admission to their program of choice. Many programs have requirements such as specific course prerequisites (particularly for those where English is not the language of instruction). Students should review the literature on the program and consult with the Director of Off-Campus Studies. Upon completing the offcampus study program, the student is responsible for arranging to have an official transcript sent to the Office of the Registrar at Wooster. When applying for an offcampus program, students should direct that the grade report be sent to the Office of the Registrar at the College. It is the responsibility of the student to know the credit system for the off-campus institution they are attending and how the earned credit from other institutions will be converted to the course credit system at Wooster. The grade for each course must be a C or higher, and course credits only—not grades— are entered on the Wooster transcript. Each course credit (1.000) at Wooster is valued at 4-semester hours or 6-quarter hours. Therefore, a 3-semester hour course transfers to Wooster as .750 course credit; and a 5-quarter hour course transfers to Wooster as .833 course credit. In addition to its own “Wooster in” programs, the College also endorses a number of off-campus study programs provided by other organizations. A complete list of all programs endorsed by the College is available on the OCS website. Many offcampus programs are available through Wooster’s membership in various organizations and academic consortia, such as the Great Lakes Colleges Association (GLCA). The GLCA collaborates on some programs with the Associated Colleges of the Midwest (ACM). Such programs of third-party providers typically involve an academic council or an advisory committee. As an institutional affiliate, the College offers direct input to the administration of these programs and shares in their assessment and evaluation. Endorsement of an off-campus program by the College signifies the College’s approval of the academic merit of the program. This formal endorsement permits the transfer of financial aid and scholarships to ONE endorsed off-campus study program (semester- or year-long) during the student’s time at Wooster. Because the costs of off-campus study programs vary, students should consult the Director of Financial Aid on the applicability of financial aid and scholarships to the costs of specific programs. Students are expected to take advantage of the opportunities provided by the College’s endorsed list of programs. Only in exceptional circumstances may other programs be endorsed through a petition to the Director of Off-Campus Studies, due February 15. All petitions are reviewed by the Director of OCS and the OCS Advisory Committee. Students must also petition for any exceptions to the above requirements. More information about petition policy for off-campus study is available on the OCS website. Off-Campus Study 214 STUDY ABROAD PROGRAMS International off-campus study, or study abroad, provides opportunities for intensive academic and cultural experiences in another country. Wooster encourages students to incorporate a study abroad program into their educational experience, and approximately one-third of each graduating class has participated in off-campus study. The College endorses programs by third-party providers in more than 60 countries and on every continent except Antarctica. International programs often require some level of facility in a foreign language, as well as a degree of intercultural sensitivity. A number of the international programs endorsed by Wooster also offer internships, field research or service learning for which students can receive academic credit. For further information, consult the OCS website and contact Jessica DuPlaga, Director of Off-Campus Studies. “Wooster in” Programs “Wooster in” programs are Wooster faculty-led programs for which students receive academic credit. They tend to be short-term programs offered during winter or spring breaks or over the summer. Recent programs include Wooster in Thailand, a semester-long fall course focusing on religion and culture in Thailand; Wooster Summer in Yunnan, a six-week intensive Chinese language program; Wooster in Kenya, a two-week spring break program offered as part of a course in anthropology; Wooster Summer in Trinidad and Tobago, a three-week tropical field biology course; Wooster Summer in Ecuador, an on-campus course accompanied by a three-week incountry conservation biology field component; and Global Social Entrepreneurship, an on-campus course accompanied by a six-week consulting internship in Bangalore, India. Upcoming programs include Wooster Summer in Tuscany, a four-week program where students earn two History credits, based in Siena, Italy; Wooster in India, a twoweek winter program focusing on the arts in India; and Wooster in Israel/Palestine, a spring break program offered as part of a course on the History of Conflict. For further information, contact Nicola Kille, Assistant Director for Global Engagement. DOMESTIC OFF-CAMPUS STUDY: PROGRAMS AND INTERNSHIPS For many students, off-campus study provides an opportunity to apply their academic work in a domestic context outside of Wooster. Many opportunities exist for off-campus study in the United States including a number of internship experiences. An internship or practicum is a supervised work situation in which students may test concepts learned in the classroom while enhancing their knowledge through experience. Internships are usually off-campus, but occasional on-campus positions may be approved by faculty members. In order to receive academic credit for an internship, the student must arrange the internship in advance through the appropriate department or program and register for it. A student may register for a maximum of two internships, for a total of no more than four Wooster course credits to count toward graduation. All internships are graded S/NC. For more information on internships, see Academic Policies – Internships. Many internship experiences are available to majors in particular departments and programs. Often they are combined with academic components as part of an off-campus program. Interested students should consult the department chairperson in their major as well as the Director of Off-Campus Studies. A complete list of domestic programs and internship opportunities is available on the OCS website. The list includes: Washington Semester Program The Department of Political Science, in cooperation with The American University, offers a one-semester program in Washington, D.C., in either the fall or spring semes- Off-Campus Study 215 ter for juniors in good academic standing. The Washington Semester program consists of a full semester of credit through a two-course seminar, featuring several weekly sessions with public and private sector decision-makers; a one-course internship in an agency or organization of the student’s choice (among such options as the Congress, executive branch, justice system, interest groups, think tanks, or trade associations); and a research project utilizing the resources of the nation’s capital. Full-time faculty members direct the program. Students reside at American University and have full access to its facilities while enrolled in the Washington Semester Program. A summer internship program is also available. For further information, contact Eric Moskowitz, Department of Political Science. United Nations Semester Students on this program live on the Drew University campus and spend two days a week in New York City, where the university maintains a center across the street from the United Nations. The seminar includes a study of the United Nations system, conferences with UN diplomats, and meetings with representatives from the various national missions accredited to the UN. Students also have opportunities to serve as interns for UN agencies, national missions, or non-governmental organizations. That seminar, a research paper, and two courses taken at Drew University combine to produce four Wooster credits. For further information, contact Kent Kille, Department of Political Science. Ethics and Society The Department of Religious Studies internship is designed to bring together the academic study of religion and a practical experience dealing with ethical and religious issues in American society. It is intended to place a student in an off-campus situation in which a conflict of values may be examined in a particularly clear way and in which the student may reflect critically upon the ethical and religious dimensions of social phenomena. Placements will be sought that enable the student to participate directly in an institution’s program (political, legal, social, religious, etc.) at a significant level of responsibility. Majors in religion will be given special consideration as applicants, although those with minors in religion and non-majors may also apply. The students are expected to work 35-45 hours a week, will receive three semester course credits, and are obligated to participate in both pre-internship preparation and post-internship reflection. For further information, contact Charles Kammer, Department of Religious Studies. Seminary Semester Wooster offers programs at Claremont School of Theology (Claremont, CA), Interdenominational Theological Center (Atlanta, GA), Chicago Theological Seminar (Chicago, IL), Louisville Presbyterian Theological Seminar (Louisville, KY), Hebrew Union College (Cincinnati, OH), or Pittsburgh Theological Seminary (Pittsburgh, PA). The program involves a two-course credit internship. Two courses of the student’s choice will also be taken at the seminary. Given the variation in course credit systems, it is recommended that students ascertain the equivalent Wooster credit they will earn prior to registering for the Seminary Semester Program. Courses that are available cover a wide range of topics, including: ethics in politics or economics, peacemaking, liberation theology, feminist approaches to theology, Islam, Hinduism, Hebrew, early Jewish history, aging, and African American contributions to theology. These programs are open to any student regardless of major. A minimum 3.0 GPA is required. For further information, contact Charles Kammer, Department of Religious Studies. Off-Campus Study 216 Professional Theatre Internships The Department of Theatre and Dance offers a variety of internships for qualified students throughout the year, including the summer. With the cooperation of a variety of professional theatres, students are placed in acting, business management/public relations, or general and technical internships for academic credit. Internships will vary in length, depending on the needs of the theatre and the availability of the student. A background of theatre course work, experience with College of Wooster Theatre productions, demonstrable talent, interest, and maturity are required. Applications must be submitted to the Department of Theatre at least one semester in advance. Internships carry two to four course credits, two of which may count toward the major in theatre. For further information, contact Dale Seeds, Department of Theatre and Dance. The Business Economics Internship Students of any major who have junior standing, a 2.75 cumulative GPA, and who have completed at least ECON 10100 and 20200 are eligible to apply for the Business Economics Internship. Additional Business Economics course work generally enhances a student’s attractiveness to participating firms. The Internship is an intensive 22-week, off-campus experience designed to acquaint students with the operations of a real-world firm, its goals, and problems. Internships normally begin during the summer following a student’s junior year and continue into the middle of the fall semester. The goal of the program is to enable students to put their academic work into practice in a real-world setting and to provide them with information that will be useful in their future course work. The College maintains an ongoing relationship with several local firms that regularly offer internships. Students may also make contact with other firms themselves, but formal internship arrangements must be made with the Internship Director in advance of the internship and must conform to the Internship’s general guidelines. Students who successfully complete the 22-week program are eligible for two courses of academic credit graded on an S/NC basis. A formal preparatory internship meeting is held during the fall semester of each year. For further information, contact John Sell, Department of Economics. Practicum in Psychology Qualified junior and senior Psychology majors have the opportunity to obtain offcampus applied experience at a clinic, agency, or institution. Local placements include the Counseling Center of Wayne and Holmes Counties, the College Nursery School, Ida Sue School, and Every Woman’s House, among others. Off-campus programs include placements at the Massillon Psychiatric Hospital in Massillon, Ohio, and University Hospitals of Cleveland. For further information, contact Michael Casey, Department of Psychology. Practicum in Women’s, Gender, and Sexuality Studies The Practicum involves supervised participation in practical efforts toward understanding and/or working for gender equity, to be undertaken through approved placement in an organization in the community or a student-defined project addressing these goals. The work will culminate in written analysis of the practicum experience in relation to coursework in WGSS. Students interested in a practicum experience are also urged to explore study abroad programs that focus on gender issues, but also domestic sites such as the GLCA-recognized Philadelphia Center Program, and make prior arrangement with a WGSS faculty member to count their off-campus work as a practicum upon submission of a reflective paper or journal entries. The prerequisites are WGSS 12000 and at least one other WGSS course. Permission of the WGSS chair is required before registration. Summer Academic Programs 217 For further information, contact Nancy Grace, Coordinator of the Women’s, Gender, and Sexuality Studies Program. The Philadelphia Center This experiential program furnishes students the opportunity to grow professionally, academically, and personally within an urban environment. Comprised of a four-days per week internship (more than 700 placements are available) and two academic seminars, the program uses the city and all its resources as a “classroom for learning.” Students design a goal-oriented document, or Learning Plan, that provides the structure for integrating work experience with educational, social, and professional development goals. Founded in 1967, this fall or spring semester program is open to students with sophomore standing in any major with a 2.5 GPA. For further information, contact James Burnell, Department of Economics. New York Arts Program The New York Arts Program is designed to provide those students seriously interested in the arts opportunities unavailable to them on their home campuses. In order to establish the highest possible standards, the program encourages participants to see themselves as novitiate professionals. The program is recognized by the GLCA. The program has two main goals: to provide experience and knowledge in highly focused areas (primarily through the apprenticeship) and to provide all participants with a broadened knowledge of all the arts. The means of achieving these goals are adapted to the requirements of individual participants. For further information, contact Marina Mangubi, Department of Art and Art History. Other Internship Opportunities Other internship programs at the College are available in chemistry, education, music therapy, sociology, communication, and physical education. Some inter - national off-campus programs also offer credit-bearing internships. For details, see Interdepartmental Courses IDPT 40600, 40700, 40800, contact the relevant department and appropriate department chairperson, or visit the Off-Campus Study office. No more than two internships and a maximum of four Wooster course credits will count toward graduation. All internships are graded S/NC. For more information on internships, see Academic Policies – Internships. SUMMER ACADEMIC PROGRAMS In 2012, the academic calendar for Summer Session is from May 21 to June 29. During the Summer Session, students may arrange for additional courses, such as tutorials, internships, off-campus programs, or Independent Study, with the approval of the Dean for Curriculum and Academic Engagement. The College provides special off-campus opportunities including “Wooster Summer In” programs at international locations. For further information about “Wooster Summer In” programs, please contact Jessica DePlaga, Director of Off-Campus Studies. 218 DEGREE REQUIREMENTS Three Baccalaureate degrees are offered: Bachelor of Arts (B.A.), Bachelor of Music (B.M.), and Bachelor of Music Education (B.M.E.). The requirements for each of these degrees are listed below. Although each student has a faculty adviser, the student is responsible for understanding the requirements for the degree and for meeting these requirements. Students should review their progress toward meeting graduation requirements with their faculty adviser each semester. DEGREE REQUIREMENTS BACHELOR OF ARTS 32 course credits are required for graduation, subject to restrictions on residency, fractional credit, transfer credit, and course load. Except where noted, individual courses may be counted toward multiple requirements. First-Year Seminar in Critical Inquiry (1 course) Students will complete the First-Year Seminar in Critical Inquiry in their first semester. Writing In coordination with the First-Year Seminar Program and the Program in Writing: Writing Proficiency (0-1 courses) Students will demonstrate basic writing proficiency in their first year, through placement examination or completion of the College Writing course. Writing Intensive Course (1 course) Students will complete a course designated as Writing Intensive (W) in any semester between the completion of the First-Year Seminar and the beginning of Junior Independent Study. Global and Cultural Perspectives Foreign Language (0-2 course) Students will demonstrate proficiency in a foreign language through the secondlevel course in a given language sequence, through placement examination or course work. Studies in Cultural Difference (1 course) Students will complete a course (C) that examines either a culture outside the United States or the culture of an American minority group (e.g., African American, Asian American, Hispanic or Latino American, Native American). Courses may be taught in English or in a foreign language. Religious Perspectives (1 course) Students will complete a course (R) from any department or program that examines the religious dimension of humankind in relation to issues of cultural, social, historical, or ethical significance. Note: A student may not use the same course in fulfillment of both the Studies in Cultural Difference requirement and the Religious Perspectives requirement. Quantitative Reasoning (1 course) Students will demonstrate basic quantitative proficiency through completion of a course (Q) that involves a substantial element of quantitative reasoning. Degree Requirements 219 Learning Across the Disciplines (6 courses) Students will complete no fewer than two approved courses in each of three academic areas: Arts and Humanities (AH), History and Social Sciences (HSS), Mathematical and Natural Sciences (MNS). [An individual course may be counted toward only one of these three areas.] Learning in the Major (10-16 courses) Students will complete a major in a department or program. The number of courses required in the major shall be no less than ten and no more than sixteen, including the Independent Study Sequence. The major shall contain no more than twelve courses in the same discipline, including the Independent Study Sequence. (In addition, a maximum of fifteen credits in any one discipline may be counted toward graduation.) Independent Study Sequence: Junior Independent Study (Research, Methodology, and Theory) (1 course) A one-semester course that focuses upon the research skills, methodology, and theoretical framework necessary for Senior Independent Study. The structure of this course depends upon the discipline and includes a variety of pedagogical formats, such as one-on-one mentoring experiences, small seminars, and labs. Senior Independent Study (2 courses) A two-semester one-on-one mentoring experience in which each student engages in independent research and creates an original scholarly work. DEGREE REQUIREMENTS BACHELOR OF MUSIC Three majors are offered under the B.M. degree: Performance, Theory/Com - position, and Music History/Literature. 32 courses are required for graduation, subject to restrictions on residency, fractional credit, transfer credit, and course load. Except where noted, individual courses may be counted toward multiple requirements. First-Year Seminar in Critical Inquiry (1 course) Students will complete the First-Year Seminar in Critical Inquiry in their first semester. Writing In coordination with the First-Year Seminar Program and the Program in Writing: Writing Proficiency (0-1 courses) Students will demonstrate basic writing proficiency in their first year, through placement examination or completion of the College Writing course. Writing Intensive Course (1 course) Students will complete a course designated as Writing Intensive (W) in any semester between the completion of the First-Year Seminar and the beginning of Junior Independent Study. Global and Cultural Perspectives Foreign Language (0-2 course) Students will demonstrate proficiency in a foreign language through the secondlevel course in a given language sequence, through placement examination or course work. Degree Requirements 220 Studies in Cultural Difference (1 course) Students will complete a course (C) that examines either a culture outside the United States or the culture of an American minority group (e.g., African American, Asian American, Hispanic or Latino American, Native American). Courses may be taught in English or in a foreign language. Religious Perspectives (1 course) Students will complete a course (R) from any department or program that examines the religious dimension of humankind in relation to issues of cultural, social, historical, or ethical significance. Note: A student may not use the same course in fulfillment of both the Studies in Cultural Difference requirement and the Religious Perspectives requirement. Quantitative Reasoning (1 course) Students will demonstrate basic quantitative proficiency through completion of a course (Q) that involves a substantial element of quantitative reasoning. Non-Music Electives (0-5 courses) Independent Study (3 courses)—see below Learning in the Major (24 courses, including Independent Study) A. BACHELOR OF MUSIC (PERFORMANCE MAJOR) 1. Applied Music (6-8 courses) This requirement includes a half recital of 25-30 minutes of music in the junior year (MUSC 40100) and a full recital of 45-60 minutes of music in the senior year (MUSC 45100-45200), each to be performed after a successful jury examination covering preparation and competence. These recitals constitute I.S. for the performance major. In the junior year, performance majors must enroll in one semester of one-hour lessons (200-level) and one semester of MUSC 40100. 2. Music Theory (8 courses) MUSC 10100 (Theory I), 10200 (Theory II), 20100 (Theory III), 20200 (Theory IV), 30100 (Theory V), 30200 (Form and Analysis), 30300 (Basic Conducting), and 30400 (Counterpoint) 3. Music History (4 courses) MUSC 21000 (Basic Repertoire), 21100 (History I), 21200 (History II), and 21300 (History III) 4. Group Music (1.25 courses) Ten semesters (.125 course credit per semester per group) of participation in the following: MUSC 15000-15700 (Small Ensemble), 16000 (Wooster Singers), 16100 (Wooster Chorus), 16200 (Wooster Symphony Orchestra), 16300 (Scot Band), 16400 (Wooster Jazz Ensemble), or 16500 (Gospel Choir), with at least two semesters of participation in Wooster Singers or Wooster Chorus and at least four semesters of participation—in the major instrument or voice—in the most appropriate of the following major ensembles: Band, Orchestra, Chorus, or Wooster Singers. 5. Pedagogy (.5 course) MUSC 37100 (Instrumental Pedagogy) for instrumental majors or MUSC 37000 (Vocal Pedagogy) for voice majors (.5 course credit) 6. Music Technology (.5 course) MUSC 28000 (Introduction to Music Technology) Degree Requirements 221 7. Music Electives (1.75-3.75 courses) To be chosen by the student and the adviser; may include additional I.S. Keyboard skills must be sufficient to satisfy the Piano Proficiency requirement. B. BACHELOR OF MUSIC (COMPOSITION MAJOR AND THEORY/ COMPOSITION MAJOR) 1. Composition (4 courses) MUSC 20800 or 20900 (Acoustic Composition or Electronic Composition; two semesters at .5 course credit each), 40100 (Junior I.S.), and 45100-45200 (Senior I.S.) 2. Music Theory (9 courses) MUSC 10100 (Theory I), 10200 (Theory II), 20100 (Theory III), 20200 (Theory IV), 30100 (Theory V), 30200 (Form and Analysis), 30300 (Basic Conducting), 30400 (Counter point), and 30500 (Orchestration) 3. Music History (5 courses) MUSC 21000 (Basic Repertoire), 21100 (History I), 21200 (History II), 21300 (History III), and 31100 (Seminar in Music Literature) 4. Applied Music (2 courses) Four semesters at .5 course credit each on the same instrument. 5. Group Music (1.25 courses) Ten semesters (.125 course credit per semester per group) of participation in the following: MUSC 15000-15700 (Small Ensemble), 16000 (Wooster Singers), 16100 (Wooster Chorus), 16200 (Wooster Symphony Orchestra), 16300 (Scot Band), 16400 (Wooster Jazz Ensemble), or 16500 (Gospel Choir), with at least two semesters of participation in Wooster Singers or Wooster Chorus and at least four semesters of participation—in the major instrument or voice—in the most appropriate of the following major ensembles: Band, Orchestra, Chorus, or Wooster Singers. 6. Music Technology (.5 course) MUSC 28000 (Introduction to Music Technology) 7. Music Electives (2.25 courses) To be chosen by the student and the adviser; may include additional I.S. Keyboard skills must be sufficient to satisfy the Piano Proficiency requirement. C. BACHELOR OF MUSIC (MUSIC HISTORY/LITERATURE MAJOR) 1. Music History/Literature (10 courses) MUSC 21000 (Basic Repertoire), 21100 (History I), 21200 (History II), 21300 (History III), 40100 (Junior I.S.), 45100-45200 (Senior I.S.), and three from AFST 21200 (African American Folklore), MUSC 21400 (History of African American Music), 21500 (Music of the United States), 21600 (The Art of Rock Music), 21700 (Survey of Jazz), 21800 (Masterpieces of Musical Theatre), 21900 (Women in Music), or 31100 (Seminar in Music Literature) 2. Music Theory (9 courses) MUSC 10100 (Theory I), 10200 (Theory II), 20100 (Theory III), 20200 (Theory IV), 30100 (Theory V), 30200 (Form and Analysis), 30300 (Conducting), 30400 (Counterpoint), and 30500 (Orchestration) 3. Applied Music (2 courses) Four semesters at .5 course credit each on the same instrument. 4. Group Music (1.25 courses) Ten semesters (.125 course credit per semester per group) of participation in the Degree Requirements 222 following: MUSC 15000-15700 (Small Ensemble), 16000 (Wooster Singers), 16100 (Wooster Chorus), 16200 (Wooster Symphony Orchestra), 16300 (Scot Band), 16400 (Wooster Jazz Ensemble), or 16500 (Gospel Choir), with at least two semesters of participation in Wooster Singers or Wooster Chorus and at least four semesters of participation — in the major instrument or voice — in the most appropriate of the following major ensembles: Band, Orchestra, Chorus, or Wooster Singers. 5. Music Technology (.5 course) MUSC 28000 (Introduction to Music Technology) 6. Music Electives (1.25 courses) To be chosen by the student and the adviser; may include additional I.S. Keyboard skills must be sufficient to satisfy the Piano Proficiency requirement. DEGREE REQUIREMENTS BACHELOR OF MUSIC EDUCATION Two majors are offered under the B.M.E. degree: Public School Teaching and Music Therapy. Because of the heavy requirements for these degrees, it is likely that the student will need to carry overloads or extend the time required to complete the degree program. Students should note that either option will probably result in additional tuition charges. Note: Completion of the degree may require more than eight semesters of full-time academic work. A. BACHELOR OF MUSIC EDUCATION (PUBLIC SCHOOL TEACHING MAJOR) 36.75 to 39 courses are required for graduation, subject to current restrictions on residency, fractional credit, transfer credit, and course load. Except where noted, individual courses may be counted toward multiple requirements. Students will complete 14.5 to 16.75 courses outside music. First-Year Seminar in Critical Inquiry (1 course) Students will complete the First-Year Seminar in Critical Inquiry in their first semester. Writing In coordination with the First-Year Seminar Program and the Program in Writing: Writing Proficiency (0-1 courses) Students will demonstrate basic writing proficiency in their first year, through placement examination or completion of the College Writing course. Writing Intensive Course (1 course) Students will complete a course designated as Writing Intensive (W) in any semester between the completion of the First-Year Seminar and the beginning of Junior Independent Study. Studies in Cultural Difference (1 course) Students will complete a course (C) in History or the Social Sciences that examines either a culture outside the United States or the culture of an American minority group (e.g., African American, Asian American, Hispanic or Latino American, Native American). Courses may be taught in English or in a foreign language. Degree Requirements 223 Religious Perspectives (1 course) Students will complete a course (R) from any department or program that examines the religious dimension of humankind in relation to issues of cultural, social, historical, or ethical significance. Note: A student may not use the same course in fulfillment of both the Studies in Cultural Difference requirement and the Religious Perspectives requirement. Quantitative Reasoning (1 course) Students will demonstrate basic quantitative proficiency through completion of a course (Q) in the Mathematical or Natural Sciences that involves a substantial element of quantitative reasoning. Arts and Humanities (1 course) Students will complete one non-music course in the Arts or Humanities (AH). [A student may not use this course in fulfillment of the Studies in Cultural Difference requirement or the Religious Perspectives requirement.] Mathematical and Natural Sciences (1-1.25 courses) Students will complete one to one and one-quarter courses in the Mathematical or Natural Sciences (MNS). [A student may not use this course in fulfillment of the Quantitative Reasoning requirement.] History and Social Sciences (1 course) Students will complete one course in History or the Social Sciences (HSS). [A student may not use this course in fulfillment of the Studies in Cultural Difference requirement or the Religious Perspectives requirement.] Psychology (1 course) Students will complete PSYC 11000. [This course may not count toward the History and Social Sciences requirement.] Education (2.5 courses) Students will complete EDUC 10000, 12000, and 30000. Non-Music Elective (1 course) Student Teaching (3 courses) The student teaching sequence satisfies the College requirement of three courses of Independent Study. Students will complete EDUC 39600-39800. All degree requirements except MUSC 39500 (Special Topics in Music Education) and the final semester of recital attendance must be completed prior to the semester in which the student registers for student teaching. Concurrent registration for MUSC 39500 and student teaching is expected; however, when student teaching is completed in the fall semester, MUSC 39500 must be completed prior to that semester. The recital attendance requirement continues through the student teaching semester. Learning in the Major (22.25 courses) 1. MUSIC THEORY (7.5 courses) MUSC 10100 (Theory I), 10200 (Theory II), 20100 (Theory III), 20200 (Theory IV), 30100 (Theory V), 30300 (Basic Conducting), 30500 (Orchestration), and 30600 (Choral Conducting) 2. MUSIC HISTORY/LITERATURE (3 courses) MUSC 21000 (Basic Repertoire), 21200 (History II), and 21300 (History III) 3. PERFORMANCE (7.75 courses) a. Group Music (1.25 courses) Degree Requirements 224 Ten semesters (.125 course credit per semester per group) of participation in group music, including at least two semesters of participation in Wooster Singers, and at least four semesters of participation—in the major instrument or voice—in the most appropriate of the following major ensembles: Band, Orchestra, Chorus, or Wooster Singers. b. Class Instruments and Voice (2.5 courses) MUSC 17000 (Class Voice), 17100 (Brass I), 17200 (Brass II), 17300 (Strings I), 17400 (Strings II), 17500 (Woodwinds I), 17600 (Woodwinds II), 17700 (Percussion), and 37200 (Functional Piano) c. The remainder is to be taken in performance areas depending upon the pre-college preparation of the student. Keyboard skills must be sufficient to satisfy the Piano Proficiency requirements. Each student is required to give a half recital of 25-30 minutes of music in either the junior or senior year (prior to the semester in which student teaching is scheduled). The recital is to be performed after a successful jury examination covering preparation and competence. 4. MUSIC TECHNOLOGY (.5 course) MUSC 28000 (Introduction to Music Technology) 5. MUSIC EDUCATION (3.5 courses) MUSC 29000 (Foundations of Music Education), 34200 (Methods and Materials for Teaching Pre-K and Elementary General Music), 34300 (Methods and Materials for Teaching Secondary Choral and General Music), 34400 (Methods and Materials for Teaching Instrumental Music), 37000 (Vocal Pedagogy), and 39500 (Special Topics in Music Education) B. BACHELOR OF MUSIC EDUCATION (MUSIC THERAPY MAJOR) 38.75 courses are required for graduation, subject to current restrictions on residency, fractional credit, transfer credit, and course load. Except where noted, individual courses may be counted toward multiple requirements. Students will complete 14.25 courses outside music. First-Year Seminar in Critical Inquiry (1 course) Students will complete the First-Year Seminar in Critical Inquiry in their first semester. Writing In coordination with the First-Year Seminar Program and the Program in Writing: Writing Proficiency (0-1 courses) Students will demonstrate basic writing proficiency in their first year, through placement examination or completion of the College Writing course. Writing Intensive Course (1 course) Students will complete a course designated as Writing Intensive (W) in any semester between the completion of the First-Year Seminar and the beginning of Junior Independent Study. Studies in Cultural Difference (1 course) Students will complete a course (C) that examines either a culture outside the United States or the culture of an American minority group (e.g., African American, Asian American, Hispanic or Latino American, Native American). Courses may be taught in English or in a foreign language. Religious Perspectives (1 course) Students will complete a course (R) from any department or program that examines Degree Requirements 225 the religious dimension of humankind in relation to issues of cultural, social, historical, or ethical significance. Note: A student may not use the same course in fulfillment of both the Studies in Cultural Difference requirement and the Religious Perspectives requirement. Quantitative Reasoning (1 course) Students will demonstrate basic quantitative proficiency through completion of a course (Q) that involves a substantial element of quantitative reasoning. History and Social Sciences (2 courses) Students will complete two approved courses (HSS) in Political Science, History, or Economics. [A student may not use these courses in fulfillment of the Studies in Cultural Difference requirement or the Religious Perspectives requirement.] Psychology (3 courses) Students will complete PSYC 10000, 21200, and 25000. Mathematical Sciences (1 course) Students will complete one course in the Mathematical Sciences. Natural Sciences (1 course) Students will complete a course dealing with human anatomy. Sociology (2 courses) Students will complete SOCI 10000 and either SOCI 20400 or SOCI 21300. [A student may not use these courses in fulfillment of the Religious Perspectives requirement.] Education (1 course) Students will complete EDUC 20000. Non-Music Electives (1-2 courses to complete 14.25 courses outside the major) Independent Study (.25 course) Students will complete MUSC 40700-40800 (a six-month, full-time clinical experience in a facility approved by the American Music Therapy Association). Learning in the Major (24.5 courses) 1. MUSIC THEORY (7 courses) MUSC 10100 (Theory I), 10200 (Theory II), 20100 (Theory III), 20200 (Theory IV), 30100 (Theory V), 30300 (Basic Conducting), and 30500 (Orchestration) 2. MUSIC HISTORY/LITERATURE (2 courses) MUSC 21000 (Basic Repertoire) and either 21200 (History II) or 21300 (History III) 3. MUSIC THERAPY (6 courses) MUSC 19000 (Introduction to Music Therapy) and 19100 (Recreational Music— Programming and Leadership), each for .5 course credit; 29100 (Music Therapy in Psychiatry and Rehabilitation); 29200 (Music Therapy with the Developmentally Disabled); 29300 (Practicum I in Music Therapy), 29400 (Practicum II in Music Therapy), and 29500 (Advanced Practicum in Music Therapy), each for .25 course credit; 39200 (Psychology of Music) and 39300 (Research Seminar in Music Therapy), each for .5 course credit; and 39400 (Program Development and Administration in Music Therapy) and 40700- Degree Requirements 226 40800 (Internship) (.25 course credit) 4. MUSIC EDUCATION (2 courses) MUSC 29000 (Foundations of Music Education), 34200 (Methods and Materials for Teaching Pre-K and Elementary General Music), and the most appropriate course from either 34300 (Methods and Materials for Teaching Secondary Choral and General Music) or 34400 (Methods and Materials for Teaching Instrumental Music) 5. MUSIC TECHNOLOGY (.5 course) MUSC 28000 (Introduction to Music Technology) 6. PERFORMANCE (7 courses) a. Group Music (1 course) Eight semesters (.125 course credit per semester per group) of participation in group music, including at least two semesters of participation in Wooster Singers, and at least four semesters of participation—in the major instrument or voice—in the most appropriate of the following major ensembles: Band, Orchestra, Chorus, or Wooster Singers. b. Class Instruments and Voice (2.5 courses) MUSC 17000 (Class Voice), 17100 (Brass I), 17300 (Strings I), 17400 (Strings II), 17500 (Woodwinds I), 17700 (Percussion), 37000 (Vocal Pedagogy), and 37200 (Functional Piano) c. Applied Music (3.5 courses) The remainder is to be taken in performance areas depending upon the precollege preparation of the student. Keyboard skills must be sufficient to satisfy the Piano Proficiency requirement. Each student is required to give a half recital of 25-30 minutes of music in either the junior or senior year. The recital is to be performed after a successful jury examination covering preparation and competence. DEGREE REQUIREMENTS MUSIC DOUBLE DEGREE DOUBLE DEGREE: BACHELOR OF MUSIC OR MUSIC EDUCATION AND BACHELOR OF ARTS A double degree enables students to make connections among fields that can enrich the study of each and expand career opportunities. Full double-counting of requirements for the two degrees is allowed. Upon graduation, the student will receive two diplomas and will participate in one Commencement ceremony. In most cases, completion of a double degree will require five years. Interested students should confer with the chairperson of the Department of Music and must have written approval from the Dean for Curriculum and Academic Engagement. Students who wish to pursue a double degree must declare their intention to do so no later than October 1 of the junior year. DEGREE REQUIREMENTS GRADUATE & PROFESSIONAL DUAL DEGREE The College has established formal programs with a number of leading universities to provide students with the opportunity to pursue a liberal arts degree from Wooster in conjunction with a graduate/professional degree from the other insti - Academic Policies 227 tution. These programs provide students with a rich liberal arts experience that focuses upon a dynamic understanding of multiple disciplines, independent and collaborative inquiry, global engagement, and social responsibilities and also facilitate their progress towards a graduate or professional degree. Graduate or professional programs in medicine, dentistry, law, physical therapy, engineering, nursing, architecture, and forestry and environmental studies are examples of eligible programs that may be approved. The Dean for Curriculum and Academic Engagement determines which graduate and professional programs are consistent with a Wooster baccalaureate degree and will set conditions for awarding the degree. The conditions for participation in a dual degree program are provided in the Catalogue under Pre-Professional and Dual Degree Programs. ACADEMIC POLICIES Each student is assigned a faculty adviser to supervise his or her academic program at the College. However, it is each student’s responsibility to make final decisions about his or her education. In addition, each student is responsible for understanding and following all academic policies. ACADEMIC POLICIES – REQUIREMENTS FOR ALL DEGREE PROGRAMS AND COMMENCEMENT GENERAL REQUIREMENTS FOR ALL DEGREE PROGRAMS A minimum of 16 course credits must be completed at The College of Wooster: • including four courses for general education requirements (foreign language, studies in cultural difference, religious perspective, learning across the disciplines); • including seven courses in the major in addition to the Senior Independent Study. RESIDENCE REQUIREMENTS • Two years of residence at Wooster are required for the B.A. degree, with one of them the senior year. • Students are required to be in residence for the two semesters preceding the fulfillment of their degree requirements and are permitted to take no more than 4.250 (including .125 course credit in music performance groups) course credits per semester in the two semesters in which they undertake the Independent Study Thesis. • The last six courses (including the two-course Senior Independent Study) counting toward graduation must be completed in the College’s curricular program. • For transfer students, at least seven of the courses in the major, including the Senior Independent Study, must be taken at Wooster. MINIMUM GRADES ACCEPTABLE IN THE MAJOR AND MINOR Only grades of C- or higher are accepted for the major or minor. In addition, a student must have a major(s) GPA of 2.000 or higher at the time of graduation. All courses taken in the major(s) are counted towards the major(s) GPA. Thus, this Academic Policies 228 includes not only courses that are taken to fulfill the minimum requirements of the major(s) but also any additional elective courses in the major(s). (Courses that are repeated cannot be counted twice when computing the 32 courses required for graduation.) COMMENCEMENT To graduate from The College of Wooster, a student must meet all College requirements, including the following: • The student has completed all requirements in the major. • The student has a minimum of 32 course credits. • The student has a cumulative GPA of 2.000 or higher. • The student has a major GPA of 2.000 or higher. • The student is in good standing under the Codes of Academic Integrity and Social Responsibility as administered through the judicial system of the College. Students who have failed to meet the requirements to graduate will be permitted to participate in Commencement (“walk at Commencement”) only if the following conditions are met: • The student has successfully completed 31 of the 32 required course credits. • All other requirements and electives, except one course, have been completed. • The student has a cumulative GPA of 2.000 or higher. • The student has a major GPA of 2.000 or higher. • The student arranges through the Office of the Registrar to complete the outstanding course credit, whether at the College or at another institution. • The student has no outstanding obligations under the Codes of Academic Integrity and Social Responsibility. • The Commencement program will include a notation that the student has not yet completed the degree. Students who have completed eight semesters of college-level coursework, including at least 16 College of Wooster course credits, and who have met all of the College’s degree requirements will be awarded the appropriate degree at the next scheduled Commencement. (See Admission – Transfer Credit and Graduation Requirements.) Students who finish degree requirements mid-year or in absentia must confirm their status for graduation and intentions for the May Commencement in writing with the Registrar by February 1. Students may participate in only one Commencement ceremony. Grade point averages at the time of graduation will be recorded on the permanent transcript. Records of any courses taken at Wooster subsequent to graduation will appear on the transcript, but grades will not affect the grade point average at the time of graduation. For information on Departmental Honors and Latin Honors at Commencement, please see Honors and Prizes. ACADEMIC POLICIES – MAJORS AND MINORS MAJORS A liberal arts education should help students to appreciate the nature of the academic disciplines—as intellectual tools that enable us to think in structured and systematic ways, and for the depth of inquiry they allow. Students will come to understand a particular field of inquiry in depth, and develop a basis of knowledge and methodological ability that will enable them to participate actively and significantly in a disciplinary community. By coming to know at least one discipline in Academic Policies 229 depth, students will equip themselves to become scholars engaged in the creation of knowledge. A student must declare a major in February of the sophomore year prior to registration for the junior year. DOUBLE MAJORS With the approval of the chairpersons of the two relevant departments and the Dean for Curriculum and Academic Engagement, students are permitted to declare double majors. Requirements for each major in a double major are the same as those for a single major with the exception that, subject to the approval of both departments, a joint Senior I.S. project may be done on a topic that incorporates materials and approaches from both disciplines and fulfills the requirements of both departments. Each major in a double major must include at least six courses (except Senior Independent Study 45100 and 45200) that do not count in the second major. Students who declare double majors must complete two separate Junior I.S. courses (40100) — one in each major department. Students who declare double majors must register for Senior Independent Study in one major during fall semester and in the second major in spring semester. Students who wish to pursue a double major must declare their intention to do so no later than October 1 of the junior year. Students enrolled in dual degree or pre-professional programs may not double major. Double majors are not permitted in: International Relations and its participating departments (Economics, History, Political Science), Urban Studies and its participating departments (Economics, Political Science, Sociology), Biochemistry and Molecular Biology and its participating departments (Biology, Chemistry, Neuroscience), Chemical Physics and its participating departments (Chemistry, Mathematics, Physics), and Neuroscience and the following programs (Biochemistry and Molecular Biology, Biology, Chemistry, Psychology). A Student-Designed Major may declare a double major (subject to approval by the Dean for Curriculum and Academic Engagement), as long as (1) there is no course overlap between any of the courses in the two majors; (2) the second major is an existing major in one of the established academic departments; and (3) Junior Independent Study is completed in each major. Senior Independent Study may be combined between the majors, if the proposal clearly demonstrates that it can be done. All other requirements and deadlines for declaration are the same as any other double major. STUDENT-DESIGNED MAJOR Some students may find their educational objectives best served in a curricular pattern other than the normal one. In such cases, after consultation with the appropriate faculty members, the student may submit a plan for a Student-Designed Major to the Dean for Curriculum and Academic Engagement. This plan must be submitted no later than March 1 of the sophomore year. The student will be expected to outline precise aims, the courses that will be taken, and the procedure for meeting degree requirements in accordance with established guidelines. In considering applications for student designed majors, the Dean shall make decisions based on the intellectual content and rigor of the proposed program, and its integrity as a major in the liberal arts. The Dean may also take into consideration preparation for graduate education, certification, or licensing, but these shall not be the determining factors. Once a major has been approved, any subsequent changes to the major must be submitted to the Dean for Curriculum and Academic Engagement in advance for approval. MINOR A student may declare one or two minors, consisting of six courses in a department or program. Each minor must include at least four courses that are distinct from Academic Policies 230 any other minor. These four courses cannot be used in fulfillment of the major(s). A student must declare a minor by March 1 of the senior year. Some major and minor combinations are not permitted or allow less overlap. These restrictions are listed under Special Notes for each department or program. ACADEMIC POLICIES – REGISTRATION, COURSES, AND GRADES REGISTRATION AND CHANGES IN COURSE REGISTRATION The Office of the Registrar is the principal source of information about registration procedures. Each student is assigned a faculty adviser to supervise his or her academic program at the College. However, it is each student’s responsibility to make final decisions about his or her education. In addition, each student is responsible for understanding and meeting all registration and graduation requirements. The faculty has established the following policies concerning registration: 1. It is the student’s responsibility to pre-register for at least 3.000 course credits, and to maintain the normal course load each semester – see Degree Require - ments. Failure to do so can result in: (i) loss of on-campus housing; (ii) loss of financial aid: and (iii) failure to be certified as a full-time student for insurance, financial aid, immigration, or other purposes. 2. A student is officially registered only after the student’s name appears on class lists and the student has confirmed his/her registration with the Office of the Registrar on return to campus each semester. 3. Students are expected to be on campus when classes begin. Students who do not attend the first meeting of a class may be dropped from the class by the instructor. In this event the Registrar will drop the student from the class, notify the student, the instructor, and the academic adviser. The add/drop form is not necessary for this single transaction. 4. A student may add course credits before the end of the second week of the semester, and only with the permission of the faculty member teaching the course and the approval of the faculty adviser. 5. A student may drop a course before the end of the sixth week of the semester after consulting with the faculty member teaching the course and with the approval of the faculty adviser. A course dropped before the end of the sixth week will be removed without record of registration. 6. A student must declare the S/NC grading option with the acknowledgement of the academic adviser and course instructor no later than the end of the sixth week of the semester. Once the S/NC option is elected, it cannot be changed back to the letter-grade option. 7. A student may add course credits for audit before the end of the second week of classes. A student may change registration status in a course from credit to audit before the end of the sixth week of classes. Once the audit status is declared for a particular course, it cannot be changed back to the credit option. 8. To make changes in their course schedules after the stated deadlines, students must petition the Dean for Curriculum and Academic Engagement. If the petition is granted, changes are subject to a late registration fee of $100 for each course change. COURSE LOAD • 4.000 course credits per semester is the normal course load. Academic Policies 231 • A minimum of 3.000 course credits is needed to maintain full-time status. • A maximum of 4.630 course credits is permitted without the approval of the Dean for Curriculum and Academic Engagement. • Students may register for up to .125 course credit in music performance groups beyond the maximum specified. • For students in the Bachelor of Music Education program (Majors: Music Therapy or Public School Teaching) the maximum course load is 4.875. THE GRADING SYSTEMS A. There are four grading systems: 1. A letter system using the marks and grade points: A = 4.000—a grade in the A range indicates outstanding performance in which there has been distinguished achievement in all phases of the course A- = 3.667 B + = 3.333 B = 3.000—a grade in the B range indicates good performance in which there has been a high level of achievement in some phases of the course B- = 2.667 C + = 2.333 C = 2.000—a grade in the C range indicates an adequate performance in which a basic understanding of the subject has been demonstrated. C- = 1.667 D = 1.000— a grade of D indicates a minimal performance in which despite recognizable deficiencies there is enough merit to warrant credit. F = 0.000—a grade of F or NC indicates unsatisfactory performance. L = satisfactory performance in an audit course. An unsatisfactory audit performance does not appear on the transcript. 2. A two-level system using the marks: S = Satisfactory Performance NC (no credit) = Unsatisfactory Performance 3. A two-level system for Senior Independent Study 45100 (see Departmental/ Program Independent Study Handbook for details) using the marks: SP = Satisfactory Progress NC = No Credit 4. A four-level system for Senior Independent Study 45200 (see Departmental/ Program Independent Study Handbook for details) using the marks: H = Honors G = Good S = Satisfactory NC = No Credit B. Each course earns one course credit toward graduation except where otherwise indicated. A course equates to four semester hours of credit or six quarter hours of credit. C. The Cumulative GPA (grade point average) includes all A-F grades, and the transcript will carry the notation that these grades are averaged in the cumulative GPA. The cumulative GPA is calculated by totaling the number of grade points acquired for all courses that are letter graded (A-F) and dividing that total Academic Policies 232 by the number of course credits. The F grade is calculated into the cumulative GPA. The marks H, G, S, and NC are not calculated into the cumulative GPA. In addition, only grades received in courses taught by Wooster faculty are included in the cumulative GPA. Grades received during off-campus study at another institution are recorded as received from the other institution but are not counted in the Wooster cumulative GPA. For transfer students, only academic work completed at Wooster is included in the Wooster cumulative GPA. D. Students are permitted to elect the equivalent of four courses (in addition to Senior Independent Study) graded S/NC out of 32 courses required for graduation. The minimum equivalent grade to earn S in courses graded S/NC is C-. First-Year Seminar in Critical Inquiry and the College Writing course will not be graded S/NC. Transfer students are permitted to have one-eighth of the courses remaining to be taken at Wooster graded S/NC. Courses taken S/NC are not permitted in the major department/program, in the minor department/program, nor in courses exceeding the number in the major or minor unless specific exceptions to this regulation are stated by individual departments/programs. E. Courses for which credit is not received are designated F or NC, except in those cases for which the designation “W” (Withdrawn) is approved. Such withdrawals require a written petition to the Dean for Curriculum and Academic Engagement and are approved only in exceptional circumstances. F. Requests for a medical withdrawal from a course (also designated “W” on the transcripts) must be submitted in writing to the Dean for Curriculum and Academic Engagement no later than the last day of classes of the semester in which the course was taken. In unusual circumstances, such requests may be submitted by the last day of classes of the semester following that for which the medical withdrawal is requested. Withdrawal for medical reasons is approved by the Dean for Curriculum and Academic Engagement after consultation with counseling and medical staff. G. An Incomplete (designated “I”) is only appropriate if a student has attended and participated in the classroom activities throughout the semester and a small portion of the work of a course is unavoidably unfinished. This work must be completed before the end of the first week of the following semester (including work for Semester II that must be completed before the end of the first week of the Summer Session). If the work is not completed by the time specified, the I automatically becomes an F or NC. Credit for a course completed at the College will not normally be awarded after the deadline for changing incomplete grades. Exceptions to this policy require a written petition to the Dean for Curriculum and Academic Engagement and are approved only in exceptional circumstances. H. Students may repeat a course one time for credit if the original grade was a D or lower. In order to repeat a course the student must first obtain approval from the academic adviser and the appropriate department. The repeated course must be taken according to the same grading system as the original course (e.g., graded A-F or S/NC). Credit for the class will be granted only one time. The original grade remains on the student’s transcript, although the credit for the original course becomes 0.0. Only the grade in the second course counts toward the cumulative GPA. A course may be repeated off-campus only with pre-approval by the appropriate department chair; the course will count as credit but the grade will not count in the student’s GPA. I. Each faculty member has the obligation to inform students at the beginning of each course of the means of evaluation for the course and the factors to be considered in the evaluation process (e.g., mastery of course material, use of evidence, ability to generalize, writing ability, verbal ability, mathematical ability, Academic Policies 233 logical ability, ability to meet deadlines, class presence). Faculty are asked to inform students throughout the term as to how they are performing with regard to the criteria of evaluation. Each student must receive a grade in one major course assignment in each course prior to the end of the sixth week of class (i.e., before the last day to “drop” a course). Grades are due at times to be announced by the Office of the Registrar. J. Final examinations or other integrating assignments are mandatory in all courses, except in Independent Study and fractional courses (i.e., courses earning less than one full credit). No more than one-half of the final grade may come from a single assignment, including the final examination. Final examinations are to be given only at those times scheduled for each particular class by the Registrar. No examinations are to be given on reading days. Exceptions to the above must be approved by the Dean for Curriculum and Academic Engagement. K. The criteria for evaluating Independent Study are contained in the Departmental/Program Independent Study Handbook. L. A change of grade in a course taken at the College will not normally be permitted more than one semester after the date of completion of the course. M. Grade reports are released online at the end of each term to students and to academic advisers. The Family Educational Rights and Privacy Act (FERPA) provides for student control over release of confidential academic information, including grades. Requests for grade information from sources other than the student must comply with FERPA guidelines for disclosure and release of academic record information. It is the student’s responsibility to share grade information. In the event that a parent requests academic information, it must first be established that the student is a dependent as defined by IRS standards. Student waiver of FERPA rights and parental verification of dependency is documented by completing the FERPA Release form posted on the web page of the Dean of Students. Prior to processing requests for grades by outside sources, including parents, the Registrar will verify authorized consent to receive confidential information and student consent to waive FERPA rights of protection. AUDITS • Full-time students are permitted to audit one course without charge in any semester. • In the case of majors in the Music Department, this course could be a regular course carrying 1.000 credit or a combination of partial credit courses adding up to 1.000 credit, with the exception that a student may not audit any more than one half-hour applied lesson in a given semester. • The deadline for adding a course for audit is the end of the second week of classes in any semester. Once the audit status is declared for a particular course, it cannot be changed to the credit option. • The deadline for changing registration in a course from credit to audit is the end of the sixth week of classes. TUITION-FREE COMMUNITY AUDIT PROGRAM The College of Wooster provides the opportunity for local residents to audit one course each semester at no cost. The purpose of this program is to provide the opportunity for the continued growth and development of community members, strengthen the relationship between the community and the College, and enrich the learning environment at the College. To be eligible to audit classes an individual must complete a brief application and be accepted as an auditor at the College, there must be room in the class after all cur- Academic Policies 234 rent students have registered, the professor’s continued approval is required, and the Dean for Curriculum and Academic Engagement’s continued approval is required. The costs of all materials and textbooks are the responsibility of the auditor. No college credit will be awarded for audited courses. Individuals are responsible for knowing and abiding by all polices and procedures outlined in The College of Wooster Catalogue, the Code of Social Responsibility, and the policies in The Scot’s Key. INTERNSHIPS Internships provide students an opportunity to extend their educational experience by applying their academic work to a context outside of the classroom, such as a community organization, non-profit organization, business, or government organization. Students work and learn under the joint oversight of a site supervisor and a faculty supervisor. The faculty supervisor will construct an educational plan and a syllabus for the course, including a reading list, a reflective writing/discussion exercise, and a summative assignment. The form for registering for an internship is available in the Office of the Registrar. The following policies apply to departmental, interdisciplinary, and multidisciplinary internship; there may be additional specific departmental requirements. • The student must arrange the internship in advance through the appropriate department or program. • The student must obtain approval in advance from both a faculty supervisor and the Dean for Curriculum and Academic Engagement. • No more than two internships, and a maximum of four Wooster course credits, will count toward graduation. (Internships receive variable course credit, 0.25 – 4.00). • All internship courses are graded S/NC. • During a summer internship, it is permissible for a student to receive both academic credit from the College and payment from the employer or organization. In order to earn academic credit for a summer internship, the student must register and pay tuition for the internship. MAXIMUM COURSE CREDITS IN PERFORMANCE, WORKSHOPS, AND PHYSICAL EDUCATION A maximum of two (2.000) course credits in performance, workshops, and physical education activities may be counted toward the minimum of 32 course credits required for graduation. • These two course credits may include at most the equivalent of one (1.000) Wooster course credit for private music instruction, music performance ensembles and groups, and workshop courses in Communication Studies and Theatre. All Music performance courses are offered for both full (1.000) and fractional (.125, .250, .500) credits. Communication Studies workshops are .250 credit courses, and Theatre workshop and performance courses that count toward the allowable 1.000 performance course are .250 credit courses. Exceptions to these regulations are made for majors or minors, and are stated by individual departments. • Four Physical Education activities classes that count for .250 credit make up the second 1.000 course credit that can be counted toward the minimum of 32 course credits required for graduation. Students who participate on intercollegiate athletic teams may count only .250 varsity sports credit, PHED 13000, toward the four allowable physical education activities courses. Academic Policies 235 TRANSFER CREDIT POLICY See Admission – Transfer Credit Policy. SCHEDULING OF CLASSES The normal times at which courses are offered are: Monday/Wednesday/Friday 8:00 - 8:50 12:00 - 12:50 9:00 - 9:50 1:00 - 1:50 10:00 - 10:50 2:00 - 2:50 11:00 - 11:50 3:00 - 3:50 Monday/Wednesday 2:00 - 3:20 Tuesday/Thursday 8:00 - 9:20 1:00 - 2:20 9:30 - 10:50 2:30 - 3:50 A number of courses meet four or five times a week, combining the time slots above. Laboratory sections are traditionally held in the afternoons from 1:00 to 3:50 p.m. A few courses may be offered in the evening hours on weekdays (TWTh), normally one evening a week (7:00 - 9:40 p.m.) or two evenings a week (7:00 - 8:20 p.m.) Some performance courses in Music and Theatre meet after 4:00 p.m. and/or in the evening. By faculty legislation, no classes are scheduled in the Tuesday, 11:00 - 11:50 a.m., time slot during the regular academic year. This time is reserved for departmental seminars, departmental Independent Study programs, and college-wide academic events. Specific information about course offerings and class hours is given in the Course Schedule available at the time of registration. The College reserves the right to withdraw courses for insufficient registration or to meet changing conditions. There will be fifteen weeks in each semester with at least fourteen weeks of classes, at least a two-day study period between the end of classes and final examinations, and a final examination or another integrating assignment in all courses except for Independent Study and fractional courses; final examinations may not be scheduled prior to the examination period except by permission of the Dean for Curriculum and Academic Engagement. TEACHER AND COURSE EVALUATIONS Each faculty member is obligated to use some form of written student evaluation of his/her course(s) at least twice each academic year. In courses in which faculty members choose not to seek such student comments, students who wish to complete an evaluation may acquire an appropriate form from the Provost and return the completed evaluation to the faculty member being evaluated. Completed student evaluations are not to be read by the faculty member until course grades have been submitted. Each faculty member then sends the evaluations, along with a summary statement, to the departmental chairperson, who is requested to add his/her comments and forward the information to the Provost to share with the Committee on Teaching Staff and Tenure. VETERAN’S EDUCATION The College is fully accredited under the laws that provide educational benefits for veterans. Specialized military courses are considered for credit on the basis of the recommendations of the American Council on Education as contained in A Guide to the Evaluation of Educational Experiences in the Armed Services. Such credit is allowed only for courses which fit into the curriculum offered by the College. The Registrar is the College’s certifying official. Academic Policies 236 ACADEMIC POLICIES – ACADEMIC STANDING, WITHDRAWAL, AND READMISSION CLASS STANDING Class standing is determined by the Registrar at the beginning of the fall semester of each academic year. The minimum number of credits which must be satisfactorily completed for class standing are as follows: Sophomore — 7 credits, Junior — 15 credits, Senior — 24 credits. Entering students (other than transfer students) who by reason of approved Advanced Placement work or other credits have completed seven or more credits will be given sophomore class standing. The same rules apply to transfer students, and the minimum number of courses needed for sophomore standing at the start of the spring semester is 11 credits. REGULATIONS CONCERNING GOOD ACADEMIC STANDING AND ACADEMIC PROBATION Satisfactory Academic Progress Policy and Need-Based Financial Aid The Committee on Academic Standards reviews the progress of each student toward graduation at the end of each semester and may place a student on academic probation or ask a student to withdraw if it is found that he or she is not making minimal progress toward graduation. To receive or continue to receive need-based financial aid, students must be making minimum satisfactory progress toward their degrees. Provisions for students to receive or continue to receive non-need based merit awards are described in the student’s merit award letter, a separate College policy statement, or both. Need-based financial aid programs affected by this policy include all Federal grant, loan, and work-study programs, state financial aid awards, and College of Wooster institutional need-based financial aid awards. Minimum Satisfactory Academic Progress To meet the number of courses necessary for graduation, the normal expectation is that a student will complete credits at the rate of four per semester for eight semesters. A student earning fewer than seven full course credits in two consecutive semesters will be judged to be making less than satisfactory progress toward graduation and will be placed under “warning.” A consistent pattern of failing to complete seven courses in two consecutive semesters may result in a student being placed on academic probation. Similarly, students are expected to demonstrate progress towards sustaining a semester and cumulative grade point average of 2.0, which is required for graduation and the student’s major area. Satisfactory Academic Progress Guide Semester Minimum Completed Completed Course Credits Expected 1 3 2 7 3 11 4 15 5 20 6 24 7 28 8* 32 Academic Policies 237 *For financial-aid purposes, a maximum of six (6) academic years will be permitted to complete a baccalaureate program. Transfer courses accepted at The College of Wooster are counted toward the total number of courses completed. Credit for repeated courses is granted only once. Good Academic Standing In order to maintain good academic standing, a student regularly enrolled as a degree candidate must meet the following criteria: • earn at least three full course credits in any semester; • earn at least seven full course credits in two consecutive semesters; and • maintain both a semester and a cumulative grade point average of 2.000. Financial Aid Standing A student’s need-based financial aid standing follows his or her academic standing as determined by the College’s Satisfactory Academic Progress Policy. Students who are placed on academic probation by the Committee on Academic Standards will be placed on financial aid probation for the following semester but will remain eligible for federal, state and College need-based student aid. The Financial Aid Office, in coordination with the Dean of Students’ Office, notifies students by e-mail and/or letter that they must re-establish satisfactory academic progress by the end of this following semester in order to maintain their need-based aid awards. Students who do not re-establish satisfactory academic progress, as determined by the Committee on Academic Standards, by the end of this following semester will lose their need-based financial aid eligibility. Mitigating Circumstances The Satisfactory Academic Progress policy can be set aside for individual students under certain mitigating circumstances as determined by the Committee on Academic Standards. Appeal Students notified that they are at risk of losing their need-based financial aid eligibility due to failure to maintain satisfactory academic progress may appeal, in writing, to the Committee on Academic Standards through the Dean of Students’ Office. The appeal must explain the special circumstances why the student failed to meet satisfactory academic progress standards—illness or injury, for instance, or the death of a close relative—and provide an academic plan showing how the student will reestablish satisfactory academic progress by the end of the next semester. If the Committee on Academic Standards accepts the student’s appeal, the student is placed on financial aid probation and remains eligible for federal, state, and College need-based financial aid during that semester. If a student wishes to appeal the Committee on Academic Standards’ decision on his or her appeal request, the student should submit the appeal, in writing, to either the Dean of Students or Dean for Curriculum and Academic Engagement. Re-Establishing Student Aid Eligibility Students will be considered in good standing in regard to need-based financial aid and non-need based merit award eligibility when they again meet the minimum satisfactory academic progress and good academic standing standards as described in this policy statement, or upon acceptance of their appeal by the Committee on Academic Standards. Withdrawal or hiatus from the College for any period of time Codes of Community and Individual Responsibility 238 will not affect a student’s satisfactory academic progress standing. Students who apply for re-admission are required to submit an appeal in order to determine financial-aid eligibility. WITHDRAWAL AND LEAVE OF ABSENCE POLICY See Expenses – Withdrawal. Also see Admission – Implications of Admission and Registration. RULES FOR READMISSION A student who has voluntarily withdrawn or has been required to withdraw from the College is eligible to apply for readmission upon completion of a formal application for readmission; the form for this may be obtained from the Office of the Dean of Students. The completed application, including any necessary transcripts, references, and/or medical/counseling recommendations, and application fee must be received by the Office of the Dean of Students prior to the semester in which the student is eligible to resume studying here. The application fee for readmission is $350. This fee will be forfeited if the student is readmitted and subsequently decides not to re-enroll at Wooster. However, if the student is readmitted, enrolls, and returns to the College, the fee will be credited as the enrollment deposit. The deadlines for readmission applications are April 15 for readmission in Semester I and November 15 for readmission in Semester II. PETITIONS FOR EXCEPTIONS TO ACADEMIC POLICIES Academic policies have been legislated by the faculty and apply consistently to all students. Exceptions are approved only in truly extraordinary and extenuating circumstances, and primarily for documented health and medical reasons. Petitions for exceptions to Academic Policies are submitted to the Dean for Curriculum and Academic Engagement. Appeals of the following academic policies are not normally accepted: • re-appeal of a previous petition, • overload credit for first semester first-year students, • overload credit in any semester for any student beyond 5.500 credits, • change in S/NC status after the established deadline, • change in audit status after the established deadline, • off-campus study application deadlines, • changes to course registration beyond one semester, • changes to academic transcript after graduation, • “walking at Commencement” (GPA and credit requirements). THE CODES OF COMMUNITY AND INDIVIDUAL RESPONSIBILITY “The College of Wooster assumes the honesty, integrity, and responsibility of its students in all areas of academic and social life. A Code of Academic Integrity shall provide the definition and operational structure for the area of academic honor, and Codes of Community and Individual Responsibility 239 a College Code of Social Responsibility shall provide in a similar way for the area of social honor. Adherence to these Codes shall be considered an understood prerequisite for acceptance to and continuance in the College.” (Introduction to the Codes) The Code of Social Responsibility and Code of Academic Integrity are the basis for our current judicial system. The College Judicial system serves as the adjudicating agency for academic violations as well as social violations. Both of these codes deal with the infractions themselves. However, the Wooster Ethic addresses the issue of character and taking responsibility for one’s actions. THE WOOSTER ETHIC “I hereby join this community with a commitment to the Wooster Ethic upholding academic and personal integrity and a culture of honesty and trust in all my academic endeavors, social interactions, and official business of the College. I will submit only my own original work, and respect others and their property. I will not support by my actions or inactions the dishonest acts of others.” COLLEGE CODE OF ACADEMIC INTEGRITY The College of Wooster has operated under an academic honor code since the beginning of 1962-1963 when it was initiated by students. The Preamble to the Code of Academic Integrity states: The academic program at The College of Wooster seeks to promote the intellectual development of each student and the realization of that individual’s potential for creative thinking, learning, and understanding. In achieving this goal, each student must learn to use his/her mind rigorously, imaginatively, and independently. An atmosphere in which each student does his/her own work, except under circumstances in which the instructor indicates that additional aid is legitimate and profitable, is necessary for genuine academic mastery. This implies that it is each student’s responsibility neither to seek nor to use aid, but to utilize his/her own mind, talent, and inner resources to the fullest extent possible. It also places on each student an obligation not to offer or make available unauthorized sources of aid to other students, knowing that such aid is detrimental to those students and to the College community. Finally, each student must be responsible for the maintenance of an atmosphere of academic integrity by confronting violators or reporting any actions that violate its principles, since such violations ultimately harm all members of the community. These principles merely carry out the general purpose of the College to be a community in which the members find it right and necessary to promote the fullest learning by everyone. In other words, a violation of the Code of Academic Integrity conflicts with the values, work and purpose of the entire College community and is not merely a private matter between an individual faculty member and a student. COLLEGE CODE OF SOCIAL RESPONSIBILITY The Preamble to the Code of Social Responsibility states: Informed by the values derived from its Judeo-Christian heritage, the College both recognizes persons in their individuality and also affirms the social dimension of human existence. An academic community in a residential setting depends upon the willingness of individuals to associate together in a common purpose in such a way that individual freedom and responsible order co-exist. As a socially responsible academic community, The College of Wooster seeks a structure within which individual freedom may flourish without jeopardizing the requirements of an academic community and without becoming so self-centered that the resulting environment finally destroys the very freedom it was intended to support. Student Academic Centers 240 The College believes that its goals are best served in an atmosphere of personal self-discipline, guided by the principle of respect for the rights of others and of the community. It also believes that in an academic setting such an atmosphere is best reinforced by a structure which represents both the limitations deemed necessary for an academic community and any other limitations which may be agreed upon in principle by a consensus of all elements of the community — students, faculty, administration, and the Board of Trustees. Within such limitations, the exercise of self-regulation by residential units shall be accepted as a means to achieve personal individuality within a socially responsible academic community. Wooster students, therefore, acknowledge the existence of such limitations and, whenever they exceed them, accept responsibility for the consequences of their actions. In most cases, this will mean a judicial hearing on specific charges. It is also understood, however, that students whose behavior clearly indicates an incompatibility with the philosophy stated herein may be asked to leave the community for another more suited to their needs. The College is required by law to refer felonies (e.g., murder, rape, sexual assault, robbery, aggravated assault, burglary, etc.) to civil authorities. The College cannot and will not offer protection if and when civil authorities become legally involved in any case. (For additional information, refer to the policy titled College Response to Alleged Felonies.) Also, the College reserves the right to take disciplinary action in such situations. As an educational institution with a past and a future, the College has the obligation to state those continuing expectations for its students that it has derived from its purposes and heritage. These mutually agreed upon expectations and those which follow compose the Code of Social Responsibility. The Code of Social Responsibility applies to all students enrolled at the College whether residing on or off campus. It is the responsibility of the members of the community to abide by all portions of the Code and to accept the obligations placed upon them not only for personal behavior but for the enforcement of the Code through the judicial system. The Codes are printed in their entirety in the student handbook, The Scot’s Key. STUDENT ACADEMIC CENTERS CAREER SERVICES Career Services helps students bridge their liberal arts education with their career journey. We offer a comprehensive range of programs, including individual advising and special group forums that assist students in understanding their skills, interests, and values while linking this knowledge with various career options. Career Services helps students from their first year through graduation, whether that means learning about internships, seeking employment, or applying to graduate school. The above services are complemented by a library and website of career exploration and occupational information. Students may browse through summer job listings, internship opportunities, graduate school and employment materials throughout the library. Students are encouraged to use our website to learn about various career fields, specific job search strategies, posting their resume on-line, and upcoming programs and news available through our office. Student Academic Centers 241 For more information, please contact Lisa Kastor, Director, or Lucinda Sigrist-Snyder, Administrative Coordinator, at 330-263-2496. CENTER FOR DIVERSITY AND GLOBAL ENGAGEMENT The Center for Diversity and Global Engagement (CDGE) is a nexus of programs and offices coordinated to encourage and foster development of intercultural competency among all campus community members. Merging student life and curricular development with programming and outreach, the Center reflects the College’s ongoing commitment to building an institution which truly reflects our social, cultural, and political heterogeneity. Liberal arts education demands a global perspective, an understanding of the local situated in a broader world context. The CDGE aims to foster such perspectives across a range of fields on campus and beyond. The Center is housed in the newly-renovated Babcock Hall and includes the following programs and offices: Ambassadors Program, located in Babcock Hall, annually selects five geographically diverse international students or global nomads to serve as Ambassadors for their homeland. The Ambassadors investigate selected topics pertinent to their home countries in order to become “student experts” in these subjects. Ambassadors receive training and funding, and create presentations covering their countries, cultures and current events that are available to the local community at no cost. These presentations occur on campus, in local primary and secondary schools, and at community events. The Ambassadors Program also provides campus-wide programming aimed at bringing the world to Wooster. For more information, please contact Nicola Kille, Assistant Director for Global Engagement, at 330-263-2074. Interfaith Campus Ministries (OICM) seeks to challenge and nurture the spiritual and religious life of the campus. For more information, see Student Life - Religious and Spiritual Life on Campus. Off-Campus Studies (OCS) promotes global events on campus, advises students on off-campus opportunities, and facilitates both domestic and international offcampus study. For more information, see Off-Campus Study. Office of International Student Affairs (OISA), located in Babcock Hall, supports international and exchange students, global nomads, and language assistants as they adjust to a new culture. OISA also encourages and celebrates their unique contribution to the campus community and beyond. OISA’s goals include: supporting the academic and social success of international students; advocating on behalf of their unique needs and interests; educating international students about their legal rights and obligations; and encouraging intellectual growth campus-wide, with a particular focus on global perspectives and competence. For more information, please contact Mariana Weyer, Administrative Coordinator, at 330-263-4530. Office of Multi-Ethnic Student Affairs (OMSA), located in Babcock Hall, provides support services and programs for students of color. These services include on-going individual advising as students encounter academic, financial, personal, and social concerns. OMSA also assists multi-ethnic student organizations in an advisory role. OMSA works to promote dialogue and positive inter-cultural experiences for all members of the College community. For more information, please contact, Susan Lee, Director, or Mariana Weyer, Administrative Coordinator, at 330-263-4530. For more information about The Center for Diversity and Global Engagement, please contact Co-directors, Susan Lee and Amyaz Moledina, or Mariana Weyer, Administrative Coordinator, at 330-263-4530. Student Academic Centers 242 EBERT ART CENTER The Ebert Art Center is home to the Department of Art and Art History, the Visual Resources Library, and The College of Wooster Art Museum. The Department of Art and Art History offers majors in Art History and in Studio Art, and courses in both majors are designed to allow students to develop a sensitive understanding of the visual arts, past and present. In studio courses, students learn to conceive and express ideas in two-and three-dimensional media, to evaluate the aesthetic character of works of art, and to become aware of art’s inherent sociopolitical implications. Art history courses are concerned with the production and reception of the visual arts within their social, religious, cultural, and political contexts. The College of Wooster Art Museum supports and enhances the College’s goals of teaching, research, and service through exhibitions, scholarship, collection preservation, and public engagement. The museum program also promotes campus-wide engagement and interdisciplinary dialogue through collaborative exhibitions, and acts a catalyst for creative engagement both on campus and between the College and regional and national audiences. For more information about The College of Wooster Art Museum and its program visit artmuseum.wooster.edu or contact Kitty Zurko, Director, or the Administrative Coordinator at 330-263-2388. EDUCATIONAL PLANNING AND ADVISING CENTER The Educational Planning and Advising Center (EPAC), located in the Lilly House, offers a range of resources to help students develop comprehensive educational plans for their years at the College. The EPAC staff is available for individual meetings with students. Through out the year, the Center offers programming to meet the challenges of college life. This includes: Peer Mentors Program pairs first-year students with upper-class students who have excelled both academically and in their co-curricular pursuits. Mentors meet one-on-one with first-year students to help them successfully transition to college life and to discover exciting opportunities both on and off campus. Sophomore Retreat, led by a diverse group of Wooster faculty and staff, is an overnight retreat, near Mohican State Park, designed to help sophomores refine their academic and career goals. Participating students also learn more about resources and opportunities that are available across campus. For more information regarding the Educational Planning and Advising Center, please contact Alison Schmidt, Associate Dean for Educational Planning and Advising, Cathy McConnell, Director, or Karen Parthemore, Administrative Coordinator, at 330-263-2428. CENTER FOR ENTREPRENEURSHIP The Center for Entrepreneurship strives to empower students to pursue their passion (regardless of academic discipline) by learning about and creating entrepreneurial ventures that generate economic (for-profit) and/or social (non-profit) value. The Center for Entrepreneurship wants students to value and understand how important e-ship is to this culture, government, and country. The Center offers workshops, programs, and summer opportunities for entrepreneurial education. The Center is located in Morgan Hall, Room 301, and students are welcome to drop in and meet with James Levin, Director, or for more information call Martha Bollinger, Administrative Assistant, at 330-263-2267. LEARNING CENTER Located in the Rubbermaid Student Services Building, the Learning Center offers academic support to any student on campus. The Learning Center is staffed by adult Student Academic Centers 243 consultants who work with individual students in scheduled sessions. The sessions focus on time management, organizational skills, and effective study strategies tailored to meet students’ academic needs in specific courses. Students may also take advantage of quiet space for study and computer use at the Learning Center. The Learning Center is also the office of support for students with disabilities. The College recognizes that students with physical or learning disabilities may have certain needs that require specific accommodations. To ensure equal access to all courses and programs at the College, students are encouraged to submit professional documentation of the disability to the Learning Center. Reasonable and appropriate accommodations will be arranged after students meet with Learning Center staff to review their documentation. The Learning Center is open from 8:00 a.m. - 4:00 p.m. Monday through Friday. There is no fee for this service and students are encouraged to schedule appointments with the Center early in the semester. For more information, please contact Pam Rose, Director (prose@wooster.edu), or Amber Larson, Assistant Director (alarson@wooster.edu), at 330-263-2595. THE LILLY PROJECT The Lilly Project for the Exploration of Vocation, located in the Lilly House, provides opportunities for the entire College community to engage in serious vocational exploration both on campus and off. The largest program is the Summer Vocational Exploration. Students compete to receive funding for fellowships primarily of their own design. The fellowships fall under a range of categories: addressing humanitarian causes in health and legal issues, civic engagement, and defining and developing one’s personal passions, to name a few. Two other offerings include: ReIntegration, a program which provides students the opportunity to put their off-campus experiences into perspective and examine how they will make use of them back on campus and beyond; and Mini Grants, an opportunity for students, faculty, and staff to receive funding for smaller vocational exploration projects which can be done during the academic year. Students are strongly encouraged to speak with the director before submitting any proposal. For more information, please contact Cathy McConnell, Director, at 330-263-2301. MATH CENTER The Math Center in Taylor Hall, Room 301, supports students in introductory level math courses. Staffed by a math professional and/or peer tutors, the Math Center provides walk-in tutoring (no appointment required). Math Center users typically ask for assistance understanding concepts and examples from the text and/or class lectures, preparing for exams, or completing homework assignments. Some students choose to complete all of their math homework at the Center to have immediate access to the Center’s resources, while others bring in problems after attempting an assignment. While the Math Center cannot explain economics, physics, astronomy, chemistry, etc., it can help students from non-math courses solve an equation or complete an integral. For more information, please contact Linda Barbu, Director, at 330-263-2490. WRITING CENTER Two ideals figure prominently in a Wooster education: successful writing and independent students. The Writing Center is essential to both. From First-Year Seminar to Senior Independent Study, from receiving a writing assignment to final editing of a paper, from constructing an argument to documenting sources, from process to product, the College Writing Center provides one-on-one guidance, Student Life 244 resources, and support for student writers as they work through their academic careers. We strive to enable student writers to make informed, successful, and independent decisions about their writing. The staff includes experienced student writers, knowledgeable professional staff, and professionals in the field of writing. Regular appointments for many Sr. I.S. students and most students working repeatedly with the Writing Center are the best indicators of its importance. Monday evenings in the fall semester include FYS-focused support; other arrangements can be made as well. There is no charge for working in the Writing Center. Writing is a process that moves from generating ideas for writing to proofreading, and the Writing Center can help at any stage of that process. Many writers rely on Consultants and Tutors for these latter stages of the writing process, and the Writing Center strives to provide educated readers who ask common-sense questions and point out issues focus, organization, and tone, as well as mechanics. The staff works from the ideal that repairing one paper is productive, but helping writers to better understand and take control of successful writing provides much greater benefit. Our goal is to help students learn to look at their writing more critically through their identifying writing strengths and our guiding their improvement elsewhere. Appointments are not required, but they are recommended. Students are encouraged to call Debbie Baker, Administrative Assistant, at 330-263- 2205. STUDENT LIFE ART The College of Wooster Art Museum, located in the Ebert Art Center, presents rotating exhibitions in the Sussel Gallery and the Burton D. Morgan Gallery. In any given year, exhibitions might include historical and contemporary art, a group Senior Studio Art Independent Study exhibition, the Five College of Ohio Juried Student Biennial (initiated by Wooster in 2001), faculty shows, and other exhibi - tions and events that support classes and interdisciplinary dialogue. For more information, please contact Kitty Zurko, Director, or the Administrative Coordinator at 330-263-2388. CAMPUS COUNCIL In the spring of 1969, a Campus Council was created, which joined in its membership students, faculty, staff, and administration to legislate in the areas of student life and extracurricular affairs and to issue advisory opinions and make recommendations to the President of the College, the Board of Trustees, and other organizations. One of the Council‘s responsibilities is to charter all student organizations and allocate their budgets. Since its creation, the Council has become an increasingly effective forum in which ideas are heard, exchanged, and coordinated into action. A contribution of major significance was the Council‘s sponsorship of the drafting and its con tinued over sight of the Code of Academic Integrity and the Code of Social Responsibility. For more information, please contact Kevin Carpenter, Student Chair, at kcarpenter12@wooster.edu. INTERCOLLEGIATE ATHLETICS AND INTRAMURAL SPORTS The College of Wooster believes that all phases of physical education (instructional classes, intramural sports, and intercollegiate athletics) are integral parts of the Student Life 245 total educational program. All intercollegiate athletics are under the direction of the Department of Physical Education and Athletics. The College is a member of the National Collegiate Athletic Association and the North Coast Athletic Conference; its conduct of men’s and women’s intercollegiate athletics is governed by the policies of these organizations. The men’s program includes eleven sports: baseball, basketball, cross-country, football, golf, lacrosse, soccer, swimming, tennis, indoor and outdoor track. The women’s varsity program includes twelve sports: basketball, cross-country, field hockey, golf, lacrosse, soccer, softball, swimming, tennis, indoor and outdoor track, and volleyball. Tuition includes free admission for students to all regularly-scheduled intercollegiate contests held in Wooster (excludes tournaments and post-season). A varied intramural program is offered for both men and women. Activities include flag football, bowling, volleyball, golf, soccer, basketball, ultimate frisbee, floor hockey, billiards, swimming, tennis, and softball, among others. The intramural department encourages individual students as well as student groups to suggest new activities. A student group desiring to use one of the College’s intercollegiate practice or game fields or facilities must obtain prior permission from the chairperson of the Department of Physical Education and Athletics. For more information, please contact Keith Beckett, Director, or Bonnie Hughes, Administrative Coordinator, at 330-263-2349. LILLY PROJECT The Lilly Project is made possible through a generous grant from the Lilly Endowment, Inc. Its goal is to provide students and the entire College community with the opportunity to engage in vocational discussion and reflection. The Project intends to create a climate of engagement that focuses on questions of meaning and value: What is worth doing, and how can our lives contribute to that which has ultimate significance? The project will support the people and communities involved as they transform their individual and collective lives and envision what is possible for them beyond what currently exists. Lilly Project initiatives include curricular, field experience, and co-curricular elements. For more information, please contact Cathy McConnell, Director of The Educational Planning and Advising Center, or Karen Parthemore, Administrative Coordinator, at 330-263-2428. MUSIC The Scot Band is an organization of about 170 musicians which plays at all home football games, one away game, and one invitational band festival. The Scot Symphonic Band (about 80 members) gives three home concerts each season and tours during a portion of the spring vacation. The Scot Pipers and Dancers perform with the Marching Band during football season, make appearances around the state during the school year, and tour with the Symphonic Band in the spring. Membership in the Marching Band is open to all students. Symphonic Band membership requires an audition. Wooster Chorus is a group of approximately 50 mixed voices which appears on campus and in nearby communities and tours during the spring vacation. Membership is open to all students upon audition. Gospel Choir is a performing organization open to any student, faculty, staff, or community person. The choir gives at least one performance each semester of African-American choral music. Auditions are held immediately prior to the beginning of the fall semester. Wooster Singers is a choir open to all without audition. This ensemble explores choral music of a wide range of styles and historic periods and develops sightsinging skills. Performances will be scheduled depending on the size and preparation of the ensemble. Student Life 246 Wooster Symphony Orchestra is a college/community ensemble of over 60 musicians, made up of students, faculty, and local citizens, which plays three subscription concerts each season. Wooster Symphony membership requires an audition. Wooster Symphony Chamber Orchestra is an advanced orchestra; participants are selected each fall from the membership of the larger orchestra. Jazz Ensemble is an organization of 18-20 players which performs three home concerts per year in addition to occasional outside appearances. A variety of musical styles is included, and there is opportunity for members to contribute original compositions and arrangements. Jazz Ensemble membership requires an audition. Jazz Combo is a performing ensemble comprised of six to ten instrumentalists devoted to the study and performance of small-group jazz (hot, swing, bebop, cool, progressive, and fusion). Ensembles are smaller groups, such as string, woodwind, brass, and percussion ensembles, which function in addition to the above groups as there is a demand or requirement. For more information on student music groups, please contact Nancy Ditmer, Department Chair, or Donna Reed, Administrative Coordinator, at 330-263-2419. RADIO WCWS (FM 90.9, 850 watts) is operated by the College with student management as a non-commercial, educational broadcast station serving Wooster and ten surrounding counties. Programming on WCWS includes a wide range of music formats, from classical to jazz to rock, as well as sports, news, and public affairs. The station also airs special programs, including a weekly showcase about Wooster’s nationally acclaimed Senior Independent Study project. Any student interested in the various fields of broadcasting — engineering, programming, news, or sports — is invited to participate. For more information, please contact Jason Weingardt, Student General Manager, at jweingardt12@wooster.edu. RELIGIOUS AND SPIRITUAL LIFE ON CAMPUS The religious community at The College of Wooster is diverse. A variety of groups, programs, and services are provided for religious expression and spiritual growth seeking to deepen the search for meaning. These include personal and communal religious and spiritual practice as well as opportunities to serve the wider community. The Chaplain coordinates religious and spiritual life on campus. Interfaith Campus Ministries (OICM) supports existing student religious groups and initiates new groups as interest and need are indicated. The Chaplain and OICM seek to challenge and nurture the spiritual and religious life of the campus. The Chaplain and Director of Interfaith Ministry, the Campus Rabbi, the Associate Chaplain, and other staff are available for individual conversations and counseling with students, for programs dealing with questions of faith and meaning, and as a resource for religious life and observances. They can be reached at extension 2602. OICM sponsors a wide variety of programs and events that are open to the campus community. A newsletter is offered regularly, and a web page is available on the college’s website under the Student Life section. Overholt house, located on the corner of Beall and Bloomington, is the home of OICM. It offers an on-campus retreat space with a kitchen and meeting areas open to the campus. The Chaplain’s office is in the Center for Diversity and Global Engagement. Throughout the year, there are opportunities for interfaith dialogue and worship. Sunday@Six, the student-led, Ecumenical campus worship service, is held in Mackey Hall each week that classes are in session. All members of the campus community are welcome to attend or contribute. Worthy Questions invites students to meet weekly to explore with others the ‘quest’ for purpose and meaning that serves to integrate diverse aspects of one’s life. Mentors from the community join the students in the Student Life 247 process of learning to “ask questions worthy of the person they may become.” The program accepts applications annually. A multifaith group of students and staff provides opportunities for members of the campus community to engage in interfaith dialogue. Christian life on campus includes a number of student groups: • The Catholic Student Association offers services, activities, retreats, social justice programs, and speakers. All are open to all students at the College. • EnRoute is a Progressive Christian Community whose focus is exploring questions of faith together. They attend Sunday@Six and meet following the service for Koinonia, a dinner and discussion group. • Fellowship of Christian Athletes meets weekly for praise music and discussion. • Sisters in Spirit is open to all women on campus. The group meets regularly to explore a variety of faith issues pertaining to college-age women. It has also engaged in a partnership with the local domestic violence shelter. • Wooster Christian Fellowship is affiliated nationally with InterVarsity Christian Fellowship, whose main purpose is to know Christ and to make Him known. The group meets weekly for worship, teaching, and fellowship as well as offering occasional retreats, conferences, and camps. Jewish life on campus centers around the Hillel group. It works to increase the appreciation and observance of Judaism, welcoming non-Jewish students who are interested. The group sponsors activities for the entire campus, including traditional Jewish religious celebrations, guest lectures, movies, and discussions. It maintains a Hillel library and keeps its members informed of Jewish activities in the area. The activities of the Hillel community are supported, in part, by the Lottie Kornfeld Endowment. Muslim life on campus centers on Noor. Noor exists to inform the campus community about the rich Islamic tradition and heritage; as such, its membership is open to all members of campus. It also provides a community for Muslim students and arranges periodic trips to the area mosque, special meal arrangements for Ramadan, and observances of major religious holidays. Peace by Peace is an organization working to promote peacemaking activities. Its members offer educational programs, link with other advocacy groups on campus and in the community, and seek to promote skills and attitudes that assist in effective conflict resolution. Westminster Presbyterian Church is the congregation-in-residence at The College of Wooster. The congregation meets for worship on Sundays at 10:45 a.m. in The Westminster Church House. Students are invited to be active in the congregation as full or associate members. Westminster sponsors various campus programs in conjunction with the Office of Interfaith Campus Ministries and other religious groups on campus. Congregations in the Wooster area welcome students to their services and to their community life. A number of congregations welcome student participation in their choirs or offer employment opportunities. A directory is available from OICM. For more information, please contact Linda Morgan-Clement, Campus Chaplain, or the Administrative Coordinator at 330-263-2602. STUDENT GOVERNMENT ASSOCIATION The Student Government Association is the formal assembly of the student body that is an advocate for student concerns and provides various campus services, transportation to/from Cleveland-Hopkins and Canton-Akron airports at College breaks, and summer storage. The officers each have specific responsibilities and participate in Senate meetings and weekly Cabinet meetings. For more information, please contact Hannah Haas, Student President, at hhaas12@wooster.edu. Student Life 248 STUDENT ORGANIZATIONS Approximately 120 student organizations are chartered by Campus Council. These special interest organizations are open to any interested students. They include departmental clubs, international/diversity groups, political organizations, club sports, and volunteer groups. Some of the groups are as follows: Black Students’ Association (BSA), Don’t Throw Shoes, International Student Association (ISA), Noor (Muslim Students), Allies and Queers, WOODS, Circle K International, Wooster Hillel (Jewish Students), Volunteer Network, South Asia Committee, Merry Kween of Skots, the COW Belles, Wooster Scottish Arts Society, Wooster Christian Fellowship, and Wooster Cricket Club. For more information, please contact Bob Rodda, Director of Lowry Center and Student Activities, or Julie Christopher, Administrative Coordinator, at 330-263-2062. STUDENT PUBLICATIONS The College of Wooster supports a variety of student publications. These publications enrich the cultural life at Wooster and provide students with a range of avenues to engage in the creative and thoughtful expression of ideas. They also offer students hands-on experience in managing, editing, and publishing. The Goliard is the College literary and art journal. It is published annually and is staffed by students from all classes. The Index is the College yearbook, published annually and staffed by students from all classes. Sapere Aude is the Wooster Journal of Philosophical Inquiry and is in its third year. The Voice is the College weekly newspaper, staffed by students from all classes. Year One Journal is the annual publication of the First-Year Seminar Program. It features prose, fiction, and visual art produced by first-year students. It is staffed by upper-class students under the guidance of the Director of the Writing Center. For more information regarding student publications, please contact Claudia Thompson or Jacob Koehler, Co-chairs of the Publications Committee. THEATRE Auditions for plays are open to all students. A balanced selection of plays is presented each season under the direction of the Department of Theatre and Dance. Musicals are presented every other year in conjunction with the Department of Music. Programs include student-directed productions, workshop productions, and off-campus professional theatre internships. During the summer, the Ohio Light Opera Company plays its festival season in Freedlander Theatre. For more information, please contact Dale Seeds, Department Chair, or Patrice Smith, Administrative Coordinator, at 330-263-2541. WOMEN’S CLUBS AND MEN’S SECTIONS (Greek Life) There are six local social clubs for women and four local social sections for men on campus. In a variety of ways, these groups function similarly to local sororities and fraternities. Any student in good academic and social standing is eligible for membership. The general functioning of the sections and clubs, including rushing, bidding, and new member education, is under the jurisdiction of the Inter-Greek Council and the Committee on Social Organizations. The latter holds the final authority for the policy affecting these organizations. Approximately twelve percent of the student body is involved in sections and clubs. Sections and clubs may apply for College housing each year. Currently, nine groups are housed as units in College housing. For more information, please contact Joe Kirk, Director, at 330-263-2342. WOOSTER ACTIVITIES CREW (W.A.C.) The Wooster Activities Crew (W.A.C.) is the campus programming board, run by Student Services 249 students for the Wooster community. The group’s purpose is to plan, promote, and produce entertaining and socially engaging events that both provide academic relief and unite the student body. W.A.C. brings innovative and novelty events to the College, as well as plans annual events such as Party on the Green, Gala, and Springfest. For more information, please contact Daniel Casto, Student President, at dcasto12@wooster.edu. WOOSTER DANCE COMPANY The College of Wooster Dance Company is a fully-chartered College organization for all students on campus interested in dance. The approximately fifty members in the Company present two concerts each year — one during the first semester and one in the second. Auditions during first and second semesters are open to all students. The Company is associated with the Department of Theatre and Dance, which includes a dance track within the major and a minor in Theatre and Dance. Additionally, the Dance Concert is presented as one of the Theatre and Dance Department‘s main stage productions. Opportunities for students to dance, choreograph, and coordinate for the Company provide excellent experiences and college credit. For more information, please contact Dale Seeds, Department Chair, or Patrice Smith, Administrative Coordinator, at 330-263-2541. WOOSTER VOLUNTEER NETWORK Wooster Volunteer Network offers a range of community service opportunities on campus. It serves as an umbrella organization to coordinate and encourage service to the Wooster community and beyond. Opportunities include service house living and serving communities, monthly service activities coordinated by the WVN executive board, an international service trip in conjunction with OICM, hurricane relief week in the New Orleans area, and an annual spring-break trip to West Virginia. The WVN office is located in Overholt house. Students who are interested in creating projects or getting involved in community service are encouraged to stop up or visit the website found through the College’s home page. For more information, please contact the student co-chairs: Amanda Collins at acollins13@wooster.edu, or Lauren Grimanis at lgrimanis12@wooster.edu. STUDENT SERVICES CAMPUS DINING SERVICES Food is provided to College of Wooster students on a meal plan by the College owned-and-operated Campus Dining Services department. Students may select the meal plan that best suits their lifestyle and their dining habits. The meal plan choices incorporate a mix of traditional, all-you-care-to-eat meals in Lowry dining hall and Flex Dollars that can be spent like cash to purchase food and drinks at campus food locations. Meal counts are expressed in number of meals per semester, and are not limited to number of times per day or week they can be used. Neither the unused dining hall meals nor the unused Flex Dollars will roll over from semester to semester or year to year. Students must present their College I.D. card in order to utilize their meal plan. Students approved to live off-campus are welcome to subscribe separately to the meal plan contract. Students may also utilize any balances they may have in their COW Card Debit account for food purchases at Kittredge, Lowry, MacLeod’s, Mom’s, Old Main Café, Pop’s, Scot Dogs, and vending machines. Student Services 250 Kittredge Dining Hall focuses on simple, fresh, and healthier foods that have a lower carbon footprint. It is open for lunch Monday through Friday; hours of operation can be viewed online at www.wooster.edu/Student-Life. Lowry Center Dining Hall is located on the top floor of Lowry Center and features an all-you-care-to-eat food-court style meal contract service for breakfast, lunch, and dinner daily. Dining hours and menus can be viewed online at www.wooster.edu/Student-Life. MacLeod’s Coffee Bar and Convenience Store is located in the Lowry Center main lounge and offers freshly brewed Starbucks drip coffee, Starbucks espressobased favorite drinks, and a host of convenience store products. Hours of operation can be viewed online. Mom’s is located on the ground floor of Lowry Center and features ala carte grill foods, cold salads, coffee, fruit smoothies, sand wiches, soups, fountain drinks, and milkshakes. Mom’s accepts cash, COW Card Debit, and meal plan Flex Dollars. Hours of operation can be viewed online. Old Main Café, located on the “Garden Level” of Kauke Hall, provides students, faculty and staff with a relaxing oasis, whether they are seeking a break between classes or a comfortable coffee-house atmosphere in the evening. The Old Main Café offers an extensive menu of coffee, featuring Starbucks espresso-based drinks as well as Starbucks drip coffee, teas, hot chocolate and bottled beverages, as well as muffins, dessert bars and cheesecakes. Freshly-prepared salads and sandwiches are available daily and include vegan and vegetarian specialties. Cash, COW card, personal/department charges and meal plan Flex dollars are accepted as forms of payment. Sorry, we do not accept credit cards or meal plan swipes. Pop’s Sub-Stop is located on the ground floor of Lowry Center and offers quick, grab-n-go convenience for lunch, Monday through Friday. Cash, COW card, personal/department charges and meal plan Flex dollars and meal plan swipes are accepted as forms of payment. Sorry, we do not accept credit cards. Hours of operation can be viewed online. Scot Dogs is a mobile sandwich cart that offers high quality, locally produced, hot sandwiches, chips, and drinks. It will be positioned at an easily accessible location on-campus most weekdays for Lunch (weather permitting). Cash, COW card, personal/department charges and Flex dollars on meal plan are accepted as forms of payment. Sorry, we do not accept credit cards or meal plan swipes. Campus Dining Services can provide Catering services and on-location catering in any campus building or on the campus grounds at a reasonable cost. The Campus Dining Services Customer Service Office processes all catering requests, orders for student Birthday Cakes and Exam Care Packages, and administers all meal plans. The Customer Service Office can answer questions and resolve problems with regards to the meal plan. More information about catering can be viewed online. The Campus Dining Services Customer Service is located on the lower level of Lowry and may be reached by calling Chuck Wagers, Director, or Donna Yonker, General Manager, at 330-263-2358. HOUSING AND RESIDENCE LIFE The College of Wooster is a residential college; all students live on campus for their entire College career. Students must be enrolled full-time (three full course credits or more) to reside in College housing. When a student’s course registration drops to fewer than three full course credits or a student’s status is changed to “Leave of Absence” or “Withdrawn,” then he/she must immediately vacate the College’s residence hall or program house. Written exceptions to this requirement may be granted by the Dean of Students or his designee. Exceptions will be granted only for compelling reasons. The College reserves the right to remove or relocate students living Student Services 251 in College housing when circumstances warrant such action. Students must live in College housing unless they are granted off-campus living permission by the Dean of Students or his designee. A variety of housing options for individuals and groups are available, including coeducational and single-gender halls, and program-oriented halls. Housing options include the International Program, the Residential Senior Program, and Club and Section Housing among many others. All College residence halls and program houses have access to the computer network. Residence hall rooms vary in size, configuration, and styles of the furnishings. Rooms have a study desk, chair, bed, mattress, dresser, and window shades. Bedding, pillows, rugs, curtains, and other equipment are provided by the resident(s). Students provide and care for their own bed linen. Washers and dryers are provided for all College housing. Students must provide their own telephones while on-campus and local service is available to all students. The College is not responsible for loss or damage to clothing and personal effects in student rooms. Consequently, students are encouraged to carry their own insurance on personal property and to lock their room doors when out of the room. In addition to living in traditional residence halls, a number of students are housed in program houses located throughout campus. These houses accommodate groups of four to thirty people. Students are required to complete a special application to be considered for residence in these units. Groups living in program houses participate in volunteer activities that serve the campus and local community. All housing options are administered by the Office of Residence Life. In each residence hall and cluster of program houses, Resident Assistants are available for the support of the students in these communities. RAs are sophomores, juniors, or seniors who are trained to provide guidance, peer advising, and referral to campus services for students. Professional staff also live within the residential community to provide assistance to the residents. The Central Office Staff include the Associate Dean and Director of Residence Life, the Associate Director, two Assistant Directors, an Administrative Assistant, and three full-time Area Coordinators. For new students, a room reservation is made when an applicant has been accepted for admission, paid the enrollment and security deposit, and submitted the appropriate housing materials. New students must maintain a residence in College housing unless they apply for an exception to live at home with a parent or guardian. Housing assignments for new students will be completed and mailed in mid-July by the Residence Life staff. College residential facilities are open to students only when classes are in session. Students who do not have special permission to engage in a special College activity (graduation, sporting events, etc.) are asked to vacate their rooms at the close of a semester, no later than twenty-four hours after their last examination. During the second semester, those who are graduating may remain on campus until commencement ceremonies have concluded. Information on fees may be found in the Catalogue section entitled Expenses. Information on housing may be acquired by calling Christie Kräcker, Associate Dean of Students and Director of Residence Life, or Angela Sponsler, Administrative Coordinator, at 330-263-2498 or by visiting the Residence Life web site at www.wooster.edu/ Student-Life/Residence-Life. LOWRY CENTER Lowry Center, the College‘s student union, opened in the fall of 1968 as a memorial to Howard Lowry, President of Wooster from 1944 to 1967. In the “Role of the College Union”, the Association of College Unions International states the following: The union is an integral part of the educational mission of the College. As the center of the college community life, the union complements the academic experience through Student Services 252 an extensive variety of cultural, educational, social, and recreational programs. These programs provide the opportunity to balance course work and free time as cooperative factors in education. Lowry Center provides students with a range of services and contains a variety of multi-purpose areas including the bookstore, post office, information desk, MacLeod’s convenience store, Scot Lanes bowling and billiards facility, main lounge, art exhibit area, meeting rooms, the gallery of international flags, faculty lounge, dining facilities, 24-hour printing center, and Mom’s Truckstop snack bar. Also located in the building are offices for the College newspaper, Student Government Association, and Wooster Activities Crew. For more information, please contact Bob Rodda, Director, or Julie Christopher, Administrative Coordinator, at 330-263-2062. SECURITY AND PROTECTIVE SERVICES The Security and Protective Services Department provides law enforcement response, crime prevention education, and security services to the campus community 24 hours a day, 7 days a week, 365 days a year. The department also works closely with the Wooster Police Department, Wooster Fire Department and other College and City offices to provide such services and resources to the Wooster community. Primary duties include the safety and security of students, grounds and facilities. The department also monitors the College’s 911 system, fire safety systems, and campus access system. The SPS department is responsible for upholding the College policies found within The Scot’s Key as well as local, state and federal laws. The office is located on Wayne Ave., just east of the Longbrake Student Wellness Center, and is staffed 24 hours a day. The Department seeks to promote and preserve the security and safety of the College community. Our philosophy is based on the concept that officers and members of the College community work together in creative ways to help solve problems related to crime and fear of crime. Our goal is to have a positive presence here on campus based on mutual understanding and respect. Foot patrols inside buildings and bike patrols around campus are opportunities to become closer to our community. Establishing and maintaining a mutual trust within the College community is used to improve our ability to prevent crime and solve problems. Policy enforcement and intervention activities will be conducted in such a way as to provide a positive learning experience when possible. The Security Department also provides numerous services to the campus community including: safety escorts, property engraving, residential education programs, fire safety programs, vehicle and bicycle registration, student security patrols, CPR/First-Aid, and other programs. The Security and Protective Services Department is also responsible for the enforcement of parking regulations on campus. All vehicles parked on the College of Wooster campus must display a valid permit. Permits can be obtained at the Security office 24 hours a day, 7 days a week. For information on parking visit www.wooster.edu/Student-Life/Security-andProtective-Services. Requests for services can be made by contacting the Security and Protective Services Department at 330-263-2590. STUDENT WELLNESS CENTER The Longbrake Student Wellness Center (LSWC) provides comprehensive health services for College of Wooster students enrolled on a full-time basis. The staff consists of physicians from the Cleveland Clinic Wooster, professional psychological counselors, certified athletic trainers, an office administrator, and registered nursing staff to maintain 24-hour service during the academic year. Services include physician appointments, GYN appointments, nurse evaluations, blood draws, medica- Admission 253 tions, EKGs, allergy injections, hydration IVs, counseling appointments, cold care and first aid center, and overnight student beds. Programs offered comprise of Students Helping Students (student peer education), First Responder training and service, Sexuality Support Network, Depression Support Group, Paws to Pet (dog therapy), and massage therapy. The Medical Director for the LSWC is a Cleveland Clinic physician. All services are administered under his/her supervision. Physicians are available to students by appointment Monday through Friday at no charge. Confidential counseling services are available at the Wellness Center (no fee for the first ten counseling sessions). The athletic trainer evaluates non-varsity student athletes at the Wellness Center by appointment. The Cold Care Center is an educational, self-treatment module for evaluation and treatment of respiratory infections. A First Aid Center is available for treatment of minor injuries. Both the Cold Care Center and the First Aid Center are accessible 24/7 on a walk in basis. The cost associated with most of these services is included in the comprehensive fee. The student is required to purchase the Student Accident and Sickness Medical Plan or furnish a waiver indicating they have health insurance coverage through a parent or individual plan. Insurance will be used if the student needs to be seen at a medical facility outside of the LSWC. Details of the plan are provided in the brochure mailed to all students by the Business Office. For more information, please contact Esther Horst, Acting Director of LSWC, or Lori Stine, Administrative Coordinator, at 330-263-2319. ADMISSION Admission to The College of Wooster is open to qualified students regardless of age, sex, color, race, creed, religion, national origin, disability, veteran status, sexual orientation, or political affiliation. In determining admission, due consideration is given to many different expressions of a student’s qualities and abilities: scholastic achievements, performance on standardized tests, extracurricular activities, and promise to benefit from and contribute to the intellectual life of the community. APPLICATION TIMETABLE Application Decision Candidates’ Due Announced Reply Date First Year Candidates Early Decision November 15 December 1 January 1 Early Action November 15 December 31 May 1 Regular Decision February 15* April 1 May 1 Transfer June 1 for Fall Term within 2 weeks within 2 weeks Candidates Dec. 1 for Spring Term of completion of notification of application of admission * Candidates may apply after this date, but they should understand that priority will be given to those who meet the application deadline. Admission 254 APPLICATION PROCEDURE 1. Application: The College of Wooster accepts the Common Application. A copy of the Common Application can be found on the Office of Admissions website at www.wooster.edu/admissions/apply or on the Common Appli - cation website at www.commonapp.org. The applicant may also contact the Office of Admissions at 800-877-9905 to request an application. Applicants can submit their materials online or through the mail by the appropriate deadlines. A fee waiver is applied for online applicants. 2. High School Transcript: A transcript should be furnished by the secon dary school at the time the student submits an application. A final transcript will be required at the end of the senior year, and an interim transcript may be requested earlier in the senior year to monitor progress. 3. School Report: The school report form must be submitted by the applicant to his or her secondary school counselor, who should send the completed form to The College of Wooster before the application deadline. 4. Teacher Evaluation: The teacher evaluation form should be given to a teacher who has taught the applicant in an academic subject within the last two years. The completed form should be returned by the teacher to The College of Wooster before the application deadline. 5. Application Fee: A non-refundable application fee of $40 must be sent to the Office of Admissions by the stated deadline for all applicants. This fee will be waived for those who apply online. Checks or money orders should be made payable to The College of Wooster. If this fee represents a financial hard ship, a guidance counselor may submit a College Board fee waiver, or a letter requesting a fee waiver, on the student’s behalf. 6. Entrance Tests: Scores from the Scholastic Aptitude Test (SAT I) of the College Entrance Examination Board or scores from the American College Testing Program (ACT) are required of all applicants. It is recommended that all applicants take one of these tests no later than November of the senior year. Infor mation about the SAT may be obtained through www.collegeboard.com. Information about the ACT may be obtained through www.act.org. 7. Financial Aid: Over ninety percent of all students at The College of Wooster receive some form of financial aid. Applicants for financial aid based on need should file the Free Application for Federal Student Aid (FAFSA) as soon after January 1 as possible. The FAFSA may be obtained from secondary school guidance offices or online at www.fafsa.ed.gov. Additional information on need-based financial aid and merit scholarships may be obtained from the Office of Admissions. Please also consult the section on Financial Aid in this Catalogue. Students are strongly encouraged to visit the campus and to talk with an admissions counselor before making a final college choice. Although not required, a visit permits the candidate to attend an information session, have an admissions interview, tour the campus, visit classes, and meet faculty and students. Visit arrangements should be made at least one week in advance of the desired visit date through the Office of Admissions by calling 800-877-9905 or going online to www.wooster.edu/admissions/visit. All admission to the first-year class or to advanced standing are under the direction of the Office of Admissions. The Admissions Committee suggests, as a minimum, the following distribution of entrance units: Admission 255 English 4 Foreign Language 2 History and Social Science 3 Mathematics 3 Natural Science 3 plus at least one elective from the above categories for a total of sixteen academic units. EARLY DECISION and EARLY ACTION Early Decision applicants will be asked to sign a statement declaring their intention to enroll at The College of Wooster if admitted. Students applying in the Early Decision process may submit Regular Decision applications to other colleges or universities, but those students offered admission to Wooster will be expected to withdraw their applications from other institutions and not to initiate any new ones. Early Decision candidates who wish to apply for financial aid should complete the CSS Financial Aid Profile available from secondary school guidance offices. All aid awards are tentative pending submission of the official aid application (FAFSA) and other required documentation. Students who decide that Wooster is their first choice college are encouraged to apply under the College’s Early Decision option: Early Decision: Candidates must submit all of the application credentials (Early Decision Agreement, the Common Application, supplement, school report, transcript, recommendations, standardized test results, and application fee) no later than November 15. By December 1, Early Decision I candidates will receive one of three responses from the Office of Admissions: an offer of admission, postponement to the Regular Decision pool, or a denial of admission. Admitted students will have until January 1 to pay a non-refundable enrollment and security deposit. Early Action: Candidates must submit all of the application credentials (Common Application, supplement, school report, transcript, recommendations, standardized test results, and application fee) no later than November 15. By December 31, Early Action candidates will receive an admissions decision. Early Action is a non-binding application, meaning applicants can consider other institutions until the May 1 National Candidates Reply Deadline. HOME-SCHOOLED STUDENTS In addition to the standard application requirements, home-schooled students are required to interview with a Wooster admissions officer. Home-schooled students should also submit detailed course descriptions and/or syllabi for academic work completed through the home-schooling program and two letters of recommendation, including one from a person who has provided academic instruction to the student and at least one from someone outside the student’s home. DEFERRED ADMISSION For a variety of reasons, some students decide to delay their plans to attend Wooster for one year after their secondary school graduation. In such instances it is recommended that these students file their application papers during their senior year in order to insure an admission decision at that time. Should the student decide to defer admission, a deferment until the following year must be requested in writing to the Vice President for Enrollment no later than May 1. To secure a place in the class, the enrollment and security deposit of $350 must be submitted at the time the student requests to be deferred. Admission 256 INTERNATIONAL STUDENT ADMISSION The College of Wooster has made a commitment to serving the needs of the international student. These students comprise approximately six percent of Wooster’s student body. International students should begin application procedures early in their senior year. They should clearly indicate their nationality in their initial correspondence with the Office of Admissions. Foreign Diplomas: The College of Wooster recognizes that successful completion of some foreign diplomas represents academic work beyond the level of the American high school diploma. In accordance with the placement recommendations approved by the NAFSA: Association of International Educators and the American Association of Collegiate Registrars and Admissions Officers (AACRAO), students presenting these diplomas may receive up to one year of college credit. The exact number and nature of course credits granted will be determined through conferences with the Registrar and appropriate academic departments after matriculation. International Advanced Placement Credit: Students who successfully pass Advanced-Level examinations with marks of A or B will automatically receive credit for one elective for subjects that are included in the Wooster curriculum. The credits will be recorded on the transcript and included as part of the elective credit required for a Wooster degree. If the student requests that the credit apply toward major, minor, or distribution requirements, a meeting must be scheduled with appropriate department chairpersons for the purpose of determining placement and competency levels. Placement tests may be used to determine levels of competency. Departments will determine Wooster equivalent courses for credits that are granted for acceptable proficiency scores. Courses that are granted for proficiency scores, if repeated, count only once toward the minimum 32 course credits required for graduation. When necessary, departmental chairpersons will determine which courses in the Wooster curriculum will be entered on the transcript as applicable toward major, minor, or distribution credits. Students who successfully pass the International Baccalaureate Higher-Level Examinations with grades of 6 or 7 will receive one course credit toward graduation in the subjects included in the Wooster curriculum. Students submitting the Inter - national Baccalaureate examination results are subject to the same procedures that govern granting of credit for A-Level results. International Student Transfer Procedures: When possible, students should submit official transfer documents before they arrive on campus. The process of evaluating documents for transfer credit should begin with the Registrar. Where there is doubt about the accreditation status of an institution granting a particular credential, the Registrar will consult with the Coordinator of International Recruitment and assist faculty in making the evaluation. Financial Aid: Financial assistance for American students living overseas is determined on the basis of the results of the Free Application for Federal Student Aid (FAFSA), just as it is for American students living in the United States. The College of Wooster has limited funds for international students and is able to offer students only partial financial assistance. International candidates must be able to contribute at least 50% of their annual expenses, not including travel, while studying at The College of Wooster. The College offers a few scholarships that exceed 50% of expenses and awards them based upon academic achievement and financial need. According to United States Immigration law, non-U.S. citizens who are not permanent residents must submit a Certificate of Finances whether or not they are applying for financial aid. English Language Proficiency: All foreign candidates must prove competency in the English language. Students may prove their proficiency in English by taking either the Test of English as a Foreign Language (TOEFL) or the International English Language Testing System (IELTS). Students must score a minimum of 555 on the Admission 257 paper-based TOEFL, or a 90 on the internet-based TOEFL (IBT) with no sub-score below 15 to be considered for admission. The minimum score on the IELTS is a 7.0 band. The Office of Admissions may waive these requirements for students who are native English speakers or have done their schooling at an English medium school for the past four years. Applicants must contact the Admission Office to ask for such a waiver. No application will be processed or evaluated without official TOEFL or IELST results, or a wavier for the tests. International Students and the Foreign Language Requirement: Inter national students whose primary language is not English may satisfy the College’s foreign language requirement by achieving an appropriate TOEFL score (at least 555 on the written test; 90 on the internet-based test) or by meeting the College requirement in Writing by placement examination or course work. International students whose primary language is English but who are proficient in a second language must demonstrate that proficiency either by taking the College’s language placement exam (in the case of languages taught at Wooster) or providing evidence by examination or other manner to the Dean for Curriculum and Academic Engagement. The same conditions apply for American students who have studied or lived in a non-English speaking environment for an extended period. ADMISSION AS A TRANSFER STUDENT Transfer students are welcome to apply for transfer admission at any time before the end of their sophomore year. Students who wish to apply as a transfer student should submit the Common Application (available from the Office of Admissions or www.commonapp.org), supplement, high school transcript, the application fee, transcript from each college or university attended, ACT and/or SAT I scores, and the College Official’s form. Applicants are required to have test scores and official transcripts of record from each institution at which they have studied sent to the Office of Admissions. Courses completed at another accred ited institution will be accepted if the grade is C or better, if the cumulative GPA is a 2.500 or better, and if the courses are equivalent to those offered at Wooster. See Transfer Credit Policy (below) for additional information on types of transfer credit and evaluation criteria. The College will accept up to a maximum of 16 Wooster course credit equivalents completed elsewhere and transfer students must complete at least 16 course credits at Wooster to graduate, including four course credits for general education requirements (foreign language, studies in cultural difference, religious perspective, learning across the disciplines), and seven course credits in the major, including Senior Independent Study. Because of the emphasis on writing at Wooster, the writing-intensive requirement must be completed in Wooster’s program. Normally the quantitative reasoning requirement will also be completed in Wooster’s program. Exceptions will be approved by the Dean for Curriculum and Academic Engagement for both the writing requirement and the quantitative reasoning requirement. The First-Year Seminar in Critical Inquiry (see Interdepartmental Courses) is a requirement for graduation for transfer students who enter with fewer than seven course credits. Transfer Credit Policy: The College of Wooster recognizes the value of transfer work, advanced placement, and proficiency tests, and will grant a maximum of eight course credits for first-year students who have satisfactorily completed acceptable transfer credits. Students who transfer to Wooster after studying full-time at another institution are classified as transfer students. Upon receipt of the official transcript or credit document, the Registrar will determine, with the assistance of appropriate departments, how the credit will be awarded. Transfer credit appears on the student’s academic record as credit without letter grade, and it is not used in the determination of academic grade point average. Transfer work that has no established Wooster equivalent must be approved by an Admission 258 appropriate chairperson before credit is granted. Transfer credit approval forms are available in the Office of the Registrar. Credit will not be granted for transfer courses or proficiency scores that are submitted for subjects that appear to be equivalent. Credits granted for transfer work, if repeated, count only once toward the minimum 32 course credits required for graduation. All transfer course work should be submitted for earned credit within one semester of enrollment at Wooster. Beyond this semester, re-testing or other means of certification may be required. Wooster does not grant credit for online, distance learning courses, and credit based on performance on the College Level Examination Program (CLEP). Nor is credit granted for participation in programs sponsored by the National Outdoor Leadership Schools (NOLS) or Semester at Sea programs. Transfer Credit and Graduation Requirements: Transfer credit submitted by a first-year student may apply to a maximum of four general education requirements. The First-Year Seminar in Critical Inquiry, the writing-intensive and the quantitative reasoning requirements must be completed at Wooster. An exception to the residence requirement on quantitative reasoning is made for students receiving credit for scores on the AP Calculus and AP Statistics tests. Transfer credit for any of the following types may meet requirements in the major with the stipulation that seven course credits in the major must be completed at Wooster. Departments reserve the right to determine how transfer credit is equated to equivalent courses at Wooster. Placement tests may be used to determine levels of competency for any of the following types of transfer credit: College-level courses taken while a high school student: Credits earned by enrolling in college courses while pursuing the high school diploma are usually acceptable toward a degree at Wooster. Credit earned for college-level courses that are taught by college instructors in the high school or dual credit program will not be accepted at Wooster. Only college classes taught on a college campus with other college students earn credit as long as the grade is C or higher. British Advanced-Level Examinations: Students who complete the British Advanced-Level Examinations with marks of A or B will receive one elective credit for each subject that is included in the Wooster curriculum. Credit will not be granted for advanced subsidiary and ordinary level scores. International Baccalaureate (IB): Students who submit scores of 6 or 7 for the International Baccalaureate Higher-Level Examinations will receive one course credit toward graduation in the subjects included in the Wooster curriculum. Caribbean Advanced Proficiency Examinations (CAPE): Students who pass the Caribbean Advanced Proficiency Examinations with scores of I, II, and III for each passed Unit will automatically receive credit for one Wooster course for subjects that are included in the Wooster curriculum. Summer School Credits: Students who attend summer school in other accredited institutions should review their curricular needs with academic advisers prior to attending summer school. All summer school transfer credit must receive prior approval by chairs of appropriate departments and the Registrar. Advanced Placement Program of the College Entrance Examination Board: Wooster participates in the Advanced Placement Program (AP) sponsored by the College Entrance Examination Board (CEEB). Students with scores of 4 or 5 on the Advanced Placement test may receive academic credit for their scores. On some tests a score of 3 will be granted credit. Information for requesting official AP grade reports is found at the following address on the College Board AP website: www.collegeboard.com/ ap/students/index.html. Wooster’s CEEB code is 1134. Admission 259 The following table shows the AP test, the required score for credit, and how credit may count when applied toward Wooster General Education (Gen Ed) and Learning Across the Disciplines (LAD) requirements. Test Score Credit Wooster Equivalents Gen Ed & LAD Art History 4,5 1 ARTD 12000 AH Art Design 2-D 4,5 1 placement-see Chair AH Art Design 3-D 4,5 1 placement-see Chair AH Art Drawing 4,5 1 placement-see Chair AH Biology 4,5 1 placement-see Chair MNS Chemistry 4,5 1 Elective or placement-see Chair MNS 3 0 placement-see Chair Computer Science A Exam 3,4,5 1 CSCI 15100 MNS AB Exam 3 1 CSCI 15100 MNS 4,5 2 CSCI 15100, 15200 MNS Economics Micro 4,5 1 placement-see Chair HSS Macro 4,5 1 placement-see Chair HSS English Language 4,5 1 Elective, waives IDPT 11000 English Literature 4,5 1 Elective, waives IDPT 11000 Environmental Science 4,5 1 GEOL Introductory course MNS ENVS Elective science course MNS French Language 4,5 1 FREN Elective, major, minor French Literature 4,5 1 FREN Elective, major, minor German Language 4,5 1 placement-see Chair History (max = 3) United States 4 1 HIST 11000 HSS 5 2 HIST 11000, 11100 HSS European 4 1 HIST 10600 HSS 5 2 HIST 10600, 10700 HSS World 4 1 HIST Elective, major, minor HSS 5 2 HIST Elective, major, minor HSS Human Geography 4,5 1 ANTH Elective HSS Latin - Vergil 4,5 1 CLST Elective Latin Literature 4,5 1 CLST Elective Vergil & Latin Literature 4,5 2 1 in CLST major or minor-see Chair Mathematics Calculus AB 3,4,5 1 MATH 11100 Q, MNS Calculus BC 3 1 MATH 11100 Q, MNS 4,5 2 MATH 11100, 11200 Q, MNS Admission 260 Calculus BC 1,2 AB Subscore 3,4,5 1 MATH 11100 Q, MNS Statistics 3,4,5 1 MATH 10200 Q, MNS Music Theory 4,5 1 MUSC Elective; AH placement-see Chair Physics C: Mechanics 4,5 1 placement-see Chair Q, MNS C: Electr. & Magnetism 4,5 1 placement-see Chair Q, MNS Physics B 4,5 1 placement-see Chair Q, MNS Political Science United States 4,5 1 PSCI 11000 HSS Comparative 4,5 1 PSCI 14000 HSS Psychology 4,5 1 PSYC 10000 HSS Spanish Language 4,5 1 SPAN Elective, major, minor Literature 4,5 1 SPAN Elective, major, minor APPEAL OF AN ADMISSIONS DECISION Any applicant who is denied admission may appeal that decision by filing a written request for review with the Vice President for Enrollment within thirty days of the decision. A review of the original credentials and any additional supporting information the applicant wishes to submit will be made, and the applicant will be informed of the results of the review within thirty days of receipt of the request for review. THE IMPLICATIONS OF ADMISSION AND REGISTRATION The community on the College campus consists of several constituencies: the students, faculty, administration, and staff employees. Of all of these, the student spends the briefest time on campus. It is helpful, therefore, to have some clarification of the nature of the relationship between the student and the College. It is understood that in applying for admission to The College of Wooster, each prospective student thereby requests the privilege of pursuing an education here in the type of academic program and social atmosphere offered by the College. By accepting a student for admission, the College agrees that the student should attend for that purpose. This is a contractual relationship between the student and the Board of Trustees. It is the policy of the College to admit as students only those for whom graduation is a reasonable expectation and who are expected to contribute positively to the College community. However, admission and registration constitute a commitment by the College only for the term for which registration is accepted. It should be emphasized that students are on the campus because they meet qualifications which indicate that there is every expectation that they will graduate. Over the years this expectation has been achieved by a significantly high percentage of students. Realistically, it is also true that for a wide variety of reasons, some students do not continue at Wooster until they are graduated. The terms under which progress toward a degree may be interrupted should be clear: 1. The student may withdraw from the College at any time for personal reasons. If withdrawal occurs during a semester, a pro rata rebate may be made in accordance with the policy outlined in the section on Expenses. If a student with- Admission 261 draws from the College without completing the full withdrawal process, he or she will forfeit the enrollment deposit. 2. It should be noted that the commitment of the College in accepting a student’s registration is for one semester only. The College may refuse subsequent registration on the basis of (a) the student’s failure to make significant progress in course work in a satisfactory manner which continues to lead to the expectation that the student will achieve graduation (for further details see Academic Policies – Academic Standing, Withdrawal, and Readmission); (b) residency may be terminated for health reasons, which in the determination of the College physician or a member of the College’s professional counseling staff are sufficient to indicate that the student should not be on campus; (c) registration for a subsequent term may be denied by the Provost upon the recommendation of faculty members or deans for sufficient reasons. 3. Students may be asked by the Provost to terminate registration for financial reasons upon the recommendation of the Vice President for Finance and Business and Treasurer after consultation with the Dean for Curriculum and Academic Engagement and the Dean of Students. 4. Students may be asked to terminate their enrollment at any time for disciplinary reasons. It should be noted that students may participate as members of agencies which may recommend suspension or dismissal from the College. There is a Judicial System which adjudicates violations of the Code of Academic Integrity and the Code of Social Responsibility. These decisions may also be made by the Deans. It is assumed that entering students and those reregistering are familiar with the various agencies which make decisions involving their stay at The College of Wooster. 5. A student may be suspended or dismissed at any time from The College of Wooster for reasons which the College deems sufficient. During the course of the semester, each student must demonstrate a good faith effort to attend class and participate in his or her own education. Failure to attend class, disruptive or threatening behavior, and other acts which undermine the educational process or pose a direct threat to the health and safety of self or other members of the campus community can result in dismissal from the College. “Disruptive behavior” is behavior which, in the judgment of the faculty or administrative staff, (i) impedes other students’ opportunity to learn, (ii) directly and significantly interferes with the mission of the College, and/or (iii) violates the Wooster Ethic, Code of Academic Integrity, and/or Code of Social Respon - sibility. Such action may be administered by the Dean of Students, Dean for Curriculum and Academic Engagement, or their designee. Academic and financial ramifications of not completing a semester, as detailed in the Catalogue, will apply in such cases. 6. In any of these matters relating to the termination of registration, the student may appeal the decision to the President of the College, whose decision shall be final. Certain other provisions of the student’s relationship to the campus community that may be unique to The College of Wooster are noted: 1. Although the provision has rarely been applied, it should be noted that if a student who is enrolled in any off-campus program and while in residence at some other place is asked to withdraw from that program by those in charge there, application for readmission to The College of Wooster is required. 2. It should be noted also that the trustees reserve the right to determine the regulations concerning residency in the residence halls and other facets of the social life of the campus, though the administration of these regulations is del- Expenses 262 egated to various student, faculty, and administrative agencies, primarily the Campus Council, all of which cooperate in their achievement. 3. The College reserves the right to enter student rooms at any time, with or without notice, for purposes of inspection, maintenance, repair, and investigation of violations of College rules or regulations. Students are required to enter into a room and dining service agreement which involves obligations as to payments and adherence to regulations. Exceptions to these contracts are made only with the knowledge and consent of the Deans. The College of Wooster reserves the right to inform parents of any violation of the College’s alcohol policies. Causes for parental notification include, but are not limited to, excessive intoxication, alcohol poisoning, and receiving more than three alcohol violation notifications within one academic year. The Scot’s Key is the students’ handbook that sets forth regulations applying to campus life, and it is part of the student-trustee contract, as is this Catalogue. However, the College Catalogue is the official document of academic requirements and regulations. The student who chooses to attend Wooster indicates by being present and by the signature on the application form, acceptance of personal responsibilities under the Code of Academic Integrity and the Code of Social Responsibility and agrees to abide by and conform to the rules and regulations of The College of Wooster and the obligations imposed by the Codes. The enrollment and security deposit of $350 is payable on or before May 1 (with the exception of Early Decision candidates), and will be held until graduation or withdrawal from the College in accordance with the policy outlined in the section on Expenses. For additional information on the admissions process, please contact: Office of Admissions The College of Wooster Wooster, Ohio 44691-2363 1-800-877-9905 E-mail: admissions@wooster.edu www.wooster.edu EXPENSES SUMMARY OF EXPENSES FOR THE YEAR 2011-2012 (Fall and Spring Semesters) Comprehensive Fee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ 47,600 Tuition and Fees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ 38,290 Room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ 4,275 Meals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .$ 5,035 The Comprehensive Fee includes tuition, room (double occupancy), and meals. Additional fees may be assessed to students with course overloads. A detailed list of semester fees may be obtained from the Business Office. Books, supplies, and other incidental and personal expenses are not included in the comprehensive fee and are estimated to be approximately $1,600 per year. With respect to private music lessons, a fee of $480 is charged for fourteen weekly one-half Expenses 263 hour lessons per semester, regardless of whether the lessons are taken for credit or audit. This fee is waived, however, for: (a) lessons in the primary instrument or voice required of declared music majors in all music degree programs, and (b) lessons required of Music Performance Scholarship winners regardless of class year or major. Students participating in off-campus study programs will be assessed an administrative charge of $476. The tuition and other fees for students participating in endorsed off-campus study programs will be equal to the relative components of the Comprehensive Fee, unless the actual program fees are greater. The Comprehensive Fee includes out-patient and in-patient care in the Longbrake Student Wellness Center, the College student health facility. Provided services are described in the section Student Services – Student Wellness Center. Enrollment in the Student Accident and Sickness Medical Plan is required if the student has no private health insurance coverage. Students are automatically charged a $350 fee for the Medical Plan on the fall semester bill; this charge, which provides Medical Plan coverage from mid-August to mid-May, will be cancelled if a properly completed waiver form is received by the Business Office by the fall semester payment due date. A brochure describing the Student Accident and Sickness Medical Plan, together with a waiver form, are mailed to all students with the July invoice. Students are admitted free of charge to most College athletic contests. Full-time students are permitted to audit one course without charge in any semester. In the case of majors in the Music Department, this course could be a regular course carrying 1.000 credit or a combination of partial credit courses adding up to 1.000 credit, with the exception that a student may not audit any more than one halfhour applied lesson in any given semester. The Comprehensive Fee may be reduced for a course-load reduction finalized during the first two weeks of a semester. No refunds will be made for a course load reduction finalized after this period. Contact the Business Office for specific details. Please note that all rates shown are for the academic year 2011-2012. The Board of Trustees reserves the right to make changes in the fees and in other charges at any time. An enrollment and security deposit of $350 is required of all students. This deposit must be paid prior to matriculation. BILLING AND PAYMENT PROCEDURES An invoice for the fall semester will be mailed by mid-July. Payment is due in full by August 10 unless enrollment in the Monthly Payment Plan (administered by Tuition Management Systems (TMS)) for some or all of the entire year’s expenses is completed prior to the August 10 due date. For families who do not enroll in the Monthly Payment Plan, full payment of the spring semester fees, to be billed midDecember, is due by January 10. Accounts which do not satisfy these payment requirements are assessed a 5% late payment fee, subject to a maximum fee of $300, as of the close of business on each semester’s payment due date. In addition, access to dining halls is not permitted, and registration for classes cannot be approved until the student account balance is paid in full and/or the student is properly enrolled in the Monthly Payment Plan option (including the remittance of all necessary back payments for late enrollment). Students who have not paid their account in full by the first day of classes will have their course registration cancelled, and a $200 re-registration fee may be assessed to students who complete their payment requirements after the first day of classes. Students will not be permitted to participate in pre-reg- Expenses 264 istration or housing selection, or receive transcripts of grades until all student account balances have been paid in full. The Monthly Payment Plan allows families to pay some or all of the entire academic year fees in ten interest-free monthly installments beginning June 1. One-half of the Monthly Payment Plan enrollment amount is applied as a credit toward each semester’s fees. To enroll in this plan, total academic year expenses, less direct payments, and applicable financial aid grants and loans, must be estimated and noted on a Tuition Management Systems Monthly Payment Plan application form; such estimates may be subsequently revised by contacting TMS, to avoid the assessment of carrying charges and/or other penalties. Although applications will be accepted by TMS until the fall semester payment due date, those received after June 1 must include the full payment of any missed monthly installments. A $70 non-refundable application fee applies for applications received by July 31; the fee increases to $125 for applications received by TMS after July 31 (a portion of the application fee is forwarded to the College to help offset some of the College’s administrative costs associated with this plan). Families may contact TMS at 1-800-356-8329 or online at www.afford.com/Wooster for applications or further details. WITHDRAWAL Students who are not returning to the College for the subsequent semester are required to contact the Dean of Students Office and make an appointment to meet with the Dean. A student wishing to contemplate future plans or deal with a medical or family situation has the option of requesting a leave of absence for one semester. If a student wishes to take a leave of absence for a semester already in progress, he or she may do so up to the sixth week of the semester without academic penalty. No reduction or remission of fees is allowed by the College for absence, withdrawal, or dismissal unless an official notice of withdrawal is received by the end of the seventh week of a semester, in which case charges will be prorated in accordance with the schedule below. A student wishing to withdraw from the College, for personal or medical reasons, or to transfer to another academic institution, must meet with a Dean of Students staff member to begin the withdrawal process. In either case, stipulations may be attached to the student’s return to campus if deemed appropriate by the Dean. The with - drawal process contains several steps that must be completed by the student prior to his or her departure from campus. This process will be clearly explained during the aforementioned appointment. Schedule of Charges for Withdrawal 1st week of a semester 10% of the full semester’s charge 2nd week of a semester 20% of the full semester’s charge 3rd week of a semester 30% of the full semester’s charge 4th week of a semester 40% of the full semester’s charge 5th week of a semester 50% of the full semester’s charge 6th week of a semester 60% of the full semester’s charge 7th week of a semester 80% of the full semester’s charge Students receiving financial assistance under Title IV of the Higher Education Act of 1965, as amended, who withdraw during the first sixty percent of the semester (measured in calendar days), will be subject to a “Return of Title IV Funds” calculation to determine the portion of their federal student aid that must be returned to the federal government. Students who withdraw after the sixty percent point are consid- Financial Aid 265 ered to have “earned” all of their federal student aid. The College has adopted this same policy with respect to College-funded aid. State aid reductions may also be required in accordance with each state’s regulations. Consequently, no adjustment to a student’s account will be made until all appropriate financial aid reductions are calculated. Please contact the Office of Financial Aid for additional information about possible loss of aid and for examples of typical calculations. An optional Comprehensive Fee refund insurance plan is available for insuring the full refund of fees in the event of a student’s early withdrawal from Wooster because of illness. Information concerning this plan is mailed to all parents prior to the beginning of the academic year. FINANCIAL AID GENERAL INFORMATION The College of Wooster has a long-standing tradition of providing financial assistance to students who might not otherwise be able to afford college and has a broad program of financial aid to assist those who demonstrate a need for such help. Analyzing each aid applicant’s specific circumstances, Wooster will, to the extent permitted by its own financial resources, assist him or her in meeting college costs. The College offers scholarships, grants, loans, and work opportunities to supplement the resources of students and their families. Wooster assumes that education has a high priority in family affairs and that our students will share in implementing this priority. In 2010-2011, financial aid from all sources, totaling over $50 million, was awarded to 1,900 Wooster students. For more detailed information on the range of financial aid opportunities offered by the College and how to apply for financial aid, go to www.wooster.edu/Admissions-and-Financial-Aid/Financial-Aid. NEED-BASED FINANCIAL AID The Office of Financial Aid awards “need-based” aid to help meet demonstrated need. Financial aid awards to students with demonstrated need may include: College of Wooster Endowed Scholarships Scholarships endowed by friends of the College are awarded to students with financial need, according to the stipulations of the donors. They are described in the section of the catalogue entitled Endowed Scholarships. College of Wooster Need-Based Grants Students demonstrating financial need may be offered institutional grant aid in addition to grants from other sources. The major portion of grant-aid received by Wooster students comes from the College itself. Federal Pell Grants Federal Pell Grants are awarded to undergraduate students according to a federal eligibility formula. The Free Application for Federal Student Aid (FAFSA) is the Pell Grant application. Other Federal Programs Wooster receives and awards Federal Supplemental Educational Opportunity Grants Financial Aid 266 to those who qualify on the basis of extreme need, and the College participates in the Federal Work-Study Program (including community service positions). Student Employment Priority for part-time student employment on campus is determined by the Office of Financial Aid. Students seeking part-time jobs should visit the Student Employment Office, located in the Human Resources Center on Wayne Avenue. This office maintains a listing of available work opportunities and coordinates employment on campus. Jobs are usually available in the library and departmental and administrative offices. Loans Several federal and private loan programs enable students and their parents to borrow money for educational expenses on favorable terms. TO APPLY FOR NEED-BASED FINANCIAL AID Applicants for financial aid should complete both the Free Application for Federal Student Aid (FAFSA) and the supplementary institutional aid application as early as possible after January 1 each year. Forms may be obtained online, from high school guidance offices, and from the Office of Financial Aid. We recommend that pros - pective students file them by February 15 and continuing students by mid-April. International applicants file a special form, available from the Office of Admissions. Prospective students will receive notification of their awards shortly after their admission to the College. Review of continuing students’ applications begins in May. Financial assistance is awarded for one year at a time (typically for a maximum of eight semesters) and must be applied for each year. All requests to reconsider financial aid should be made in writing to the Office of Financial Aid. The policy limiting financial aid to eight semesters may be waived for students completing teacher certification requirements in a ninth semester on campus, for participants in a Wooster off-campus study program outside the regular semesters which has received special grant funding apart from the College operating budget, upon special appeal to the Dean for Curriculum and Academic Engagement. Assistance received at any time from sources other than The College of Wooster must be reported to the Office of Financial Aid as part of a student’s financial resources. As a member of National Collegiate Athletic Association (NCAA) Division III, Wooster does not grant athletic scholarships. Financial aid is available to student athletes on the basis of financial need and academic achievement. The amount of aid awarded is not related to athletic performance or degree of participation. Students who accept a College of Wooster scholarship or grant agree to room and board in College facilities, unless they live at home or have special permission from the Dean of Students to live or board elsewhere. The College reserves the right to revoke any grant or scholarship in the case of a student who violates the rules of the College. For further information please contact: Office of Financial Aid The College of Wooster Wooster, Ohio 44691-2363 Telephone: 330-263-2317 Toll free: 1-800-877-3688 FAX: 330-263-2634 E-Mail: financialaid@wooster.edu Internet: www.wooster.edu/Admissions-and-Financial-Aid/Financial-Aid Financial Aid 267 PRESBYTERIAN CHURCH GRANTS National Presbyterian College Scholarships To be eligible, a student must be a high school senior planning to enter one of the colleges related to the Presbyterian Church (USA) and must be a communicant member of the Presbyterian Church (USA). Awards are made by the national office of the Presbyterian Church (USA). The criteria for the award include academic achievement, as evidenced by the student’s secondary school record; academic aptitude, as determined by the Scholastic Aptitude Test (SAT); personal qualities—leadership in church, school and community; promise of usefulness; character and personality. Scholarship amounts range up to $1,400 where need is demonstrated. More information is available from local church offices or from: Presbyterian Church (USA), Office of Financial Aid for Studies, 100 Witherspoon, Room M052, Louisville, KY 40202- 1396. [www.pcusa.org/financialaid/program_finder.htm] MERIT SCHOLARSHIPS AWARDED TO FIRST-YEAR STUDENTS Merit scholarships are administered by the Office of Admissions and awarded through the Office of Financial Aid. They can be applied only to tuition charges, either at the College or an approved off-campus program. Merit scholarships are renewable for up to four years of study at Wooster and subject to review by the Committee on Academic Standards. Typically a student may hold only one academic merit scholarship from the College (performing arts scholarships can be awarded in addition to an academic merit scholarship), and may not receive more than $30,000 in merit awards. Scholarship amounts range from $2,000 to two-thirds tuition. The following information regarding scholarships is relevant for the 2011-2012 application cycle. Specific details such as scholarship application deadlines, dollar amounts, and minimum criteria are subject to change annually. For up-to-date information, including scholarship deadlines, please contact the Office of Admissions at 1-800-877-9905 or www.wooster.edu/Admissions/scholarships. College Scholar Awards College Scholar Awards are awarded to exceptionally promising students who participate in a scholarship competition. To compete for the College Scholar award students must submit a completed application for admission and an essay from one of the College Scholar Award questions found on our website by January 15. Clarence Beecher Allen Scholarships Honors the first African American graduate of the College, a member of the class of 1892. Awarded to entering African-American students with a demonstrated record of academic achievement and promise of continued success in college. Students who would like to be considered for Clarence Beecher Allen Scholarships must submit their application for admission and scholarship essay, postmarked by January 15. National Merit Scholarships Wooster participates in the National Merit Scholarship Program. Students named National Merit Finalists by the National Merit Scholarship Corporation are eligible for awards of $2,000. Detailed information regarding application procedures and selection criteria is available from secondary school counselors. Performing Arts Scholarships Music Scholarships Music scholarships are awarded on the basis of academic achievement and an audition with the Music Department. These scholarships are renewable for four years based on the recommendation of the Music Department. Students must submit their completed application for admission along with the music scholarship application, postmarked by January 15. Honors and Prizes 268 Theatre and Dance Scholarships Students with demonstrated interest and experience in theatre and/or dance may audition for scholarships in performance and technical areas. These scholarships are awarded based upon academic achievement and the audition. Awards are renewed on the basis of continued participation in College theatre and dance productions. Students must submit their completed application for admission along with the theater/dance scholarship application, postmarked by January 15. Scottish Arts Scholarships Pipers, dancers, and drummers are eligible to audition for these scholarships which are awarded on the basis of academic achievement and an audition. Awards are renewed on the basis of participation in Scot Band activities. Students must submit their completed application for admission along with the Scottish arts application, postmarked by January 15. Covenant Scholarships Wooster applicants who are members of the Presbyterian Church (USA) are eligible for this scholarship. Awards are made on the basis of a recommendation by the student’s minister or youth minister and academic achievement. Students must submit their completed application for admission along with letter of recommendation, postmarked by January 15. Dean’s Scholarships Scholarships are awarded based on overall academic achievement, recommendations, writing ability, and extracurricular activities. Alice Powers Scholarships Students from Trumbull and Mahoning Counties in Ohio may be eligible for scholarships of $5,000 per year, which are awarded on the basis of academic achievement and extracurricular activities. Byron Morris Awards Scholarships honoring Wooster’s former long-time Director of Admissions are awarded based on an applicant’s demonstration of significant community service activities and/or leadership within an academic or co-curricular setting. Consid - eration is based upon completed application for admission. *All merit scholarships have a postmarked deadline of February 15 unless otherwise noted. HONORS AND PRIZES ACADEMIC HONORS The Dean’s List includes students meeting the following criteria during a sem - ester: enrollment for at least four credits in letter-graded courses, a semester GPA of 3.650 or higher, and no final grade of I (Incomplete) or NC (No Credit). Students who demonstrate satisfactory progress in I.S. 451 or completion of I.S. 452 are eligible for the Dean’s List with three courses that are letter-graded. Students enrolled in a course other than an internship that is required to be graded by policy solely on an S/NC basis are eligible for the Dean’s List with three courses that are letter-graded, or two letter-graded courses and satisfactory progress in I.S. 451 or completion of I.S. 452. Honors and Prizes 269 Departmental Honors are awarded at graduation to students who meet the following standards: (1) a grade of “H” on the Senior I.S. Thesis or unanimous vote of the department; (2) a major GPA of 3.500 for all courses taken in the major department even if a specific course is not counted toward the major; (3) a cumulative GPA of 3.200 for four years at Wooster. Latin Honors, first awarded in 1998, are awarded at graduation based on overall grade point average in Wooster-graded courses: summa cum laude for 3.900 to 4.000; magna cum laude for 3.750 to 3.899; and cum laude for 3.500 to 3.749. To graduate summa cum laude, a student must receive a grade of “H” on the Senior I.S. Thesis. Latin Honors are not a substitute for Departmental Honors. HONOR SOCIETIES Phi Beta Kappa, the oldest national society for the recognition of high scholarship, has a chapter, the Kappa of Ohio, at Wooster. The student membership is made up of those seniors who are first in academic rank, a few being elected at the beginning of the senior year on junior standing, and others at the end of the year. Other national honorary societies that have chapters at Wooster are Alpha Kappa Delta (Sociology); Beta Beta Beta (Biology); Delta Phi Alpha (German); Eta Sigma Phi (Classics); Lambda Alpha (Anthropology); Lambda Pi Eta (Communication); Omicron Delta Epsilon (Economics); Phi Alpha Theta (History); Phi Sigma Iota (Foreign Languages); Phi Sigma Tau (Philosophy); Pi Kappa Lambda (Music); Pi Sigma Alpha (Political Science); Psi Chi (Psychology); Sigma Delta Pi (Spanish). PRIZES The Vonna Hicks Adrian Poetry Prize was established in 1988 by Arthur Adrian to honor his wife’s memory at Wooster. Mrs. Adrian, a member of the class of 1928, was a poet, and her work, A Gaggle of Verses, was published posthumously. Two awards will be made each year, one for an outstanding poem and one for an original critical analysis of a poem or poems. The Mary Sanborn Allen Prize is given in memory of Mary Sanborn Allen, class of 1905, and is awarded to a student majoring in a foreign language who has benefited most from study outside the United States. The J. Arthur Baird Prize Fund honors the memory of J. Arthur Baird, Synod Professor of Religious Studies and a member of Wooster’s faculty from 1954 until his retirement in 1986. This prize is awarded annually to the student who, in the estimation of the Department of Religious Studies, has demonstrated the greatest aptitude in the area of New Testament studies. The Willis C. Behoteguy Prize in French was established in 1970 in memory of Willis C. Behoteguy, a graduate of the class of 1912 and a trustee of the College. It is awarded annually to that major student who has the highest standing in French at the end of the junior year. The William Z. Bennett Prize in Chemistry, established in 1924, is given at graduation to the student who has the highest standing in chemistry. The Robert G. Bone History Prize was established in 2001 and is awarded to that person having completed the junior year with a major in history who best exemplifies the qualities of Robert G. Bone, ’28: enthusiasm for learning, unbridled curiosity about life, and unbounded kindness toward others. The Robert James Brown Memorial Peace Prize was established in 1986 by Dr. and Mrs. Lowell Brown in memory of Dr. Brown’s brother, a Wooster student who served as a Paratrooper in World War II and gave his life saving a wounded fellow soldier. It is awarded each year to the student who has been most effective in working to promote world peace and human understanding. The David L. Carpenter Pre-Law Prize was established in 1999 by the Figgie Family Foundation of Cleveland. The prize is awarded to a senior at Commencement. The recipient must be accepted by an accredited law school and must exhibit academic excellence, leadership ability, and values that will serve the profession with competence and integrity. This prize honors David L. Carpenter, class of 1965, who himself demonstrated these qualities throughout his career as an attorney. The Vivien Chan Prize in Interdisciplinary Sciences was established in 2007 by Vivien W. Chan, a chemistry graduate of the class of 1989. The prize is awarded each year to a student who has demonstrated academic excellence in the sciences and has an interest in pursuing an advanced degree in interdisciplinary sciences. First preference should be given to students pursuing biochemistry, computational biology, computational chemistry, medicine or bio-informatics. The William Wallace Chappell-Elizabeth Dalton Memorial Prize is awarded each year to the Section President who has exhibited the outstanding characteristics of scholarship, leadership, fraternity, and integrity as exemplified by both of these students. The John W. Chittum Prize in Chemistry, established in 1969, honors Dr. Chittum, who taught in the Department of Chemistry for forty years; during the last nine years, he served as Chairman. The prize is awarded in recognition of a student’s outstanding work in organic chemistry and potential as a chemist. The Thomas D. Clareson Prize in English was established in 1995 by Alice Clareson, with additional contributions from former students, in memory of Professor Thomas D. Clareson who taught in the Department of English from 1955 until 1993. The prize is awarded each year to the junior who, in the judgment of the Department of English, has written the best junior Independent Study thesis. The Arthur H. Compton Prize in Physics honors Dr. Compton, who received the Nobel Prize in Physics in 1927. This prize was established in 1928 by members of the class of 1913 and is awarded to the senior physics major attaining the highest standing in that subject. The Elias Compton Freshman Prize, established in 1926, honors the first Dean of The College of Wooster and recognizes academic excellence in the first-year class. The prize is awarded to the student who has achieved the highest standing in scholarship during the first year. The Frank Hewitt Cowles Memorial Prize in Classics honors Dr. Cowles, who was Professor of Latin from 1926 through 1947. The prize recognizes the senior student who has performed exceptional work in the study of the Greek and Latin languages. The William C. Craig Theatre Prize honors the memory of William C. Craig, longtime Chairperson of the Department of Speech and Director of the Little Theatre. It is awarded annually to the graduating senior who, in the judgment of the members of the Department of Theatre, has made the greatest contribution to all areas of the Little Theatre program. The Andrew Dearborn Cronin Emerging Leader Award was established in 1994 and endowed with a gift from Edmund B. and Leslie Cronin in 2006. This prize is awarded annually to a member of the sophomore class who has exhibited emerging leadership in a campus organization and the campus community as a whole. To encourage emerging leaders to fulfill their potential and better serve the community, the recipient of this award will direct the College to distribute income from this fund Honors and Prizes 270 271 Honors and Prizes to College-sponsored community service and volunteer programs that are the most meaningful to them. The Karen Diane Cross Memorial Award, established in 1984, is presented at graduation to a female chemistry or biology major who plans to pursue graduate work in medical or biochemical research and whose Senior Independent Study Project was noteworthy and creative. Special consideration is given to individuals whose years at the College were characterized by interest in other people and a determination to meet personal goals. The Joseph Albertus Culler Prize in Physics, established in 1942, recognizes excellence in the field of physics. The prize is awarded to the first- or second-year student who has attained the highest rank in general physics. The Cummings-Rumbaugh Prizes honor the memory of Mildred Rumbaugh Cummings and Clarence W. Cummings. Mr. Cummings was a member of the class of 1912. Three prizes are provided through a bequest: The Cummings-Rumbaugh History Prize, The Cummings-Rumbaugh Speech and Dramatics Prize, and The Cummings-Rumbaugh Government Prize. These prizes are awarded to seniors with high academic standing. The James Kendall Cunningham Memorial Prize was established in 1935 and is awarded to the pre-medical student who, at the end of the junior year, is adjudged to be the most likely to succeed in the medical profession. The Raymond R. Day Prize in Urban Studies, established in 1983, honors the founder and director of the Urban Quarter who was a cornerstone of the Urban Studies Program for fifteen years. The prize is awarded annually to the senior Urban Studies major who is adjudged to possess those scholarly and personality traits indicative of superior leadership potential in the field of Urban Studies. The Roland H. del Mar Prize in Spanish, established in 1973, is given annually at Commencement to the graduating senior with highest achievement as a major in Latin-American studies, conducted in the Spanish or Portuguese language. The Donaldson Prizes were created by the Department of English with the support of the Donaldson Fund. The Donaldson Fund was established in 1984 by Stephen R. Donaldson, a member of the class of 1968, to support creative writing at Wooster. The prizes are awarded annually to students who, in the judgment of the Department of English, have submitted the best piece of publishable fiction, the best personal essay, the best critical essay, and the best creative Independent Study. The Aileen Dunham Prize in History was established in 1965 in honor of Professor Aileen Dunham, Chairperson of the Department of History, 1946-1966. It is awarded annually to the senior major who has attained the highest rank in history. The Waldo H. Dunn Prize in English honors Dr. Dunn, class of 1906, who was Professor of English for twenty-seven years. The prize is awarded to the English major adjudged to have written the most distinguished critical Senior Independent Study Thesis. The John D. Fackler Award (formerly the John D. Fackler Medals) is made each year to the College debater who, in the estimation of the Department of Communication, has accomplished the most effective debating during the year. The Josh Farthing Endowed Prize was established in 2003 by friends and family to honor the memory of Josh Farthing, a member of the class of 1992. The prize is awarded annually to a non-music major who demonstrates a strong interest in music. First preference should be given to a member of the Wooster Chorus. The Foster Prize in Mathematics was established in 2001 with gifts from Walter D. Foster and Richard S. Foster, ’71. Income from the fund is awarded annually to the senior mathematics major who has demonstrated the most improvement in mathematics during his or her college years, as judged by the Mathematics Department faculty. The William A. Galpin Awards, established in 1927, include two first prizes and two second prizes for the two men and the two women in the senior class who may be adjudged outstanding personalities from the point of view of scholarship, social and religious leadership, and athletic ability. These students will also possess qualities which contribute significantly to the College community and to the world in future years. The Mahesh K. Garg Prize in Physics is awarded annually to an upper-class physics major who has displayed interest in and potential for applying physics beyond the classroom. The recipient will have demonstrated the qualities embodied in the saying “it is better to light a candle than to curse the darkness” and is judged to have the scholarly and personality traits for using science to serve society. The Peter H. Gore Prize Fund was established in 2002 by Jane S. Gore in memory of her husband, Peter H. Gore, a member of the class of 1964. The fund is awarded annually to an upper class student interested in international relations. The prize is used to support the student’s research expenses or travel to a professional conference. The Grace Prize in Poetry was established in 1963 in memory of A. Grace Long ’20, by Mrs. Mary Long Shoemaker and William E. Long. It is awarded during National Poetry Week to the student submitting the best poem, approximately thirty to one hundred words in length, not previously published, on the topic of the best aspects of modern living in this country. The Ronald E. Hustwit Prize was established in 2007 by students, colleagues, and friends of Ron Hustwit and is awarded annually to a senior philosophy major who, in the judgment of the Department of Philosophy, has shown great love of both the subject and the practice of philosophy. This prize honors Professor Ronald Hustwit for his life-long commitment to the students at The College of Wooster and for his contributions to the cultivation of philosophical skills, dispositions, and enthusiasm for philosophy among those students. The International Paper Company Foundation Business Economics Prize was endowed in 1985 with funds from the International Paper Company. It is awarded annually at Commencement to the outstanding senior Business Economics major as judged by the Department of Economics faculty. The Ralph L. Kinsey Poetry Award was established in 1983 by Mrs. Kinsey and honors the memory of her husband, Ralph L. Kinsey, a member of the class of 1933. It is awarded annually to the student who, in the judgment of a panel of judges composed of members of the Department of English, submits the best single poem. The Lyman C. Knight, Sr. Prize in Physical Education and Mathematics was established in 1978 and honors Professor Knight’s thirty years of service from 1910 to 1940 as a member of the Department of Mathematics. It is awarded to a sophomore who has demonstrated both outstanding promise in high school and first-year mathematics and has superior physical skills. The Maud Knight Prize in Religion honors Mrs. Knight for her devotion to the needs of others and to Westminster Church. It is awarded to a junior who plans to pursue a vocation in religion and who has excelled in academic studies. The G. Julian Lathrop Memorial Award, established in 1953 by Gayle J. Lathrop, ’32, and his wife, Jane Baughman Lathrop, ’31, in memory of their son Julian, is given 272 Honors and Prizes 273 Honors and Prizes to that graduating senior, who has already been accepted by an accredited medical school and, in the opinion of a faculty committee, is likely to make the greatest contribution in the field of medicine. Aptitude for a medical career, motivation for service, and commitment to the Christian way of life are the most important qualities to be considered. The David A. Leach Memorial Prize in Psychology was established in 1973 in memory of David A. Leach, member of the department from 1966 through 1972. It is awarded annually to a senior psychology major for excellence in experimental psychology. The Delbert G. Lean Prize in Speech, established in 1968, honors Dr. Lean, who was a member of the faculty for thirty-eight years. It is awarded annually to that senior speech major who, in the judgment of the Department of Communication, has contributed most significantly to the Communications program. The Jack Lengyel Courage, Character, and Commitment Endowed Award is presented each year to the senior football player who best exemplifies the attributes of former head coach Jack Lengyel. The award was established in 2009 on behalf of Coach Lengyel by his players, managers, and trainers in appreciation of his years at the College and of his career commitment to intercollegiate athletics. The Stuart J. Ling Jazz Award was established by family and friends in memory of Stuart J. Ling, the Neille O. and Gertrude M. Rowe Professor of Music and Professor of Education (1949-1984), and Emeritus Professor (1984-2008). Dr. Ling was founder and director of The College of Wooster Jazz Band, and also directed the Scot Marching Band and the Scot Symphonic Band. Income from the fund is awarded each year to a graduating senior who has been chosen by the music faculty as outstanding in jazz performance. The Dan F. Lockhart Outstanding Senior Award honors the memory of Dan Forrest Lockhart, class of 1974. It is awarded at Commencement to a senior who has made an outstanding contribution to the life of the College. The award recognizes high academic achievement, participation in extra-curricular activities, and demonstrated leadership in campus affairs. The Alice Hutchison Lytle Biology Award was established in 1972 by Mrs. Lytle, a graduate of the class of 1915. The prize is given annually in the fall to the female student who has received the highest grades in biology courses during her first three years of college work. The Donald R. MacKenzie Prize in Art was established in 1981 in memory of Professor Donald R. MacKenzie, who taught in the Department of Art from 1949 to 1981. The prize, whose purpose is to promote interest in the field of ceramics, sculpture, or Japanese culture, is awarded each year to a junior or senior art student who has excelled in one or more of these areas. The Julia Quinby McCleary Prize was established in 1996 by Mrs. McCleary, a member of the class of 1926 and a direct descendant of Ephraim Quinby, who made the first grant of land on which the College was established. The prize recognizes unusual initiative and achievement and is awarded annually to a sophomore or junior woman who is working to support her College attendance and who has maintained a 3.0 GPA. The Edward McCreight Prize in Dramatics, established in 1939, is awarded to the senior who is judged by the Department of Theatre to have contributed the most in dramatics. The Robert W. McDowell Prize in Geology was established in 1945 by Philip C. and Sarah Wright McDowell, members of the class of 1914, in memory of their son, a member of the class of 1945, who lost his life in World War II. It is awarded to the geology major who has the highest general standing during the junior and senior years. The Manges Athletic Prize, established in 1925 by Monroe Manges, of the class of 1888, is awarded annually at graduation to the member of the senior class most proficient in vigorous physical activity. The Horace N. Mateer Prize in Biology was established in 1926 in honor of Dr. Mateer, a Wooster alumnus who served as the first Chairperson of the Department of Biology from 1886 to 1926. This prize is awarded at graduation to the major student who has the highest standing in biology. The Barbara Ward McGraw Memorial Prize was established in 2006 by her family and friends. Barbara McGraw was a member of the class of 1953 and was certified to teach at both the secondary and elementary levels, spending most of her professional life as a successful high school teacher in the fields of English and speech communication. Income from the fund is awarded annually to a student who is planning a career in teaching secondary education and who, in the judgment of the Department of Education, best exhibits the qualities needed to be an effective educator. The Emerson Miller Memorial Prize in Speech was established in 1960 in his memory by Mrs. Garnett Miller Smith and the family. Mr. Miller was a member of the Department of Speech from 1925 until his death in 1943. The prize is awarded annually at graduation to the senior who is judged by the Department of Communication to have contributed the most to the department’s program during his or her college career. The Frank Miller Prize is awarded annually to a senior judged by the Political Science Department to have performed outstanding work in comparative politics and area studies. The John F. Miller Prize in Philosophy, established in 1913, is given at graduation to the major student who has the highest standing in philosophy. The Charles B. Moke Prize is given in honor of Dr. Charles B. Moke, who retired in 1972 after thirty-six years of teaching in the Geology Department. Funds for the prize were donated by friends and former students of Dr. Moke. The prize consists of a Brunton Compass awarded to the graduating senior who plans to make geology a vocation and who, in the judgment of the geology staff, has shown the greatest improvement during his or her college career. The Parker Myers Memorial Award honors the memory of Parker Myers of the class of 1962 and is awarded at Commencement to a senior who has maintained a high scholastic standing and who has taken a prominent part in a wide range of extracurricular activities. The Tom Neiswander Memorial Award honors the memory of Thomas Neiswander, class of 1952, and is awarded to a member of the senior class who has maintained a high scholastic standing and who has taken a prominent part in extracurricular activities. The Jonas O. Notestein Prize, established in 1923, honors Dr. Notestein, who taught Greek and Latin at Wooster from 1873 to 1928. The prize is awarded to the student who has graduated with the highest scholarship for the whole college course. This prize is awarded only to students who have done all of their college work at Wooster. The George Olson Prize in Art was established in 2000 to honor Professor Olson, an internationally recognized artist who taught at the College from 1963 until his 274 Honors and Prizes 275 Honors and Prizes retirement in 2000. The prize, created by gifts to the College in his honor and by the sale of the department’s collection of student prints, is awarded annually to a senior art major who has excelled in printmaking, painting, or drawing. The John W. Olthouse Prize in French, established in 1963, honors Dr. Olthouse who taught in the Department of French for forty-four years and he served as Chairperson for thirty-five years. The prize is awarded annually to that major student who has the highest standing in French at the end of the senior year. The Daniel and Clarice Parmelee Endowed Prize Fund was created by a gift from Alfred F., ’38, and Betty Hofacker Foster, ’40, to honor Daniel and Clarice Parmelee, members of Wooster’s music faculty from 1915-1960 (Daniel) and 1924- 1960 (Clarice). The prize is presented to a graduating senior for participation and outstanding contribution to The Wooster Symphony. The Lauradell Amstutz Peppard Prize was established in 2008 by the family of Lauradell Amstutz Peppard, a member of the class of 1934. The prize is awarded annually to the female sophomore or junior who has shown the most outstanding promise in piano performance. First preference should be given to a music major. The Theron L. Peterson and Dorothy R. Peterson Award for Outstanding Academic Achievement was established in 2008 in memory of Theron L. Peterson, a member of the class of 1936, and his wife, Dorothy R. Peterson. Income from the fund is awarded annually to a student who, as of the date of consideration, has satisfactorily completed his or her sophomore year and is majoring in biology, chemistry, physics, or mathematics and has earned the recognition as an outstanding scholar. In the event two or more students are considered for this award, the selection committee may take into account the student’s extracurricular activities and his or her role in student leadership. The Phi Beta Kappa Prize, established in 1976, is awarded annually by the Wooster chapter to a student elected to membership on the basis of junior standing who has a broad range of course work, a demonstrated concern for quality of life on campus, and leadership ability. The Pi Kappa Lambda Prize in Music, established in 1946, is awarded to the graduating senior music major (B.A., B.M., or B.M.E.) who has been selected by the faculty committee of the Upsilon chapter for highest academic standing and able performance in the major field. The Eleanor J. Pope Prize was established in 1999 by family and friends to honor the memory of Eleanor J. Pope, class of 1943. This prize is awarded annually to a nontraditional, female student who has a minimum grade point average of 2.5 and who demonstrates leadership ability. The Presser Undergraduate Scholar Award is given annually to an outstanding music major, at or after the end of the student’s junior year. The recipient is chosen by the President of the College and the Chairperson of the Department of Music, who are guided solely by considerations of merit. The Procter & Gamble Economics Prize was endowed in 1986 with funds from the Procter & Gamble Company. It is awarded annually at Commencement to the outstanding senior economics major, as judged by the Economics and Business Economics faculty. The William Byron Ross Memorial Prize in Chemistry was established in 1952 by Mrs. William B. Ross, ’15, in memory of her husband, a member of the class of 1914. It is awarded to the chemistry major who has shown outstanding originality, resourcefulness of thought, and initiative in Independent Study during the junior and senior years. The Netta Strain Scott Prizes in Art, established in 1944, include annual senior and undergraduate prizes. One prize in each category is awarded to the student who has shown outstanding ability in creative studio work, and one prize in each category is awarded to the student who has achieved the highest record in art history. The Maria Sexton Award was established by the Women’s Athletic Association (WAA) in 1969 in honor of Dr. Maria Sexton for her work with the WAA, her dedication to her profession, and her many contributions to girls’ and women’s sports. The award is given annually to the junior or senior woman of the preceding year who has demonstrated qualities of adaptability, conscientiousness, responsibility, and resolution; who has shown prominent interest and participation in WAA sponsored activities (not necessarily on the WAA board); and who has maintained an adequate academic standing during her years at the College. The Sharp Family Prize honors the memory of William G. Sharp, Jr., a member of the class of 1942. Mr. Sharp served as a judge for nineteen years, fourteen of them as Judge of Common Pleas in Wayne County, Ohio. This prize recognizes a student who is majoring in political science and who has shown improvement in his or her grade point average in the junior year. The Sisodia-Williams Prize in Biochemistry was established in 1997 by Dr. Sangram Sisodia, a member of the class of 1977, and honors Dr. Theodore Williams, a professor in the Department of Chemistry. The prize is awarded to an outstanding senior majoring in chemistry or biochemistry who plans to pursue a career in biomedical research. The Whitney E. Stoneburner Memorial Prize in Education, established in 1970, honors the memory of Mr. Stoneburner, who was a professor of education from 1926 to 1955, and is awarded annually to a senior who has prepared for a teaching career and is adjudged to have achieved distinction in the field of professional education. The Swan Prize Fund was established by the Swan family in honor of Alfred W. Swan, ’17, and Eva Castner Swan, ’18. All three of their daughters are Wooster alumnae: Jeanne, class of ’45; Ruth, class of ’47, and Dorothy, class of ’49. The prize is to be awarded annually to a student or students whose work in the field of social ethics has been outstanding. The Edward Taylor Prizes were established in 1876 by A. A. E. Taylor, President from 1873-1883. The prizes are awarded to students who have attained the highest and second highest academic standing during their first year and sophomore year. The James R. Turner Prizes in History and Women’s Studies were established in his memory in 1986. Professor Turner was a member of the History Department from 1969 to 1986. The History Prize is awarded to the student with the most distinguished Junior Independent StudyThesis. The Women’s Studies Prize is awarded to the student completing the most distinguished Senior Independent Study Thesis relating to women and women’s concerns. The Paul DeWitt Twinem Bible Award was established in 1925 by Mrs. Mary Fine Twinem in memory of her husband, Paul D. Twinem, a member of the class of 1915, and is given at graduation to the senior who, in the opinion of the staff of the Department of Religious Studies, has shown the highest degree of excellence in Biblical studies. The Ricardo Valencia Prize for Excellence in the Department of Spanish was established in 1974 and is awarded annually to the major in the Department of Spanish who, in the judgment of the department‘s staff, has done the best work in three of the four areas taught in the department. First preference should be given to a junior and the prize used to purchase materials needed for Senior Independent Study. 276 Honors and Prizes 277 Honors and Prizes The Karl Ver Steeg Prize in Geology and Geography, established in 1958, honors Dr. Ver Steeg, who taught in the Department of Geology from 1923 to 1952. The prize is awarded to the major student who has the highest general standing at the middle of the junior year. The Cary R. Wagner Prize in Chemistry, established in 1966 by Dr. and Mrs. Cary R. Wagner, is awarded annually at the beginning of the senior year to that student who showed during the junior year the greatest aptitude and, in the opinion of a jury, seems most likely to succeed in chemistry. The Elizabeth Sidwell Wagner Prize in Mathematics, established in 1966 by Dr. and Mrs. Cary R. Wagner, is awarded annually at the beginning of the senior year to that student who showed the greatest aptitude during the junior year and, in the opinion of a jury, seems most likely to succeed in mathematics. The Joseph E. Weber Premedical Award was endowed through the generosity of Joseph E. Weber. The award is given at graduation to a senior chemistry major who has been accepted by an M.D. degree-granting institution and who, in the opinion of the Department of Chemistry prehealth adviser(s), will become a compassionate and effective physician. The Miles Q. White Prize, established in 1931 by Paul Q. White of the class of 1910 in memory of his father, is offered annually to that junior biology major attaining the highest standing in the introductory biology curriculum. The Paul Q. White Prize in English was established in 1944 and honors the memory of Mr. White, class of 1910. The prize is awarded annually to that senior major student who, in the opinion of the Chairperson of the Department of English and the staff, has made the best record in English during the college course. The Theodore R. Williams Prize in Music was established in 2005 by a gift from Kenneth E. Shafer, a member of the class of 1975, and Jill Wahlgren Shafer, a member of the class of 1976, who believe that music has an impact on the development of personal character, productive work habits, and a sense of global citizenship. The prize honors the memory of Dr. Theodore (Ted) Williams and recognizes his keen appreciation of music and music performance as well as his devotion to promoting local and amateur performers and his encouragement of students to participate in music as an avocation. The prize is awarded annually to a senior who has contributed most significantly to the Department of Music as a non-music major. The recipient of the prize is selected by a committee recommended by the Music Department faculty in consultation with the Dean for Curriculum and Academic Engagement. The William H. Wilson Prize in Mathematics was established in 1926 in memory of William H. Wilson, of the class of 1889, professor of mathematics in the College from 1900 to 1907. The prize is awarded annually to that member of the senior class who has shown the greatest proficiency in mathematics. ENDOWED RESOURCES ENDOWED CHAIRS The Victor J. Andrew Professorship of Physics was established in 1992 by a gift from his son and daughter-in-law, Ed and Edie Andrew. The Chair honors the memory of Victor Andrew, who graduated from Wooster in 1926 and received an honorary degree from the College in 1949, and whose life and work demonstrated his unflagging commitment to scientific inquiry and his appreciation of the value of higher education. The Aylesworth Professorship of Classical Studies was established in 1904 by a gift from Mrs. Ann E. Aylesworth in memory of her husband, Warren Aylesworth. The Brown Professorship of Chemistry was established in 1886 by a gift from Benjamin S. Brown of Columbus, Ohio, for the endowment of a professorship in the sciences. In 1908 the alumni of the College established the Alumni Professorship of Philosophy, with Elias Compton ’81, Professor and Dean of the College, as the first appointee. In 1937 the name of the professorship was changed to The Compton Professorship of Philosophy, honoring Professor Compton’s forty-five years of service to the College (1883-1928). The endowment was completed during the years 1928 to 1965 by the Compton sons, Karl ’08, Wilson ’11, and Arthur ’13. The Henry J. and Laura H. Copeland Chair of European History was established in 1995 by a gift from Robson and Carolyn Walton of Bentonville, Arkansas. The Chair reflects the Waltons’ esteem for Henry and Lolly Copeland and recognizes their singular contributions to Wooster throughout the eighteen years of Mr. Copeland’s presidency. The Robert Critchfield Chair of English History was established in 1981 by the Board of Trustees from a gift by Robert Critchfield. The Chair is named in honor of Mr. Critchfield in gratitude for his invaluable service as a Trustee, his steadfast commitment to the College and to the values transmitted in its classrooms and on its zplaying fields, and his abiding love for the law and respect for its discipline. The Danforth Professorship of Biology was established in 1960 by the Danforth Foundation of St. Louis, Missouri, as a memorial to William H. and Adda B. Danforth. The Raymond and Carolyn Dix Chair of Spanish was established in 2002 by bequests from Mr. and Mrs. Dix. The Chair recognizes their long interest in Latin America and their conviction that international relations must be founded on knowledge and understanding. Mr. Dix was the Publisher of the Wooster Daily Record from 1953 to 1975, at which time he became the Co-Publisher with his son Victor. Mr. Dix retired in 1985. Mrs. Dix served as a member of the College’s Board of Trustees from 1944 to 1950 and from 1960 to 1985 and then served as an emerita life trustee until 2000. She was Dean of Women from 1966 to 1968 and Vice Chairman of the Board of Trustees from 1974 to 1985. The John Garber Drushal Distinguished Visiting Professorship was established in June 1977 by members of the Board of Trustees in honor of Dr. Drushal, who was President of the College from 1967 to 1977. This Chair permits the College each year to appoint as a visiting member of the faculty an individual whose achievements 278 279 Endowed Resources reflect a high standard of excellence. The appointment may be in any department or program of the College and is usually at a senior rank. At the discretion of the President, the occupant of the Chair may serve as a replacement for a member of the faculty on a research or study leave. The Aileen Dunham Professorship in History honors Aileen Dunham, who taught at Wooster from 1924 to 1966 and who also chaired the Department of History for two decades. The Dunham Professorship, established in 1988 by a bequest from Professor Dunham and by the gifts of Trustees and her students, recognizes her exceptional distinction as a teacher, the affection of generations of students, and her leadership in establishing the Department of History in the first rank among undergraduate departments in the nation. The Frank Halliday Ferris Chair of Philosophy was established in 1964 by friends of Dr. Ferris in honor of his years of service as pastor of the Fairmount Presbyterian Church in Cleveland, Ohio, and as Visiting Professor of Religion at the College from 1952 to 1956. The Michael O. Fisher Professorship of History was founded upon a gift by Mr. and Mrs. Michael O. Fisher in 1915. The Walter D. Foss Lectureship was established in 1993 by William Foss Thompson, an emeritus member of the Board of Trustees, in honor of his grandfather, Walter Foss, a Trustee from 1902 to 1917 and President of The Wooster Brush Company from 1879 to 1938. The endowed position brings to the College an Assistant Professor of exceptional promise to teach in an appropriate department or program. In addition to the salary, the endowment provides support to assist individuals entering the profession to establish themselves as exemplary teachers and scholars early in their careers. The individual receiving the appointment may serve as a replacement for a member of the faculty on leave. The Fox Professorship of Biblical Instruction was created in 1941 by Andrew M. Fox ’89 and his wife, Finette Fox. The Inez K. Gaylord Chair of French Language and Literature was established in 1984 by Edward L. Gaylord and Edith Gaylord Harper of Oklahoma City. The Chair is named in honor of Inez Kinney Gaylord, a 1903 graduate of the College, in recognition of her lifelong interest in Wooster and in the language and culture of France. The Gillespie Visiting Professorship was endowed in 1958 by Miss Mabel Lindsay Gillespie of Pittsburgh, Pennsylvania, in memory of her parents, Anna Randolph Darlington Gillespie and David Lindsay Gillespie. This endowment brings to Wooster each academic year a professor from outside the United States, taking the place of some member of the faculty who is on research or sabbatical leave. In recognition of the long tie with Scotland that The College of Wooster has had from its own founding, the professorship will normally be held by a member of one of the Scottish universities. The Gingrich Professorship of German was established in 1941 by a bequest of Gertrude Gingrich, Professor of German during the years 1893 to 1920 and 1924 to 1935. The Willard A. Hanna Chair in Southeast Asian Studies was established in 2008 through a bequest from Marybelle B. Hanna in honor of her husband, Willard A. Hanna, a member of the class of 1932. Mr. Hanna spent much of his career working in Southeast Asia and the Pacific Rim during his service in the U.S. Navy (1942-1946) and his career with the U.S. Department of State (1946-1954) and the American Universities Field Staff (1954-1976). Endowed Resources 280 The William F. Harn Professorship of Physics was established in 1958 by Miss Florence O. Wilson of Oklahoma City, Oklahoma, in honor of her uncle, William F. Harn, an 1880 graduate of the College and pioneer Oklahoman. The Hoge Professorship of Economics, in honor of the Reverend James C. Hoge, was established in 1886 by the Synod of Ohio. Mr. Hoge, who died in 1864, was an itinerant missionary and the founder of The First Presbyterian Church of Columbus. He was an early and staunch advocate of a synodical college in Ohio. The D. Willis James Foundation of the Presidency was established in 1910 by his son, Arthur Curtiss James. The Johnson Professorship of Mathematics was the gift of W. D. Johnson of Clifton, Ohio, during the second decade of the College and is one of the oldest of the endowed chairs. The Olive Williams Kettering Professorship of Music is a memorial gift (1953) by Charles F. Kettering, former Trustee of the College, his son, Eugene Kettering, and his grandson, Charles F. Kettering II. Mrs. Kettering, who died in 1946, was a former student in the Conservatory of Music and was an accomplished musician. The Lincoln Chair of Religion was established in 1980 by the Board of Trustees from a gift to the College from Mr. and Mrs. J. Howard Morris and is named in honor of Mr. and Mrs. James F. Lincoln of Cleveland, whose gift of love and precept in the home, wisdom in industrial relations, and discernment as to the future made visible the power and influence of Christianity. The Chair is for the study of religion and social ethics. The Horace N. Mateer Professorship of Biology was established in 1963 by a bequest from Dr. William E. Henderson ’91. Dr. Henderson was Professor of Chemistry at The Ohio State University from 1899 to 1940. This endowment honors Dr. Horace N. Mateer, Professor of Biology at the College from 1886-1926. The Mercer Professorship of Religion was established during the early years of the College by Boyd Mercer. The Moore Professorship of Astronomy was endowed in 1899 by the gift of the Reverend Robert B. Moore of Vineland, New Jersey, previously of Toledo, and a Trustee of the College from 1871 to 1874. The Josephine Lincoln Morris Professorship of Black Studies was established in 1999 by the Board of Trustees with a bequest to the College from Mr. and Mrs. J. Howard Morris. The Chair is named in memory of Mrs. Morris, whose deep and abiding concern for the education and advancement of African Americans benefited generations of students at Wooster. The Virginia Myers Professorship in English honors C. Virginia Myers, a Phi Beta Kappa graduate of Wooster in 1929. The Chair, established in 1990 by a bequest from Virginia Myers, recognizes her distinction as a teacher and her devotion to English literature and drama. After graduating from Wooster, Miss Myers earned advanced degrees from Radcliffe College and from Newnham College, the University of Cambridge, England, and for many years she was a member of the faculty at Geneva College for Women in Switzerland (then affiliated with Mount Holyoke College) and at Bowling Green State University. The Lewis M. and Marian Senter Nixon Professorship in the Natural Sciences was established in 2002 through a bequest from Marian Senter Nixon. A native of Canton, Ohio, Mrs. Nixon graduated from Wooster in 1927 with a degree in Latin. She taught Latin in Florida and Ohio and did graduate work in psychology and speech therapy at Syracuse University. During World War II, she was a caseworker Endowed Resources 281 with the Red Cross. She and Lewis were married in 1938, and Mr. Nixon spent 32 years working for the federal government in Washington, D.C. After his retirement in 1972, they moved to Florida where Mr. Nixon died in 1990 and Mrs. Nixon in 2001. This Professorship honors their deep and abiding respect for The College of Wooster. The Marian Senter Nixon Chair in Classical Civilization was established in 2002 and honors Marian Senter Nixon of the class of 1927. At Wooster, Mrs. Nixon was the student of Jonas Notestein, and after graduating she taught Latin in the public schools of Canton, Ohio, and Winter Haven, Florida. Mrs. Nixon developed a unique method of teaching Latin verbs. She and her husband, Lewis Nixon, were the donors of the Senter-Nixon Chair in the Natural Sciences at Wooster, and in giving this Chair in Classical Civilization, Mrs. Nixon intended to sustain the study of classical civilization and languages, Latin and Greek, at the College. The Theron L. Peterson and Dorothy R. Peterson Professorship in Biology was established in 2008 through the Dorothy R. Peterson Trust in memory of Theron L. Peterson, a member of the class of 1936, and his wife, Dorothy R. Peterson. The Peterson Chair in Biology shall be awarded to a tenured faculty member, or, in the case of a visiting appointment, the recipient of the Chair will hold the professorship for the duration of his or her appointment. The Theron L. Peterson and Dorothy R. Peterson Professorship in Chemistry was established in 2008 through the Dorothy R. Peterson Trust in memory of Theron L. Peterson, a member of the class of 1936, and his wife, Dorothy R. Peterson. The Peterson Chair in Chemistry shall be awarded to a tenured faculty member, or, in the case of a visiting appointment, the recipient of the Chair will hold the professorship for the duration of his or her appointment. The Pocock Family Distinguished Visiting Professorship was established in 2001 through a substantial bequest from Arthur F. Pocock ’41, geologist, world traveler, prize-winning author, and entrepreneur. In making this bequest to Wooster, he honored his parents, Eugene and Bess Livenspire Pocock ’11 and ’12, his brother, John William Pocock ’38, and several nieces and nephews who also attended the College. The Pocock Chair permits the College each year to appoint as a visiting member of the faculty a professor at the senior rank who is an outstanding teacher and scholar. At the discretion of the President, the occupant of the Chair may serve as a replacement for a member of the faculty on research or study leave. The Purna, Rao, Raju Chair of East-West Philosophy was established in 1993 by a gift from Dr. and Mrs. P. T. Raju. Widely regarded as one of the world’s foremost comparative philosophers, Professor Raju was a member of Wooster’s faculty from 1962 until his retirement in 1973. A President of the Indian Philosophical Congress, he was the recipient in 1958 of the Order of Merit, “Padma Bhushan,” in recognition of his contributions to philosophy and East-West understanding. The Chair recognizes Professor Raju’s international reputation, his scholarly achievements, and his years of service to Wooster. The Neille O. and Gertrude M. Rowe Professorship of Music, honoring the former head of the Department of Music and his wife, was established by friends and funds from the Laura B. Frick estate. In 1998, their daughter, Evelyn Rowe Tomlinson, added a generous gift to the professorship. Professor Rowe, a Fellow of the American Guild of Organists, directed the Conservatory of Music and served as Memorial Chapel organist from 1914 to 1945 while Mrs. Rowe taught piano, harmony, history, and appreciation of music from 1915 to 1953. The Merton M. Sealts, Jr. Visiting Professorship honors the memory of Merton M. Sealts ’38, H ’74, distinguished scholar and teacher. The Sealts Chair will serve Endowed Resources 282 each year as a replacement for a member of the faculty on research or study leave. The appointment may be made in any department, at the discretion of the President, to a junior faculty member who has demonstrated excellence in teaching and research. The Severance Professorship of Old Testament and The Severance Professor - ship of Missions represent a gift of Louis H. Severance, Cleveland industrialist and benefactor of the College. Mr. Severance served as Chairman of the Board of Trustees of the College from 1901 to 1914. The Ross K. Shoolroy Chair of Natural Resources was established in 1981 by the Board of Trustees from gifts to the College by Ross K. Shoolroy and the Ashland Oil Company. The purpose of the Chair is to provide students with the academic background necessary to enter the field of exploration for petroleum and for other of the Earth’s natural resources. The Chair is named in honor of Mr. Shoolroy to recognize his contributions to the College as a Trustee, to the community of Wooster as a patron of the arts and sciences, and to society as one of the leaders of the petroleum industry. The Lawrence Stanley Chair of Medieval History honors the memory of Lawrence D. Stanley of Columbus, Ohio. An attorney, Mr. Stanley had a lifelong interest in the development of the English common law and representative government. The Chair was established by his daughter, Laura Stanley Gunnels, to recognize Mr. Stanley’s distinguished achievements as a lawyer and civic leader. The Synod Chair of Religion represents a continuing concern for the life of the College by the church that founded it. The accumulation of funds for a professorship of religion has, since 1950, been a project of the Presbyterian Churches of the Synod of the Covenant, the Presbyterian Church, U.S.A. The Juliana Wilson Thompson Lectureship was established in 1993 by William Foss Thompson, a member of the Board of Trustees, in honor of his wife, Julie Thompson. The endowed position brings to the College an Assistant Professor of exceptional promise to teach courses in an appropriate department or program. In addition to the salary, the endowment provides support to assist individuals entering the profession to establish themselves as exemplary teachers and scholars early in their careers. The individual receiving the appointment may serve as a replacement for a member of the faculty on leave. The Mildred Foss Thompson Chair of English Language and Literature honors the memory of Mildred Foss of the class of 1914. A music major at the College, Mrs. Thompson had a lifelong interest in the English language and its literature. The Chair was established in 1986 by her son, William F. Thompson, to recognize the contributions of the Foss family and of Mildred Foss, in particular, to The College of Wooster. The Whitmore-Williams Professorship of Psychology was established in 1998 by a gift from A. Morris and Ruth Whitmore Williams ’62 of Gladwyne, Pennsylvania. Mrs. Williams has been a member of the Board of Trustees since 1994. The James R. Wilson Chair in Business Economics was established in 2001 by a gift from James R. and Linda R. Wilson. Mr. Wilson, a member of the class of 1963 and a Trustee of the College since 1980, was elected Chairman of the Board in 2000. The Robert E. Wilson Professorship in Chemistry was established in 1965 by the friends of the late Robert E. Wilson ’14, who was Chairman of the Board of Trustees from 1953 to 1964; he was a member of the Board for thirty-four years. The James N. Wise Visiting Professorship in Theatre was established in 2002 by a bequest from James N. Wise, a member of the class of 1941. Jim Wise had a brilliant career as a composer for musical theatre and as a teacher. The Chair permits the Endowed Resources 283 College each year to appoint as a visiting member of the faculty an individual whose achievements in theatre, playwriting, musical theatre, or dramaturgy are truly outstanding. At the discretion of the President, the Chair may serve as a replacement for a member of the faculty on research or study leave. ENDOWED FUNDS The Roy W. Adams Endowment Fund was established in 1999 through a bequest from Roy Adams, a member of the class of 1951. Income from the fund supports the acquisition of library materials and other teaching tools used by the Department of Political Science. The Mary E. Armstrong Memorial Book Fund was established by J. Gaylord and Betty Armstrong in memory of their daughter, who was a member of the class of 1974. Mary was a Russian studies major and was fluent in five languages. She received her Library Science degree from Rutgers University in 1978, a year before her death. Income from the fund is used to purchase books for the libraries, with preference given to acquisitions in the humanities. The James R. Baroffio Fund for Geologic Research was established in 1998 by Dr. James R. Baroffio, a member of the class of 1954. Income from the fund is awarded annually to geology majors engaged in Senior Independent Study to help defray expenses for analytical work (e.g., major element, trace element, isotopic, or geo - chronologic studies) required for their I.S. research. Allocation of funds will be determined by the Chair of the Geology Department. The Bell Distinguished Lectureship was endowed in 1999 by Jennie M. Bell and Samuel H. Bell, a member of the class of 1947 and a Federal Judge of the United States District Court for the Northern District of Ohio. The Bell Lectureship was established to engage students, faculty, members of the legal profession, and members of the community in legal issues that have broad implications for society. The Eric H. and Inge P. Boehm Library Fund was established in 2001 with a gift from Dr. Eric H. Boehm, a member of the class of 1940. Dr. Boehm received an honorary Doctor of Letters degree in 1973 in recognition of his accomplishments in the field of historical bibliography. Income from the fund is used to support the College’s libraries. The Jean M. and Malcolm C. Boggs Endowed Library Fund was established in 2001 by their children, in honor of Jean and Mal’s devoted service to the College. Income from the fund is used at the discretion of the Librarian for the purchase of books and library materials associated with Wooster’s academic program. The Roscoe and Dorothea Breneman Library Fund was established in 2001 with proceeds from a gift annuity. Dr. Breneman graduated from Wooster in 1932 and went on to become a physician in the Akron, Ohio area. Mrs. Breneman wished to honor her husband with this permanent endowment that will support the College’s libraries in perpetuity. The Shirley and Donald Buehler Endowment Fund was established by the Board of Trustees in 1988 from a gift by Shirley and Donald Buehler of Wooster. Income from the fund is awarded annually, at the discretion of the President, to support programs which enrich the cultural life of the College and the community. The Carruth Humanities Endowment is an endowment for lectures, performances, commissions, or exhibitions in the humanities commemorating the contri- bution of John R. Carruth, Professor of Music (1952-1972), to The College of Wooster and celebrating the human values he exemplified: the pursuit of intellectual and artistic excellence and a delight in the human imagination and its ability to transcend, unite, and transform the diversity of common experience. The Henry E. Carter Art Fund was endowed in 2007 by Mary Carter in memory of her husband, Henry E. Carter, a member of the College’s Board of Trustees from 1975-1983. Income from the Fund will be used to support the study of art and design. The Class of 1938 Endowed Library Fund was established in 1988 by members of the class of 1938 at the time of their 50th reunion. Income from the fund is used to support the College’s libraries. The Class of 1960 Endowed Fund was established in 2010 by members of the class of 1960 at the time of their 50th reunion. The fund honors the memory of Howard F. Lowry, Wooster’s President from 1944 to 1967. The President of the College may direct income from the fund to the following purposes: academic disciplines and special programs; excellence in teaching and faculty professional activities; support for student scholarships; or initiatives supporting student learning including, but not limited to, internship experiences, investigation of other cultures, special commitments to environmental protection, national economic needs, and efforts to further social justice. The Henry Jefferson Copeland Endowment for Campus Ministry was established in 1995 by a grant of $1 million from the Henry Luce Foundation to support the position of Campus Minister or Chaplain at the College, which is held by an ordained Presbyterian minister, and in support of the College’s goal of drawing upon the faith of the Reformed tradition for the benefit of its students. The Fund is named for Henry Jefferson Copeland, Wooster’s ninth President, and recognizes his commitment to the religious dimension of campus life and the College’s tradition of service. The Henry J. Copeland Fund for Independent Study was established in 1995 by members of the Board of Trustees in recognition of Mr. Copeland’s leadership as President from 1977 to 1995 and his commitment to the College’s goal of supporting students in meeting the highest standards of achievement. All members of the Board contributed to the Fund, and major gifts were provided by Ed and Edie Andrew, Stan and Flo Gault, and Fran Shoolroy. Income from the $1 million endowment is used to assist students with unusual research expenses associated with their Independent Study projects. The Dorothy Horn Cox Endowed Library Fund was established in 2007 by Holly V. Humphreys ’67 and her mother, Neva I. Humphreys, from the proceeds of a trust from Dorothy Horn Cox, a member of the class of 1923. Income from the fund is used to purchase books and resource materials for the College’s libraries. The W. R. “Ted” Danner Endowed Fund in Geology was established by Mr. Danner in 1996. Income from the fund is used to defray expenses for students and faculty members engaged in geological fieldwork. The Dewald Endowed Fund for Academic Excellence was established in 1986 by Dr. and Mrs. Donald W. Dewald of Mansfield, Ohio, on the occasion of the fiftieth anniversary of Dr. Dewald’s class of 1936. The annual income from this fund is used to strengthen Wooster’s academic program and to provide both incentives and rewards for those who have excelled. The Becky DeWine Endowment Fund was established in 1995 by Frances and Michael DeWine to honor the memory of their daughter, a member of the class of Endowed Resources 284 Endowed Resources 285 1993, who was killed in an automobile accident shortly after her graduation from Wooster. The fund is to provide a summer internship opportunity in journalism, preferably with a newspaper, during the summer between a student’s junior and senior years. The recipient will be selected by the Chairperson of the English Department or his or her designate. The Dillon Art Fund was established in 1984 by a gift from Mr. and Mrs. David T. Dillon of San Antonio. The fund is used to bring visiting artists to the campus. The Dix Family Endowment was established by Raymond and Carolyn Dix in 1982. Income from the fund is awarded annually, at the discretion of the President, to support programs in the area of communications and public events. The Donaldson Fund was established in 1984 by a generous gift from Stephen R. Donaldson, class of 1969, to support opportunities for creative writing at Wooster. Mr. Donaldson is a writer of literary fantasies and science fiction. The Bette Cleaveland Ewell Endowment for the Arts was established in 2004 by a generous bequest from Mrs. Ewell, a member of the class of 1946. Income from the fund is awarded annually, at the discretion of the President, to support programs of particular distinction in art and music, including visiting artists or performers, which may be beyond the range of normal operating budgets. The Farina Endowed Library Fund was established in 2001 by Louis J. and Jane Warner Farina ’70. Income from this fund is to be used for the acquisition of catalogued items and equipment for the College’s libraries. The Helen Murray Free Endowment was established by her children through the Al and Helen Free Foundation. Income from the Fund brings to the campus each year a renowned woman or man who is a practitioner in the chemical sciences (materials science, nanotechnology, and molecular biology). This scientist will interact with chemistry students at a technical level and will present an all-College convocation on the contributions of science to the quality of life. The Daniel C. Funk Endowment for Communications, established in 2001 by proceeds from several gift annuities, honors Daniel C. Funk, a member of the class of 1917. Mr. Funk was a Wooster Trustee from 1937 until 1972 and served as the College’s solicitor for many years. His wife, Elizabeth Reese Funk ’23, also was active in Wooster affairs. Mr. Funk established Wooster’s annuity program, making the first such gift in 1969. As a long-time advocate of the Speech and Communications programs on campus, he wished to bolster this department for future generations. The Elizabeth R. Funk Endowment for Music, established in 2001 by proceeds from several gift annuities, honors Elizabeth Reese Funk, a member of the class of 1923. Mrs. Funk was a staunch advocate of the College’s music programs and enjoyed attending concerts, recitals, and other performances over the many years she and her husband, Daniel C. Funk ’17, were Wooster residents. Through this fund, she wished to support the College’s music programs for future students, faculty, and members of the surrounding community. The Margaret Hemphill Gee Library Fund was established in 2003 by a bequest from Margaret Hemphill Gee, a member of the class of 1933. Income from the Fund is used to purchase books for the College’s libraries. The Gerstenslager Music Endowment Fund was established in 2000 by The Gerstenslager Company. Income from the Fund is used to benefit the Department of Music and the students who participate in the Music program. The R. Stanton Hales Presidents’ Discretionary Endowment Fund was established in 2007 by members of the Board of Trustees in honor of R. Stanton Hales, who served as Wooster’s 10th President from 1995-2007. Income from this fund is to be used at the discretion of the President to respond to new ideas or unforeseen opportunities. The Grace Elizabeth Hall Endowed Library Fund was established in 1996 by Grace Elizabeth Hall, a member of the class of 1944. Income from the fund is used to purchase books for the College’s libraries. The Julia Shoolroy Halloran Fund is an endowment to support excellence in theatre and the visual arts. Income from the fund will be used to support projects of unusual artistic merit or to bring to the campus performers and programs of high quality, which would be beyond the range of annual operating budgets. Allocation of the fund will be made annually by the Provost and two persons appointed by the President. The Jane A. Hanna Library Fund was established in 1995 by gifts from family and friends in memory of Jane Atkinson Hanna, a member of the class of 1944. An additional gift was added to the fund in 2008 from the proceeds of a planned gift. Income from the fund is used to purchase books, periodicals, and other materials for the College’s libraries. The Willard A. and Marybelle B. Hanna Library Fund was established in 1992 by Mr. and Mrs. Willard Hanna of Hanover, New Hampshire, to build and sustain a core collection of books and resource materials of permanent value to undergraduates interested in Asia. Mr. Hanna was a graduate of The College of Wooster in the class of 1932, and Mrs. Hanna was a graduate of the University of Michigan in the class of 1936. Mr. Hanna maintained a lifelong interest in Asia, first as a teacher in China and later with the U.S. Navy, the U.S. Foreign Service, and the American Universities Field Staff. Mrs. Hanna spent part of her childhood in China, and together Mr. and Mrs. Hanna spent many years in Indonesia and other Southeast Asian countries. Over the course of his career, Mr. Hanna prepared hundreds of field reports on Southeast Asian affairs and authored a dozen books on Asia. The Asian Studies Collection created by the Hanna Library Fund is a principal resource for students in Independent Study. The Deborah P. Hilty Endowment Fund was established in 2007 to support the College’s curriculum in writing. Professor Hilty was a member of the Department of English from 1964-67 and 1970-2003 and served as Secretary of the College and of the Board of Trustees from 1976-2001. The teaching and practice of non-fictional writing were at the core of her life’s work. Annual income from the Fund, which will be administered by the Chair of the English Department and the Provost, in consultation with faculty members in English and in the Program in Writing, will be used to underwrite programs beyond the scope of the annual operating budget and may include such activities as conferences and symposia on campus for students, faculty and staff members with special focus on new horizons in the teaching of writing at a liberal arts college. The Hans Jenny Memorial Research Fund was established in 2001 by alumni from the classes of 1964-1966 to honor the memory of Hans Heinrich Jenny, Professor of Economics from 1949-1982, Vice President for Finance and Business from 1966- 1982, and Professor Emeritus from 1982 until his death in 1998. Income from the fund is used at the discretion of the Department of Economics to support Senior Independent Study projects, faculty research, and collaborative research between faculty and students. The Kate-Gerig Endowed Fund was established in 1989 by Frederick H. “Fritz” Kate and Lois Gerig Kate of Oklahoma City, Oklahoma, in memory of their parents, Endowed Resources 286 Endowed Resources 287 Henry and Lida Kate and Christian and Erma Gerig, and in recognition of the long association of the Kate and Gerig families with the College and the Wooster community. The Kate-Gerig Fund is to support public events at the College, and income from the fund is awarded annually, at the discretion of the President, to bring speakers and performers to the campus for the benefit of the College and the local community. The John Kauffman Endowed Fund for Economics was established as an endowment in 2008 by John H. Kauffman, President of Kauffman Tire, Inc. Income from the fund is used to support the programs in economics and business economics as determined by the faculty in the Department of Economics. The Kendall-Rives American Research Grant was established in 1995 by Paul L. Kendall ’64 and Sharon K. Rives of Randolph, Vermont. The fund supports research projects conducted in a Latin American country as part of, or in preparation for, a Senior Independent Study project on some aspect of U.S.-Latin American relations. Allocation of these funds to a sophomore or junior who is proficient in Spanish or Portuguese is made by the Provost. The Lottie Kornfeld Endowment Fund was established in 1996 by Ms. Kornfeld, a graduate of the class of 1945. Income from the Fund is used to support on- and offcampus programs and activities consistent with the objectives of the Jewish Students’ Organization, but beyond the range of annual operating budgets. The Muriel Kozlow Endowment for The College of Wooster Art Museum was established in 2005 by a bequest from Muriel Mulac Kozlow, a member of the class of 1948. Income from the fund is used annually, at the discretion of the Museum Director, to mount an exhibition that would be beyond the scope of normal operating budgets. The Lindner Endowment was established in 2007 by Carl H. Lindner of Cincinnati, Ohio, to benefit the Department of Philosophy. Income from the fund will be used by the Department to support the teaching of ethics. The use of these funds is to be designated by the Chair of the Department. The Henry Luce III Fund for Distinguished Scholarship was established in 1980 through a gift from The Henry Luce Foundation to honor an esteemed Trustee and to permit Wooster to recognize exceptional scholar-teachers in its faculty by enabling them to bring to completion works of major scholarly significance. The Paul McClanahan Family Endowed Library Fund was established in 2008 with the proceeds of a gift annuity by Paul H. McClanahan, Sr., a member of the class of 1937, and Ruth Kempton McClanahan, a member of the class of 1940. The fund also honors their children: Neal K. McClanahan ’62, Paul H. McClanahan, Jr. ’64, and Alice J. McClanahan ’67. Income from the fund is used to purchase books and other materials for the College’s libraries. The McCoy Library Fund was established in 1991 by Margaret Stockdale McCoy, class of 1939, and Richard H. McCoy of Pittsburgh, Pennsylvania. Income from this fund is to be used to strengthen the information retrieval systems of the library and thereby to enhance the library’s services to students and faculty members. Mrs. McCoy served as a member of Wooster’s Board of Trustees from 1983 to 1989 and in 1991 received the John D. McKee Award for Outstanding Service. The Jean McCuskey Endowed Library Fund was established in 2005 with the proceeds of several planned gifts by Jean McCuskey, a member of the class of 1931. Income from the fund is used to purchase books and resource materials for the College’s libraries. The Robert Meeker Endowment Fund was established in 1981 by Robert B. Meeker, class of 1951, of Troy, Ohio, to support the study of business at the College. The fund provides assistance for faculty development, seminars, library materials, and visiting faculty members in areas of the curriculum associated with Business Economics and related programs. The Walter Meeker Endowment Fund was established in 1981 by Dr. Walter B. Meeker, class of 1950, of Troy, Ohio, to support the study of chemistry and biology at the College. The fund provides assistance for faculty development and research, seminars, library materials, and visiting faculty members in areas of the curriculum associated with chemistry and biology. The Dorothy E. Morris Endowment Fund was established in 1999 by a bequest from Dorothy E. Morris, class of 1926. Income from the endowment is used to support projects within the Department of Music that are over and above the normal operating expenses. The Jo and Howard Morris Fund for Programs in Religion and Society was established by the Board of Trustees in 1980 through a gift from Mr. and Mrs. J. Howard Morris of Cleveland. The fund supports programs each year that make religious and ethical questions an integral part of the education program of the College. The Peter Mortensen Endowed Lecture Fund was established in 2006 with a gift from Peter Mortensen, class of 1956, with gratitude for the contribution of The College of Wooster to the success and happiness of three generations of the Mortensen family. Income from the fund is used to support one or more public lectures and/or performances related to the First-Year Seminar, or for similar purposes directly related to the academic program. The John and Marianne Olthouse Endowed French Department Fund was established in 2010 by a bequest from Mary Katherine Olthouse Davis, a member of the class of 1940, in memory of her mother, Marianne Reinicke Olthouse, a member of the class of 1915, and her father, John W. Olthouse, Professor of French at the College from 1911-1955 and Professor Emeritus from 1955-1977. Income from the fund is used for books, outside lecturers, movies, or other needs determined by the department. The John and Marianne Olthouse Endowed German Department Fund was established in 2010 by a bequest from Mary Katherine Olthouse Davis, a member of the class of 1940, in memory of her mother, Marianne Reinicke Olthouse, a member of the class of 1915, and her father, John W. Olthouse, who served as an Instructor of German (1911-1918) and Professor of French (1911-1955; Emeritus 1955-1977) at the College. Income from the fund is used for books, outside lecturers, movies, or other needs determined by the department. The John and Marianne Olthouse Endowed Spanish Department Fund was established in 2010 by a bequest from Mary Katherine Olthouse Davis, a member of the class of 1940, in memory of her mother, Marianne Reinicke Olthouse, a member of the class of 1915, and her father, John W. Olthouse, who was Professor of French (1911-1955; Emeritus 1955-1977), Instructor of German (1911-1918), and Assistant Professor of Spanish (1918-1919) at the College. Income from the fund is used for books, outside lecturers, movies, or other needs determined by the department. The Richard G. Osgood, Jr. Lectureship in Geology was endowed in 1981 by his three sons in memory of their father, an internationally-known paleontologist who taught at Wooster from 1967 until 1981. Funds from the endowment will be used to bring a well-known geologist interested in paleontology and stratigraphy to the campus each year to lecture and meet with students. The Marjorie Owen Endowed Fund for Psychology was established by a gift from Marjorie L. Owen, a member of the class of 1942. Income from the Fund is awarded annually to benefit the Department of Psychology. Endowed Resources 288 Endowed Resources 289 The Harriet A. Painter and Walter E. Painter Fund in Music was established in 2002 through a bequest from their daughter, Sarah J. Painter, a member of the class of 1925. Income from the fund supports the acquisition of compact discs, tapes, records, and books for the Department of Music. The Alma J. Payne Library Fund was established in 2003 by a bequest from Alma J. Payne, a member of the class of 1940. At the time of her death, Miss Payne was professor emerita of English and American studies at Bowling Green University where she taught from 1946 to 1978. Income from the fund is used for the purchase of library materials in the field of American studies. The Theron L. Peterson and Dorothy R. Peterson Biology Research and Expense Fund was established in 2008 through the Dorothy R. Peterson Trust in memory of Theron L. Peterson, a member of the class of 1936, and his wife, Dorothy R. Peterson. Income from the Fund shall be used to support the recipient of the Theron L. Peterson and Dorothy R. Peterson Professorship in Biology with research and to assist in purchasing and maintaining current equipment and supplies for the Peterson Biology Chair, for the Peterson Biology Chair’s colleagues in the Department, and for the Department’s teaching program. The Theron L. Peterson and Dorothy R. Peterson Chemistry Research and Expense Fund was established in 2008 through the Dorothy R. Peterson Trust in memory of Theron L. Peterson, a member of the class of 1936, and his wife, Dorothy R. Peterson. Income from the Fund shall be used to support the recipient of the Theron L. Peterson and Dorothy R. Peterson Professorship in Chemistry with research and to assist in purchasing and maintaining current equipment and supplies for the Peterson Chemistry Chair, for the Peterson Chemistry Chair’s colleagues in the Department, and for the Department’s teaching program. The Theron L. Peterson and Dorothy R. Peterson Partners in Excellence Endow - ment Fund was established in 2008 through the Dorothy R. Peterson Trust in memory of Theron L. Peterson, a member of the class of 1936, and his wife, Dorothy R. Peterson. Income from the Fund shall be used to help financially support an outstanding faculty member’s professional development, with a particular emphasis on collaborative faculty-student research. The John W. Pocock Fund was established by Mr. Pocock in 1987. It is an endowment to underwrite the activities of the Board of Trustees and reflects Bill Pocock’s dedication to the strength of Wooster’s Board. Mr. Pocock was a member of the Board from 1957 to 1992 and served as its Chairman from 1970 to 1987. The Margaret M. Pollock Library Fund was established in 2000 by her bequest to The College of Wooster. She graduated in the class of 1938 and was a librarian with the Akron-Summit County Public Library for many years. Income from the Fund is used to support the College’s libraries. The Sarah Diane Purdum Book Fund was established in 1990 in honor of Sarah Purdum, a graduate of the class of 1984, by her parents Clarence W. Purdum, Jr. and Patricia L. Purdum. Income from the Fund is used annually at the discretion of the librarian to purchase nonfiction books for the College’s libraries. The Isabel and Elizabeth Ralston Presidential Endowment Fund for Faculty Development was established in 1987 by a bequest from Isabel Ralston of the class of 1934 and by a grant from The George Gund Foundation of Cleveland to assist newly appointed members of the faculty to establish ongoing research programs that will advance their professional careers as productive scholars and able teachers. The income from the fund is administered by the Provost. The Margaret Ann Record Endowed Fund for Student Activities was established in 1990 by her mother, Mrs. Paul R. Record. The fund honors the memory of Margaret Ann Record, a member of the class of 1949, who died while a student at Wooster. Income from the fund is awarded annually, at the discretion of the President, to support programs and activities that enrich the cultural lives of students beyond the classroom. The Margaret Beck Renner Library Fund was established in 1998 with a gift from Margaret Renner, a member of the class of 1947. Income from the fund is used to purchase books in the fields of foreign languages, mathematics and physics to honor Margaret, who was a French major, and her late husband, Dale W. Rinehart ’37, who majored in physics. The Bruce and Mary Rigdon Library Endowment was established in 2006 by Bruce Rigdon, a member of the class of 1958, and his wife, Mary. Income from the fund is used to purchase books and resource materials for the College’s libraries. The Seele Fund for Andrews Library was established in 1988 by Diederika M. Seele, in memory of Keith C. Seele, ’22. Income from the endowment is used, at the discretion of the President, for materials and programs that benefit the students and faculty who use the library. The Silber Fund honors the memory of Edith and Erwin Silber. Established in 1987 by Elizabeth Grant Silber, income from the endowed fund is used to support the study of German language and literature. The Charles and Rachel Smith Fund was established in 1987 by Dr. Charles A. Smith, class of 1929, in memory of his parents Edward James and Anna May Smith. The income from this fund is used annually in programs designed to assist students of the College to understand the functioning of the American economy, especially the important role that freedom of choice, capital formation, the profit motive, and individual initiative and responsibility play in our economic system. The Lawrence D. Stanley Summer Research Program Endowment in History was established in 2007 by Laura Stanley Gunnels in honor of her father and the 50th anniversary of the class of 1958. Mrs. Gunnels shared with her father an interest and passion for history and established the summer research endowment to support student research in history. Her intent is for students to develop the passion and habits of mastery that will advance the study of history in perpetuity. First preference is given to students following their first-year at Wooster. Each student is partnered with a faculty research advisor. The Leah Stoner Stevens Library Fund was established in 1982 by a bequest from Leah Stoner Stevens, a member of the class of 1915. The bequest was given in memory of Mrs. Steven’s sister, Jean Stoner, a member of the class of 1912; Mrs. Stevens and Mrs. Steven’s husband, Ernest C. Stevens. The Fund is used to purchase books and other materials for the College’s libraries. The John Mercer Syverud II Memorial Fund was established in 1994 by the family and friends of John Syverud II, a graduate of the class of 1990. Income from the endowed fund is assigned annually to the German Department and is used to support and enhance the teaching and learning of students in the department. The Grace Tompos Endowed Tree Fund was established in 2001 at the time of Grace’s retirement as Executive Director of Development at the College. Income from the fund is used to support special tree related publications and expenses, including the purchase of rare trees. The Totten Geology Student Research Fund was established in 2007 by Dr. Stanley M. Totten and Susan March Totten, members of the class of 1958, in honor of Endowed Resources 290 291 Endowed Resources the 50th anniversary of their graduation from the College. Income from the fund is used to assist geology majors in research related to their major. The Tree Conservation Endowment Fund was established in 1987 as a result of a challenge grant received by the College from an anonymous foundation to establish an endowed fund to support a tree conservation maintenance and replacement program at Wooster. Through gifts from generous alumni and friends, the College continues to add to this special fund. The value of the trees to Wooster’s campus goes well beyond the numerical figures of the endowment. Many of the oldest trees were standing when the College was founded in 1866. In fact, the beauty of the wooded hilltop is what led to the selection of the College’s present site. This unusual tree endowment permits Wooster to preserve the beauty of its campus. The Karl Ver Steeg and Charles B. Moke Fund for Geologic Research was established in 1991 by Frederick H. “Fritz” Kate of Oklahoma City. The fund supports the Independent Study research of Wooster students and faculty in the Department of Geology. Allocation of these research funds is made by the faculty of the Geology Department. The Sherman A. and Florence M. Wengerd Department of Geology Endowment Fund was established in 1996 by Florence M. Wengerd in memory of her husband, Sherman A. Wengerd, a 1936 Wooster graduate and internationally-known petroleum geologist. Income from this fund is used to purchase equipment and supplies for undergraduate teaching and research in the areas of sedimentology and stratigraphy. The Whitmore-Williams Endowed Fund for the Nursery School was established in 1998 by a gift from A. Morris and Ruth Whitmore Williams, a member of the class of 1962 and Trustee of the College. The fund benefits The College of Wooster Nursery School, which was established in 1947 and provides field experience for psychology students at the College. Mrs. Williams served as Co-Director of the Gateway School, a noted pre-school in suburban Philadelphia. The Ronald C. Wilcox Endowed Library Fund was established in 1995 by Ron Wilcox, a member of the class of 1975. Income from the fund is used to purchase books and other resources that pertain to modern American social history. The James R. Wilson Fund for Business Economics was established in 2001 by a gift from James R. ’63 and Linda R. Wilson. Income from the fund is used to support programs that enhance the study of Business Economics by bringing to campus distinguished business and financial leaders for seminars, forums, and guest lectures. The William H. Wilson Research Awards were established in 1945 through a gift from Mr. and Mrs. M. H. Frank and Dr. and Mrs. Robert E. Wilson in honor of William H. Wilson, a member of the class of 1889 and Johnson Professor of Mathematics and Astronomy from 1900-1907, for the encouragement of faculty research in the natural sciences and mathematics. The Women’s Advisory Board Hardship Fund for International Students was endowed in 2007 by members of The Women’s Advisory Board. Short-term financial support from this fund is available, through a formal application process, to upperclass international students who demonstrate severe economic hardship due to unforeseen circumstances that are beyond their control. Preference is given to seniors with strong academic records who have made significant contributions to the campus community. The Arthur Bambridge Wyse Endowed Lectureship was established in 2006 through a planned gift by Marylyn Crandell Wyse, a member of the class of 1929, in memory of her husband, Arthur B. Wyse, a member of the class of 1929. A scientific researcher for the Navy, Arthur Wyse was killed in a blimp accident in 1942. Income 292 from this fund is used to help compensate a visiting assistant professor serving as a replacement for a Wooster faculty member on research or study leave. The Richard D. Yoder Fund for Music was established in 2001 through a generous bequest from Dr. Yoder, a member of the class of 1947. Income from the fund is used to support music performance programs within the Department of Music. ENDOWED SCHOLARSHIPS The Paul R. Abbey Endowed Scholarship was established in 2009 by gifts from Paul R. Abbey ’73, a member of Wooster’s Board of Trustees. His great-grandfather, Dwight C. Hanna, graduated from the College in 1883. His grandfather, Dwight C. Hanna, Jr., graduated in 1912, and his uncle, Dwight C. Hanna III, graduated in 1944 and was a Trustee of the College from 1968-1974 and 1982-1996. His daughter, Kendall G. Abbey, graduated from the College in 2009. Income from the fund is awarded annually to students in good academic standing who have demonstrated financial need. The Allardice-Wise Scholarship in Theatre was established by James N. Wise ’41, in memory of his friend and classmate James B. Allardice. The scholarship is awarded, on the basis of a competitive audition, to a senior Theatre major who has demonstrated exceptional talent and interest in theatre and who has financial need. The Clarence B. Allen Scholarships were established in 1988 by Mr. and Mrs. J. Howard Morris and are named in honor of Wooster’s first black graduate, Clarence Beecher Allen, class of 1892. The Reverend Mr. Allen attended McCormick Theological Seminary and was ordained by the Presbytery of St. Louis in 1895. These merit scholarships are awarded to African American students who demonstrate high academic achievement. The David L. Allen Endowed Scholarship was established in 2006 through a bequest from David L. Allen, a member of the class of 1965. Dr. Allen taught sociology at The University of Findlay from 1970 until his death in 2005. In recognition of his involvement in student and community life, Dr. Allen was named “Outstanding Educator” by the Findley-Hancock County Chamber of Commerce in 1995. Income from this fund is awarded to students who demonstrate financial need, with first preference given to students planning to major in sociology. The Joseph Sanborn Allen and Grace Allen Scholarship was established in 2005 by a bequest from Grace H. Allen, widow of Joseph Sanborn Allen, a member of the class of 1934. Income from this scholarship is awarded annually to sophomore, junior, and senior students in good standing who have demonstrated financial need. The Margaret and Louise Amstutz Scholarship Fund was established in loving memory by The Reverend Platt Amstutz, class of 1905. The fund provides scholarship aid to students with demonstrated financial need. Preference is given those students planning a career of full-time Christian service. The Margaret Neely Anderson Scholarship Fund was established in 2000 by Margaret Neely Anderson, a member of the class of 1944, and her husband, Thomas F. Anderson. Income from the fund is awarded annually to students who are in good academic standing and who have demonstrated financial need. The Joyce Elaine Andrews Scholarship was established in 2007 by Joyce Andrews, a member of the class of 1959. Income from the fund is awarded annually to students in good academic standing with demonstrated financial need. Endowed Resources 293 Endowed Resources The Angerman Family Scholarship was established in 2007 by a gift from Thomas W. Angerman in recognition of the class of 1953 and the Independent Minds Campaign. Mr. Angerman served on the Class of 1953 50th Anniversary Committee. The scholarship honors the relationship the Angerman family has maintained with the College over two generations. Mr. Angerman’s son Michael, a member of the class of 1986, is also a graduate of the College. Income from the fund is awarded annually, with first preference given to students from Western Pennsylvania. The Harold G. and Helen F. Arnold Scholarship Fund was established in 1966 by their daughter and son-in-law, Dr. and Mrs. Richard E. Garcia. The scholarship will be awarded annually to a young woman recommended by The Women’s Advisory Board. In this recommendation, consideration will be given to general need and wholesome Christian character. The John Robert Arscott Memorial Scholarship was established in 1992 in memory of Dr. Arscott, a 1926 graduate of Wooster, by his son, David Arscott, a member of the class of 1966. The scholarship is awarded annually by the Director of Financial Aid to a junior or senior with financial need. Preference is given to a student majoring in English who has demonstrated both academic achievement and leadership while at Wooster, and who has an interest in later pursuing graduate study. The Ralph D. Au Scholarship Fund was established in 1999 by Marjorie B. Au in loving memory of her husband, who was a 1933 graduate of The College of Wooster. The scholarship is to be awarded annually to junior or senior chemistry and/or physics majors. Selection is made by the Scholarship Committee. The James E. Aust Scholarship Fund was established in 1982 by Mrs. L. A. Klages of Akron in loving memory of her son, James E. Aust. The scholarship is awarded annually to students requiring financial assistance to attend Wooster. The Mary Jane Smirt Bachtell Scholarship, established by Sam Bachtell in 2001 in memory of his wife, is awarded annually to a student with financial need. The Bachtells were both graduates of the class of 1951. The Bachtell/Lewis Scholarship Fund was established in 1987 by W. A. (Web) and Nancy Bachtell Lewis, members of the class of 1951. The scholarship is awarded annually to students with financial need. The Martha W. Bain Endowed Scholarship was established in 2005 by a bequest from Martha Weimer Bain, a member of the class of 1938. Income from this scholarship is awarded annually to students who demonstrate financial need, with first preference given to students from Loudonville, Ohio. The Douglas B. Ball Endowed Scholarship was established in 2007 through a bequest from Douglas Bleakly Ball, a member of the class of 1961. Income from the fund is awarded annually to students who have demonstrated financial need, with first preference given to students who are majoring in history. The Dr. Ralph and Margaret Bangham Scholarship in Biology was established in 2007 by Jean W. Bangham, a member of the class of 1953, and honors the memory of Jean’s father, Dr. Ralph V. Bangham, Wooster’s Danforth Professor of Biology from 1923-1963, and Jean’s mother, Margaret Williams Bangham. Income from the fund is awarded annually to students who have demonstrated financial need and are majoring in biology. The Bank One Scholarship was established in 1983 through the generosity of Bank One of Wooster. The income from the endowment is used for scholarships for deserving juniors and seniors who are majoring in economics or business economics at the College. Preference is given to students from the Wayne County area. 294 The Malcolm and Sue Basinger Scholarship was established by members of the class of 1951 following Mac’s death in 1998 after a courageous battle with ALS disease. The income from this fund is used to provide scholarships for students who have financial need and who have demonstrated academic achievement and service to their communities. First preference will be given to students whose high school classes had fewer than 300 members. The Martin John, Lois and Jeffrey Bender Endowed Scholarship Fund was established in 2004 by Dr. M. John Bender, a member of the class of 1944, his wife Lois, and his son Jeffrey Bender, a member of the class of 1978. Income from this scholarship fund is awarded annually to a student who has financial need and is a U.S. citizen, with first preference given to a student who is preparing for a career in medicine. This scholarship is renewable through the senior year provided the student remains in good academic standing. The Ernest N. and Ellen B. Bigelow Scholarship was established in 2007 with proceeds from a pooled income fund and gifts from their children, Bruce ’66, Ann ’68, Mark, and Gail. Mr. and Mrs. Bigelow were members of the class of 1939 and 1937 respectively. Income from the fund is awarded annually to students in good academic standing with demonstrated financial need. First preference is given to deserving chemistry or philosophy majors. The Jennifer Kay Blair Endowed Scholarship was established in 2006 in recognition of and appreciation for the profound and lasting effect of the Wooster experience on Jennifer, a graduate of the class of 1989. Income from the fund is awarded annually to one or more female students who have a strong academic record and demonstrated financial need. First preference is given to a history major who is also an active participant in the College’s musical programs. Recipients are selected by the College and The Women’s Advisory Board. The Blanchard Scholarship in honor of Werner J. Blanchard was established by Juliet Stroh Blanchard as an expression of their deep concern for education under Christian auspices. The scholarship is made available each year to a student who comes from a foreign country and possesses unusual leadership qualities, and who is able to fit into a Christian environment and contribute to the international spirit of the College community. The David G. Blanchard Memorial Scholarship Fund was established by gifts from his family and friends. David was a member of the class of 1958. First preference for the scholarship is given to a student from Africa. The Joan Blanchard Scholarship was established in 2009 by Joan Blanchard ’78, a member of the Board of Trustees. Income from the fund is awarded annually to female students with demonstrated financial need. First preference is given to a female student who has an interest in geology or science. The Jay Blum and Mary Blum Scholarship Fund was established in 2003 by Marian N. Blum from the proceeds of several gift annuities and honors the memory of her husband, Jay W. Blum, a member of the class of 1929, and her daughter, Mary N. Blum, a member of the class of 1964. Income from the fund is awarded to students in good academic standing who demonstrate financial need. First preference is given to students majoring in economics or business economics. The Jean “Bunny” Bogner Endowed Scholarship was established in 1998 by her husband, Robert P. Bogner, and other friends. The scholarship recognizes Bunny’s warm affection for the College and her years of service on The Women’s Advisory Board. Awards are made annually to one or more students who have strong academic records and financial need. Recipients are selected by the College and The Women’s Advisory Board. Endowed Resources Endowed Resources 295 The Bonsall-Braund Scholarship is the gift of Nancy Braund Boruch, a member of the class of 1964. The scholarship is given in memory of Nancy’s mother, Ann Bonsall Braund ’37, and Nancy’s father, the Reverend Eric T. Braund. Income from the fund is awarded annually to students with financial need, with first preference given to minority students or students who are planning a career in the ministry or in social service. The Bourns Family Scholarship Fund was established in 1990 to recognize the long association of the Bourns family with The College of Wooster. The fund was made possible by gifts in memory of Lowell B. Bourns, class of 1927, who served in the administration of the College from 1959 to 1972, and by other gifts from family members. The income from this fund is awarded annually to students requiring financial assistance, with preference given to persons representing cultural minorities in America. The Ruth L. Bower Scholarship Fund was established in 2002 through a generous bequest from Mrs. Bower, a member of the class of 1939. Income from this scholarship is awarded annually to students on the basis of academic achievement and financial need. The Nancy Brown Endowed Scholarship was established in 2010 by a bequest from Nancy Campbell Brown, a member of the class of 1952. Income from the scholarship is awarded annually to students who have demonstrated financial need. The Urlene F. Brown Scholarship in the Performing Arts honors the memory of Urlene Fern Brown, class of 1969, and was established by her many friends in 1974. It is awarded at the annual Recognition Banquet to an African American student who has contributed significantly to the performing arts at Wooster. The Robert M. Bruce Memorial Scholarship was established in 1983 by his family and friends. Robert Bruce was a member of the class of 1939 and served as Professor of Physical Education for eighteen years and as Athletic Director for nine years. This scholarship recognizes students who have demonstrated outstanding growth in writing skills during the First-Year Seminars. The scholarship is awarded to a first-year student who has shown the most improvement in writing skills. The John Bruère Scholarship was established during The Campaign for Wooster and honors the memory of Dr. John Bruère, who was a minister at Calvary Presbyterian Church and a professor of religion at Wooster from 1936 to 1944. The scholarship is awarded to a student or students from the greater Cleveland area. The John D. Brush, Jr. Endowed Scholarship was established in 2007 with a generous gift from Douglas F. Brush, a member of the class of 1977 and a Trustee of the College, and from The Brush Family Foundation. The scholarship honors the memory of Doug’s father, a businessman who, throughout his life, pursued the study and creation of art as an avocation. It is awarded annually to students with financial need and demonstrated interest in the fine arts who intend to concentrate their studies in fine arts at Wooster. First preference is given to a student from the greater Rochester, New York, area and/or any descendants of John D. Brush, Jr. The Elizabeth Hazlett Buchanan Scholarship Fund was established by the Women’s Synodical Society in the Synod of Ohio, the net income from which is paid to The College of Wooster for the purpose of granting scholarships to students at that institution, the beneficiary each year to be determined by a joint committee from The Women’s Advisory Board and the College. The Bunn Scholarship Fund was established by the bequests of George W. Bunn and Louise Craft Bunn ’26. The income from this fund provides one or more scholarships to be awarded annually, on the basis of financial need and potential ability, to students planning to enter business careers or other professions. 296 The Robert R. Cadmus, M.D. Scholarship was established by his family in 1995 to honor the memory of Robert R. Cadmus, M.D., class of 1936. Income from the fund is awarded annually to a student who demonstrates financial need. The Camp Family Scholarship was established in 1971 by a gift from the Camp Family in memory of Howard E. Camp, a member of the class of 1916. This fund was augmented in 1998 and 2001 through bequests from Christine Camp Birkenstock, a member of the class of 1951 and the daughter of Howard E. and Florence Camp, and Christine’s husband, Jack Birkenstock. The scholarship is awarded annually to students with demonstrated financial need. The Ralph F. Carl Scholarship was established in 2005 through the proceeds of several planned gifts by Dr. Ralph F. Carl, a member of the class of 1938. Income from the fund is awarded annually to students who have demonstrated financial need. The David L. Carpenter Scholarship was established in 1984 by David L. Carpenter, class of 1965. After his death in 1999, the partners of his law firm, Calfee, Halter & Griswold, made a generous additional gift as a memorial to David. His mother, Myra Schweininger Carpenter, class of 1937, also contributed to this scholarship. The scholarship is awarded to one or more students with financial need from the Northern Ohio area. The recipient(s) should demonstrate superior academic achievement, and consideration shall be given to non-academic or extracurricular activities or interests in awarding this scholarship. The Anderson Bogardus Cassidy Scholarship was established in 2008 by Phoebe Anderson Cassidy ’58 in honor of the 50th anniversary of her graduation from the College. The scholarship also honors her grandmother, Ruth E. Bogardus, and her four great-aunts, Grace, Laura, Mary, and Sarah Anderson, all of whom were graduates of the College. Income from the fund is awarded to a female chemistry major who has completed one year of study at the College. The Annarie Peters Cazel Scholarship was established in 1991 in her memory by Fred A. Cazel, Jr. The scholarship is awarded to students with high academic standing in the areas of Classical Studies, Archaeology, or Art History. Mrs. Cazel was a Greek and Latin scholar at Wooster in the class of 1941; she received her doctorate from Johns Hopkins in Art and Archaeology. The John W. Chittum Scholarship Fund, established in 1985, honors Dr. Chittum, an outstanding teacher, who taught his students the importance of organization, clear thinking, and integrity in all facets of one’s life. The scholarship recipient must be a chemistry major who has completed one year of study at Wooster. The Chopin Music Scholarship Fund was established in 1986 through the generosity of Nancy Gould. Miss Gould’s sister, Sybil, was Professor of Art at Wooster from 1944 to 1972, and this scholarship honors her and the donor’s lifelong interest in music. Scholarships are awarded each year to a junior and senior music major, with preference given to women concentrating in piano performance and having financial need. The Mildred B. and Glenn J. Christensen Scholarships were established in 1993 by a bequest from their estate. Dr. Christensen, a Wooster graduate in the class of 1935, was a member of the Department of English at Lehigh University and served as Dean, Vice President, and Provost of the University’s College of Arts & Sciences. These scholarships are awarded annually to deserving students with demonstrated financial need. The Lou Cramblett Christianson Scholarship was established in her memory as a teacher in 1995 by her husband, Paul Christianson, and by her family and friends. Additional gifts were added to the fund in 2008 and 2009 by Dr. Christianson. Income Endowed Resources 297 Endowed Resources from the scholarship is awarded each year to a junior or senior student of high academic achievement and financial need who is planning a teaching career in elementary education. First preference is given to a student who has career intentions to deal with the educational needs of children globally. The Clarke Family Scholarship Fund was established in 1996 through the generosity of James T. Clarke, Trustee and member of the class of 1959, and Patricia Kemp Clarke. Income from the endowed fund is awarded annually to students who have financial need. First preference is given to African American men from inner city areas of the United States who have demonstrated leadership ability. The Margaret Reed and John O. Clay Scholarship was established in 1985 by their son John R. Clay. Mr. and Mrs. Clay are graduates of the classes of 1943 and 1945, respectively. The scholarship is awarded annually to a student who has demonstrated academic achievement and financial need. The Cleveland Scholarship Fund was established by alumni and friends living in Cleveland during The Campaign for Wooster. These scholarships are awarded to students from the greater Cleveland area. Named scholarships in this special fund include the John Bruère Scholarship, the Howard Lowry Scholarship, the Bess and Eugene Pocock Scholarship, and the William E. and Maryan Fuhrman Smith Scholarship. The Colbrunn Family Scholarship Fund was established by Ethel B. Colbrunn, class of 1934. This memorial fund honors her parents, W. W. and Anna Colbrunn, as well as her brother and sister, Earl and Florence Colbrunn. The Bertha Margaret Lear Colclaser Aid Fund was established by L. A. Colclaser in memory of his beloved wife, to provide scholarships and loans to assist students preparing for the ministry or for missionary service, who have maintained a scholastic standing and a position of activity in the life of the College, which makes them worthy of such aid, and who without aid would be unable to continue their college education. The Gordon D. Collins Scholarship was established in 2001 by his family, friends and former students. It honors Gordon Collins, the first Whitmore-Williams Pro - fessor of Psychology, who retired in 2000 after 37 years as a member of the Psychology Department. The scholarship is awarded to a junior psychology major at the College who, in the evaluation of the Psychology faculty, has made a significant contribution to the College and has demonstrated financial need. The Karl T. Compton Scholarship is the gift of the Alfred P. Sloan Foundation. It is awarded each year to a member of the first-year class who has demonstrated ability in mathematics and in physics, and who gives evidence of continuing interest in these subjects. The scholarship is awarded for both the first year and sophomore year. The Marjorie A. Compton Endowed Scholarship was established in 2003 by a bequest from Marjorie A. Compton, a member of the class of 1947. Income from this scholarship is awarded annually to students in good academic standing who have demonstrated financial need, with first preference given to students who are majoring in philosophy or psychology. The Martha Granger Cooper Scholarship Fund was established in 2000 by Joe and Martha Cooper in recognition of the 50th anniversary of her graduation from Wooster in the class of 1950. The scholarship is awarded to a student with financial need. The Correll Family Music Scholarship Fund was established from the proceeds of a gift annuity from Virginia W. and Arthur G. Correll, members of the class of 1940. 298 Income from the fund is awarded to a student or students with demonstrated financial need who are majoring in music. The Ralph Cottle Scholarship Fund was established in 2002 by a bequest from Dr. Ralph I. Cottle, a physician and a founder of The Wooster Clinic. Dr. Cottle began his practice in Wooster in 1952, and he and his family have been associated with the College for fifty years. The scholarship is awarded annually to a student needing financial assistance. The Covenant Scholarship Fund was established through Major Mission Fund gifts to the College. Congregations of churches within the Synod can nominate high school seniors for admission to the College. Students with financial need are eligible for scholarship assistance through the Covenant Scholarship Fund, which is administered by the Director of Financial Aid at Wooster. The Alexander and Florence Cowie Memorial Scholarship Fund was established in 1995 by a bequest from Florence Rapp Cowie, a member of the class of 1925. Income from the fund is awarded annually to students majoring in the sciences who have demonstrated academic achievement and financial need. The Cyrus Burns Craig Scholarship was established in 1963 by a bequest of Mrs. Martha White Craig Frost, in memory of Dr. Cyrus Burns Craig, physician and Associate Medical Director, New York Neurological Institute. The Frederick W. and Ruth Perkins Cropp Scholarship was established in 1978 by family and friends to honor Mrs. Cropp, class of 1925, and the late Dr. Cropp, class of 1926, for their lifetimes of Christian service. It is awarded annually to a deserving student with demonstrated financial need. The Custer Scholarship Fund was established by a gift from Monford D. and Vesta M. Custer of Coshocton, Ohio, and has been supplemented by contributions from their daughter, Eleanor W. Custer, a member of the class of 1926. The income from this fund is used for scholarships for three or more worthy students each year. First preference is given to students preparing for full-time Christian service. Selections are made by the Scholarship Committee based on scholarship, Christian character, leadership, and financial need. The Donald D. and W. Rebecca Custis Scholarship Fund was established by Don Custis, a member of the class of 1958, and Becky Custis in appreciation of the importance of The College of Wooster in the lives of three generations of the Custis family. Income from the fund is awarded by the College on the basis of financial need. Preference is given to students preparing for careers in health care professions. The Harold Alden Dalzell Memorial Scholarship Fund was established by family and friends as a memorial to Dr. Dalzell, Vice President of the College from 1948 to 1954. Income from this fund is used to provide scholarships for deserving students who have exhibited qualities of leadership, Christian character, and financial need. The Arthur Vining Davis Foundations Scholarship was established during Independent Minds: The Campaign for Wooster by a grant from The Arthur Vining Davis Foundations of Jacksonville, Florida. Income from this fund is awarded annually to students in good academic standing who have demonstrated financial need. The D. D. Davis Scholarship Fund is the gift of the Davis Foundation of Oak Hill, Ohio. The income from this fund is used for scholarships for not fewer than three men each year. Selections are made by the Scholarship Committee based on scholarship, Christian character, leadership, and financial need. The F. Lyle Davison Student Aid Fund was established in 1952 by Mr. Davison, who was a graduate of The College of Wooster in the class of 1932. The income from Endowed Resources 299 Endowed Resources this trust fund is awarded annually to a deserving student on the basis of need by the Committee on Scholarships and Student Aid. The Charles and Roland del Mar Scholarship Fund is a gift of The Charles Delmar Foundation, Washington, D.C. It honors the memory of Charles Delmar and Roland H. del Mar, Trustee of the College from 1964-1982. In the award of scholarships from the fund, preference is given to students from Puerto Rico, Central and South America, and Mexico. The George H. Deuble, Jr. Memorial Scholarship was established in 1967 by The Deuble Foundation of Canton, Ohio. George H. Deuble, Jr., graduated from The College of Wooster in 1947 and died November 26, 1965. The income from this fund is awarded annually to deserving students who are graduates of Stark County high schools and who have financial need. The Dilley Family Scholarship was established in 1977 by the Dilley family. It is awarded to students who have financial need and whose parent(s) are in service occupations such as teaching and the ministry. The J. Garber and Dorothy W. Drushal Scholarship was established in 1977 by students of the College and other friends in honor of J. Garber Drushal, Wooster’s eighth President. The scholarship was augmented in 2004 and 2005 by gifts from family and friends in memory of President Drushal’s wife, Dorothy W. Drushal. Income from the scholarship is awarded annually to one or more students who partic ipate in campus activities and exhibit leadership and academic achievement. The Aileen Dunham Scholarship in History was established in 1965 in honor of Professor Aileen Dunham, Chairman of the Department of History from 1946-1966. It is awarded annually to a student who has achieved excellence in history courses and who demonstrates financial need. The Eberhart Family Scholarship Fund was established in 2002 from the proceeds of a gift annuity. This scholarship honors the memory of Lola G. Eberhart and her husband, E. Kingman Eberhart, Hoge Professor of Economics from 1938 to 1971. The scholarship also recognizes their children, grandchildren, and great-grandchildren who have attended the College. Income from this scholarship is awarded annually to students who are in good academic standing and who have demon - strated financial need. The Horatio and Lyda Ebert Scholarship Fund was established in 1969 by Horatio and Lyda Ebert, who were longtime friends of the College. Their son, Robert O. Ebert, was a member of Wooster’s Board of Trustees from 1991-2000 and was an emeritus life member of the Board from 2000-2009. This scholarship is awarded annually to students with demonstrated financial need. The Eckles-Wong Scholarship was established in 2005 with a gift from David Eckles and Allene Wong, in honor of their son Ryan, a member of the class of 2006. Income from the fund is awarded annually to students in good academic standing who have financial need. The Esther Edgar Scholarship Fund was established in 2001 by a bequest from Esther Edgar. This scholarship is awarded annually, with first preference given to Christian students of Assyrian descent from Iran or the United States. The Linda Smith Edgecomb Endowed Scholarship was established by Franklin K. Smith, class of 1947, and Jean Horn Smith, class of 1948, in memory of their daughter who graduated from Wooster in 1976. Income from the fund is awarded annually to students in good standing who have financial need. First preference will be given to students who are planning careers in the health care field. 300 The Harry V. and Donna D. Eicher Endowed Scholarship Fund was established in 2007 by planned gifts from Harry V. Eicher, a member of the class of 1943, and his wife, Donna Doerr Eicher, a member of the class of 1942. Income from the fund is awarded to students with demonstrated financial need with first preference given to students who are active in extracurricular activities. The Endowed Faculty Scholarship Fund is supported by contributions from individual members of the faculty. Established in 1970, following a proposal by Professor John D. Reinheimer, the fund has grown substantially over the years and is awarded to students who demonstrate financial need. The Raymond L. Falls, Jr. Scholarship Fund was established in his memory by his family and friends. Mr. Falls, a member of the class of 1950, had a distinguished career as an attorney in New York City. The scholarship is awarded annually on the basis of financial need and academic achievement to a member of the junior class who has declared a major in Philosophy. The Nels F. S. and Katharine P. Ferré Scholarship Fund was established by family and friends to honor the lives of Nels F. S. Ferré and Katharine Pond Ferré. Dr. Ferré, a distinguished educator, theologian, and philosopher, was the Frank Halliday Ferris Professor of Philosophy at The College of Wooster from 1968 until his death in 1971. Mrs. Ferré, a reader, translator, editor, and poet, remained engaged in Wooster activities until 1988. The scholarship is awarded annually to students who have achieved academic excellence and demonstrate financial need. The Howard V. (Bus) and Elleanor R. Finefrock Scholarship honors the memory of Howard V. Finefrock, a member of the class of 1936, and Elleanor Reinhardt Finefrock, a member of the class of 1940. It was established in 1974 by members of the Finefrock family and friends. The income from the fund is awarded annually with first preference given to men and women who are members of the swimming teams. The FirstMerit Scholarship Fund was established in 1994 through the generosity of Peoples National Bank of Wooster. Income from the fund is awarded annually to a student who has demonstrated academic achievement and financial need and who is living in an Ohio region served by FirstMerit. The Fletcher-Brown Scholarship was established in 1998 by a bequest from Mary E. Fletcher, class of 1932, in appreciation of her parents, David H. and Clara Brown Fletcher, and her grandparents, Frank and Elizabeth Stauffer Brown. This scholarship is awarded annually to a student who demonstrates financial need. Preference may be given to a student from a farming community. The Melcher P. Fobes Scholarship was established in 1994 by Dr. Fobes, Emeritus Professor of Mathematics. Dr. Fobes taught in Wooster’s Department of Mathematics for forty-one years. The scholarship is awarded annually to students who demonstrate high academic achievement and who have financial need. The Walter D. Foss Scholarship Fund was established in 1994 by Donald J. Thompson and his brother, William Foss Thompson, a member of the Board of Trustees. This fund honors their grandfather, Walter D. Foss, who served as a Trustee of the College from 1902 to 1917, and who was President of The Wooster Brush Company from 1879 to 1938. The income from this scholarship is awarded by the Director of Financial Aid at Wooster. The Doon, John, and Julia Foster Family Scholarship was established in 2005 by John S. and Doon Allen Foster, members of the class of 1980, in honor of their 25th reunion. Income from this scholarship is awarded annually to students in good academic standing who have demonstrated financial need. Endowed Resources 301 The Edward S. and Emily K. Foster Endowed Scholarship was established by their family to honor Mr. and Mrs. Foster. Mr. Foster, class of 1935, was a professor of physics at the University of Toledo (Ohio) and served on the board of the Toledo Public Schools for fourteen years, including four years as President. Mrs. Foster, class of 1933, was a reference librarian in the Toledo Public Library. Income from the fund is awarded to a graduate of the Toledo Public Schools or to a student from the surrounding geographic area who demonstrates academic promise and financial need. The Herman Freedlander Student Aid Fund, established in 1946, provides a scholarship each year for a member of the junior or senior class who looks forward to a career in merchandising in the field of business administration with a preference for retailing. The Laura Frick Endowed Scholarship Fund was established in 1983 by The Laura B. Frick Trust. The scholarship is awarded to a student with financial need from Wayne County. The Frueauff Foundation Scholarship Endowment Fund was established in 1965 by the Charles A. Frueauff Foundation, Inc., of New York City. The income from the fund provides scholarships that are awarded annually by the Committee on Scholarships and Student Aid. The Daniel and Elizabeth Funk Endowed Scholarship was established in 2003 by a bequest from their son, Edward R. Funk, a member of the class of 1946. This scholarship honors the memory of Daniel C. Funk, a member of the class of 1917 and a Wooster Trustee from 1937 until 1972, and his wife, Elizabeth Reese Funk, a member of the class of 1923. Income from this scholarship is awarded annually to students in good academic standing who have demonstrated financial need. First preference is given to students majoring in physics. The Martin Evan Galloway Scholarship was established by members of his family, friends, and members of the First United Presbyterian Church of Middletown, Ohio, in memory of Martin who would have graduated with the class of 1965. It is to be awarded to an average student, or students, on the basis of general need and wholesome Christian character. The Stanley C. Gault Scholarship Fund was established in 1996 by The Goodyear Tire and Rubber Company in recognition of Mr. Gault’s extraordinary leadership as Chairman of the company from 1991-1996. The scholarship provides annual financial assistance to first-year students on the basis of academic qualifications and financial need. The Gisinger-Steiner Memorial Scholarship was established by the family and friends of Scott and Mabel Gisinger and Ivan and Lillian Gisinger Steiner. The income from this fund is awarded annually to deserving students with financial need. The Marjory Steuart Golder Appreciation Scholarship was established in 1998 by Malcolm and Jean Malkin Boggs, members of the class of 1948. This scholarship honors the memory of Marjory Golder, who served as the College’s Dean of Women from 1946 to 1960, and is awarded annually to students with financial need. The Harold and Ruth Goldman Scholarship was established in their memory by the Fagans family. Both of the Goldmans were active in community service during their retirement years. The scholarship is awarded annually to a student with financial need who plans to major in the humanities and has demonstrated service to the community. The Arlo G. and E. Velma Graber Scholarship was established in 1984 by Mrs. Graber, class of 1924, and the late Mr. Graber, class of 1926. The award is made annuEndowed Resources 302 Endowed Resources ally to students who have achieved academic excellence and who have financial need. The Minnie K. and Errett M. Grable Scholarship Fund was established in their honor in 1982 by the Grable family of Pittsburgh, Pennsylvania. The income from this scholarship is awarded annually to students with financial need who have demonstrated qualities of leadership and academic development. The Roy I. Grady Scholarship was established in 2005 by his former student, Ellyn Palmer Jones, a graduate of Wooster in the class of 1955, and by his daughter, Dorothy Grady Bland, from the class of 1939. Roy Grady was a 1916 graduate of the College and served as professor in the Department of Chemistry from 1923 until 1959. Dr. Grady was part of a family legacy at the College that can be traced back to an uncle, U.L. Mackey, from the class of 1891. Known for his wit, genuine care and concern for students, and a life-long love of learning, Dr. Grady gave valuable leadership to the department and served as a trusted mentor for students seeking careers in the sciences. Income from this fund is awarded annually to a junior or senior with financial need who is majoring in chemistry. The Frances Guille-Secor Memorial Fund was established in 1975 in memory of Dr. Guille-Secor, class of 1930, who served The College of Wooster for thirty years as a teacher in the Department of French. Dr. Guille-Secor had a keen interest in French language and civilization and in high standards of excellence for the generations of students she taught at the College. The fund provides scholarship income for a student who has demonstrated high competence in French language, literature, and civilization, as well as in other studies. The Gurney Family Scholarship Fund was established in 2002 by a gift from the Gurney family, in honor of the nine members who attended the College. The scholarship is awarded annually to a student needing financial assistance. The Gustafson Scholarship Fund honors the memory of Mr. and Mrs. V. E. Gustafson, long-time friends of the College. The fund was established in 1971, and recipients will be selected by the financial aid officer on the basis of financial need. The Margaret E. Hadley Endowed Scholarship was established in 2007 by a bequest from Margaret Hadley, a member of the class of 1940. Income from the fund is awarded annually to students with demonstrated financial need. The Ethel R. and Homer E. Haines Scholarship Fund was established in 2002 by a bequest from Ethel R. Haines, a member of the class of 1926. Tuition grants are awarded for one year to rising sophomores, juniors, and seniors in good academic standing. A recipient of a one-year grant may reapply for one additional year if he or she remains in good academic standing at the College. The Hanke Family Scholarship was established in 2007 by the Paul G. Duke Foundation of Troy and Columbus, Ohio. The scholarship honors the long term service of Paul Hanke ’58 to the Foundation, his devotion to the merits of a liberal arts education, and the 50th anniversary of his graduation. Three generations of the Hanke Family have called Wooster home. Paul’s mother, Frances, graduated from Wooster in 1926. His sons, Mark and Doug, extended the family legacy and graduated from the College in 2004. Income from the fund is awarded annually to a junior or senior student or students who exhibit academic achievement and demonstrate active participation in the “Wooster Experience.” First preference should be given to students with a major or minor in chemistry, biochemistry, or economics. The Dwight C. Hanna Scholarship was established in 2008 from the proceeds of a gift annuity from Dr. Dwight C. Hanna III, a member of the class of 1944. Dr. Hanna served as an Alumni Trustee from 1968-1972 and in 1982 joined Wooster’s Board of 303 Endowed Resources Trustees on which he served as an Active Trustee from 1982-1996 and as an Emeritus Trustee from 1996 until his death in 2007. Dr. Hanna received a Distinguished Alumni Award in 1974 in recognition of his accomplishments as a plastic and reconstructive surgeon and as a humanitarian. Income from the fund is awarded annually to students who have financial need, with first preference given to students who are planning careers in medicine and/or in public health. The Hauschild Family Scholarship was established in 2006 by a gift from Lester P. Hauschild in recognition of the 50th anniversary of the class of 1957 and Independent Minds: The Campaign for Wooster. Mr. Hauschild served on the Board of Trustees as an Alumni Trustee from 1996-2002 and as chair of the Class of 1957 Fiftieth Anniversary Committee. The scholarship honors the memory of his parents, Margaret McKee Hauschild ’22 and Lester P. Hauschild, Sr. Income from the fund is awarded annually to students with financial need, with first preference given to students from Lawrence County or Western Pennsylvania. Additional preference is given to students majoring in economics who have demonstrated success in extracurricular activities and service to others. The Phyllis Johnson and William H. Havener Endowed Scholarship Fund was established in 1986 by Dr. and Mrs. William H. Havener. The income from the endowed fund is used for scholarships to students with financial need who have a record of academic achievement and a demonstrated capacity for leadership. The Paul and Eillene McGrew Hawk Scholarship Fund was established in 1998 by a gift from David and Patricia Hawk Clyde of the class of 1950. The scholarship honors the memory of Mrs. Clyde’s parents, Paul C. Hawk of the class of 1925 and A. Eillene McGrew Hawk of the class of 1926, both loyal and dedicated alumni of the College. Paul served as President of the Alumni Association from 1951 to 1953. In 1962, they were the first couple to receive Wooster’s Distinguished Alumni Award. The scholarship is awarded annually to a student with financial need. The William Randolph Hearst Endowed Scholarship Fund was established in 1988 by a grant from the William Randolph Hearst Foundation. Income from the Fund is awarded annually to minority students at Wooster who have financial need. The Helen M. Heitmann Scholarship Fund was established in 2006 through a bequest from Helen Heitmann, a member of the class of 1948. Income from this fund is awarded to students with demonstrated financial need who are pursuing a course of study in anatomy, physiology, neurology or related disciplines, with special emphasis on human movement. The Heitman-Goetter Scholarship was established in 1999 in loving memory of Karl William Goetter and Viola Heitman Goetter, grandparents of an alumna from the class of 1993 and an alumnus from the class of 1994. This scholarship is awarded annually to a rising senior majoring in history, foreign language or music. The recipient must demonstrate academic excellence, be recommended by the department of his or her major, and demonstrate financial need. The Hendrickson Family Scholarship Fund was established in 1959 to honor the memory of The Reverend Edward S. Hendrickson. It was augmented in 1986 by gifts from the family and friends in memory of Jane Leber Hendrickson ’52. The scholarship is awarded to students who have financial need. The Herr Family Scottish Arts Scholarship was established in 2007 by a gift from James W. Herr. The endowed award honors Mr. Herr’s parents, Wesley and Margaret McMurray Stanners Herr, and his brother, Thomas J. Herr, a member of the class of 1971. The scholarship is awarded annually to a student who plays the bagpipes and who participates in the Pipe Band at the College. First preference is given to students in good academic standing who also have demonstrated financial need. Endowed Resources The William P. Hilliker Scholarship Fund was established in 2001 by a bequest from William P. Hilliker, a member of the class of 1925. This scholarship is awarded annually to a member of the junior or senior class who has demonstrated financial need. First preference will be given to students in good academic standing who are also actively involved in extra-curricular activities and campus life beyond the classroom. The Helen M. Hoagland Endowed Scholarship Fund was established in 1998 through the generous bequest of Miss Hoagland, a member of the class of 1925. Income from the Fund is awarded annually to students based upon education, talent and need. The Marge and Larry Hoge Endowed Scholarship Fund was established in 2007 by Margery Neiswander Hoge, a member of the class of 1947, in memory of her husband, Lawrence A. Hoge, a member of the class of 1949. Income from the fund is awarded annually to students with demonstrated financial need. Recipients are selected by the College and The Women’s Advisory Board. The Beatrice and Ernest M. Hole Scholarship honors Beatrice Beeman Hole, a member of the Class of 1919, and Ernest “Mose” Hole, a member of the class of 1918. Mr. Hole taught and served as a coach and Athletic Director at Wooster for 46 years. The income from this scholarship is awarded to students with financial need. The Alice Joanne Holloway Fund was established by her parents, Mr. and Mrs. Harold S. Holloway, in 1981. The endowed fund honors Ms. Holloway, a graduate of the class of 1954 at Wooster. Income from the fund is awarded annually to entering students who have demonstrated outstanding academic achievement and financial need. The Donald P. Holloway Scholarship Fund was established in 2002 by Mr. Holloway, a friend of the College, as a “living memorial” to the importance of education in the liberal arts and sciences. Income from the fund is awarded annually to sophomore students who have demonstrated outstanding academic achievement and financial need, with first preference given to students who received the Alice Joanne Holloway Scholarship in their first year at Wooster. The Dorothy G. Holloway Scholarship was established by gifts and a bequest from Donald P. Holloway, a friend of the College, in memory of his mother. Income from the fund is awarded annually to junior students who have demonstrated outstanding academic achievement and financial need. First preference is given to students who received the Alice Joanne Holloway Scholarship and/or the Donald P. Holloway Scholarship during their first or second year at Wooster. The Harold S. Holloway Scholarship was established by gifts and a bequest from Donald P. Holloway, a friend of the College, in memory of his father. Income from the fund is awarded annually to senior students who have demonstrated outstanding academic achievement and financial need. First preference is given to students who received the Alice Joanne Holloway, Donald Holloway, and/or Dorothy Holloway Scholarships during their first, second, or third year at Wooster. The Mildred Eckert Hommel Student Aid Fund was established in 1968 by a bequest of Mildred Eckert Hommel of Cleveland, Ohio. Income from the fund is awarded as student aid grants by the Scholarship Committee on the basis of scholarship, Christian character, outstanding qualities of leadership, and financial need. The W. Dean and Harriet P. Hopkins Scholarship Fund was established in 1989 by Mr. Hopkins’s law firm — McDonald, Hopkins, Burke & Haber Co., L.P.A. — to honor his many years of service and leadership. Alumni and friends also contributed to this endowed fund. Mr. Hopkins served on Wooster’s Board of Trustees for 50 304 Endowed Resources years. The scholarship also honors Mr. Hopkins’ wife, Harriet Painter Hopkins, a member of the class of 1932. The scholarship is awarded to a student who has financial need and exemplifies the scholarly qualities that Mr. Hopkins demonstrated throughout his life. The C. Dale and Mary A. Horner Scholarship was established in 2010 by a bequest from Betty J. Horner, a member of the class of 1943, in memory of her parents, C. Dale Horner and Mary A. Horner. Income from the scholarship is awarded annually to a female pre-med or chemistry major who has demonstrated financial need and high moral character. The Frank C. Howland Scholarship Fund was established in 1963 by Mame E. Howland, in loving memory of her husband. It was endowed in 1974 through a bequest from Mrs. Howland’s estate, and since then gifts have been added through the Howland Memorial Fund. The recipients are selected by the financial aid officer of the College. The Lois Howland Memorial Scholarship Fund is the gift of Mr. and Mrs. Fred B. Howland of Titusville, Pennsylvania, in memory of their daughter, Lois Howland. The income of this fund is used for scholarships for not fewer than three women each year. Selections are made by the Scholarship Committee based on scholarship, Christian character, outstanding qualities of leadership, and financial need. The Jean Waterbury Howlett Endowed Scholarship was established in 1988 to recognize Mrs. Howlett’s devotion to the College and its students and her years of service on The Women’s Advisory Board. The scholarship is awarded annually to students with superior academic records who have financial need, with preference given to students of the humanities. Recipients are selected by the College and The Women’s Advisory Board. The G. Pauline Ihrig Fund in French was established in 1970 to honor Dr. Ihrig’s 47 years in the Department of French. She served as Chairperson for 14 of those years. In 1990, the fund was increased through a bequest from Dr. Ihrig and became a scholarship. The scholarship is to be awarded annually to a student who has demonstrated both academic achievement and financial need. First preference is to be given to a student who is studying French and who has demonstrated outstanding scholarship in the language. The International Paper Endowed Scholarship Fund was established in 1999 through the generosity of the International Paper Company Foundation. Income from this fund is awarded annually to a student who has demonstrated academic achievement and who has financial need. The Annie B. Irish Scholarship Fund was established in 1994 by The Women’s Advisory Board in honor of Annie B. Irish (1857-1886). Dr. Irish was the first woman to receive a Ph.D. degree from the University of Wooster and was Wooster’s first female professor. This scholarship is awarded annually to a young woman who has financial need and displays qualities of general excellence. The Mary Z. and Rachael Johnson Memorial Fund was established by the family and friends in honor of Mary Z. Johnson, Professor of Political Science, 1926-1955, and her sister, Rachael Johnson. The income from this fund provides a student aid grant to a student who is majoring in one of the social sciences. The Willard H. Johnson Family Scholarship was established in 2000 by a gift from Willard Johnson, class of 1966. Income from the fund is awarded annually to a student who has demonstrated academic achievement and financial need. The Herrick L. Johnston Scholarship in Chemistry honors the memory of Herrick L. Johnston, class of 1922, Sc.D. 1943, and was established by Margaret 305 Endowed Resources Vanderbilt Johnston Dettmers in 1982. Income from the fund is awarded annually on the basis of merit to an incoming first-year student who plans to major in chemistry. The John C. and Marie W. Johnston Endowed Scholarship Fund was established in 1999 by Johnston family members and friends to honor John and Marie’s involvement in the life of the College. John, a graduate from the class of 1938, received Wooster’s Distinguished Alumni Award in 1963 and served on the Board of Trustees from 1977-1989. Marie was a member of The Women’s Advisory Board from 1981 until her death in 2003. Scholarship awards are made annually to one or more students who have demonstrated academic achievement and financial need. The Richard and June Johnston Endowed Scholarship Fund was established in 2009 from the proceeds of several planned gifts by June Weber Johnston to honor the memory of her husband. The Johnstons were members of the class of 1940. Income from the fund is awarded annually to first-year students who demonstrate financial need. The Walter O. and Anna Jones Endowed Scholarship was established in 1999 by their daughter, Elizabeth “Betsy” Jones Hayba, class of 1949. The scholarship is awarded to students in good standing who have demonstrated financial need, with first preference given to the extended family of Elizabeth “Betsy” Jones Hayba. The Gregory Paul Julian Scholarship was established in 1998 by Colonel Russell E. Julian, class of 1941, and Jeanne E. Julian to honor the memory of their son. The scholarship is awarded annually to a student with financial need. First preference is given to students from military service families. The Sally Comin Kaneshige Scholarship Fund was established in her memory in 1975 by her family and friends. Mrs. Kaneshige was a 1955 Wooster graduate and at the time of her death a member of the Ohio University music faculty. The income from this fund is used each year to help worthy students, with preference shown to those majoring in music. The Harry A. and Eva K. Kauffman Scholarship was established in 1986 by their son, John H. Kauffman. The scholarship is awarded to children of employees of Kauffman Tire Service, Inc. The Kauffman Organization has stores in Georgia, Florida, and Ohio. Eligible students must have financial need and demonstrate academic excellence. The Ethel J. Keeney Scholarship Fund was established in 2001 through a generous bequest from Miss Keeney, a member of the Class of 1924. This scholarship is awarded annually to students who demonstrate financial need. The Carolyn Verlie Kent Scholarship Fund was established in 1991 by E. Joseph and Elizabeth Verlie in memory of their daughter, Carolyn Verlie Kent, a Phi Beta Kappa member of the class of 1976. Carolyn was a civic leader in Greater Cleveland until her death in 1988. This scholarship gives first preference to African American students from Cuyahoga County. The KeyBank Scholarship Fund was established in 1997 through the generosity of KeyBank in Cleveland, Ohio. Income from the Fund is awarded annually to a student who has demonstrated financial need. The Frances H. and Frank R. Kille Student Assistance Fund was established in 2006 from the proceeds of a planned gift and honors the memory of Frances H. Kerby Kille, a member of the class of 1926, and her husband, Frank R. Kille, a member of the class of 1926 and a 1954 honorary degree recipient. Income from the fund is awarded annually to students who have demonstrated financial need and who have completed their first year at Wooster. 306 Endowed Resources The William M. Kittredge and the Louise Irwin Kittredge Clark Scholarship was established from the proceeds of a gift annuity by Louise Irwin Kittredge Clark, a member of the class of 1928. The scholarship honors the memory of Mrs. Clark’s first husband, William M. Kittredge ’28, and is awarded to students of good character who have demonstrated academic achievement and financial need. The Paul Evans Lamale Scholarship in the Social Sciences was established in 1947 by Charles E. Lamale, a member of the class of 1907, and Mrs. Lamale in memory of their son, a member of the class of 1941 and a U.S. Marine Corps aircraft pilot and captain who fell in the American attack at Rabaul, New Britain, on January 30, 1944. The scholarship is awarded annually to a student of outstanding personal worth majoring in one of the social sciences, as an aid to completing the senior year, and is applied against the charge for tuition. The Barbara Burkland Landes Endowed Scholarship was established in 1997 by Mrs. Landes, class of 1941. This scholarship recognizes her longtime membership on The Women’s Advisory Board and is awarded annually by the Board and the College to a young woman who has financial need and demonstrates academic promise. The David Goheen Leach Memorial Scholarship Fund was established in 1998 from the estate of David G. Leach, class of 1934 and Sc.D. 1966. Mr. Leach was a plant geneticist and a leading authority on rhododendrons. This scholarship is awarded annually with preference given to a male biology major who has an interest in botany or horticulture. The Delbert G. Lean Memorial Scholarship was established in June 1971 through the generosity of the members of the class of 1921. Dr. Lean was a professor in the Department of Speech from 1908 until he retired in 1946. Known for his oratory, Dr. Lean gave valuable leadership to the department, serving as chairman for 38 years. The income from this fund is used for scholarships for deserving students with preference given to a student majoring in speech but not necessarily limited thereto. The Henry Lee Scholarship Fund was established in 1987 by C. Dennis and Margaret Lee Scott ’69 to honor Mrs. Scott’s father, a 1936 graduate of the College and one of the first students from China to attend Wooster. The scholarship is awarded annually to international students with financial need. The Janice Lynn Potter Lee Endowed Scholarship was established in her memory by her father, John J. Potter. A graduate of the class of 1973, Janice Lynn Potter majored in French and later was a secondary-school teacher of French and Spanish. Income from the scholarship is awarded annually to students who have financial need, with first preference given to students who are majoring in French. The Abraham Lincoln Memorial Scholarship was established in 1964 by a committee representing students, faculty, and members of Westminster Presbyterian Church. It is awarded annually, and will normally be given to an African American student on the basis of need and ability. The Maxine R. Loehr Piano Scholarship is awarded annually by the Scholarship Committee to a promising piano student. Evidence of financial need is also considered in making the award. The Longbrake Scholarship was established in 1990 by Martha and Bill Longbrake ’65, in honor of the many members of the Longbrake and Barr families who have attended The College of Wooster. The scholarship is awarded to a first-year student and is renewable for four years. The award is made on the basis of demonstrated academic achievement, success in extracurricular activities, and service to others. First preference will be given to children or grandchildren of Wooster alumni. 307 Endowed Resources The Howard Lowry Scholarship was established during The Campaign for Wooster and honors the memory of Howard Lowry, who was the President of the College from 1944 to 1967. The scholarship is awarded to a student or students from the greater Cleveland area. The Lewis and Daisy V. Lowry Scholarship Fund was established by their son, Dr. Howard F. Lowry ’23, seventh President of The College of Wooster. It provides scholarships that are awarded annually by the Scholarship Committee. The Gertrude Lum Scholarship was established in 2001 through a bequest from Gertrude Sheva Lum, a member of the class of 1950. Dr. Lum conducted viral research for the World Health Organization in Asia and South America from the 1960s to the 1980s. Income from this scholarship is awarded to students with demonstrated financial need. First preference is given to students majoring in biology. The Sara Wishart MacMillan Scholarship Fund was established in 1987 in her memory by her family and friends. A daughter of Wooster’s sixth president, Charles F. Wishart, Mrs. MacMillan graduated from the College in 1932 and had a lifetime interest in and devotion to Wooster. The scholarship is awarded annually to students with financial need, with preference given to students with promise in speech communication. The Mariska P. Marker Scholarship Fund was established in 2005 by Mariska P. Marker in honor of Pamela Frese, Wooster’s Professor of Anthropology, and in recognition of Dr. Frese’s excellent teaching and her sincere interest in her students’ welfare. The recipient of the scholarship is selected during his or her junior year and awarded the scholarship in his or her senior year. The recipient must be an anthropology major, have a grade point average of 3.8 or higher, and plan to further pursue the study of anthropology, either professionally or in graduate school, after graduation. In the event that more than one student is qualified for this scholarship, first preference shall be given to the student who has the greatest demonstrated financial need. The Alexandra Babcock Marshall Scholarship honors the memory of Alexandra Babcock Marshall, class of 1935. Born in Russia, she fled Communism to come to America in 1922. This scholarship is awarded annually with first preference given to a student who is studying Russian language, culture or history. The Anne Mayer Music Scholarship was established by Anne Mayer in honor of the 50th anniversary of her graduation from Wooster. Anne Mayer, a member of the class of 1957, blossomed under the guidance of faculty members Stuart Ling and Richard Gore. She shared her own passion for music and genuine care with a generation of students at Carleton College, where she was a member of the music faculty. Income from this fund is awarded annually, with first preference given to students majoring in music. The Eleanor H. and Bernard A. Mazurie Scholarship Fund was established in 2004 by a bequest of Eleanor Herold Mazurie. The scholarship honors the memory of Eleanor and her husband Bernard Mazurie, a member of the class of 1927. Income for the fund is awarded annually to students in good academic standing who demonstrate financial need. The McClenahan Scholarship Fund was established in 1994 by estate gifts from Sallie Phillips McClenahan, a former trustee of the College, and her husband, Robert Wallace McClenahan. The fund honors the memory of their parents, ZeBarney T. and Sallie Hews Phillips and Robert Stewart and Jeanette Wallace McClenahan. The Harrold and Hazelyn Melconian McComas Scholarship Fund was established in 1999 by gifts from the Hazelyn and Harrold McComas Charitable Trust and 308 Endowed Resources 309 the Melitta S. Pick Charitable Trust. This scholarship honors Hazelyn Melconian McComas, a Phi Beta Kappa graduate of the class of 1948, and her husband, Harrold J. McComas. Income from this fund is awarded annually to a student who demonstrates financial need, with first preference given to a major in history, religion, political science, English, or speech. The J. Robert and Abigal Welch McConnell Scholarships were established in 1979 through gifts from Mr. McConnell, a friend of the College. They are awarded annually to worthy students in pursuit of education at Wooster, with first preference given to pre-medical students. The Richard and Margaret Stockdale McCoy Scholarship is awarded by the Director of Financial Aid upon the recommendation of the Office of Admissions to an incoming student with strong academic credentials and a notable record of achievement in high school activities. The scholarship is based upon merit and is awarded annually to a student who might otherwise not have chosen to attend the College. The award is made available for each of the student’s four years at Wooster on the condition of satisfactory progress toward a degree. The Robert and Billye McCracken Endowed Scholarship Fund, established in 2003 from the proceeds of several planned gifts, honors Dr. Robert L. McCracken, a member of the class of 1934, and his wife Billye Newman McCracken. Income from the fund is awarded to students in good standing who demonstrate financial need. First preference is given to students who plan careers in medicine. The McCullough Scholarship was established in 2004 by Hugh McCullough, a member of the class of 1954, and honors the memory of his parents, Martin McCullough ’26 and Evrell Bennett McCullough ’28. Income from the scholarship is awarded annually to a junior or senior who is a U.S. citizen with demonstrated financial need. First preference is given to a student majoring in economics, music, or chemistry who is a member of the Scot Marching Band. The Elizabeth J. McElhinney Scholarship Fund is the gift of Elizabeth J. McElhinney Hay and Walter M. Hay. Established in 1960, the fund provides scholarship assistance for both male and female students. The Albert Gordon McGaw Memorial Scholarship was established by Wilbert H. McGaw in memory of his father. Recipients are selected by the College and The Women’s Advisory Board. The Marion M. and Ellen M. McGrew Scholarship was established in 1991 by Mac and Pat McGrew of the classes of 1929 and 1931. The scholarship is awarded annually to students who have demonstrated financial need and high academic achievement and who are majoring in one of the natural sciences. The Ola Weygandt McKee Scholarship Fund was established by Dr. Milton C. Oakes, of Mansfield, Ohio. Dr. Oakes studied Latin under her instruction during her first year of teaching. Mrs. McKee and her husband John D. McKee, both from the class of 1917, supported the College in many ways during their lifetimes. The scholarship is awarded to students who demonstrate financial need. The Robert A. McMillan Scholarship was established in 1999 with a gift from Mary Elizabeth Remsburg, class of 1946, in honor of her brother Robert, class of 1947. Income from the fund is awarded annually to a junior or senior who excels in music performance. Preference is given to a student who plays a keyboard instrument, with first preference given to a student who excels in organ performance. The Homer G. and Della W. McMillen Scholarship Endowment Fund was established through a gift by Mr. and Mrs. McMillen. Preference is given to students Endowed Resources 310 from Weirton, West Virginia, St. Clairsville, Mt. Vernon, and Worthington, Ohio. The Mary Bonsall Mikkelsen Endowed Scholarship was established in 2005 from the proceeds of a planned gift and honors the memory of Mary Bonsall Mikkelsen, a member of the class of 1942. Income from this scholarship is awarded annually to students who have demonstrated financial need. The Jean Pollock Milburn and Joseph W. Milburn Fund was established in 1968 by a bequest of Martha P. Milburn ’42, of New York City, in honor of her mother and father. Scholarships provided by the fund are awarded by the Committee on Scholarships and Student Aid, with preference given to students of music. The Don J. Miller Memorial Fund was established by the family and friends of Don J. Miller, class of 1940. In recognition of Mr. Miller’s devotion to the science of geology, the scholarship is awarded annually to a student who is majoring in geology and who demonstrates both scholarly ability and financial need. The Emerson W. Miller and Garnett Miller Smith Memorial Scholarship was established in 1993 by Virginia Miller Reed, class of 1945, in memory of her parents. The scholarship is awarded annually by the Director of Financial Aid to a member of the first-year class who has a strong record of academic achievement and participation in co-curricular activities. The Fannie and Rollie Miller Memorial Fund Scholarship was given in memory of his parents by their son, Dr. Robert C. Miller. The income from this fund is used for scholarships for worthy students selected by the Scholarship Committee. The James A. and Mary Alice Lehman Miller Scholarship Fund was established in 1994 by estate gifts from James A. Miller and his wife, Mary Alice Lehman Miller. Dr. Miller grew up in China and was a cum laude graduate of the College in the class of 1928. He received an honorary degree from Wooster in 1962 and the Distinguished Alumni Award in 1977. The Willis W. and Mildred S. Miller Scholarship Fund was established in December 1974 by Mr. and Mrs. Miller. The scholarship is awarded annually, with first preference given to young people from River View High School in Warsaw, Ohio. The selection is made by the high school principal and staff and the financial aid officer at The College of Wooster. The Thomas J. Mills Scholarship Fund was established in 2005 from the proceeds of several planned gifts and honors the memory of Thomas J. Mills, a member of the class of 1930. Income from this scholarship is awarded annually to students in good standing who have demonstrated financial need, with first preference given to students majoring in economics. The Charles Burdette and Margaret Kate Moke Scholarship Fund was established in 1984 by Frederick H. Kate ’38 in recognition of their long service to Wooster and sincere interest and concern for students and alumni. Dr. Moke graduated from Wooster in the class of 1931 and taught geology at the College for 36 years. The scholarship is awarded annually, with first preference given to a geology major who has demonstrated self-reliance and dedication to academic achievement. The Kathleen McNiece Moore Scholarship in Flute and Voice is given in memory of Jessie and Wilder Ellis in recognition of their long association with The College of Wooster and their deep interest in students of flute and voice. The scholarship is awarded to a student who is in his or her junior year and is specializing in flute or voice. The Morley-Hall Endowed Scholarship Fund was established in 2010 by the William and Ruth Urban Estate and a gift from Elizabeth and Joseph Morley. Ruth Endowed Resources 311 Hall Urban was a member of the class of 1936, and Joseph and Elizabeth Burton Morley are members of the class of 1968. Income from the fund is awarded to one or more full-time residential students who have demonstrated financial need, are United States citizens, and are in good academic standing at the College. First preference is given to students whose families are in the ministry or teaching professions. The Isabell Demboski Moses Endowed Scholarship was established in 2006 by the Sam and Kathy Salem Philanthropic Fund and honors Isabell Demboski Moses, a member of the class of 1990. Income from the fund is awarded annually to a student majoring in economics or business economics who demonstrates significant financial need. The James B. Munson Memorial Scholarship was established in 1995 by his family and friends. A member of the class of 1939, he was a participant in intercollegiate athletics and was the son of Carl B. Munson, coach at Wooster for 41 years. The scholarship is awarded to a student with financial need, with first preference given to graduates of Wooster High School. The Nell Murray Scholarship Award and The Arthur Murray Scholarships were established through gifts from Mr. and Mrs. George Pope of the class of 1941. Each scholarship is awarded to a deserving student, with first preference given to young people from Wooster and Wayne County. The Marilyn Myers Memorial Scholarship Fund honors the memory of Marilyn Myers of the class of 1972. She died during her sophomore year, and her family and many friends established this memorial fund. Selection of the recipient is made by the Scholarship Committee. The Neill Family Scholarship was established in 1984 by Ronald H. Neill, class of 1966. The scholarship is to be awarded to one or more students from the Greater Cleveland-Akron area. The recipient(s) should demonstrate superior academic achieve ment, and consideration shall be given to non-academic or extracurricular activities or interests in awarding this scholarship. The David H. Nelander Scholarship Fund was established in 2000 by his wife, Julie Talbot Nelander, a member of the class of 1960. Dr. Nelander graduated from Wooster in 1959 as a chemistry major. He served as a member of the Board of Trustees from 1994 until his untimely death in April 2000. First preference is given to a student planning to major in chemistry. The John J. Newberry Endowed Scholarship was established in 2004 with a gift from John J. Newberry, a Wooster parent and member of the Board of Trustees for 42 years. Income from this fund is awarded to students who demonstrate financial need. The Donald L. Noll Book Scholarship was established in 1999 by Jane Noll and other family members and friends. It honors the memory of Don Noll, who served as Manager of the College Bookstore from 1968-1983. Income from the Fund is awarded to students with financial need to assist them in purchasing books from the Florence O. Wilson Bookstore to be used in their academic courses. The Lucy Lilian Notestein Endowed Scholarship Fund was established in 1977 by DeWitt Wallace in honor of Miss Notestein, a member of the class of 1911. Miss Notestein, for many years an editor with Reader’s Digest, served as a trustee of the College for thirty-two years before becoming an Emeritus Life Member of the Board in 1972. This scholarship is awarded annually to students who have attained at least sophomore standing. The Helen K. and Ernest S. Osgood Scholarship Fund was established in 1984 by Helen Kaslo Osgood, a member of the History Department for thirty years beginning Endowed Resources 312 in 1951. In 1958 she married Ernest S. Osgood, Emeritus Professor of History, University of Minnesota. His activities at Wooster included advising Independent Study students in history, which gave him much pleasure. The scholarship is awarded annually, in consultation with the Department of History, to a junior or senior history major who has demonstrated high academic achievement and financial need. The Stephen E. and Katharine Greenslade Palmer Memorial Scholarship was established in 1993. The scholarship honors the memory of the Reverend Dr. Stephen E. Palmer, Sr., class of 1917, and his first wife, Katharine Greenslade Palmer, class of 1919. Dr. Palmer served forty-nine years as a Presbyterian minister and held pastorates in Wisconsin, Ohio, Wyoming, and New York. He received an honorary Doctor of Divinity degree from the College in 1942. The Pancoast Family Scholarship Fund was established in 1995 by the Pancoast family. This fund recognizes the long ties the family has had with the College: John R. ’35 and Katherine Wick ’50, David W. ’64 and Carol Stromberg ’64, and David Wick, Jr. ’93. The income from this fund is used to provide scholarship assistance to a student who has demonstrated financial need with preference to a student who is from at least a second generation Wooster family. The Ruth Frost Parker Endowed Scholarship was established in 1987 by Mrs. Ruth Parker ’45 of Sandusky, Ohio. The income from this fund is awarded annually by the Director of Financial Aid to worthy students who have demonstrated financial need. The Parkhurst Family Scholarship was established in 1944 with a gift from Jeannette Parkhurst ’31. This scholarship honors her parents, Ralph and Myrtle Williams Parkhurst of Bellevue, Ohio. The income from this fund is awarded annually by the Director of Financial Aid to students who have achieved high academic standards and who have demonstrated financial need. The William Parsons Scholarship was established in 1984 by Margaret Parsons Critchfield ’27 and Katherine Parsons Junkin ’23 in memory of their father, William Parsons, D.D. Dr. Parsons was born in England as one of eleven children, was a cowhand on the Western range as a youth, and later homesteaded in Kansas. As a Presbyterian minister, he held pastorates in Oregon, lowa, Ohio, Pennsylvania, and New Jersey and insisted that his daughters attend The College of Wooster. Preference for this scholarship is given on the basis of financial need and academic promise to the sons and daughters of Presbyterian ministers. The William Albert Patterson Endowed Scholarship was established in 2005 by a bequest from William A. Patterson, a member of the class of 1969. An avid reader and historian, Mr. Patterson was an ordained Presbyterian minister from 1975 until his death in 2003. Income from the fund is awarded to students with demonstrated financial need. The Sara L. Patton Performing Arts Scholarship and Activities Fund was established in 2007 by generous gifts from Richard J. Bell and David H. Schwartz, members of the class of 1963. The Fund honors Sally’s service to Wooster as Vice President for Development and her abiding love for the theatre. The scholarship portion of the Fund (80%) is awarded annually to students with financial need who participate in theatre or dance. The activities portion of the Fund (20%) is used to support special opportunities in theatre that may be beyond the reach of normal operating budgets. The Theron L. Peterson and Dorothy R. Peterson Scholarship for Outstanding Academic Achievement was established in 2008 in memory of Theron L. Peterson, a member of the class of 1936, and his wife, Dorothy R. Peterson. Income from the fund is awarded annually to a junior or senior student majoring in biology, chemistry, Endowed Resources 313 physics, or mathematics on the basis of his or her academic performance. First preference is given to a student with both a very high academic achievement and demonstrated financial need. The Robert H. and Susan M. Pfeil Fund was established in 1991 through a bequest from Mr. Pfeil. After a business career in Cleveland, Ohio, Mr. Pfeil retired to Wayne County. He believed in private, church-related education and wanted to provide scholarships for future generations of young people who attend The College of Wooster. Scholarships are awarded on the basis of need and academic credentials. The J. Robert and Mary P. Pfouts Scholarship was established in 2008 through a generous bequest from Mr. and Mrs. Pfouts. Bob Pfouts was a member of the class of 1933. Income from the fund is awarded annually to students who are in good academic standing and have demonstrated financial need. The John Plummer Memorial Scholarship for Contributing to a More Welcoming Campus for LGBT People was established in 2008 through the support of Hans P. Johnson, a member of the class of 1992, and many other friends and colleagues of John B. Plummer. John was a member of the class of 1964 and served in the College’s business office for 36 years. He also worked closely with students as treasurer of Campus Council and as a mentor to the College’s lesbian, gay, bisexual and transgender (LGBT) student community. In his memory, income from the scholarship is awarded annually to a sophomore, junior, or senior student who promotes a more open and respectful campus environment, regardless of the student’s sexual orientation. Recipients will be selected based upon the recommendation of the scholarship’s advisory committee. The Bess and Eugene Pocock Scholarship was established during The Campaign for Wooster and honors the memory of Bess Livenspire Pocock, class of 1912, and Eugene W. Pocock, class of 1911. Mrs. Pocock taught school in Shaker Heights and was a recognized speaker; the Reverend Mr. Pocock served Calvary and Noble Road Presbyterian Churches in Cleveland. The scholarship was established by their son, John W. Pocock, class of 1938, a member of Wooster’s Board of Trustees for 35 years; he served as Chairman of the Board for 17 years. The scholarship is awarded to a student or students from the greater Cleveland area. The John William Pocock Endowed Scholarship Fund was established in 1992 by a gift from Booz-Allen & Hamilton, Inc., in memory of Bill Pocock. It honors Mr. Pocock’s career at Booz-Allen as well as his commitments and service to higher education. Income from the fund is awarded annually to a student who has demonstrated academic achievement and the promise of becoming a leader in his or her chosen profession. The Pocock Family Endowed Scholarship was established in 2006 by Elizabeth Snavely Pocock. This scholarship honors the memory of her husband, John W. Pocock, class of 1938, who was a member of Wooster’s Board of Trustees for 35 years and served as Chairman of the Board for 17 years. Income from the fund is awarded to students who have demonstrated academic achievement and financial need. The Katharine West Pratt Scholarship was established in 1996 by Beth and Mikael Salovaara in honor of Mr. Salovaara’s grandmother, a member of Wooster’s class of 1915. One or more scholarships are awarded annually on the basis of financial need and academic achievement, with preference given to African American students from Ohio, Oregon, or West Virginia. The Elizabeth Prestel Endowed Scholarship was established in 2004 by a bequest from Elizabeth Hainer Prestel, a member of the class of 1932. Income from this scholarship is awarded annually to students who are in good academic standing and have demonstrated financial need. Endowed Resources 314 The Charles and Elma Rapp Memorial Scholarship Fund was established in 1995 by a bequest from Florence Rapp Cowie, class of 1925, in memory of her parents. Income from the fund is awarded annually to female students majoring in language or the humanities who have demonstrated academic achievement and financial need. The Hans W. Regenhardt Scholarship was established in 1990 through a bequest from Mr. Regenhardt. A long-time resident of Wooster, he was a self-made entrepreneur who believed that the education process should continue throughout one’s lifetime. Scholarships will be awarded to highly qualified students who need financial assistance. The John D. Reinheimer Scholarship Fund was established in 2006 by a gift from his family in memory of Dr. Reinheimer, Professor in the Department of Chemistry from 1948 to 1985. Preference is given to students with strong academic records and demonstrated financial need. The Peter James Renfrew Scholarship was established in 2005 by Tracy and Joyce Kempf Renfrew in memory of their son, Peter J. Renfrew, a member of the class of 1982. Income from the scholarship is awarded annually to a student who is in good academic standing and who has demonstrated financial need. First preference is given to a student from River View High School in Warsaw, Ohio, or from Coshocton County, or from the State of Ohio, in that order of priority. The Paul H. Resch Scholarship Fund honors the memory of Paul H. Resch of Youngstown, Ohio. The income from the endowed Resch Fund is awarded annually to deserving students who require financial assistance to attend the College. The C. Kirk Rhein, Jr., Scholarship Fund was established in his memory in 1997 by his family and friends. Kirk, a member of the class of 1976, died in the crash of TWA Flight 800. The scholarship is awarded annually to students who have financial need and who have demonstrated academic achievement and the ethic of service to others. The Kate Risley Endowed Scholarship was established in 2002 by a gift from her family. Kate was a member of the class of 1997 who died in the summer prior to her senior year. The scholarship is awarded annually in her memory to a student sharing Kate’s personal qualities and interests. The Darel Jay Robb Scholarship was established in 1994 by the Robb family and many friends. The fund is a memorial to Darel Jay Robb, class of 1970; before his untimely death in 1994, Darel was a medical librarian at the University of Illinois. The income from this endowed scholarship is awarded to a student who demonstrates financial need and academic achievement. The John M. Robinson, M.D. Scholarship was established in 2001 and honors John M. Robinson, M.D., a member of the class of 1941. This scholarship is awarded to students who are preparing for a career in medicine and who demonstrate financial need. The Robinson-Saunders Family Scholarship is awarded annually based on academic achievement and financial need. The fund honors Margaret A. and James G. Robinson and their daughter Ruth Robinson Saunders, class of 1942, and was established in recognition of the value of a liberal arts education in developing character and intellect. The Ronald T. and Josette Rolley Endowed Scholarship Fund was established in 2009 by a gift from Ronald T. Rolley, a member of the class of 1959, and his wife, Josette Roling Rolley. Income from the fund is awarded to students with demonstrated financial need who are in good academic standing at the College. Endowed Resources 315 The Harry G. and Lucy A. Romig Memorial Scholarship Fund was established in 1990 by their children and grandchildren to recognize the Romigs’ dedicated service as Presbyterian missionaries in the Shantung Province of China. The scholarship is awarded annually to an international student, with preference given to students from the People’s Republic of China. The Jane C. Ross Endowed Scholarship was established in 2006 through a bequest from Jane C. Ross, a member of the class of 1942. Income from the fund is awarded to students who have demonstrated financial need, with first preference given to students majoring in sociology. The John William and Elizabeth Scott Roudebush Scholarships were established in 1978 by their son, Rex S. Roudebush. A friend of the College, Mr. Roudebush wished to honor the memory of his parents who were graduates of Wooster in the 1880s. The scholarships are awarded annually to deserving students, with first preferences given to those in pre-law and international studies. The Neille O. and Gertrude M. Rowe Scholarship honors Professor Rowe, who directed the Conservatory of Music and served as Memorial Chapel organist from 1914 to 1945. Mrs. Rowe taught music appreciation and other music courses at the College from 1915 until 1953. The scholarship was established by their daughter, Evelyn Rowe Tomlinson, class of 1931. The Rowe Scholarship is to be awarded to a student who demonstrates financial need and who is a music major. The Francis H. Rutherford Scholarship, established by a bequest from Mr. Rutherford in 1984, is awarded to one or more students on the basis of Christian character, leadership, financial need, and sense of community responsibility. Consideration is given to the student’s own efforts toward financing his/her college education. The George H. Rutherford Scholarships are given in memory of George H. Rutherford, an alumnus in the class of 1921 and trustee of the College from 1961 to 1966. Income from this fund is used for scholarships for one or more students on the basis of Christian character, leadership, financial need, and sense of community responsibility. Consideration is given to a student’s own efforts toward financing his/her college education. The Shirley Snider Ryan Endowed Scholarship was established in 2005 by Shirley Snider Ryan, a longtime member of The Women’s Advisory Board. Mrs. Ryan wished to honor the dedication of The Women’s Advisory Board through the creation of this scholarship. Income from the fund is awarded annually to one or more junior or senior women who are residents of Wayne County, Ohio, and who demonstrate financial need. Recipients are selected by the College and The Women’s Advisory Board. The Mary Sager and William Dean Sager Endowed Scholarship was established in 1988 by Alexander E. Sharp, class of 1923. This fund is used to provide scholarship assistance to students at the College with financial need. The John H. and Harriet Hurd Scheide Scholarships are the gift of Mr. John H. Scheide of Titusville, Pennsylvania, and grew out of the interest that Mr. Scheide had for years in the children of missionaries. The scholarships are awarded for one year to children of missionaries, ministers, and educators of the Presbyterian Church, U.S.A. Selections are based on high scholarship, outstanding qualities of leadership, and financial need. The William I. Schreiber Scholarship was established in 1998 by friends and family. It honors William Schreiber, Gingrich Professor of German, 1937-1975. Mr. Schreiber founded and directed the Wooster-in-Vienna program from 1960-1985. The Endowed Resources 316 scholarship is awarded to students who demonstrate financial need, with a first preference given to those who are majoring in German or plan to study during their school years in a German speaking country. The Scotland Family Scholarship was established in 2007 through a bequest from James Scotland, Jr., a member of the class of 1937. Income from the fund is awarded annually to a student who is in good academic standing and who has demonstrated financial need. The Scott Family Scholarship honors four Wooster women: Esther Scott Galloway ’20, Agnes Elizabeth Scott ’22, Eleanor Scott Evans ’25, and Mary Catharine Scott Hunt ’27. This scholarship is to be awarded annually to students who demonstrate financial need and academic excellence. The Robert Ellsworth Scott Scholarships were established in 1971 by the Louise Orr Scott Foundation. The income from this trust fund is awarded annually to deserving students, with preference given to student athletes with financial need from Cambridge or Guernsey County, Ohio. The Merton and Ruth Sealts Scholarship was established in 2000 through a bequest from Merton M. Sealts, Jr., a member of the class of 1937. The scholarship honors the memory of Merton M. Sealts, Jr., and his wife, Ruth Mackenzie Sealts, and is awarded annually to students with financial need. First preference is given to students majoring in English. The Helen Secrest Scholarship, was established by The Women’s Advisory Board in memory of Helen Secrest. The scholarship is awarded annually to a young woman on the basis of scholarship, potential leadership, and need for financial assistance. The Helen Colville Sevitts Scholarship Fund was established in 1986 by her sister, Ruth Colville Stewart. The scholarship will be awarded to students who demonstrate financial need, and first preference will be given to students who are studying foreign languages. The Gretchen H. Shafer Endowed Scholarship was established in 2007 by Gretchen Shafer, a member of the class of 1949. Income from the fund is awarded annually to students who demonstrate academic achievement and financial need. First preference should be given to students who are graduates of Brighton High School in Rochester, New York. Second preference should be given to graduates of other Monroe County high schools. Should no qualified students be available from Monroe County, the scholarship may be awarded to any qualified student at the College. The Coe Shannon Endowed Scholarship was established in 2007 with proceeds from an annuity by Coe Shannon, a member of the class of 1949. Income from the fund is awarded annually to students who have demonstrated financial need, with first preference given to students majoring in English. The Jack and Betty Shuster Scholarship Fund was established in 2007 by Betty and Jack Shuster, a member of the class of 1959. Income from the fund is awarded annually to students in good academic standing with demonstrated financial need who are involved in co-curricular activities. The Dale and Frances Shutt Scholarship Fund was established in 1992 by Frances Shutt. Mr. and Mrs. Shutt were longtime residents of Wooster, and their daughter, Frances E. Pratt, is a graduate of the College. The scholarship is awarded to students with financial need. The Margaret Skinner Scholarship Fund was established in 1991 through a generous bequest from her adopted daughter, Emma Kish Skinner, class of 1927. Income Endowed Resources 317 from the fund is awarded annually to students who have demonstrated qualities of leadership at Wooster and who have financial need. The William E. and Maryan Fuhrman Smith Scholarship was established in 1982 by Maryan Fuhrman Smith, a member of the class of 1938, and William E. Smith during The Campaign for Wooster. This scholarship is awarded to a student or students from the greater Cleveland area. The J. M. Smucker Scholarship is the gift of The J. M. Smucker Company and members of the Smucker family. It is awarded each year to a member of the incoming class who has financial need. Other criteria for the award include a superior academic record in high school and evidence that the student has made a contribution to the quality of life in his or her community. The scholarship is for four years, and first preference will be given to a student from Wayne County in Ohio. The Allen W. Snyder Memorial Scholarship was established in June 1971 through the generosity of the members of the class of 1921 in honor of their classmate Allen W. Snyder. The income from the fund is used for scholarships for deserving students selected by the Scholarship Committee. The Stephen Markham Stackpole Endowed Scholarship Fund was established in 1999 by a gift from Stephen M. Stackpole, a member of the class of 1951. The scholarship is awarded annually to students who are majoring in a science and who demonstrate financial need. The David L. Steiner, M.D. Scholarship was established in 1995 by a gift from David L. Steiner, M.D., a graduate of the class of 1924 from Lima, Ohio. Income from the fund is awarded annually to a student who has demonstrated financial need. The Frances Whitney McCuskey Stewart Scholarship was established in 2005 by a gift from George H. Stewart. Income from this fund is awarded annually to female students with first preference given to single female students who are working their way through college. The Craig T. and Jean I. Stockdale Scholarship was established in 2000 by Craig T. Stockdale, a member of the class of 1931, and his wife, Jean Ingram Stockdale. Income from this fund is awarded to students who demonstrate academic achievement, leadership, and financial need. The Whitney E. and Edna S. Stoneburner Scholarship Fund was established in 1994 in their memory and in recognition of their dedication and commitment to Wooster by their son, Roger W. Stoneburner of the class of 1944, and his wife, Jean Kelty Stoneburner of the class of 1947. The scholarship is awarded to a member of the junior class who has financial need and is planning to pursue a teaching career in elementary or secondary education. Whitney E. Stoneburner served the College as Professor of Education from 1926 to 1955. The Mildred E. Swanson and Jennifer G. Beagle Endowed Scholarship was established in 2007 by the Mildred E. Swanson Foundation. The scholarship honors Jennifer Giesecke Beagle, a member of the class of 1991, and the memory of her aunt, Mildred E. Swanson. Income from the fund is awarded to a student with demonstrated financial need. First preference is given to a sociology major who is an active participant in the Wooster Volunteer Network. The Synod of the Covenant Endowed Scholarship for Minority Students was established by the Synod in 1989 in conjunction with its Bicentennial Fund drive. Income from the fund is awarded annually to one or more African American students who have financial need. The Leslie Gordon Tait Scholarship was established in 1982 by the Tait and Edwards families in honor of Professor Leslie Gordon Tait of the Department of Endowed Resources 318 Religion. It is awarded to a junior or senior, preferably a senior, who has shown a keen interest in the academic study of religion and who has demonstrated outstanding academic ability, especially in a broad range of religion courses. The Clara Albright Talbot Scholarship was given by her son, John C. Talbot, class of 1924, and several members of the Talbot family. Clara Albright Talbot graduated from Wooster in 1888 with a degree in music. She was the first of 36 family members who have attended the College over the past century. The scholarship is awarded to a student with academic achievement and demonstrated financial need. The Maude E. Taylor Scholarship, established under the will of Curtis N. Taylor in memory of his sister, provides a scholarship for a worthy and needy student at the College. Preference is given to those majoring in business economics. The Sallie J. Taylor Scholarship, established under the will of Curtis N. Taylor in memory of his sister, provides a scholarship for a worthy and needy student at the College. Preference is given to those majoring in religion. The Thurston Family Scholarship was established in 1996 by a gift from Max A. and Eleanor Linden Thurston ’51. Income from the Fund is awarded annually to students who have demonstrated service to their community and who have financial need. First preference should be given to students from small towns or rural communities. The Mary Coffman Tilton Scholarship Fund was established in 2008 by her children, Andrew Robert Tilton and Anna Tilton Daniel, from the proceeds of a trust from Mary Coffman Tilton, a member of the class of 1964. Income from the fund is awarded annually to students who are in good academic standing and have demonstrated financial need. First preference is given to students studying music performance and/or music education. The John M. Timken, Jr. and Polly M. Timken Endowed Scholarship Fund was established in 1998 by John M. Timken Jr. a member of the class of 1973. The annual income derived from the endowment is awarded annually to one or more students who demonstrate financial need, with first preference given to residents of Stark or Wayne County, Ohio. The Totten Scholarship was established in 2004 and honors Paul Totten, a member of the class of 1942, and Enid Robinson Totten, a member of the class of 1944. The Tottens share a passion for community service for which they have received many leadership awards. Income from this scholarship is awarded annually to students with financial need who have demonstrated their own commitment to community service. The Arthur B. and Margaret S. Towne Scholarship was established in 1998 by Edgar A. Towne, a member of the class of 1949, in memory of his parents. Income from the fund is awarded annually to a deserving student with demonstrated financial need. The Ralph H. Triem Scholarship was established in 2005 by his son, Edward T. Triem. Income from this scholarship is awarded annually to students with financial need, with first preference given to students majoring in either philosophy or religious studies. The Karl R. Trump Scholarship was established in 1976 by his family and friends in memory of Mr. Trump, who served twenty-seven years as a member of the voice faculty. The scholarship is awarded annually to a deserving music student, with first preference given to a voice major. The Anne F. Trupp and Nelle F. Fisher Scholarship was established in 2002 by bequests from Nelle F. Fisher and Anne Fisher Trupp, a member of the class of 1945. Endowed Resources 319 Income from this scholarship is awarded annually to students in good academic standing who have demonstrated financial need. First preference is given to students majoring in English. The Francis and Elizabeth Twinem Scholarship was established by Dr. Twinem, class of 1917. The scholarship is awarded at the beginning of the senior year to a premedical student who, during his or her first three years at Wooster, has shown promise for a distinguished career in medicine. The J. Lawrence Vodra Scholarship was established in 2009 by Larry and Nancy Morning Vodra, members of the classes of 1961 and 1963. Income from the fund is awarded to a junior or senior majoring in the sciences who has a grade point average of 2.8 or higher and has demonstrated financial need. The Nancy Morning Vodra Scholarship was established in 2009 by Larry and Nancy Morning Vodra, members of the classes of 1961 and 1963. Income from the fund is awarded to a student at the College who has a grade point average of 2.8 or higher and has demonstrated financial need. The Joseph F. Vojir Scholarship was established in 1987 by Mr. and Mrs. Stewart R. Massey in honor of Mr. Massey’s grandfather. The scholarship is awarded annually to a junior or senior with financial need who has demonstrated excellence in music performance. The Jack and Sue Reed Wakeley Scholarship in Political Science and Psychology was established in 2000 by Jack and Sue Reed Wakeley, members of the class of 1954. This scholarship is awarded annually to a junior majoring in political science and a junior majoring in psychology. The recipients must have an overall GPA of at least 3.0, a GPA of at least 3.2 in their majors and be judged by the faculty members of their respective departments to have potential for excellence as scholars. The Geraldine Ann Walklet Scholarship Fund was established in 1988 by Marie Cummings Walklet, class of 1934, and M. Donald Walklet, class of 1933, in memory of their daughter, Gerrie, a member of the class of 1963. This scholarship is awarded annually to students who demonstrate financial need. The James Wallace Endowed Scholarship Fund was established in 1965 by Mr. Dewitt Wallace as a memorial to his father. The fund is administered by the Scholarship Committee. The Jeanette Sprecher Walter Endowed Scholarship was established in 2010 by a bequest from Jeanette Walter, a member of the class of 1944. Income from the scholarship is awarded annually to students who are in good standing and have demonstrated financial need. The Walton Family Foundation Scholarship was established in 2003 through a gift from the Walton Family Foundation, Inc. as a part of Wooster’s Independent Minds Campaign. Income from this scholarship is awarded to students with financial need. The Frederic Kent Warner Endowed Scholarship Fund was established in 1986 by family and friends in memory of Frederic, who was a 1976 graduate of The College of Wooster. The income from this fund is awarded to students who have demonstrated financial need, with first preference given to students majoring in geology. The Margaret G. Warner Scholarship was established in 1999. Miss Warner was a member of the class of 1926 and taught high school history and social studies throughout her professional career. This scholarship is to be awarded to students with financial need, and first preference should be given to recipients in the field of history. Endowed Resources 320 The Wayne County Scholarship Fund was established in 1978 through the generosity of area business, industry, and individuals to indicate their commitment to the young people of this area. Income from the fund is awarded annually to students from Wayne County who attend the College and who have demonstrated financial need. A named scholarship in this fund honors Edward L. Buehler. The Gale H. and Mildred J. Weaner Endowed Scholarship was established in 2008 through the generous bequest of Gale H. Weaner, a member of the class of 1941, and his wife, Mildred June Weaner. Income from the fund is awarded annually to students who are in their sophomore, junior, or senior years and are in good academic standing with demonstrated financial need. First preference is given to students who are interested in a career in education. The Dr. John Gardner Weeks Scholarship Fund was established in 1962 in memory of John Gardner Weeks of the class of 1955 by his family and friends. Dr. Weeks died in an automobile accident while he was investigating opportunities for the practice of Radiology upon completion of his residency in June 1962 at the University of Michigan Hospital. The scholarship is awarded each year to a senior who is planning a medical career. Selection is made by the Scholarship Committee. The Douglas F. Weiler Scholarship, established in 1985, honors the memory of Douglas Weiler, class of 1986, from Crookston, Minnesota, who was killed in a farmrelated accident following his sophomore year at Wooster. The scholarship is awarded annually to a junior active in a theme-dorm or house (not a section or club) who demonstrates an exceptional attitude of concern toward others. The Carleton and Elma Weimer Endowed Scholarship was established in 2006 by their daughter, Sarah Weimer Bitzer, a member of the class of 1958. The scholarship honors the memory of Sarah’s parents, Carleton Earle Weimer, a member of the class of 1927, and Elma Hostetler Weimer. Income from the fund is awarded annually to students in good academic standing who have demonstrated financial need. The L. C. Weiss Memorial Scholarship was established in 2001 by a gift from the Clara Weiss Fund to honor the memory of Louis Carl Weiss, Hon. LLD ‘59, who was a Trustee of the College from 1949 to 1969. The scholarship is awarded to students in the sophomore or junior years who have demonstrated academic achievement with a grade point average of 3.5 or better. The Stanley R. Welty Family Scholarship was established in 2009 by Janet N. Welty, an honorary member of the Women’s Advisory Board, to honor the memory of her husband, Stanley R. Welty, Jr., and his father, Stanley R. Welty, Sr. Stan Welty, Jr. served as a member of Wooster’s Board of Trustees from 1991 to 2002, and as Trustee Emeritus from 2003 until his death in 2007. His father, Stan Welty, Sr., was a member of the class of 1924 and an honorary life member of the Board of Trustees from 1977 until his death in 1991. Income from the fund is awarded annually to one or more female students with financial need who have demonstrated qualities of leadership and academic achievement, especially in economics or business economics, and are active in community service at Wooster or in their hometowns. Recipients are selected by the College and The Women’s Advisory Board. The Welty Endowed Scholarship Fund was established in 1986 by Stanley R. Welty ’24, an honorary life member of the Board of Trustees. Income from the Welty Scholarship Fund is awarded annually to students with financial need who have demonstrated qualities of leadership and academic achievement. The Edward B. Westlake, Jr. Scholarship Fund was established in 1975 by Mr. E. B. Westlake, Jr., of the class of 1925. The scholarship is awarded annually to an outstanding member of the first-year class having financial need. First preference is Endowed Resources 321 given to graduates of Marysville (Ohio) High School. The scholarship is renewable each year for four years. The Whitaker Family Scholarship was established in 1994 by Ronald C. Whitaker and Susan Schweikert Whitaker to honor the three generations of family members who have attended the College. The scholarship is awarded annually by the Director of Financial Aid, with preference given to a male student who has demonstrated leadership ability and financial need. The Jack M. White Scholarship Fund was established in 2003 by a bequest from Jack M. White, an economics major from the class of 1951. Income from the scholarship is awarded to students in good academic standing who demonstrate financial need. The Whitmore-Williams Scholarship Fund was established in 1978 by A. Morris and Ruth Whitmore Williams ’62. Scholarships are awarded on the basis of academic achievement and Christian character. Two recipients, one male and one female, are selected annually from the first-year class, and the scholarships are renewable for three additional years. The Theodore Williams Scholarship was dedicated in 1996 through the efforts of the Black Alumni Council. The scholarship was named in honor of Theodore R. Williams, Robert E. Wilson Professor of Chemistry from 1959-2001 and Professor Emeritus from 2001-2005, and is given annually to an African American student who, after his or her junior year, has demonstrated exceptional academic achievements. A second scholarship is awarded to an African American student who, after his or her first year at Wooster, has demonstrated financial need and potential for academic achievement and leadership. The Edgar M. Wilson Scholarships are awarded to lineal descendants of the Robert W. Wilson family and to children of professional men who by reason of adversity are not able to educate their children. The Florence Ogden Wilson Scholarship was established through a bequest from Miss Wilson of Oklahoma City, Oklahoma. Miss Wilson was a long-time friend of the College; the College bookstore also bears her name. The income from this scholarship is awarded annually to students who demonstrate financial need and high academic achievement. The Clarence H. and Ruth Hagerman Winans Endowed Scholarship Fund was established by a planned gift from Marcia Winans Pinney, a member of the class of 1933, in memory of her father, Clarence H. Winans, a member of the class of 1899, and her mother, Ruth Hagerman Winans, a member of the class of 1903. Income from the fund is awarded to a student who has demonstrated financial need and who possesses qualities of scholarship and integrity. The Daniel W. and Dorothy V. Winter Scholarship was established in 1987 by a friend of the College. Income from the endowed fund will be awarded annually to a student recommended by the Music Department who has demonstrated excellence in music performance and who also has financial need. The Women’s Advisory Board Scholarship was established in 1987 by members of the Board. Awards are made annually to young women who have financial need. It is administered by the Executive Committee of The Women’s Advisory Board. The Jeannette McGraw Woodring Scholarship Fund was established in 1995 by a bequest from Paul D. Woodring in memory of his wife, Sarah Jeannette McGraw Woodring, class of 1936. Income from the fund is awarded annually to a junior or senior woman of outstanding academic achievement who plans to attend graduate school in preparation for college teaching. Endowed Resources 322 The Rear Admiral Robert DuBois Workman Scholarship Fund was established in 1987 through the generosity of Dr. William F. and Patricia Workman Foxx, ’44 and ’46, in memory of Mrs. Foxx’s father, a 1913 graduate of the College, who was Chief of Chaplains in the U.S. Navy during World War II. The scholarship is awarded annually to students with financial need. The Norman L. Wright Endowed Scholarship was established in 1999 by Helen Agricola Wright to honor the memory of her husband, Norman L. Wright, M.D., a member of the class of 1947. The scholarship is administered by The Women’s Advisory Board. The Wright Scholarship was established in 1971 to aid a talented string player pursuing a professional degree in music. Selection is made by the faculty of the Department of Music, based on their knowledge of the student’s talents and serious interest in stringed instrument study. The Arthur Bambridge Wyse Endowed Scholarship was established in 2006 through a planned gift by Marylyn Crandell Wyse, a member of the class of 1929, in memory of her husband, Arthur B. Wyse, a member of the class of 1929. Income from this fund is awarded to students with financial need. First preference is given to students majoring in mathematical science. The Ralph A. Young Appreciation Scholarship was established in 1998 by Malcolm and Jean Malkin Boggs, members of the class of 1948. This scholarship honors the memory of Racky Young, who served as the College’s Dean of Men from 1950 until 1970, and is awarded annually to students with financial need. The Ralph A. Young Scholarship, established in 1975 by his friends and colleagues in honor of his thirty-nine years of service to Wooster, is awarded each year to a junior religion major who demonstrates financial need and who, in addition to outstanding academic work, has demonstrated ability in co-curricular activities. The Jean Zapponi Scholarship was established in 1997 by members of her family. The scholarship honors the memory of Jean Zapponi, a lifelong Wooster resident who believed in the value of education throughout her lifetime. The scholarship is awarded to students who demonstrate financial need. The Mortimer H. Zinn Scholarship was established by the Fagans family to commemorate Mr. Zinn’s love of physics, his dedication to Judaism, and his service to others. The scholarship will be awarded annually to a student with financial need who plans to major in math or the physical sciences and who is involved in community service. Other scholarships of varying amounts are available for students from specified localities or classes, the principal ones being shown below: The Jerry and Bette Ashley Scholarship Fund The George F. Baker Scholarship The C. Glenn-Barber, M.D. Scholarship The Cynthia Barr Memorial Scholarship Fund The Byal-Patterson Scholarship The Class of 1937 Scholarship Fund The Class of 1958 Merit Scholarship Fund The John R. Crosser Fund The Lester S. Evans Scholarship The John and Olive Firmin Scholarship The Earl R. Gamble Scholarship The Joseph C. Gindlesperger and Dora Wynn Gindlesperger Memorial Fund Endowed Resources 323 The James Sylvester Gray Memorial Scholarship The Caroline Pfouts Harrold and Maude Harrold Better English Scholarship The William G. McCullough Scholarships The Esther M. Martin Memorial Fund The Elsa U. Pardee Scholarship The C. W. Patterson Scholarship The H. Lincoln and Alice C. Piper Students’ Fund The Lloyd and Ruth Sanborn Scholarship Fund The Kathryn and William Small Scholarship Fund The Boyd W. Smith Scholarship The Robert E. Stevenson and Helen Stevenson Scholarship Fund Other scholarships and student aid funds, not designated by the donors, are as follows: The Isabel Shaw Adams Scholarship The Katherine McCurdy Albright Memorial Scholarship Fund The James Allardice Memorial Scholarship Fund The Mary Sanborn Allen Scholarship Fund The Almendinger Scholarship The Sidney R. and Orietta E. Archer Scholarship Fund The Mary C. Arnold Endowed Scholarship Fund The Myron A. and Marie Dunlap Bachtell Memorial Scholarship Fund The Alice M. Bailey Scholarship Fund The Mrs. Alva C. Bailey Endowed Scholarship Fund (Recipients selected by the College and The Women’s Advisory Board) The Arthur C. Baird Scholarship The Grace Bascom Memorial Fund The J. C. Beardsley Fund The Josephine Volker Bennett Scholarship Fund The Helen Bentz Scholarship The Emma Bigelow Scholarship The Peter Bissman Scholarship The Patricia E. Blosser Endowed Scholarship Fund The Elizabeth B. Blossom Scholarship The Mary Metz Booher Scholarship Fund The Helen Brice Scholarship The Edward Brown Memorial Fund The Irvin H. and Dorothy M. Brune Loan Fund The Mary E. Caldwell Scholarship Fund The Campaign for Wooster Scholarship The Adelaide Campbell Scholarship The Wilson F. Cellar Scholarship The Roy V. and Cora Craig Chapin Scholarship The William Wallace Chappell Scholarship Fund The Class of 1900 Memorial Scholarship The Class of 1905 Selby Frame Vance Scholarships The Class of 1924 Scholarship Fund The Class of 1925 Scholarship Fund The Class of 1926 Scholarship Fund The Class of 1927 Scholarship Fund The Class of 1928 Alfred D. McCabe Memorial Fund The Class of 1929 Scholarship Fund Endowed Resources 324 The Class of 1930 Scholarship Fund The Class of 1932 Scholarship Fund The Class of 1933 Scholarship Fund The Class of 1934 Scholarship Fund The Class of 1935 Scholarship Fund The Class of 1937 Scholarship Fund The Class of 1939 Scholarship Fund The Class of 1940 Scholarship Fund The Class of 1941 Scholarship Fund The Class of 1942 Scholarship Fund The Class of 1943 Scholarship Fund The Class of 1944 Scholarship Fund The Class of 1945 Scholarship Fund The Class of 1946 Scholarship Fund The Class of 1947 Scholarship Fund The Class of 1948 Scholarship Fund The Class of 1949 Scholarship Fund The Class of 1950 Scholarship Fund The Class of 1951 Scholarship Fund The Class of 1956 Scholarship Fund The Class of 1957 Scholarship Fund The Class of 1958 Scholarship Fund The Class of 1961 Scholarship Fund The Class of 1966 Scholarship Fund The Ray and Ada Cofman Scholarship Funds The Wilson Compton, Jr., Scholarship The Wilson Compton, Sr., Scholarship The Wilson Compton Recognition Scholarships The Reverend and Mrs. Hubert F. Craven and Howard T. Craven, M.D., Scholarship Fund The Ralph E. Crider, Jr., Endowed Scholarship Fund The Harry W. Crist Memorial The George D. Crothers Scholarship The O.D. Culler Scholarship The Edwin George Cuthbertson Scholarship The Dale Scholarship Fund The Nelle A. Davis Scholarship Fund The Nancy M. Dickens Endowed Scholarship Fund The Esther H. and Robert W. Dobbins Scholarship The Robert McMorran Donaldson Memorial Fund The Elmer H. and Carrie A. Douglass Memorial Scholarship The Edward E. Ehret and Etta Gingrich Ehert Scholarships The Helen E. Enlow Scholarship Fund The Elizabeth Stevenson Ferson Scholarship The Theodore and Lillie Fetter Endowed Scholarship Fund The Sanford E. and Grace W. Fisher Scholarship The Jean R. and J. Calvin Fleming Endowed Scholarship Fund The R. J. Frackelton Memorial The John D. Frame, Sr. Scholarship Fund The Berenice R. France Scholarship Fund The Carl F. Funk Scholarship Fund The Gabbert Scholarship Endowed Resources 325 The Galpin Memorial Fund The Galpin-White Fund The Inez K. Gaylord Scholarship The Gee Family Scholarship Fund The Z. Montgomery Gibson Memorial Scholarship The Mabel Lindsay Gillespie Scholarship Fund The Goodyear Heights United Presbyterian Church Scholarship The James A. Gordon Scholarship The Sybil J. Gould Endowed Scholarship Fund for Drawing and Graphic Arts The Grand Lodge of Free and Accepted Masons of Ohio Scholarship The Frances A. Hallock Endowment Fund The Mary M. Hampu Scholarship Fund The Edna J. Harding Scholarship Fund The Arthur Harrison and Pearl K. Halley Scholarship Fund The Lemuel D. and Howard G. Harrold Athletic Scholarship The John M. Hastings Scholarship The Ronald B. Hendee Scholarship Fund The Marian Hood Huey Memorial Scholarship The Ashley J. Huffman Memorial The Winona Alice Hughes Fund The Huguenot Society Scholarship The John Schuyler Husted Scholarship The Frank Hyde Memorial Scholarship Fund The Lillian Illenberger Memorial Scholarship The J. Earl Jackman Scholarship The Japan Association Scholarship The Ernest A. and Marian Wellman Jones Scholarship Fund The Flora R. Jordan Memorial Scholarship Fund The Frieda Bull Jump Scholarship Fund The Charles E. Juneman Memorial Scholarship The Elizabeth Kahrl Memorial Scholarship The George F. Karch Loan Fund The Albert Kasten Family Memorial Scholarship The Emma B. Kennedy Fund The James E. Kennedy Memorial Scholarship Fund The A. Catherine Kidd Scholarship Fund The Mattie M. Kilcawley Scholarship Fund The Pauline Kindig Scholarship The Bertha M. Kitchen Scholarship Fund The Helen Kley Memorial Scholarship The Lewis L. LaShell Memorial Scholarship The Lauretta and George Laubach Scholarship Fund The Stevenson P. Lewis Memorial Fund The James Paxton and Bessie Swan Leyenberger Memorial The Hazel Perkey Love Memorial Scholarship The Josephine Lowrie Scholarship Fund The Lubrizol Awards The Kenneth H. MacKenzie Scholarship Fund The Christina A. MacMillan Scholarship The B. R. Maize Scholarship The Dorotha M. and Benjamin D. Marshall Scholarship The John McClellan Scholarship Endowed Resources 326 The Adeline and Jay McDowell Scholarship The Leon McDowell Endowed Scholarship The John McSweeney Memorial Scholarship The Ersie E. Miller Fund The Dora U. Morgan Scholarship The James W. Morgan Scholarship The Ada E. Morrett Scholarship The Mary V. Muhlhauser Scholarship The Alta B. Murray Endowed Scholarship The Mary H. Myers Scholarship The Mary V. Myers Scholarship The Minnie E. Myers Scholarship The Richard Proctor Nelson Memorial Scholarship The NOW Scholarship Fund The Elizabeth Nydegger Scholarship Fund The Frances and Grace Oviatt Scholarship Fund The George B. Owens Endowed Scholarship The Perpetual Scholarship The Della G. Plants Scholarship Fund The Platter Scholarship The Daniel Poling Scholarship The Pomerene Memorial Fund The Presbyterian Book Store Founders’ Scholarship The Nelson and Emma B. Randles Scholarship The Peter Rapp Scholarship Fund The Reader’s Digest Foundation Scholarship Fund The Michael V. Ream Scholarship The William and Elva Reither Loan Fund The Laura Steigner Relph Scholarship Fund The Stevens and Elaine Rice Scholarship Fund The Paul Robson Endowed Scholarship Fund The Eve Roine Richmond Memorial Scholarship The Yale K. Roots-Erwin Scholarship Fund The Schell Foundation Student Loan Fund The Schwartz Memorial Scholarship Fund The Albert B. Scofield Memorial Scholarship The Lois G. Scott Scholarship Fund The Self-Help Scholarship Fund The Frederick K. Shibley Scholarship The Katherine Silvis Endowed Scholarship The Clara A. Smith Scholarship The Margaret and Esther Smith Scholarship Fund The Marjorie F. Snider Scholarship Fund The Ruth E. Stephan Scholarship Fund The Ethel M. Stonehill Endowed Scholarship Fund The Jack D. Strang Endowed Scholarship The Surdna Foundation Scholarship Fund The Mary F. Sweyer Memorial Scholarship The Julia Steiner Taylor Memorial Scholarship The Susannah B. Taylor Scholarship The Wade A. Taylor Memorial Scholarship The Vesta A. Thomas Memorial Scholarship Fund Buildings and Facilities 327 The Estella Welty Thompson Class of 1912 Scholarship Fund The Alice Engle Thurston Scholarship Fund The Frederick A. Tice Loan Fund The Kenneth D. Trunk Scholarship The Elizabeth Wood Vance Scholarships The Karl Ver Steeg Memorial Scholarship The Charles N. Vicary Scholarship The Louis E. Ward Fund The Louis E. and Margaret C. Ward Loan Fund The Helen Waugh Scholarship Fund The Lenore Welsh Memorial Scholarship The Marguerite White Talbot and Gretchen R. White Endowed Scholarship Fund The Laura B. Wiley Scholarship The J. Robert Wills Memorial Scholarship The Robert E. Wilson Scholarship The J. T. M. Wilson Memorial Scholarship The Forest C. Wineland Loan Fund The John F. and Martha Winter Scholarship Fund The Isabel A. Yocum Scholarship The Joe Herman Yoder and Ruth A. Yoder Memorial Scholarships Further information regarding these scholarships may be obtained from the Director of Financial Aid. BUILDINGS AND FACILITIES Campus planning has been followed at Wooster since 1900; all buildings now in use have been constructed since that time. Forty buildings, many of the English collegiate Gothic type of architecture, are located on approximately 240 acres. They are constructed principally of cream-colored brick and buff brick; two are of stone. Most buildings are trimmed with Indiana limestone or Ohio sandstone. Quinby Quadrangle, the square about which the College grew, was formally named at the 75th Anniversary in honor of the donor, Ephraim Quinby, of Wooster. The Quadrangle is a part of the 21-acre tract that constituted the original campus. ACADEMIC BUILDINGS The Andrews Library (1962) was made possible largely through the gift of the late Mabel Shields (Mrs. Matthew) Andrews of Cleveland, Ohio. Along with the adjoining Flo K. Gault Library for Independent Study and the nearby Timken Science Library in Frick Hall, its five floors are air-conditioned and house a collection of about one million items, including books, periodicals, microforms, recorded materials, newspapers, and government publications. The Writing Center and the offices of the Registrar and Financial Aid are also housed here. The libraries provide seating Buildings and Facilities 328 for nearly 800 library users, including over 350 carrels for seniors engaged in Independent Study. Eight group study rooms allow small groups of students to work collaboratively. All libraries have secure wireless access to the Internet. The libraries are a selective depository for United States government publications. There are also several special collections, including the Wallace Notestein Library of English History, and extensive microtext collections which include the Atlanta UniversityBell & Howell Black Culture Collection, the Library of American Civilization, Herstory, and the Greenwood Science Fiction Collection. Wooster’s library catalog is part of CONSORT, an electronic catalog shared with Denison University, Kenyon College, and Ohio Wesleyan University. CONSORT, in turn, is part of OhioLINK, a network of 88 academic and public libraries throughout the state. Interlibrary loan of periodical articles or books from out-of-state libraries is also available. The libraries include classrooms, computer labs, and the Media Library, which houses the libraries’ collection of recorded materials and listening and viewing stations. The Armington Physical Education Center (1968, 1973) houses the Timken Gym - nasium, which provides intercollegiate basketball seating for 3,420 and serves as a multi-station area for classes, intramural sports, and recreational activities. The Swigart Fitness Center of approximately 3,700 square feet was modernized and expanded in 1998 and reconditioned in 2011 as a varsity weight training facility. Armington also contains a 75’ by 45’, six-lane swimming pool, with seating for 450, a multi-purpose gymnasium, an exercise physiology laboratory, a coeducational training room, and equipment and laundry rooms. Herman Freedlander Theatre (1975) honors the memory of a long-time resident and friend of the Wooster community. This thrust-proscenium theatre seats 400. A stage lift and costume and dressing rooms are included in the facility. The Effie Shoolroy Arena Theatre was given in her memory by Ross K. Shoolroy. The Arena Theatre is designed for experimental productions and seats up to 135 patrons. The Flo K. Gault Library for Independent Study (1995) is named for Flo Kurtz Gault, ’48, who, with her husband, Stanley C. Gault, ’48, contributed the principal gift for the building. The 32,000-square-foot structure is connected to Andrews Library and serves as a focus for the College’s commitment to the Independent Study program. (See The Andrews Library for resources and services available in the two buildings.) Kauke Hall (1902; remodeled in 1961-1962 and 2005-2006), the central building of the Quadrangle, was a gift of the citizens of Wooster and Wayne County and was named in honor of Captain John H. Kauke, long-time College trustee and benefactor. It houses the following departments and programs: Africana Studies, Archaeology, Chinese, Classical Studies, Comparative Literature, Cultural Area Studies, English, French, German, History, International Relations, Political Science, Religious Studies, Russian Studies, Sociology and Anthropology, Spanish, Urban Studies, and Women’s, Gender, and Sexuality Studies. Facilities for sociology laboratory studies, computing for the humanities and social sciences, and offices for faculty are here as well. During the renovation of 1961-1962, the Delmar Archway (named after its donor, Charles Delmar) was added to the center of the building. In 2005-2006, Kauke underwent an extensive renovation to recapture its distinctive architectural character and to bring it up to modern technological standards. New additions included a student commons and a ground floor café and courtyard, featuring a brick “Donor Wall.” Hundreds of alumni, parents, and friends, as well as members of Wooster’s corporate community, made this $18 million renovation possible. Principal gifts were received from The Walton Family Foundation, The Timken Foundation of Canton, Stanley C. and Flo K. Gault, Edward J. and Edith G. Andrew, and the Donald and Alice Noble Foundation. Buildings and Facilities 329 John Gaston Mateer Hall (1968) is a building dedicated to the study of biology and its related fields. It was made possible primarily through the help of Mr. and Mrs. Ward Canaday, and is named in honor of the late Dr. John G. Mateer, alumnus and trustee of the College. The air-conditioned facilities include classrooms, laboratories, the biology library, departmental offices, individual independent study labs, a greenhouse, and a 250-seat lecture hall. A passageway connects Mateer to the renovated and expanded Severance Hall (Chemistry). Burton D. Morgan Hall (2002) bears the name of Burton D. Morgan, founder of The Burton D. Morgan Foundation of Hudson, Ohio. The building was a gift from the Foundation and houses the Departments of Economics (including Business Economics), Education, and Psychology. The office of the Chief Information and Planning Officer is here as well as the Center for Entrepreneurship and the College’s Office of Information Technology. The Scheide Music Center (1987) bears the name of William H. Scheide, Princeton, New Jersey. It houses the Department of Music. Among its facilities are the Gault Recital Hall and the Timken Rehearsal Hall. The Noble Atrium contains a commissioned bronze relief sculpture, “The Four Seasons” by Michelle Stuart of New York City, and “The Guitar Player” by Harry Marinsky. The Scot Center (2012), which is connected to Armington Physical Education Center, will open in January 2012 and includes four intramural courts for basketball, tennis and volleyball, an NCAA regulation 200-meter running track, indoor long jump, triple jump and pole vault, a fitness center with a full array of circuit training stations, traditional as well as elliptical running machines, rowing machines, and free weights, batting cages and other equipment for indoor baseball practice, locker rooms, and athletic department offices. The ground level includes a large meeting room and a multi-purpose flexible space for student activities. Scovel Hall (1902) was renovated in 1983-1984. The building bears the name of Dr. Sylvester F. Scovel, the third President of the College, and houses the Departments of Geology and Philosophy. Among its facilities are the Charles B. Moke Lecture Hall, the Ross K. Shoolroy Lecture Hall, and the Julia Shoolroy Halloran Humanities Seminar Room. Severance Art Building (1973), formerly Severance Gymnasium (1912), was given to the College by Louis H. Severance, one of the leading benefactors of the College. It housed the Department of Physical Education until 1973. From 1973-1996, it housed the studio program of the Department of Art. In 1979, additional renovation provided space for the Office of Publications. The Ebert Art Center (1997) represents a major renovation and expansion of this facility, which now accommodates the art studio, art history, and art museum programs. The College of Wooster Art Museum, which is located in an addition to the original gymnasium building, includes the Charlene Derge Sussel Art Gallery, Burton D. Morgan Gallery, a storage area for the College’s Permanent Art Collection, and museum preparatory areas in addition to lecture and seminar rooms. The Office of Publications is located in the Severance portion of the renovation, along with the art studios, slide library, and art department faculty offices. The principal gift for this project came from the Horatio B. Ebert Foundation and honors Horatio and Lyda Ebert, Robert O. Ebert, and Adrienne and Cecile Ebert. The garden area south of the Ebert Art Center was given by members of the Board of Trustees and other friends in honor of Laura (Lolly) Harper Copeland and in recognition of her exceptional contributions to the College, especially her interest in its buildings and grounds. The garden sculpture, “Girl with Doves” by Richard Hallier, was the gift of the Robert O. Ebert family. Buildings and Facilities 330 Severance Hall (1902; remodeled in 1960, 1999), a companion building to Scovel Hall, houses the Department of Chemistry. It was named in honor of its donor, Louis H. Severance. Gifts from Trustees, alumni, and friends made possible the $11.2 million renovation and expansion of laboratories, classrooms, and offices completed in 1999, the largest capital project of Wooster’s Campaign for the 1990s. Taylor Hall (1902; renovated in 1985) bears the name of Dr. A.A.E. Taylor, the second President of the College. It houses the departments of Mathematics and Computer Science and Physics. Facilities in Taylor Hall include the Timken Computer Center, the Rubbermaid Mathematics Center, and the Andrew Lecture Room. The Timken Science Library in Frick Hall (1900, 1998) was the original University of Wooster Library, 1900-1915, and The College of Wooster Library, 1915- 1962 (after which the library collections in the humanities and social sciences were moved into Andrews Library). Its construction and expansion were made possible by gifts from Henry Clay Frick of Pittsburgh, Pennsylvania, in memory of his parents. Following more than three decades as the College Art Museum, its grand neo-classical reading room was lovingly restored and its three floors of book stacks were completely rebuilt in 1998-1999, and the building was reopened as the Timken Science Library in Frick Hall, consolidating four branch libraries. The principal gift for the renovation was made by The Timken Foundation of Canton, Ohio. The building includes study carrels, two group study rooms, and a computer laboratory, in which science majors learn to search the major professional scientific databases online. Wishart Hall (1966) located on the corner of University and Bever Streets, was designed for the Department of Communication (formerly Speech) and contains the Freedlander Speech and Hearing Clinic, WCWS, dance studio, general classrooms and faculty offices for the Departments of Communication and Theatre and Dance, the Delbert G. Lean Lecture Room (remodeled in 2008), and the Craig Theatre Library, which houses the collection of the late Professor William Craig. This building, a gift of the citizens of Wayne County, was named in honor of Charles Frederick Wishart, sixth President of the College. ADMINISTRATIVE BUILDINGS Gault Admissions Center (2002) was a gift to the College from Stanley C. ’48 and Flo K. Gault ‘48 and their children, Stephen Gault ’73, Christopher Gault, and Jennifer Gault Marsh. The building is dedicated to the memory of Mr. Gault’s sister, Donna Jean Gault Bauman, a member of the Class of 1941. The Gault Center houses the Office of Admissions. Gault Alumni Center (1941; completely remodeled in 1993), at the corner of Beall Avenue and Pine Street, was designed and constructed by H.C. Frick and is the former Overholt residence. The building housed the Department of Music from 1941 until 1987 and bore the name of Karl Merz, the first director of Wooster’s Conservatory of Music (1882-1890). From 1987 until 1992, it served as a residence for students. Generous gifts from alumni in the 50-year reunion classes of 1936, 1938, 1940, and 1941, as well as a major gift from Mr. and Mrs. Gault, made it possible to renovate the building completely during the 1992-1993 academic year to serve as a home for the Alumni Association and for use by the Alumni and Development Offices. Galpin Memorial Building (1931), given by William A. Galpin and named in honor of his father, is the headquarters of the administrative staff. On the lower level is the Business Office. On the first floor are the offices of the Provost, Dean for Curriculum and Academic Engagement, Dean for Faculty Development, and Dean of Buildings and Facilities 331 Students. The offices of the President, Secretary of the College, Vice President for Finance and Business and Treasurer, and the Director of Administrative Services are located on the second floor. On the third floor are the offices for the Vice President for Development. RESIDENCE HALLS AND HOUSES Matthew Andrews Hall (1954), housing 94 students, is the gift of Mrs. Matthew Andrews in memory of her husband. The accommodations include double and triple rooms, comfortable public areas, laundry facilities, kitchen area with stove and refrigerator, and a lounge. Armington Hall (1966) is an all single-room facility housing 84 students. The residence hall, located on Wayne Avenue, was made possible through a gift from Mr. and Mrs. George Armington of Austinburg, Ohio. The accommodations include laundry facilities, kitchen area with stove and refrigerator, and formal and informal lounges. Aultz House (1987), 575 and 575 1⁄2 East University Street, accommodates 7 students. Residents participate in the Lincoln Way Reads program. Avery House (1990), 558 Stibbs Street, accommodates 7 students. Residents volunteer with the Wayne Center for the Arts and provide several on-campus events such as monthly poetry readings and a spring concert. Babcock Hall (1935, renovated in 2008-2009), the gift of Birt E. Babcock, of the class of 1894, accommodates approximately 90 students involved in the CrossCultural Living and Experiences Program. Accommodations include laundry facilities, kitchen area with stove, microwave, and refrigerator, lounge, and a dining/study room. On the first floor are the offices of The Center for Diversity and Global Engagement, including the Office of International Student Affairs (OISA), the offices of Interfaith Campus Ministries (ICM), the Ambassadors Program, and International Scholar Services. Bissman Hall (1966) is a residence hall for 138 students. This residence hall, located on Beall Avenue, was made possible through the principal gift of the late Elizabeth Bissman Martin, of the class of 1897, in memory of her mother and father, Anna and Peter Bissman. Bissman Hall is currently undergoing minor renovations and will house first-year students. The accommodations include laundry facilities, kitchen areas with a stove and refrigerator, and formal and informal lounges. Bornhuetter Hall (2004), located on the northwest corner of Beall and Wayne Avenues, accommodates 184 students. The residence hall was made possible by a principal gift from Ronald L. Bornhuetter ’53, his wife, Carol, and their children, as well as substantial gifts from members of the Board of Trustees and the Classes of 1952 and 1953. The building’s facilities include a multi-purpose room with kitchen facilities, study lounges on each floor, and a central courtyard. Bryan House (1987), 1439 Beall Avenue, accommodates 12 students. Residents volunteer with the students who attend the Ida Sue School. Calcie House (1971), 823 College Avenue, accommodates 10 students. Residents are members of the Dream Program, promoting awareness and facilitating multicultural interactions and understanding of different cultural practices and beliefs of College of Wooster students. Colonial House (2001), 809 Beall Avenue, accommodates 10 students. Residents are members of the Women of Images program and serve as a support network for incoming and current women of color. Women of Images also seeks to further cultural awareness on campus and in the community. Buildings and Facilities 332 Otelia Compton Hall (1955), at the corner of Beall and Wayne Avenues, is a residence hall housing 113 students. Built in honor of Mrs. Elias Compton through funds in a large measure given by citizens of Wooster and Wayne County, the building has a formal lounge, two social rooms, and a kitchen. In 1995, the Compton Hall Guest Room was refurbished and renewed by the Women’s Advisory Board of the College and dedicated in honor of Laura (Lolly) Copeland, wife of Wooster’s ninth President, in appreciation of her contributions to the College and the community. The accommodations also include laundry and kitchen facilities, a formal and informal lounge, and two large multipurpose rooms. Corner House (2001), 819 Beall Avenue, accommodates 10 students. Residents are part of Gallows, which provides substance-free entertainment through regular comedy routines. Crandall House (1963), 326 and 3261⁄2 Pearl Street, accommodates 6 students. Residents oversee Bursting the Wooster Bubble, designed to encourage strong and healthy town relations between College students and the larger Wooster community. Bursting the Wooster Bubble also utilizes the Bridges of Hope program as a base project for their volunteer work. Douglass Hall (1929), the gift of E. P. Douglass of the class of 1877, houses 114 students. Located on the north end of campus, Douglass Hall is centrally located to most residential and academic buildings. Douglass has undergone exterior renovations in the past two years and interior furniture replacement. The facilities include laundry and kitchen facilities, multipurpose room, and an informal and large formal lounge. East End Apartments, 723-725 East University Avenue, accommodate 6 students in two units of three persons each. Fairlawn Apartments, 1025 East Wayne Avenue, accommodate 18 students in six units of three persons each. Residents manage The Honey House program, which aims to educate the College community on hive collapse disorder, raise honeybees, and sell honey. Gable House (1959), 836 College Avenue, accommodates 9 students. Residents of Gable House work with residents of Shearer House to operate a substance-free coffee house on Friday and Saturday nights called Common Grounds. Gault Manor (2008), located on the northeast corner of Beall and Wayne Avenue, contains 35 double rooms with private bathrooms and 3 single rooms to accommodate 75 students. The residence hall was fully funded by a major gift from Stanley C. and Flo K. Gault, both members of the class of 1948. There are large common rooms on the first and second floors, as well as four smaller lounges distributed throughout the building. Gault Manor also has a recreational space on the ground floor that is equipped with state-of-the-art audio-visual equipment for programs and activities. Other accommodations include laundry facilities and convenience areas with sinks and microwaves. Grossjean House (1976), 657 East University Street, accommodates 6 students. Hider Apartments (1985), 561 and 5611⁄2 East University Street, accommodate 7 students. Hider House (1985), 567 East University Street, accommodates 11 students. Residents belong to Men Working for Change, where residents seek to grow as men of character and competence as they volunteer their time raising awareness of domestic violence issues. Men Working for Change work in the community at local Wooster high schools. Holden Hall Main Building (1907), Wooster’s largest residence hall, accommodates 310 students. Named for Dr. Louis E. Holden, fourth President of the College, Buildings and Facilities 333 it has several common and recreation areas. Holden Wing was added to the main building in 1961 and was renovated in the summer of 2004. Ground and First floor Holden “L” were renovated in the summer of 2005. The Holden Annex (1921) houses 50 students. The accommodations include laundry and kitchen facilities, and formal and informal lounges. Iceman House (1987), 1455 Beall Avenue, accommodates 9 students. Residents volunteer with Habitat for Humanity to assist in building homes in the Wooster community and also spend time at the Re-Store site. Johnson House (1972), 1419 Beall Avenue, accommodates 9 students. Residents of the Johnson House (or the Peace and Justice House) work with the Red Cross’s new after school program in the Wooster community. Kate House (1968) accommodates 14 students. Residents plan and implement activities for children and youth at the Attention Center, a Wayne County facility for pre-trial delinquent youth. Kenarden Lodge (1911; completely remodeled in 1991-92) is a residence hall built and named by Mrs. John S. Kennedy. It accommodates 141 students. The accommodations include laundry and kitchen facilities, convenience areas with sinks, a workout room and informal lounge, and a formal lounge with fireplace and spiral staircase. Kennedy Apartments (1987), 1433 Beall Avenue, accommodate 12 students in four units of three persons each. Kieffer House (1965), 829 College Avenue, accommodates 8 students. Residents manage a program called Homework for Hoops, which entails tutoring students from FIAT and the Human Resource Center at Mom’s Truck Stop, a small eatery on campus. After tutoring for an hour the men take the students to the Physical Education Center (PEC) for an hour of basketball. Lewis House (1965), 829 College Avenue, accommodates 8 students. Residents enrich and enhance the ability of students in the Cornerstone Reading Enrichment Program to read at a more proficient level. Henry Luce III Hall (1990) is located just south of Scheide Music Center on Beall Avenue. The residence hall, housing 98 students, was made possible through the principal gift of The Henry Luce Foundation of New York City. Students live in suites which accommodate six to twelve residents. Six Language Suites (Chinese, Classics, French, German, Spanish & Russian) provide students with a living/learning environment focusing on developing foreign language skills. The building’s facilities include a formal lounge with fireplace, recreation and meeting rooms, a fitness room, kitchen area, and an informal lounge. Miller Manor (1872), 909 Beall Avenue, gift of Mrs. Alice Miller Eberbach of Ann Arbor, Michigan, houses 29 students and one staff member. For many years this building was the President’s Home. Reed House (1930), 1447 Beall Avenue, accommodates 9 students. Residents of Reed House crochet hats and scarves each month for Every Woman’s House, Children’s Christian Home of Ohio, and patients at The Cleveland Clinic-Wooster. Scot Cottage (1941), 902 Beall Avenue, accommodates 15 students. Residents provide support to local farmers who use organic methods and educate the campus through dinners and meetings; students also volunteer with Autumn Harvest Farm. Shearer House (1929), 835 College Avenue, accommodates 9 students. Residents of Shearer House work with residents of Gable House to operate a substance-free coffee house on Friday and Saturday nights called Common Grounds. Buildings and Facilities 334 Stadium House (2001), 629 and 629 1⁄2 East University Street, accommodates 8 students. Residents of Stadium House’s first floor help the Humane Society socialize animals in preparation for adoptions, as well as organize opportunities for College of Wooster students to volunteer with them at the shelter. Residents of Stadium House’s second floor oversee the Charter program called Women of Dene, which seeks to bring together women of all backgrounds, ethnicity, cultures and races to challenge prejudices and intolerance and promote interracial relations on campus. Stevenson Hall (1966), located on Wayne Avenue, is the gift of the late Miss M. Maude Stevenson of Lancaster, Ohio, as a memorial to her mother and father, Helen and Robert E. Stevenson, and accommodates 54 students. Accommodations include laundry facilities, kitchen areas, formal and informal lounges, and a discussion/study lounge. Wagner Hall (1957), East Wayne Avenue, a residence for 131 students, was the gift of alumni Dr. Gary Richard Wagner and Mrs. Elizabeth Sidwell Wagner in honor of their mothers: Ella Blue Wagner and Margaret Sutton Sidwell. In 1991, Ruth Frost Parker (‘45) generously funded the addition of a gabled roof and new windows and lights. Accommodations include laundry and kitchen facilities, and an informal and formal lounge. Weber House (1999), 574 Stibbs Street, accommodates 10 students. Residents teach English as a Second Language to individuals in the Wooster community. Westminster Cottage (1944), 904 Beall Avenue, accommodates 31 students. Yost House (1971), 817 College Avenue, accommodates 7 students. Residents belong to Men of Harambee, an organization of black males who attempt to assist other men of color in learning and examining who they are culturally, socially and politically. OTHER BUILDINGS Culbertson/Slater House (1965), 602 E. Wayne Avenue, houses the offices of Security and Protective Services, Keys and IDs, Environmental Safety, and Residence Life. Kittredge Hall (1966) adjoins Otelia Compton Hall on Wayne Avenue. This airconditioned dining hall, which seats 320 people, was made possible by the principal gift of Mrs. Thomas J. Watson, Sr. (Jeanette Kittredge Watson), formerly a student at The College of Wooster, and was given her family name. Another substantial gift for the building was made by Mr. and Mrs. George Armington. Lilly House (2002), 1452 Beall Avenue, houses the Lilly Project for the Exploration of Vocation and the Center for Academic Advising. Built in 1910, the house was sold to the College in 2002 and completely renovated in 2003. Longbrake Student Wellness Center (2002), located on Wayne Avenue south of the Service Center and across from Wagner Hall, was made possible through the principal gift of William ’65 and Martha Longbrake of Seattle, Washington. The facilities include six treatment rooms, seven offices, eight in-patient beds, a pharmacy, lounge, and medium-sized conference room. Lowry Center (1968) is named in honor of Howard Lowry, Wooster’s seventh President. It contains the following facilities: snack bar, convenience store and coffee shop, Wired Scot, dining room, ballroom, lounges, meeting rooms, student government, activities and publications offices, pool tables, bowling lanes, postal center, The Florence O. Wilson Bookstore, H. William Taeusch Faculty Lounge, Office Services, and Campus Dining Services. Buildings and Facilities 335 McGaw Chapel, completed in the fall of 1971, is the gift of Mr. and Mrs. Foster G. McGaw in memory of his parents, Francis A. McGaw, class of 1885, and Alice S. Millar McGaw. The seating capacity is 1,600. The Holtkamp organ, built in 1953, is the gift of the Davis family and other friends of the College as a memorial to David D. Davis. Extensive renovation and refurbishing of the instrument, funded by the Davis family, was completed in 1993. Olderman House (1998), 807 College Avenue, was renovated in 2008 and houses the Departments of College Relations and Public Information. Overholt House (2001), 1473 Beall Avenue, provides space for the Office of Interfaith Campus Ministries and the Wooster Volunteer Network. Papp Stadium at Severance Field includes the football field and the Carl Munson All-Weather Track. Originally erected in 1915, the stadium was restored in 1991 with a substantial gift from Dr. John P. Papp ’60 and gifts from other alumni and friends. The football stands seat 5,000. The Stadium also houses the grounds maintenance equipment. The track was completely rebuilt in 1993 through gifts of alumni and friends of the College, including the family of Grant E. Rose ’39. In the summer of 2009, the College replaced the natural grass surface of Severance Field with a synthetic multi-use turf field. The renovation also included the installation of lights and the resurfacing of the track. The renovation was made possible by a naming gift from Edith G. and Edward J. Andrew ’61. All-weather facilities for tennis are provided by the General Dudley J. Hard Memorial Tennis Courts (1965) near Bissman Hall. The courts were rebuilt in 1993, with an endowment established for their maintenance through the generosity of Donald and Alice Noble. Practice and playing fields are also provided at other locations on the campus for touch football, softball, soccer, lacrosse, baseball, field hockey, and archery. These include the Art Murray Baseball Diamond, the Carl W. Dale Soccer-Lacrosse Field, the Cindy Barr Hockey-Lacrosse Field, and the Rick Mueller Practice Field. The L.C. Boles Memorial Golf Course is located on campus east of Papp Stadium. The President’s Home (1928) is located on the campus at 433 East University Street. It was completely remodeled in 1969 and significantly refurbished in 1996. Additional renovations were made in 2007. Rubbermaid Student Development Center (1989). Originally erected in 1876 as the College observatory on the northwest corner of Beall Avenue and University Street, this facility was redesigned in 1941 as a Student Union and Campus Bookstore. It was moved to its present site on University Street east of Holden Hall prior to construction of Andrews Library, and renamed the Temporary Union Building. From 1968 until 1987, it was occupied by the Department of Music. Renovated in 1989 by a grant from Rubbermaid Incorporated, it houses the Career Services Office and the Learning Center. The Service Center (1960), gift of Mr. and Mrs. George E. Armington, located at 580 East Wayne Avenue, houses the offices of Human Resources (Payroll and Student Employment), the Director of Physical Plant Services, Administrative Purchasing, Custodial Services, the building maintenance craft shops, the transportation department, and equipment. The Grace E. Smith Memorial Walk (1955) is the gift of an alumna ’08 of Toledo, Ohio. The walk extends from Galpin Memorial Building to Beall Avenue. The Herman Westinghouse Memorial Power Plant (1939) is the central station from which steam heat is supplied to all buildings on the campus. The Directories 336 Westminster Church House (1965), located on the corner of College Avenue and Pine Street, was built largely through funds provided by members of the congregation of Westminster Presbyterian Church and is used by the church for its offices, meetings, Church School, and worship services. It also houses the College nursery school for preschool children, which provides in-service teaching experience for college students, and the administrative offices of the Ohio Light Opera. The Wooster Inn (1959), gift of the late Robert E. Wilson ’14, provides overnight accommodations for thirty-three guests. Additional facilities include a dining room, a conference room, and lounges. The Inn is located at the southeast corner of Wayne Avenue and Gasche Street, adjacent to the Boles Memorial Golf Course. THE DIRECTORIES PRESIDENTS Willis Lord, D.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1870-1873 Archibald Alexander Edward Taylor, D.D., LL.D. . . . . . . . . . . . . . . . . . . . . . . 1873-1883 Sylvester Fithian Scovel, D.D., LL.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883-1899 Louis Edward Holden, D.D., LL.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1899-1915 John Campbell White, LL.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915-1919 Charles Frederick Wishart, D.D., LL.D., Litt.D., L.H.D. . . . . . . . . . . . . . . . . . 1919-1944 Howard Foster Lowry, Ph.D., Litt.D., LL.D., D.C.L., L.H.D. . . . . . . . . . . . . . . 1944-1967 J. Garber Drushal, Ph.D., LL.D., L.H.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1967-1977 Henry Jefferson Copeland, Ph.D., LL.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1977-1995 Raleigh Stanton Hales, Jr., Ph.D., Sc.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1995-2007 Grant H. Cornwell, Jr., Ph.D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007- Emeritus Henry Jefferson Copeland, Ph.D., LL.D., President of the College. 1966. President Emeritus since 1995. Raleigh Stanton Hales, Jr., Ph.D., Sc.D., President of the College. 1990. President Emeritus since 2007. BOARD OF TRUSTEES The Board of Trustees shall consist of not more than forty-three members, onethird of whom are elected annually for a three-year term. Six members of the Board are nominated to membership by the alumni of the College. The President of the College is a trustee ex-officio. Emeritus/a Life Trustees are those who have been elected after serving three or more terms on the Board or whose service terminated after seventy years of age. Honorary Life Trustees are those named whose great service to the College clearly merits exceptional recognition, whether or not they have been members of the Board. The Directories 337 Officers David H. Gunning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chairman Douglas F. Brush . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vice Chairman Stanley C. Gault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chairman Emeritus James R. Wilson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Chairman Emeritus Members The year of first election to the Board is shown after the name. Trustees nominated by alumni are indicated by an asterisk. (*) Paul R. Abbey, B.A., M.B.A., Cleveland. 2003. Richard J. Bell, B.A., M.B.A., Alexandria, Virginia. 2008. * Sandeep Bhatia, B.A., Shaker Heights. 2009. Joan Blanchard, B.A., Boston, Massachusetts. 2004. Richard A. Bowers, B.A., M.D., Sewickley, Pennsylvania. 2011. Douglas F. Brush, B.A., Pittsford, New York. 2000. Marjorie M. Carlson, B.A., M.A., Bratenahl. 1974. Joan P. Carter, B.A., Haddonfield, New Jersey. 1986. Jayne H. Chambers, M.B.A., McLean, Virgina, 1996-02; 2006. Doon A. Foster, B.A., Wilton, Connecticut. 2007. * Donald R. Frederico, B.A., J.D., Boston, Massachusetts. 2007. Stephen C. Gault, B.A., M.B.A., Louisville, Kentucky. 2002. David H. Gunning, B.A., J.D., Cleveland Heights. 1989-92; 1994. Diane L. Hamburger, B.A., Hinsdale, Illinois. 1992-98; 2000. Jennifer A. Haverkamp, B.A., M.A., J.D., Silver Spring, Maryland. 1997. J. C. Johnston III, B.A., J.D., Wooster. 2003. Donald L. Kohn, B.A., Ph.D., Takoma Park, Maryland. 2011. * Karen M. Lockwood, B.A., J.D., Arlington, Virginia. 2011. * Lee E. Limbird, B.A., Ph.D., Nashville, Tennessee. 2010. Gregory A. Long, B.B.A., Wooster. 2006. William A. Longbrake, B.A., M.A., M.B.A., D.B.A., Seattle, Washington. 1988-94; 1995. H. Christopher Luce, B.A., New York, New York. 2000. Stewart R. Massey, B.A., Morristown, New Jersey. 1987-89; 1993; 2011. Lynne D. McCreight, B.A., M.A.T., Moscow, Idaho. 1990-96; 1998. E. Blake Moore, Jr., B.A., J.D., Toronto, Ontario. 2008. Mary A. Neagoy, B.A., Shaker Heights. 1997-2003; 2009. Solomon Oliver, Jr., B.A., J.D., M.A., Cleveland. 1991-97; 2000. John P. Papp, B.A., M.D., East Grand Rapids, Michigan. 1994. Dale C. Perry, B.A., Ph.D., Welch, Minnesota. 1971-76; 1984. Mikael Salovaara, A.B., B.A., J.D., M.B.A., Bernardsville, New Jersey. 1994. Robert L. Savitt, B.A., New York, New York. 2007. Richard N. Seaman, B.S., M.B.A., Wooster. 1994. * Kenneth E. Shafer, B.A., M.D., Wooster. 2009. Marianne M. Sprague, B.A., M.A., Santa Barbara, California. 2006. Sally J. Staley, B.A., M.I.A., Rocky River. 2006. Peter E. Sundman, B.A., New York, New York. 2003. * Nancy Wilkin Sutherland, B.A., M.A.T., Western Springs, Illinois. 2008. Clarence R. Williams, B.A., M.Ed., San Antonio, Texas. 1997-03; 2004. Ruth W. Williams, B.A., Gladwyne, Pennsylvania. 1994. James R. Wilson, B.A., M.B.A., Park City, Utah. 1980. The Directories 338 Emeritus and Emerita Life Members The first date indicates the year of first election to the Board; the second, the year of election to Emeritus or Emerita Life membership or to Honorary Life membership. Edward J. Andrew, B.A., Orland Park, Illinois. 1982; 2008. Eugene C. Bay, B.A., B.D., D.Min., Rochester, New York. 1988; 2009. Laura Bornholdt, B.A., M.A., Ph.D., Bloomington, Indiana. 1967; 1977. James T. Clarke, B.A., M.B.A., Park City, Utah. 1977-83; 1985; 2011. John J. Compton, B.A., M.A., Ph.D., Nashville, Tennessee. 1975; 2001. T. William Evans, B.A., M.Sc., D.D.S., M.D., Columbus. 1989, 2010. Julia A. Fishelson, A.B., Wooster. 1989; 1998. David D. Fleming, B.A., M.B.A., Sherborn, Massachusetts. 1999; 2008. Jerrold K. Footlick, B.A., J.D., Durham, North Carolina. 1978-84; 1988; 2006. Stanley C. Gault, B.A., LL.D., Wooster. 1972; 2000. George Ingram, Jr., B.S., M.S., North Branford, Connecticut. 1971; 1986. Max A. Lauffer, B.S., M.S., Ph.D., Middletown, Pennsylvania. 1967; 1987. James A. McClung, B.A., M.A., Ph.D., Winnetka, Illinois. 1997; 2011. Steven A. Minter, B.A., M.S., Shaker Heights. 1978-80; 1981; 2004. Ruth F. Parker, B.A., Sandusky. 1989; 1995. Timothy P. Smucker, B.A., M.A., Orrville. 1970; 1985. William F. Thompson, B.A., M.B.A., Boston, Massachusetts. 1987; 1996. Robert L. Tignor, B.A., M.A., Ph. D., Princeton, New Jersey. 1998; 2006. S. Robson Walton, B.S., J.D., Bentonville, Arkansas. 1986; 2001. Earl Wade Wendell, B.A., Nashville, Tennessee. 1997; 2002. Honorary Life Members Elizabeth S. Pocock (Mrs. J.W.), B.A., Lake Forest, Illinois. 1993. Frances G. Shoolroy, Wooster. 1987. ADMINISTRATION PRESIDENT’S CABINET Grant H. Cornwell, Jr., Ph.D., President, 2007. Ellen F. Falduto, Ed.D., Chief Information and Planning Officer, 2008; 2010. Heather M. Fitz Gibbon, Ph.D., Dean for Faculty Development, 1990; 2009. W. Scott Friedhoff, Ph.D., Vice President for Enrollment and College Relations, 2010. Kurt C. Holmes, M.A., Dean of Students, 2001. John L. Hopkins, B.A., Associate Vice President for College Relations and Marketing, 2002. Laurie K. Houck, B.A., Vice President for Development, 2011. Angela K. Johnston, B.S., Secretary of the College and Chief of Staff, 2010. Henry B. Kreuzman, Ph.D., Dean for Curriculum and Academic Engagement, 1990; 2009. Carolyn R. Newton, Ph.D., Provost, 2010. Laurie L. Stickelmaier, M.Acc., Vice President for Finance and Business and Treasurer, 2009. ADMINISTRATIVE STAFF OFFICE OF THE PRESIDENT Grant H. Cornwell, Jr., Ph.D., President, 2007. Bettye Jo Mastrine, A.S., Executive Assistant to the President, 1979; 1984. Marguerite K. Cornwell, M.A., Associate to the President, 2008. The Directories 339 OFFICE OF THE SECRETARY Angela K. Johnston, B.S., Secretary of the College and Chief of Staff, 2010. Sally A. Whitman, B.A., Executive Assistant to the Secretary and Administrative Assistant to the President, 1985; 1998. ACADEMIC AFFAIRS Carolyn R. Newton, Ph.D., Provost, 2010. Dottie S. Sines, A.S., Executive Assistant to the Provost, 2005; 2010. Lisa A. Crawford, Academic Budget and Data Analyst, 2002; 2011. Heather M. Fitz Gibbon, Ph.D., Dean for Faculty Development, 1990; 2009. Darlene G. Berresford, Administrative Coordinator for the Dean for Faculty Development, 2004; 2010. Henry B. Kreuzman, Ph.D., Dean for Curriculum and Academic Engagement, 1990; 2009. Connie L. Pattin, A.S., Executive Assistant to the Dean for Curriculum and Academic Engagement, 1982; 1984. Art Museum Kathleen McManus Zurko, M.A., Director and Curator of the Art Museum, 1989. Athletics Keith Beckett, Ph.D., Director of Physical Education, Athletics, and Recreation, 1984. Brenda Meese, M.S., Assistant Director of Physical Education, Athletics, and Recreation, 1989. Phillip P. Basile, M.S., Physical Education Center Equipment Room Manager/Equipment Purchasing Manager, 2002. Meghan E. Horn, Assistant to the Athletic Director, 2009. Thomas Love, B.S., Head Athletic Trainer, 1982. Nathaniel W. Whitfield, M.B.A., Assistant Director of Physical Education, Athletics, and Recreation Facilities and Operations, 2008; 2010. Center for Diversity and Global Engagement Susan E. Lee, M.A., Assistant Dean for Multi-ethnic Student Affairs and Co-Director of the Center for Diversity and Global Engagement, 2002; 2009. Amyaz Moledina, Ph.D., Co-Director of the Center for Diversity and Global Engagement, 2003; 2010. Nicola Kille, B.A., Assistant Director for Global Engagement, 2010. Linda Morgan-Clement, D.Min., Henry Jefferson Copeland Campus Chaplain and Director of Interfaith Campus Ministry, 1996. Jessica E. DuPlaga, M.A., Director of Off-Campus Studies, 2011. Ruth Lopez, Director of the Office of International Student Affairs, 2011. Jill A. Munro, M.A., Director of International Student and Scholar Services, 2008; 2009; 2011. Center for Entrepreneurship James A. Levin, B.A., J.D., Director of Center for Entrepreneurship, 2009. Educational Assessment Theresa Ford, M.A., Director of Educational Assessment, 2006. Educational Planning and Advising Center Alison H. Schmidt, M.Ed., Associate Dean for Educational Planning and Advising, 1989; 2011. Cathy L. McConnell, M.A., Director of the Lilly Project and the Educational Planning and Advising Center, 2009; 2010. The Directories 340 Learning Center Pamela Rose, B.A., Director of the Learning Center, 1987. Amber Larson, Assistant Director of the Learning Center, 2008. Linda Marion, M.A., Learning Disabilities Tutorial Consultant, 1985. Libraries Mark A. Christel, M.I.L.S., Director of Libraries, 2008. Patricia McVay Gorrell, Media Library Manager, Andrews Library, 1984. Gwen Short, B.S., Science Library Associate, Timken Science Library, 2007. Alyssa Bender, M.L.I.S., Access Services Evening Manager, Libraries, 2007. David M. Wiebe, B.A., Access Services Daytime Manager, Libraries, 2007. Registrar Suzanne Bates, M.Ed., Registrar, 2007. Kristine M. Kleptach Jamieson, M.L.I.S., M.A., Associate Registrar, 2009. Julia Rhind Chisnell, Assistant to the Registrar, 2009. Writing Center To be announced, Director of Writing. Writing Center Consultants: Leann Bertoncini-Thelin, M.S., 2007; Jennifer Derksen, M.A., 2007; Jessica Marie Jones, B.F.A., 2007. STUDENT AFFAIRS Kurt C. Holmes, M.A., Dean of Students, 2001. Paula J. Akins, Executive Assistant to the Dean, 2005. Carolyn L. Buxton, Ed.M., Senior Associate Dean of Students, 1993. Anne M. Gates, Ed.D., Associate Dean of Students for Academic Success and Retention, 2007; 2009. Christie Bing Kräcker, M.S., Associate Dean of Students, 2006. Robyn Laditka, M.A., Assistant Dean of Students and Director of Special Programs, 2007. Career Services Lisa Kastor, M.Ed., Director of Career Services, 1996. Marylou E. LaLonde, B.Ed., Assistant Director of Career Services, 2007. Interfaith Campus Ministries Linda Morgan-Clement, D.Min., Henry Jefferson Copeland Campus Chaplain and Director of Interfaith Campus Ministry, 1996. Health Services & Counseling Esther B. Horst, B.S.N., Acting Director, Longbrake Student Wellness Center, 2010. Ray R. Tucker, M.A., Counselor, 2000. Residence Life Krista A. Kronstein, M.A., Director of Residence Life, 2011. Joe Louis Kirk, B.A., Associate Director of Residence Life and Director of Greek Life, 1996; 2009. Lauren H. Dyer, M.Ed., Assistant Director of Residence Life, 2011. Security and Protective Services Steven D. Glick, B.A., Director of Security and Protective Services, 2011. Student Activities & Summer Programs Robert Rodda, M.A., Director of Lowry Student Center, Student Activities, and Summer Programs, 1990. The Directories 341 Michael Gorrell, Manager, Post Office, 1986. Julia C. Metcalf, Assistant Director of Student Activities, 2008. Santha Schuch, B.S., Facilities Scheduling and Summer Conferences Coordinator, 1998. ENROLLMENT AND COLLEGE RELATIONS W. Scott Friedhoff, Ph.D., Vice President for Enrollment and College Relations, 2010. Sandi Kiser, Executive Assistant to the Vice President for Enrollment, 1993; 1998. Admissions Jennifer D. Winge, B.A., Dean of Admissions, 2011. Brian D. Atkins, M.A.T., Director of International Admissions, 2009. Catherine Finks, B.S., Senior Associate Director of Admissions, 2002. James H. Williams, B.A., Senior Associate Director of Admissions and Coordinator of Multicultural Recruitment, 2010. Tamara Bergert, B.A., Associate Director of Admissions, 2007; 2010. Lindsey Duerr, M.Ed., Associate Director of Admissions, 2005; 2010. Scott Jones, B.A., Associate Director of Admissions and Systems and Communications Analyst, 2008; 2010. Rebecca Keiser, B.A., Assistant Director of Admissions, 2010; 2011. Erin Kelly, B.A., Senior Assistant Director of Admissions, 2010; 2011. Charles N. Laube, M.Ed., Associate Director of Admissions, 2007; 2011. Mollie E. Conley, B.S., Coordinator for Admissions Operations, 2008; 2010. College Relations John L. Hopkins, B.A., Associate Vice President for College Relations and Marketing, 2002. Cally Gottlieb-King, Director of Publications, 2011. Matthew P. Dilyard, College Photographer, 1987. John Finn, M.A., Director of Public Information, 1984. Hugh Howard, M.B.A., Director of Sports Information, 1999. Dwight J. Nagy, B.S., Assistant Director of Web Communication, 2010. Melissa Y. Schultz, B.S., Director of Web Communication, 2008. Nicholas T. Stroud, B.S., Assistant Director of Sports Information, 2010. Alexander C. Winkfield, B.A., Videographer/Multimedia Specialist, 2010. Financial Aid David B. Miller, Ph.D., Director of Financial Aid, 1991. Nancy Porter, Associate Director of Financial Aid, 1990. Joe Winge, B.A., Associate Director of Financial Aid, 2011. Katie Davis., Assistant Director of Financial Aid, 2005; 2011. DEVELOPMENT AND ALUMNI RELATIONS Laurie K. Houck, B.A., Vice President for Development. Jessica E. Armstrong, A.A.S., Executive Assistant to the Vice President, 1998; 2010. Alumni Relations and The Wooster Fund Heidi McCormick, B.A., Director of Alumni Relations and The Wooster Fund, 2010. Carolyn Ciriegio, B.A., Assistant Director of The Wooster Fund, 2008; 2010. Sara Dresser, B.A., Alumni Relations Assistant, 2009. Meena Ghaziasgar, M.A., Assistant Director of Development for Alumni and Parents, 2007; 2009. Landre Kiser McCloud, B.A., Assistant Director of Alumni Relations for Mentor Programs, 2008. Meret E. Nahas, B.A., Development Assistant, 2010. Sharon Rice, M.Ed., Assistant Director of Alumni Relations, 1995. The Directories 342 Development Sara L. Patton, M.A., Senior Director of Development, 1973, 2011. Bradley L. Cors, M.B.A, Director of Development, 2004. Moses P. Jones-Lewis, M.A., Director of Development, 2003. Sandra E. Nichols, B.A., Director of Development, 2002; 2010. R. Eileen Fitzgerald, Director of Research and Development, 1986. Rebecca R. Schmidt, M.A.T., Director of Stewardship, 1992. Pamela Stanley, B.S., Director of Support Systems, 1998. Erin M. Yoder, B.A., Assistant Director of Development, 2011. Wooster Alumni Magazine Karol J. Crosbie, M.S., Senior Editor, Wooster, 2005. FINANCE AND BUSINESS Laurie L. Stickelmaier, M.Acc., Vice President for Finance and Business and Treasurer, 2009. Laurel Rooks, B.A., Executive Assistant to the Vice President for Finance and Business, 2009. Business Office Peggy R. DeBartolo, B.S., Controller, 2011. Sue R. Bennett, M.B.A., Manager of Budget, 1996. Krista Way, Accounts Receivable Manager, 2006; 2011. Marlene Kanipe, Senior Financial Analyst, 2009. Desiree Lutsch, Accounts Payable Manager, 2009. Heather L. Veney, Staff Accountant, 2010. Gloria M. Wilson, B.A., Assistant Controller, 2006. Campus Grounds Beau B. Mastrine, A.A.S., Director of Campus Grounds, 1996. College Investment John W. Sell, Ph.D., Director of College Investment, 1981; 2009. Custodial Services Kenneth M. Fletcher, Director of Custodial Services, 2010. Facilities and Auxiliaries Jacqueline S. Middleton, Associate Vice President for Facilities and Auxiliaries, 1995; 2010. Bookstore Kathy Jerisek, Manager of The Florence O. Wilson Bookstore, 1973; 2004; 2010. Rogera Flack, Assistant Manager of The Florence O. Wilson Bookstore, 1989; 2004; 2010. Campus Dining Services Charles E. Wagers, Director of Campus Dining Services, 1982. Carol Berkey, Service Manager, 2001. Madison Chastain, Food Production Manager, 2006. Gladys Keegan, Purchasing Manager, 1998. Richard Keyes, Campus Dining Chef, 2002. William Stephen Marshall, Campus Dining Sous Chef, 2007. Molly E. Sanchez, General Manager of Campus Dining Student Services, 2006. Marjorie Shamp, B.A., Food Safety, Sanitation, and Training Specialist, 2000. Rebecca Underwood, Service Manager, 2004. Kristie Ward, Catering and Retail Operations Manager, 2001. Donna Yonker, General Manager of Campus Dining Support Services, 1980. The Directories 343 Physical Plant Douglas Laditka, B.A., Director of Physical Plant Operations, 2004; 2009; 2010. James Davis, Manager of Structural Trades, 2000; 2010. William Doll, Supervisor, Plumbing Shop, 2008; 2010. Fred Horst, Supervisor, Electric Shop, 1981. Thomas Lockard, Supervisor, Paint Shop, 1991. Jeffery J. Mori, Facilities Procurement and Projects Manager, 2006; 2010. Lanny Whitaker, Supervisor, Power Plant, 1993. Human Resources Gary Thompson, M.Ed., Director of Human Resources, 1972. Benefits Administrators: Holly Lantis, M.P.A., 2001; Carol Rakoczy, B.A., 2001. Jacqueline Hamilton, Manager of Employment, 2008. Kimberly Parr, B.A., Payroll Manager, 2002. Natalie Richardson, B.S., Human Resources Information System Specialist, 1989; 2007. Emily Seling, B.A., Student Employment Payroll Administrator, 2009. Purchasing and Contracts Sheila T. Wilson, M.S., Director of Purchasing and Contracts, 1991; 2006; 2010. Tracy A. Holtz, Procurement Specialist, 2010. INFORMATION AND PLANNING Ellen F. Falduto, Ed.D., Chief Information and Planning Officer, 2008; 2010. Applications Development Tabitha Conwell, Director of Applications Development, 1996. Matthew Baker, B.S., Programmer/Analyst, 2008. Deborah Kilbane, Senior Programmer/Analyst, 2005. Connie J. McCarty, A.A.B., Senior Programmer/Analyst, 2005. Michael J. Thompson, B.S., Programmer/Analyst, 2002; 2006. Digital Infrastructure Vincent T. DiScipio, B.S., Director of Digital Infrastructure, 1996. Roger Dills, B.S., Senior Systems Administrator, 2002. Daniel J. Krites, A.A., Presentation Technology and Event Support Specialist, 2008. John M. McCreight, B.A., Systems Administrator, 2006. Gerald C. McMillen, Jr., M.Ed., Web Systems Administrator, 2009. Michael V. Naylor, B.A., Assistant Director of Digital Infrastructure for Systems and Media Services, 2006. Instructional Technology Matthew K. Gardzina, M.A., Director of Instructional Technology, 2008. Joseph Benfield, Instructional Technology Support Specialist, 2008. Jon W. Breitenbucher, Ph.D., Instructional Technology Specialist, 2001; 2005. User Services Mary Schantz, B.A, Director of User Services, 2004. Michael Benchoff, B.S., Senior User Support Specialist, 2003. Joel Bosler-Kilmer, Senior User Support Specialist, 2007; 2011. David Edwards, B.S., User Support Specialist, 2007. John Shatzer, B.A., Student Technology Assistant Program Coordinator and User Support Specialist, 2000; 2011. OHIO LIGHT OPERA Laura M. Neill, B.A., Executive Director, 1993. The Directories 344 EMERITUS Henry J. Copeland, Ph.D., L.L.D., President, 1966. Emeritus since 1995. Raleigh Stanton Hales, Jr., Ph.D., Sc.D., President, 1990. Emeritus since 2007. Kenneth R. Plusquellec, M.Div., Dean of Students, 1967. Emeritus since 2001. William H. Snoddy, Vice President for Finance and Business, 1959. Emeritus since 1999. FACULTY As of July 1, 2011 In the groups below, the names following that of the President are arranged alphabetically, with the date of first appointment. A second or third date indicates a reappointment. Graduate training is indicated by the names of the institutions where such training was received and the dates. A double dagger (‡) before the name denotes deceased. Grant H. Cornwell, Jr., President. 2007. B.A. St. Lawrence 1979; M.A., Ph.D. Chicago 1982, 1989. Emeritus William M. Baird, Professor of Economics. 1968. Emeritus since 1997. B.A. Wittenberg 1957; M.A., Ph.D. Ohio State 1959, 1968. James Edgar Bean, Associate Professor of French and Physical Education. 1965. Emeritus since 1987. B.A. Wooster 1942; B.D., S.T.M. Union Theological Seminary 1945, 1966; M.A. Kent State 1975. Donald Gene Beane, Professor of Mathematical Sciences. 1962. Emeritus since 1994. B.A. Iowa Wesleyan 1951; M.A., Ph.D. Illinois 1958, 1962; M.S. Ohio State 1966. Barbara L. Bell, Government Information Librarian. 1980. Emerita since 2004. B.S., M.S. Florida State 1963, 1965. Richard H. Bell, Frank Halliday Ferris Professor of Philosophy. 1969. Emeritus since 2004. B.A. Vanderbilt 1960; B.D., M.A., Ph.D. Yale 1964, 1966, 1968. Robert B. Blair, Professor of Sociology. 1971. Emeritus since 1999. Registrar 2002-2007. Emeritus since 2007. B.A. Juniata 1960; B.D. Bethany Theological Seminary 1964; M.A., Ph.D. Northwestern 1967, 1974. Charles Lamonte Borders, Jr., Professor of Chemistry. 1968. Emeritus since 2002. B.A. Bellarmine College 1964; Ph.D. California Institute of Technology 1968. Richard Hayden Bromund, Professor of Chemistry. 1967. Emeritus since 2006. A.B. Oberlin 1962; Ph.D. Pennsylvania State 1968. Dale Allen Brown, Professor of Mathematical and Computer Sciences. 1987. Emeritus since 2008. B.A. Hiram 1967; M.S., Ph.D. Syracuse 1969, 1973. Daniel Fairchild Calhoun, Aileen Dunham Professor of History. 1956. Emeritus since 1994. B.A. Williams 1950; M.A., Ph.D. Chicago 1951, 1959. Clayton Paul Christianson, Mildred Foss Thompson Professor of English Language and Literature. 1963. Emeritus since 1997. B.A. St. Olaf 1955; M.A. State University of Iowa 1956; Ph.D. Washington University 1964. Gordon Dixon Collins, Whitmore-Williams Professor of Psychology. 1963. Emeritus since 2000. B.A. Tarkio 1957; M.A., Ph.D. Ohio State 1960, 1965. Henry J. Copeland, Professor of History. 1966. Emeritus since 1999. President 1977- 1995. Emeritus since 1995. A.B. Baylor 1958; Ph.D. Cornell 1966. The Directories 345 Frederick William Cropp III, Professor of Geology. 1964. Emeritus since 1998. B.A. Wooster 1954; M.S., Ph.D. Illinois 1956, 1958. W. Lee Culp, Registrar. 1947, 1963. Emeritus since 1985. B.A. Wooster 1941. Margaret W. Curl, Collection Services Librarian. 1988. Emerita since 2011. B.A. Central Washington 1971; M.A. Oregon 1974; M.L.I.S. California, Berkeley 1984. Floyd Leslie Downs, Professor of Biology. 1963. Emeritus since 1998. A.B. Cornell 1958; M.S., Ph.D. Michigan 1960, 1965. Brian James Dykstra, Neille O. and Gertrude M. Rowe Professor of Music. 1969. Emeritus since 2007. B.S. Juilliard 1964; M.M. Eastman 1965; D.M.A. Eastman 1969. Richard C. Figge, Gingrich Professor of German. 1974. Emeritus since 2004. B.A. Carleton College 1964; M.A., Ph.D. Stanford 1966, 1970. Susan G. Figge, Professor of German. 1977. Emerita since 2004. B.A. California, Santa Barbara 1964; M.A., Ph.D. Stanford 1966, 1974. Joanne S. Frye, Professor of English. 1976. Emerita since 2009. B.A. Bluffton 1966; Ph.D. Indiana 1974. John Morgan Gates, Aileen Dunham Professor of History. 1967. Emeritus since 2002. A.B., M.A. Stanford 1959, 1960; Ph.D. Duke 1967. Paul L. Gaus, Benjamin S. Brown Professor of Chemistry. 1977. Emeritus since 2008. B.S. Miami, Ohio 1971; Ph.D. Duke 1975. Raleigh Stanton Hales, Jr., Professor of Mathematical Sciences. 1990. President 1995- 2007. Emeritus since 2007. B.A. Pomona 1964; M.A., Ph.D. Harvard 1965, 1970. Charles R. Hampton, Johnson Professor of Mathematical Sciences. 1972. Emeritus since 2008. B.S. Michigan 1967; M.A., Ph.D. Wisconsin 1968, 1972. Ishwar C. Harris, Synod Professor of Religious Studies. 1981. Emeritus since 2009. B.A. Lucknow (India) 1961; M.Div. Howard 1967; S.T.M. Pacific, Berkeley 1969; Ph.D. Claremont 1974. Peter L. Havholm, Professor of English. 1971. Emeritus since 2009. B.A. Shimer 1962; M.S., Ph.D. Connecticut 1970, 1972. LeRoy Wilbur Haynes, Professor of Chemistry. 1961. Emeritus since 1999. B.A. Drew 1956; Ph.D. Illinois 1961. Henry Dunham Herring, Professor of English. 1969. Emeritus since 2007. B.A., M.A. South Carolina 1961, 1964; Ph.D. Duke 1968. Damon D. Hickey, Director of Libraries. 1991. Emeritus since 2008. B.A. Rice 1965; M.Div. Princeton Theological Seminary 1968; M.L.S. North Carolina, Chapel Hill 1975; M.A. North Carolina, Greensboro 1982; Ph.D. South Carolina 1989. Claude W. Hinton, Horace N. Mateer Professor of Biology. 1968. Emeritus since 1990. A.B., M.A. North Carolina 1948, 1950; Ph.D. California Institute of Technology 1954. James Andrew Hodges, Michael O. Fisher Professor of History. 1962. Emeritus since 1998. B.S. North Alabama 1955; M.A., Ph.D. Vanderbilt 1959, 1963. Vivian Loyrea Holliday, Aylesworth Professor of Classical Studies. 1961. Emerita since 1999. A.B. Winthrop 1957; M.A. Missouri 1959; Ph.D. North Carolina 1961. John Louis Hondros, Henry J. and Laura H. Copeland Professor of European History. 1969. Emeritus since 2001. A.B. North Carolina, Chapel Hill 1959; M.A., Ph.D. Vanderbilt 1963, 1969. Linda C. Hults, Professor of Art. 1987. Emerita since 2011. B.A. Indiana 1971; Ph.D. North Carolina, Chapel Hill 1978. The Directories 346 Charles E. Hurst, Professor of Sociology. 1970. Emeritus since 2008. B.S., M.S. Wisconsin, Milwaukee 1963, 1965; Ph.D. Connecticut 1972. Donna K. Jacobs, Science Librarian. 1989. Emerita since 2009. B.A. South Florida 1969; M.L.S. Kent State 1989. Michael D. Kern, Horace N. Mateer Professor of Biology. 1976. Emeritus since 2003. B.A. Whittier 1962; M.S., Ph.D. Washington State 1965, 1970. William Franklin Kieffer, Robert E. Wilson Professor of Chemistry. 1940, 1946. Emeritus since 1980. B.A. Wooster 1936; M.Sc. Ohio State 1938; Ph.D. Brown 1940. Rodney J. Korba, Associate Professor of Communication. 1987. Emeritus since 2010. B.F.A. Denison 1972; M.A. Denver 1973; M.A. Arizona State 1974; M.A. Southern California 1982; Ph.D. Denver 1986. Dudley Arnold Lewis, Professor of Art. 1964. Emeritus since 1996. B.A. Allegheny 1952; M.A., Ph.D. Wisconsin 1954, 1962. Henry Bernard Loess, Professor of Psychology. 1958. Emeritus since 1988. B.S., M.S. Northwestern 1949, 1950; Ph.D. Iowa 1952. David Franklyn Moldstad, Professor of English. 1957. Emeritus since 1991. A.B. Hiram 1947; M.A. Brown 1949; Ph.D. Wisconsin 1954. Robert E. Nye, Professor of Physical Education. 1964. Emeritus since 1995. B.S., M.S. Springfield 1958, 1959. George William Olson, Professor of Art. 1963. Emeritus since 2000. B.A. Augustana 1958; M.A., M.F.A. Iowa 1962, 1963. Virginia B. Pett, Robert E. Wilson Professor of Chemistry. 1981. Emerita since 2009. B.A. Wooster 1963; M.A., Ph.D. Wayne State 1972, 1979. Kenneth R. Plusquellec, Dean of Students. 1967. Emeritus since 2001. B.A. Wooster 1957; M.Div. McCormick Theological Seminary 1960. Gene Edward Pollock, Hoge Professor of Economics. 1961. Emeritus since 1998. B.S., M.S., Ph.D. Ohio State 1960, 1961, 1971. David L. Powell, Benjamin S. Brown Professor of Chemistry. 1964. Emeritus since 2001. A.B. Oberlin 1958; Ph.D. Wisconsin 1962. Margaret S. Powell, Government Information Librarian. 1980. Emerita since 2001. A.B. Oberlin 1958; M.S. Illinois 1959. Richard Dale Reimer, Hoge Professor of Economics. 1962. Emeritus since 1996. B.A. Bethel 1957; M.S. Kansas State 1958; Ph.D. Michigan State 1962. John Merrill Russell, Professor of Music. 1974. Emeritus since 2008. B.Mus. Oberlin 1965; M.Mus. Boston 1968. William B. Scott, Associate Professor of Psychology. 1984. Emeritus since 2008. B.A. Franklin and Marshall 1970; M.A. Simon Fraser 1972; Ph.D. McGill (Canada) 1979. Gordon Lichty Shull, Professor of Political Science. 1955. Emeritus since 1993. A.B. Manchester 1946; B.D. Yale Divinity School 1951; M.A., Ph.D. Illinois 1952, 1955. Robert Houston Smith, Professor of Religious Studies. 1960. Emeritus since 1996. B.A. Tulsa 1952; B.D. Yale Divinity School 1955; Ph.D. Yale 1960. Elena Sokol, Professor of Russian Studies. 1987. Emerita since 2009. B.A. Colorado 1965; M.A., Ph.D. California, Berkeley 1967, 1974. Atlee LaVere Stroup, Professor of Sociology and Education. 1948. Emeritus since 1989. B.S. in Ed. Kent State 1942; M.A., Ph.D., M.S.W. Ohio State 1946, 1950, 1982. Leslie Gordon Tait, Mercer Professor of Religious Studies. 1956. Emeritus since 1991. B.A. Harvard 1948; M.Div. Pittsburgh Theological Seminary 1951; Ph.D. Edinburgh 1955; Honorary Fellow, Edinburgh 1991. The Directories 347 Karen J. Taylor, Associate Professor of History. 1986. Emerita since 2010. B.A. Utah 1980; M.A. Clark 1982; Ph.D. Duke 1988. Pablo Valencia, Professor of Spanish. 1961. Emeritus since 1992. B.A., M.A., Ph.D. Michigan 1959, 1960, 1966. Alvin James Van Wie, Professor of Physical Education. 1960. Emeritus since 1991. B.A. Wooster 1952; M.A. Northern Michigan 1963. John Ward Warner, Jr., Professor of Mathematical Sciences and Education. 1958. Emeritus since 1983. A.B. Taylor 1940; M.S. Iowa 1958; Ph.D. Ohio State 1964. Andrew A. Weaver, Professor of Biology. 1955. Emeritus since 1986. B.A. Wooster 1949; M.S., Ph.D. Wisconsin 1951, 1955. Margaret Barnes White, Associate Professor of Psychology and Education. 1961. Emerita since 1984. B.A. Wooster 1960; M.A. Kent State 1961. David Jordan Wilkin, Professor of French. 1964, 1971. Emeritus since 2002. B.A. Hope 1961; M.A. Pittsburgh 1964; Ph.D. Brown 1974. Yvonne Williams, Professor of Black Studies and Political Science. 1973, 1978. Emerita since 2000. B.A. Pennsylvania State 1953; M.A. Connecticut 1955; Ph.D. Case Western Reserve 1981. Donald Louis Wise, Danforth Professor of Biology. 1958. Emeritus since 1994. B.A. Wabash 1951; M.S., Ph.D. New York 1954, 1958. Elbridge Carl Zimmerman, Associate Professor of Mathematical Sciences and Director of Academic Computing Services. 1968. Emeritus since 1994. B.A. Wooster 1954; M.S. Drexel Institute of Technology 1962; Michigan 1963, 1964. Active Abigail Adams, Visiting Assistant Professor of Anthropology. 2011. B.A. Wooster 1993; M.A., Ph.D. New Mexico 1996, 2009. Mary Addis, Associate Professor of Spanish. 1984. B.A. St. Mary’s (Notre Dame) 1971; M.A., Ph.D. California, San Diego 1977, 1984. Elissa Alzate, Visiting Assistant Professor of Political Science. 2011. B.A. Florida International 2003; M.A., Ph.D. California 2006, 2010. Judith C. Amburgey-Peters, Associate Professor of Chemistry. 1996. B.S. Georgetown College 1988; Ph.D. North Carolina, Chapel Hill 1993. Heath Anderson, Juliana Wilson Thompson Visiting Assistant Professor of Anthropology. 2010. B.A. Boston 1999; M.A., Ph.D. Pennsylvania State 2003, 2009. Ahmet Atay, Assistant Professor of Communication. 2010. B.A. Marmara (Turkey) 1998; M.A. Ohio 2001; M.A. Northern Iowa 2003; Ph.D. Southern Illinois 2009. Mary Anna Bader, Associate Professor of Religious Studies. 2002. B.S. Houghton 1983; M.Div. Lutheran Theological Southern Seminary 1987; Th.M., Ph.D. Lutheran School of Theology, Chicago 1993, 2002. Suzanne Bates, Registrar. 2007. B.S., M.S. Indiana 1971, 1975. Kabria Baumgartner, Instructor in History. 2011. B.A., M.A. California, 2003; Ph.D. Massachusetts (expected). Keith D. Beckett, Professor of Physical Education and Director of Physical Education, Athletics, and Recreation. 1984. B.S., M.S. Indiana University of Pennsylvania 1979, 1982; Ph.D. Pittsburgh 1989. Katharine Beutner, Visiting Instructor in English. 2011. B.A. Smith 2003; M.A. Texas, Austin 2006; Ph.D. Texas at Austin (expected). Paul A. Bonvallet, Associate Professor of Chemistry. 2004. B.A. Kenyon 1996; Ph.D. Wisconsin 2001. The Directories 348 Angela Bos, Assistant Professor of Political Science. 2007. B.A., M.A., Ph.D. Minnesota 2001, 2005, 2007. Denise M. Bostdorff, Professor of Communication. 1994. B.S. Bowling Green 1982; M.A. Illinois 1983; Ph.D. Purdue 1987. Daniel Bourne, Professor of English. 1988. B.A., M.F.A. Indiana 1979, 1987. Jennifer L. Bowen, Assistant Professor of Mathematics and Computer Science. 2007. B.A. Boston College 1998; M.S., Ph.D. Virginia 2001, 2005. Matthew W. Broda, Assistant Professor of Education. 2007. B.S. Kent State 1999; M.Ed. Ashland 2004; Ph.D. Kent State 2007. Barbara S. Burnell, Professor of Economics. 1977. B.A. Connecticut 1973; M.A., Ph.D. Illinois 1975, 1977. James D. Burnell, Professor of Economics. 1977. B.A. Western Illinois 1973; M.A., Ph.D. Illinois 1975, 1977. Denise D. Byrnes, Associate Professor of Mathematical Sciences. 1991. B.S., M.S., Ph.D. Ohio State 1985, 1987, 1992. Lisa Campanell Komara, Coaching Staff with Adjunct Teaching Duties in Physical Education. 1997. B.A. West Liberty State 1982; M.S. West Virginia 1988; West Virginia. Michael B. Casey, Associate Professor of Psychology. 2000. B.A. Michigan State 1982; M.S., Ph.D. Virginia Tech 1993, 1995. Mark A. Christel, Director of Libraries. 2008. B.A. Wisconsin 1990; M.A. Rutgers 1992; M.I.L.S. Michigan 1994. Susan D. Clayton, Whitmore-Williams Professor of Psychology. 1992. B.A. Carleton 1982; M.S., Ph.D. Yale 1984, 1987. Jessica Clemons, Science Librarian. 2009. B.S. SUNY, Syracuse 2006; M.S.L.S. Clarion 2008. Sibrina Collins, Assistant Professor of Chemistry. 2008. B.A. Wayne State 1994; M.S., Ph.D. Ohio State 1996, 2000. Brian J. Cope, Associate Professor of Spanish. 2005. B.A. Washington, St. Louis 1995; M.A., Ph.D. California, Irvine 1998, 2004. Grant H. Cornwell, Jr., President and Professor of Philosophy. 2007. B.A. St. Lawrence 1979; M.A., Ph.D. Chicago 1982, 1989. Yuri F. Corrigan, Assistant Professor of Russian Studies. 2009. B.A. Saskatchewan (Canada) 2001; M.A., Ph.D. Princeton 2005, 2008. Christa C. Craven, Assistant Professor of Anthropology and Women’s, Gender, and Sexuality Studies. 2006. B.A. New College, Florida 1997; M.A., Ph.D. American 2000, 2003. Patrick Crittenden, Visiting Assistant Professor of Biology, 2011. B.S., M.S., Ph.D. Mississippi 1991, 1994, 2008. Lisa W. Crothers, Instructor in Religious Studies. 2010. B.A. Muhlenberg 1990; M.A. Indiana 1998; Ph.D. Emory (expected). Carrie deLapp Culver, Associate Professor of Music. 2008. B.Mus., M.M. Missouri 1987, 1989; D.A. Northern Colorado 1998. James R. Daehn, Visiting Assistant Professor of Mathematics and Computer Science. 2010. B.S. SUNY, Oswego, 1992; M.S., RIT, 1994. Suzanne Daly, John Garber Drushal Distinguished Visiting Assistant Professor of English. 2011. B.A. Pennsylvania 1993; M.A., M.Phil., Ph.D. Columbia 1995, 1999, 2002. The Directories 349 Nancy E. Ditmer, Professor of Music. 1984. B.M. Capital 1972; M.A. Iowa 1982. Theodor Duda, Professor of Music. 1990. B. Mus. Baldwin-Wallace 1973; M.M. Michigan State 1975; A.Mus.D. Illinois, Urbana-Champaign 1995. Carolyn A. Durham, Inez K. Gaylord Professor of French Language and Literature. 1976. B.A. Wellesley 1969; M.A., Ph.D. Chicago 1972, 1976. Marion Duval, Gillespie Visiting Assistant Professor of French. 2011. B.A., M.A. Picardie (France) 2000, 2004; M.A. Iowa, 2005; Ph.D. Iowa 2011. Paul L. Edmiston, Professor of Chemistry. 1997. B.S. Pepperdine 1993; Ph.D. Arizona 1997. Karl J. Feierabend, Assistant Professor of Chemistry. 2009. B.S. Furman 2001; Ph.D. Colorado, Boulder 2006. Heather Moir Fitz Gibbon, Dean for Faculty Development and Professor of Sociology. 1990. A.B. Kenyon 1981; M.A., Ph.D. Northwestern 1983, 1988. Monika J. Flaschka, Visiting Assistant Professor of History and Women’s, Gender, and Sexuality Studies. 2011. B.S. Arizona, 1997; M.A., Ph.D. Kent State, 2001, 2004, 2009. Monica Florence, Assistant Professor of Classical Studies. 2007. B.A. Reed 1994; Ph.D. Boston University 2004. Stephen Flynn, Emerging Technologies Librarian. 2011. B.A. Lawrence 2009; M.S.I. Michigan 2011. Travis M. Foster, Assistant Professor of English. 2009. B.A. Amherst 1999; M.A., Ph.D. Wisconsin 2003, 2009. Dean Fraga, Professor of Biology. 1994. B.S. Cincinnati 1982; Ph.D. Wisconsin 1990. Pamela R. Frese, Professor of Anthropology. 1986. B.A. Maryland 1974; M.A., Ph.D. Virginia 1979, 1982. Joan S. Friedman, Assistant Professor of History and Religious Studies and Campus Rabbi. 2004. B.A. Pennsylvania 1974; M.A.H.L. Hebrew Union-Jewish Institute of Religion 1977; M.Phil., Ph.D. Columbia 1986, 2003. Joan E. Furey, Assistant Professor of Communication. 2007. B.A. Chicago 1990; M.A. Pittsburgh 1994; Ph.D. Illinois 2003. John P. Gabriele, Raymond and Carolyn Dix Professor of Spanish. 1986. B.A., M.A. Connecticut 1975, 1977; Ph.D. North Carolina, Chapel Hill 1981. Jack B. Gallagher, Olive Williams Kettering Professor of Music. 1977. B.A. Hofstra 1969; M.F.A., D.M.A. Cornell 1975, 1981. Harry Y. Gamble, Associate Professor of French. 2002. B.A. Wake Forest 1989; M.A., Ph.D. New York 1996, 2002. Amber L. Garcia, Assistant Professor of Psychology. 2006. B.A. St. Mary’s, California 1996; M.A. Claremont 2000; Ph.D. Purdue 2005. Shila Garg, William F. Harn Professor of Physics. 1987. B.S. Madras (India) 1970; M.S. Sussex (U.K.) 1972; Ph.D. Kent (U.K.) 1975. David Gedalecia, Michael O. Fisher Professor of History. 1971. B.A. CUNY, Queens 1965; M.A., Ph.D. Harvard 1967, 1971. Gary Gillund, Associate Professor of Psychology. 1989. B.S. North Dakota State 1977; Ph.D. Indiana 1982. Donald M. Goldberg, Professor of Communication. 1996. 2010. B.S. Lafayette College 1977; M.A., Ph.D. University of Florida 1979, 1985. Mark D. Gooch, Technology and Government Information Librarian. 2001. B.A. Wooster 1990; M.L.S. Kent State 1994. The Directories 350 Jennifer L. Graber, Assistant Professor of Religious Studies. 2006. B.A. Goshen 1995; M.T.S. Emory 1999; Ph.D. Duke 2006. Nancy M. Grace, Professor of English. 1987. B.A. Otterbein 1973; M.A., Ph.D. Ohio State 1981, 1987. Mark W. Graham, Associate Professor of Religious Studies. 2002. B.A., M.A. Ohio 1979, 1988; Ph.D. Indiana 2003. Simon J. M. Gray, Associate Professor of Computer Science. 1987. 2003. B.A., M.A. Virginia 1982, 1984; M.S. North Texas 1987; Ph.D. Kent State 1998. Raymond Gunn, Assistant Professor of Sociology. 2006. B.A. Hunter 1984; M.A. Long Island 1997; Ph.D. Pennsylvania 2009. Julia Chance Gustafson, Access Services Librarian. 1982. B.A., M.S. Western Michigan 1979, 1981. James L. Hartman, Professor of Mathematical Sciences. 1981. B.S. Manchester 1975; M.S., Ph.D. Michigan State 1977, 1981. Jennifer Poole Hayward, Professor of English. 1992. B.A. Wesleyan 1983; M.A. San Francisco State 1987; Ph.D. Princeton 1992. Julie Heck, Visiting Assistant Professor of Biology. 2011. B.A. Wooster 1997; Ph.D. Cornell 2006. Mareike Herrmann, Associate Professor of German. 2001. Magisterstudium ChristianAlbrechts (Germany) 1988; M.A. Bowling Green 1989; Ph.D. Massachusetts 1995. Madonna J. Hettinger, Lawrence Stanley Professor of Medieval History. 1989. A.B. St. Francis 1977; M.A., Ph.D. Indiana 1979, 1986. Kurt C. Holmes, Dean of Students. 2001. B.S. Allegheny 1988; M.A. West Virginia 1997. Katherine Holt, Associate Professor of History. 2005. B.A. George Washington 1995; M.A. New York 1999; M.A., Ph.D. Princeton 2001, 2005. Matt Hooley, Visiting Instructor in English. 2011. B.A. Carleton, 2004; M.A. Wisconsin-Madison 2006; Ph.D. Wisconsin-Madison (expected). Shirley A. Huston-Findley, Associate Professor of Theatre. 1999. B.A. Indiana 1986; M.A. Miami, Ohio 1989; Ph.D. Missouri, Columbia 1998. Ronald Earl Hustwit, Frank Halliday Ferris Professor of Philosophy. 1967. B.A. Westminster 1964; M.A. Nebraska 1965; Ph.D. Texas 1970. Donald Thomas Jacobs, Victor J. Andrew Professor of Physics. 1976. B.A., M.A. South Florida 1971, 1972; Ph.D. Colorado 1976. John G. Jewell, Visiting Assistant Professor of Psychology. 2009. B.S. Lebanon Valley 1992; M.S. Bucknell 1994; Ph.D. Kent State 1998. Michelle L. Johnson, Associate Professor of Communication. 1997. B.A. Northeast Missouri State 1990; M.A., Ph.D. Arizona 1992, 1996. Shelley A. Judge, Assistant Professor of Geology. 2008. B.S. Mount Union 1991; M.A.T. Kent State 1993; M.S., Ph.D. Ohio State 1998, 2007. Charles L. Kammer III, James F. Lincoln Professor of Religious Studies. 1990. B.A. Colgate 1968; M.Div., Ph.D. Duke 1971, 1977. Bryan T. Karazsia, Assistant Professor of Psychology. 2009. B.S. Denison 2003; M.A., Ph.D. Kent State 2005, 2009. Paul Nick Kardulias, Professor of Anthropology and Archaeology. 1996. B.A., M.A. Youngstown State 1974, 1977; M.A. SUNY, Binghamton 1980; Ph.D. Ohio State 1988. Kent J. Kille, Associate Professor of Political Science. 1998, 2000. B.A. North Carolina, Chapel Hill 1991; M.A., Ph.D. Ohio State 1995, 2000. 351 The Directories Shannon King, Assistant Professor of History. 2007. B.A., M.A. North Carolina Central 1996, 1998; Ph.D. Binghamton 2006. Ronda Kirsch, Visiting Instructor of Mathematics. 2011. B.S., M.A. Kent State 2008, 2010. Stacia Kock, Visiting Instructor in Women’s, Gender, and Sexuality Studies. 2011. B.A. Wooster 2004; M.A. Louisville 2006. Jacob E. Koehler, Visiting Acquisitions Librarian. 2005. B.A. Ohio State 2002; M.L.I.S. Kent State 2008. Matthew Krain, Associate Professor of Political Science. 1998. B.A. SUNY, Binghamton 1992; Ph.D. Indiana 1998. Henry B. Kreuzman III, Dean for Curriculum and Academic Engagement and Associate Professor of Philosophy. 1990. B.S. Xavier 1981; M.A., Ph.D. Notre Dame 1984, 1990. Mary Joan Kreuzman, Visiting Assistant Professor of Mathematics. 1990. B.S. Xavier 1980; M.S., Ph.D. Notre Dame 1982, 1985. Jeffrey S. Lantis, Professor of Political Science. 1994. B.A. Bethany 1988; M.A., Ph.D. Ohio State 1991, 1994. Elys L. Law, Reference and Instruction Librarian. 2003. B.A., M.L.I.S. Wisconsin, Milwaukee 1988, 1991. Cody Leary, Assistant Professor of Physics. 2011. B.S. Puget Sound 2003; M.S., Ph.D. Oregon 2004, 2010. Susan Y. Lehman, Clare Boothe Luce Associate Professor of Physics. 2003. B.A. Goshen 1993; M.S., Ph.D. North Carolina, Chapel Hill 1996, 1999. Michele Leiby, Instructor in Political Science. 2011. B.A. Moravian 2002; M.A. New Mexico 2004; Ph.D. New Mexico (expected). Richard M. Lehtinen, Associate Professor of Biology. 2003. B.S. Winona 1995; M.S. Minnesota 1997; Ph.D. Michigan 2003. James A. Levin, James N. Wise Visiting Associate Professor of Theatre and Dance and Director of Arts Management and Entrepreneurship. 2009. B.A. Michigan 1975; J.D. Case Western Reserve 1979. Karen T. Lewis, Assistant Professor of Physics. 2010. B.S. Wisconsin-Madison 1999; Ph.D. Pennsylvania State 2005. Jeffrey Lindberg, Professor of Music. 1986. B.S., M.S. Illinois, Urbana-Champaign 1976, 1978; Vienna Academy of Music. John Lindner, Moore Professor of Astronomy. 1988. B.S. Vermont 1982; Ph.D. California Institute of Technology 1989. Dan Liu, Visiting Instructor in Chinese. 2011. B.A. Yunnan 2005. Marilyn D. Loveless, Horace N. Mateer Professor of Biology. 1987. B.A. Albion 1971; Ph.D. Kansas 1984. Sharon E. Lynn, Associate Professor of Biology. 2004. B.S. South Carolina 1996; Ph.D. Washington 2002. John R. Lytle, Visiting Assistant Professor of French. 2010. B.A. Wooster 1999; M.A., M.Phil. Yale 2002, 2004; Ph.D. Yale 2007. Marina Mangubi, Associate Professor of Art. 2000. A.B. California, Berkeley 1988; M.F.A. Michigan 1993. Matthew Mariola, Visiting Assistant Professor of Environmental Studies. 2009. B.A. Wooster 1998; M.S. Wisconsin 2004; Ph.D. Ohio State 2009. Terri Mason, Visiting Instructor in Education. 2011. B.A. Oklahoma 1974; M.A. Central Oklahoma 1982. Setsuko Matsuzawa, Assistant Professor of Sociology. 2008. B.A. Sophia 1987; M.A. Duke 1991; M.A. Georgetown Law Center 1994; Ph.D. California 2007. The Directories 352 Lee A. McBride III, Assistant Professor of Philosophy. 2006. B.A. St. Mary’s, California 1997; M.A. Claremont 1999; Ph.D. Purdue 2006. David L. McConnell, Professor of Anthropology. 1992. B.A. Earlham 1982; M.A., Ph.D. Stanford 1991. Brenda L. Meese, Associate Professor of Physical Education and Assistant Director of Physical Education, Athletics, and Recreation. 1989. B.A. Wooster 1975; M.S. North Carolina, Greensboro 1981. Philip Mellizo, Assistant Professor of Economics. 2010. B.A. Wyoming 2002; M.A. Massachusetts (Amherst) 1979; Massachusetts (Amherst). Bridget J. Milligan, Associate Professor of Art. 2001. B.F.A. Miami, Ohio 1997; M.F.A. Indiana 2001. Sarah Zubair Mirza, Visiting Fellow. 2011. B.A. William Paterson 2000; M.A., Ph.D. Michigan 2004, 2010. Amyaz A. Moledina, Associate Professor of Economics. 2003. B.A. Macalester 1993; Ph.D. Minnesota 2002. Denise D. Monbarren, Special Collections Librarian. 1985. B.A., M.L.S. Kent State 1982, 1983. Nicole J. Moore, Visiting Assistant Professor of Physics. 2011. B.S. Harvey Mudd 2003; Ph.D. Rochester 2009. Steve L. Moore, Coaching Staff with Adjunct Teaching Duties in Physical Education. 1987. A.B. Wittenberg 1974; M.S. Ohio 1976. Kara Morrow, Assistant Professor of Art and Art History. 2010. B.F.A., Auburn 1993; M.A. Alabama 1998; Ph.D. Florida State 2007. William R. Morgan, Professor of Biology. 1991. B.S. Cornell 1982; Ph.D. Yale 1988. Eric Moskowitz, Associate Professor of Political Science. 1984. B.A. Earlham 1965; M.A., Ph.D. Indiana 1974, 1979. Peter C. Mowrey, Associate Professor of Music. 1997. B.Mus., M.Mus. Georgia 1989, 1991; D.Mus. Indiana 1995. Beth Ann Muellner, Associate Professor of German. 2004. B.A. Minnesota 1985; M.A. Maryland 1995; Ph.D. Minnesota 2003. Mazen Naous, Assistant Professor of English. 2009. B.F.A. Boston Conservatory 1996; M.A. Massachusetts, Boston 2001; Ph.D. Massachusetts, Amherst 2007. Boubacar N’Diaye, Associate Professor of Black Studies and Political Science. 1999. B.A., M. Public Admin. Sangamon State 1991, 1993; Ph.D. Northern Illinois 1998. John G. Neuhoff, Associate Professor of Psychology. 2000. B.A. Baldwin Wallace 1991; M.A., Ph.D. Kent State 1994, 1996. Carolyn R. Newton, Provost and Professor of Biology. 2010. B.S. Colorado State 1973; Ph.D. SUNY Buffalo 1979. Jimmy A. Noriega, Instructor in Theatre and Dance. 2011. B.A. Missouri Southern 2005; M.A. Cornell 2009; Ph.D. Cornell (expected). Anne M. Nurse, Associate Professor of Sociology. 1999. B.A. California, Berkeley 1990; M.A., Ph.D. California, Davis 1993, 1999. Cynthia L. Palmer, Associate Professor of Spanish. 2000. B.S. Northern Arizona 1987; M.A., Ph.D. Arizona 1993, 2000. R. Drew Pasteur, Assistant Professor of Mathematics. 2008. B.S., M.Ed. Florida 1996, 1997; M.S., Ph.D. North Carolina State 2004, 2008. Sheryl M. Petersen, Visiting Instructor in Biology. 2011. B.A. Hiram 2000; M.S. Case Western Reserve 2006; Ph.D. Case Western Reserve (expected). Charles F. Peterson, Associate Professor of Black Studies. 2000. B.A. Morehouse 1992; M.A., Ph.D. SUNY, Binghamton 1995, 2000. The Directories 353 Timothy C. Pettorini, Coaching Staff with Adjunct Teaching Duties in Physical Education. 1981. B.A., M.A. Bowling Green 1973, 1980. Pamela B. Pierce, Professor of Mathematical Sciences. 1994. B.A. Amherst 1985; M.Ed. Massachusetts 1986; M.S., M.Phil., Ph.D. Syracuse 1988, 1992, 1994. Meagen A. Pollock, Assistant Professor of Geology. 2008. B.S. Marshall 2001; Ph.D. Duke 2007. Peter C. Pozefsky, Professor of History. 1994. A.B. Harvard 1984; M.A., Ph.D. UCLA 1986, 1993. Maria Teresa Micaela Prendergast, Associate Professor of English. 1999. 2007. B.A. Yale 1977; M.A., Ph.D. Virginia 1983, 1990. Thomas A. Prendergast, Associate Professor of English. 1997. B.A. Marquette 1981; M.A., Ph.D. Virginia 1984, 1992. Diana B. Presciutti, Assistant Professor of Art and Art History. 2011. B.A. Dartmouth 1998; M.A. Syracuse 2003, Ph.D. Michigan 2008. John R. Ramsay, Professor of Mathematical Sciences. 1987. B.A. Berea 1980; M.S., Ph.D. Wisconsin 1984, 1987. Jeremy Rapport, Visiting Assistant Professor of Religious Studies. 2009. B.A., M.A. Kansas 1992, 2002; Indiana 2009. Evan P. Riley, Visiting Assistant Professor of Philosophy. 2009. B.A. Louisville 1995; B.A. Oxford (U.K.) 1997; Ph.D. Pittsburgh 2008. J. Morgan Robison, Visiting Instructor in Spanish. 2011. B.A., M.A. Ohio State 1986, 1990. Jeff Roche, Associate Professor of History. 2001. B.A., M.A. Georgia State 1992, 1995; Ph.D. New Mexico 2000. John Rudisill, Associate Professor of Philosophy. 2005. B.A. Coe 1991; Ph.D. Iowa 2001. Jake Rundall, Visiting Assistant Professor of Music. 2011. B.A. Carleton 2002; M.M., Ph.D. Illinois 2004, 2010. William Albert Hayden Schilling, Robert Critchfield Professor of English History. 1964. B.A. Southern Methodist 1958; M.A., Ph.D. Vanderbilt 1961, 1970. Elizabeth A. Schiltz, Associate Professor of Philosophy. 2003. B.A. Ohio Wesleyan 1993; Ph.D. Duke 2000. Alison H. Schmidt, Associate Professor of Education and Associate Dean of Advising. 1989. B.A. Wooster 1975; M.Ed. Ashland 1992; Kent State. Sarah J. Schmidtke, Assistant Professor of Chemistry. 2006. B.S. Marquette 2000; Ph.D. Minnesota 2005. Michael L. Schmitz, Coaching Staff with Adjunct Teaching Duties in Physical Education. 1995. B.S. Bowling Green 1974. Melissa M. Schultz, Assistant Professor of Chemistry. 2006. B.S. Creighton, 1999; Ph.D. Oregon 2004. Dale E. Seeds, Professor of Theatre. 1984. B.S. Bowling Green 1971; M.A., Ph.D. Kent State 1975, 1990. John W. Sell, James R. Wilson Professor of Business Economics. 1981. B.S. Pennsylvania State 1973; M.A., Ph.D. UCLA 1975, 1981. Ibra Sene, Assistant Professor of History. 2008. B.A., M.A., M.I.L.S. Université Cheikh Anta Diop (Dakar, Senegal), 1996, 1998, 1999; Ph.D. Michigan State 2008. Nicholas Shaw, Visiting Assistant Professor of Chemistry. 2011. B.A. St Olaf 2002; Ph.D. Clemson 2010. Gregory K. Shaya, Associate Professor of History. 2001. B.A., M.A., Ph.D. Michigan 1988, 1993, 2000. The Directories 354 Josephine Shaya, Assistant Professor of Classical Studies. 2001. B.A., M.A., Ph.D. Michigan 1988, 1994, 2001. Debra Shostak, Professor of English. 1987. B.A. Carleton 1975; M.A., Ph.D. Wisconsin 1977, 1985. John Siewert, Associate Professor of Art. 2000. B.A. Minnesota 1985; M.A., Ph.D. Michigan 1987, 1995. Laura K. Sirot, Assistant Professor of Biology. 2010. B.S., M.A. Michigan 1992, 1993; M.S., Ph.D. Florida 1999, 2004. Mark J. Snider, Associate Professor of Chemistry. 2001. B.A. Capital 1997; Ph.D. North Carolina, Chapel Hill 2001. Amy Jo Stavnezer, Associate Professor of Psychology. 2002. B.S. Allegheny 1994; M.S., Ph.D. Connecticut 1998, 2000. Larry L. Stewart, Mildred Foss Thompson Professor of English Language and Literature and Professor of Education. 1967. B.A. Simpson 1963; M.A., Ph.D. Case Western Reserve 1964, 1971. Stephanie S. Strand, Assistant Professor of Biology. 2008. B.S. Nevada 1995; Ph.D. Washington, St. Louis 2004. Wendy Teo, Merton M. Sealts, Jr. Visiting Instructor in Classical Studies. 2010. B.A. Reed 2000; Ph.D. Brown (expected). Claudia Thompson, Associate Professor of Psychology. 1982. B.A. Delaware 1975; Ph.D. Brown 1981. Garrett Thomson, Elias Compton Professor of Philosophy. 1994. B.A. Newcastle on Tyne 1978; Ph.D. Oxford 1984. Thomas F. Tierney, Associate Professor of Sociology. 1999. B.A. Moravian 1979; Ph.D. Massachusetts 1990. Kimberly Tritt, Professor of Theatre. 1983. B.F.A., M.A. Ohio 1975, 1983. Diane Ringer Uber, Professor of Spanish. 1989. B.A. Wooster 1974; M.A. Penn - sylvania State 1977; Ph.D. Wisconsin 1981. Bas van Doorn, Assistant Professor of Political Science. 2007. B.S. Michigan 1992; M.A. Amsterdam (The Netherlands) 2000; Ph.D. Minnesota 2008. Lisa L. Verdon, Assistant Professor of Economics. 2009. B.S. Lake Superior State 1998; M.A. North Carolina, Greensboro 2004; Ph.D. Florida State 2009. Sophia Visa, Assistant Professor of Computer Science. 2008. B.S. Sibiu (Romania) 1998; M.S., Ph.D. Cincinnati 2002, 2007. Rujie Wang, Professor of Chinese. 1995. B.A. Wabash 1983; M.A., Ph.D. Rutgers 1986, 1993. James M. Warner, Associate Professor of Economics. 1999. B.A. Bates 1983; M.A. California, Riverside 1985; M.A. New Hampshire 1992; Ph.D. Utah 1996. Mark R. Weaver, Professor of Political Science. 1978. B.A. Ohio 1970; M.A., Ph.D. Massachusetts 1973, 1979. Megan Wereley, Associate Professor of Education. 2002. B.A. Wooster 1994; M.A., Ed.D. Teachers College, Columbia 1996, 2005. James D. West, Assistant Professor of Biochemistry and Microbiology. 2008. B.S. Campbell 1999; Ph.D. Vanderbilt 2005. Margaret L. Wick, Visiting Assistant Professor of Communication. 2001. 2007. 2010. B.S., M.A., Ph.D. Southern Illinois 1971, 1974, 1978. The Directories 355 Gregory C. Wiles, Associate Professor of Geology. 1998. B.S. Beloit 1984; M.S. SUNY, Binghamton 1987; Ph.D. SUNY, Buffalo 1992. Craig Willse, Visiting Assistant Professor of Sociology. 2011. B.A. New College of Florida 1997; M.A. Queens 2005; Ph.D. City University of New York 2010. Mark A. Wilson, Lewis M. and Marian Senter Nixon Professor of Natural Sciences and Geology. 1981. B.A. Wooster 1978; Ph.D. California, Berkeley 1982. Leslie E. Wingard, Assistant Professor of English. 2008. B.A. Spelman 1999; M.A., Ph.D. UCLA 2002, 2006. Thomas G. Wood, Professor of Music. 1991. B.M., M.M. Cincinnati 1980, 1985; D.M.A. Wisconsin 1992. Robert L. Woodward, Jr., Walter D. Foss Visiting Assistant Professor of Chemistry. 2010. B.A., B.S., Ph.D. Ohio State 2006, 2010. Josephine R. B. Wright, Professor of Music and Josephine Lincoln Morris Professor of Black Studies. 1981. B.M. Missouri 1963; M.M. Pius XII Academy (Italy) 1964; M.A. Missouri 1967; Ph.D. New York 1975. Lisa Wong Yozviak, Assistant Professor of Music. 2009. B.S. West Chester 1991; M.Mus. D.M. Indiana 1997, 2010. Jingjing Yang, Visiting Assistant Professor of Economics. 2011. B.A. Xiamen University (China) 2001; M.A. Fudan University (China) 2004; Ph.D. Michigan State 2010. Walter Zurko, Professor of Art. 1981. B.A. Wisconsin 1977; M.F.A. Southern Illinois, Edwardsville 1980. Adjunct Teaching Staff Jerri Lynn Baxstrom, French. 2010. B.A., M.Ed. Ashland 1974, 1992. Katie E. Boes, Biology. 2010. B.S. Bucknell 2002; Ph.D. Michigan State 2008. Carol M. Bucher, Education. 1998. B.A. Ashland 1967; M.A. Connecticut 1985. April Contway, FYS. 2010. B.A. Wooster 1999; M.A. Kent State 2010. Jay Gates, Pocock Family Distinguished Visiting Scholar. 2011. B.A. Wooster 1968; M.A. Rochester 1970; Ph.D. Illinois 2008. Charlene A. Gross, Theatre and Dance. 2007. B.A. Ashland 1997; M.F.A. New York 2000. Manon Grugel-Watson, Physics. 2009. B.A. Wooster 1999; M.S. Case Western Reserve University 2001. Mary Kilpatrick, Chemistry. 1995. B.A. Wooster 1964. Denise Rotavera-Krain, FYS, Music. 2004. B.A. Binghamton University 1994; M.A. Temple Uni versity 1997. Tommy D. Love, Physical Education. 1982. M.Ed. Cincinnati 1979. Carol A. Mapes, Economics. 2005. B.S. Ashland 1986. Steven McCallum, Art and Art History. 2011. B.F.A., M.F.A. Kent State 1973, 1976. Mary Mullen, English. 2011. B.A. Notre Dame 2004; M.A., Ph.D. WisconsinMadison 2005, 2010. Emily A. Sullivan, Art and Art History. 2010. M.F.A. Kent State 2010. The Directories 356 COMMITTEES OF THE FACULTY 2011-2012 The President is a member of all faculty committees except the Committee on Conference with Trustees and the Committee on Teaching Staff and Tenure. ELECTED COMMITTEES Teaching Staff and Tenure: Provost, Daniel Bourne (3), Beth Ann Muellner (3,) Shirley Huston-Findley (2), Alison Schmidt (2), Elizabeth Schiltz (1), Mark Snider (1) Faculty Grievance: Mareike Herrmann (3), Madonna Hettinger (3), Charles Kammer (3), Denise Bostdorff (1), Mark Graham (1) Educational Policy: Dean for Curriculum and Academic Engagement, Provost, Travis Foster (3), Walter Zurko (3), Jennifer Bowen (2), Matthew Broda (2), Katherine Holt (1), Gregory Wiles (1) Conference with Trustees: Paul Bonvallet (3), Angela Bos (3), Shannon King (3), Mark Wilson (2), Dean Fraga (1), Amy Jo Stavnezer (1) Financial Advisory: Provost, Vice President for Development, Vice President for Finance and Business, Dean of Students, Mark Christel (3), Erik Moskowitz (3), Melissa Schultz (3), Jennifer Graber (2), Susan Lehman (2), James Warner (1) Committee on Committees: Dean for Faculty Development, Yuri Corrigan (2), Shelley Judge (2), Matthew Krain (1), Sarah Schmidtke (1) Biology Marilyn Loveless Stephanie Strand (fall semester) Biochemistry and Molecular Biology James West (fall semester) Chemistry Judy Amburgey-Peters Classical Studies Josephine Shaya Computer Science Sofia Visa (fall semester) English Jennifer Hayward Thomas Prendergast Debra Shostak French Harry Gamble Geology Meagen Pollock (fall semester) History Gregory Shaya (spring semester) Music Peter Mowrey Physics Susan Lehman Psychology Susan Clayton (spring semester) Political Science Kent Kille (fall semester) Mark Weaver (spring semester) Sociology/Anthropology Christa Craven David McConnell Anne Nurse (spring semester) Spanish John Gabriele (fall semester) Cynthia Palmer (spring semester) Faculty Members on Leave 2011-2012 The Directories 357 APPOINTED COMMITTEES Academic Standards: Dean for Curriculum and Academic Engagement, Dean of Students, Associate Dean of Students, Registrar, Vice President for Enrollment and College Relations, Denise Byrnes, Lisa Crothers, Drew Pasteur, Leslie Wingard Alumni Board: Peter Pozefsky, John Siewert Assessment: Carrie deLapp-Culver, Michelle Johnson, Laura Sirot Campus Council: Provost, Vice President for Finance and Business, Dean of Students, Julia Gustafson, Lisa Verdon, Megan Wereley Campus Sustainability: Richard Lehtinen, Karen Lewis, Matthew Mariola Henry J. Copeland Fund for Independent Study: Sibrina Collins, Sharon Lynn, Setsuko Matsuzawa, Boubacar N’Diaye, Dale Seeds Cultural Events: Monica Florence, Rujie Wang, Lisa Yozviak GLCA: Madonna Hettinger, Bas van Doorn Honorary Degrees: Amber Garcia, Jeffrey Lantis, Bridget Milligan, Denise Monbarren Judicial Board: Brian Cope, Joan Friedman, Raymond Gunn, Charles Kammer, Brenda Meese, Robert Woodward Library, Information Resources, and Technology: Provost, Dean for Faculty Development, Chief Information Technology Officer, Director of Instructional Technology, Director of User Services, Director of the Libraries, Joan Furey, Mark Gooch, Ibra Sene Publications: Jacob Koehler, Claudia Thompson Research and Study Leaves: Provost, Dean for Faculty Development, Marina Mangubi, John Neuhoff, Charles Peterson APPOINTMENTS TO GREAT LAKES COLLEGES ASSOCIATION COMMITTEES AND COUNCILS Academic Council: Madonna Hettinger, Bas van Doorn Committee for Institutional Commitment to Educational Equity (ICEE): Susan Lee, Amyaz Moledina Women’s Studies Committee: Nancy Grace ACM/GLCA Chinese Studies: Rujie Wang ACM/GLCA Newberry Library Program: Maria Teresa Prendergast GLCA New York Arts Program: Marina Mangubi ACM/GLCA Oak Ridge Science Semester (Natural): Melissa Schultz GLCA Philadelphia Center: James Burnell GLCA Borders Program: Pamela Frese COMMITTEES APPOINTED BY THE DEANS Appointed by the Dean for Faculty Development Faculty Research and Development Committee: Dean for Faculty Development, John Lindner, Maria Teresa Prendergast, John Sell Appointed by the Dean for Curriculum and Academic Engagement Educational Planning and Advising Steering Committee: Alison Schmidt, Anne Gates, Lisa Kastor, Cathy McConnell, Pamela Rose The Directories 358 Graduate Scholarships and Fellowships Committee: Davies-Jackson Scholarship: Denise Bostdorff Fulbright-Hays Scholarship: John Siewert Barry M. Goldwater Scholarship: Sarah Schmidtke Kendall-Rives Latin America Research Scholarship: Pamela Frese Luce Scholars Program: Thomas Wood James Madison Fellowship: Shannon King British Marshall Scholarship: Madonna Hettinger Earl F. Morris Memorial Scholarship: John Rudisill National Science Foundation Graduate Research Fellowship: Sarah Schmidtke Rhodes Scholarship: Madonna Hettinger Truman Scholarship: Bas van Doorn Morris K. Udall Scholarship: Matthew Mariola Off-Campus Study Advisory Committee: Director of Off-Campus Studies, Dean for Curriculum and Academic Engagement, Mareike Herrmann, Madonna Hettinger, Jeffrey Lantis, Melissa Schultz COMMITTEES APPOINTED BY THE COMMITTEE ON TEACHING STAFF AND TENURE Interdepartmental Curriculum Committees: Archaeology: P. Nick Kardulias, J. Heath Anderson, Josephine Shaya, Gregory Wiles Biochemistry and Molecular Biology: William Morgan, Paul Edmiston, Dean Fraga, Mark Snider, Stephanie Strand, James West Chemical Physics: Paul Edmiston, John Lindner Comparative Literature: Mary Addis, Yuri Corrigan, Carolyn Durham, Monica Florence, Beth Ann Muellner, Mazen Naous, Rujie Wang East Asian Studies: Mark Graham, David Gedalecia, Setsuko Matsuzawa, Rujie Wang Environmental Studies: Richard Lehtinen, Charles Kammer, Matthew Mariola, Melissa Schultz Film Studies: Mareike Herrmann, Brian Cope, Carolyn Durham, Dale Seeds, John Siewert International Relations: James Warner, Katherine Holt, Kent Kille, Matthew Krain, Jeffrey Lantis, Amyaz Moledina, Peter Pozefsky, Ibra Sene, Gregory Shaya Latin American Studies: Katherine Holt, Cynthia Palmer Neuroscience: Amy Jo Stavnezer, Dean Fraga, Gary Gillund, Sharon Lynn South Asian Studies: Mark Graham, Lisa Crothers, Shirley Huston-Findley, Elizabeth Schiltz Urban Studies: James Burnell, Heather Fitz Gibbon, Eric Moskowitz Women’s, Gender, and Sexuality Studies: Nancy Grace, Travis Foster, Amber Garcia, Raymond Gunn, Katherine Holt 359 WOMEN’S ADVISORY BOARD The Women’s Advisory Board of The College of Wooster was established in 1892. The Board serves to provide financial support for eligible women and to encourage interaction with international students. The primary goals of the Board are to continually fund and administer scholarships as well as to promote interest in The College of Wooster among alumni and the local community. Associate Members Honorary Members Wendy Barlow Mona Buehler Ann Cicconetti Marie Cross Cheryl Gooch Mary Beth Henthorne Jackie Kiefer Beth Ladrach Sue Mathur Laura Neill Cheryl Shapiro Mary Jane Beem, Wooster Jane Snyder Black, Mansfield Mim Blair, Medina Carol Lower Brenner, Kirtland Nancy Winder Carpenter, Cleveland Peg Reed Clay, Canal Fulton Christabel Dadzie, Laurel, MD Virginia Estrop, Springfield Connie Arnold Garcia, Bellaire, MI Ellen Bergantz Hunter, Medina Nova Brown Kordalski, Brecksville Elizabeth Van Cleef Lauber, Dover Joan Leasure McAnlis, Wadsworth Joanne McAnlis, Fishers, IN Sandra Moser McIlvaine, Wadsworth Carol Stromberg Pancoast, Bay Village Ruth Mast Steimel, Millersburg Mary Ann Keibler Taylor, Shaker Heights Bonnie Trubee, Millersburg Lucinda Weiss, Akron Helen Agricola Wright, Coshocton Pat Bare, Wooster Carol Briggs, Wooster Geri Rice Burden, Wooser Lolly Copeland, Montreat, NC Marguerite (Peg) Kelsey Cornwell, Wooster Marian Cropp, Wooster Julia Fishelson, Wooster Lois Freedlander, Wooster Flo Kurtz Gault, Wooster Catherine Graves, Wooster Diane Moore Hales, Santa Rosa, CA Marge Neiswander Hoge, Wooster Elizabeth Hooker, Wooster Gennie Johnston, Wooster Beverly Lacey, Nellysford,VA Kathy Long, Wooster Gayle Tinlin Noble, Wooster Dr. Vi Startzman Robertson, Wooster Shirley Ryan, Wooster Judy Seaman, Wooster Jill Henley Shafer, Wooster Fran Shoolroy, Wooster Jenny Coddington Smucker, North Lawrence Heidi Steiner, Wooster Mina Ramage Van Cleef, Wooster Janet Welty, Wooster Buzz Williams, San Antonio, TX Kathy Zink, Wooster Executive Committee Officers Heather Nicolozakes, President Pam Matsos, First Vice President Jennifer Reynolds, Second Vice President Karin Wiest, Recording Secretary Elsa Boen, Corresponding Secretary Mary Zuercher, Treasurer 360 THE CALENDAR 2011-2012 SEMESTER I August 25 - 28 Thurs.-Sun. — New Student Orientation 27 - 28 Sat.-Sun. — Upperclass students arrive 29 Mon. — Classes begin at 8:00 a.m. 30 Tues. — 142nd Convocation at 11:05 a.m. September 23 - 25 Fri.- Sun. — Family Weekend September 30 - Oct. 2 Fri.- Sun. — Homecoming Weekend October 14 Fri. — Fall break begins at 4:00 p.m.* 19 Wed. — Classes resume at 8:00 a.m. November 22 Tues. — Thanksgiving recess begins at 4:00 p.m.* 28 Mon. — Classes resume at 8:00 a.m. December 9 Fri. — Classes end at 4:00 p.m. 10 - 11 Sat.-Mon. — Reading Days 12 - 15 Mon.- Thurs. — Examinations* SEMESTER II January 16 Mon. — Classes begin at 8:00 a.m. March 9 Fri. — Spring break begins at 4:00 p.m.* 26 Mon. — Classes resume at 8:00 a.m. 26 Mon. — Senior Independent Study Thesis due at 5:00 p.m. April 27 Fri. — Senior Research Symposium May 4 Fri. — Classes end at 4:00 p.m. 5 - 6 Sat.-Sun. — Reading Days 7 - 10 Mon.-Thurs. — Examinations* 13 Sun. — Baccalaureate 14 Mon. — Commencement SUMMER SESSION May 21 Mon. — Classes begin at 8:00 a.m. June 29 Fri. — Classes end at 4:00 p.m. *Classes/examinations must be held at scheduled times. Plan travel arrangements accordingly. 361 THREE-YEAR CALENDAR 2012-2013 2013-2014 2014-2015 SEMESTER I Fall Classes Begin 8:00 a.m. — 8/27 (Mon) 8/26 (Mon) 8/20 (Wed) Fall Break Begins 4:00 p.m.* — 10/12 (Fri) 10/4 (Fri) 10/3 (Fri) Classes Resume 8:00 a.m. — 10/17 (Wed) 10/9 (Wed) 10/13 (Mon) Thanksgiving Recess Begins — 11/20 (Tues) 11/26 (Tues) 11/25 (Tues) 4:00 p.m.* Classes Resume 8:00 a.m. — 11/26 (Mon) 12/2 (Mon) 12/1 (Mon) Fall Classes End 4:00 p.m. — 12/7 (Fri) 12/6 (Fri) 12/5 (Fri) Fall Reading Days Begin — 12/8 (Sat) 12/7 (Sat) 12/6 (Sat) Fall Reading Days End — 12/9 (Sun) 12/8 (Sun) 12/7 (Sun) Examination Days Begin* — 12/10 (Mon) 12/9 (Mon) 12/8 (Mon) Examination Days End — 12/13 (Thurs) 12/12 (Thurs) 12/11 (Thurs) Winter Break 4 weeks 4 weeks 4 weeks SEMESTER II Spring Classes Begin 8:00 a.m. — 1/14 (Mon) 1/13 (Mon) 1/12 (Mon) Spring Break Begins 4:00 p.m.* — 3/8 (Fri) 3/7 (Fri) 3/6 (Fri) Classes Resume 8:00 a.m. — 3/25 (Mon) 3/24 (Mon) 3/23 (Mon) Senior I.S. Due 5:00 p.m. — 3/25 (Mon) 3/24 (Mon) 3/23 (Mon) Spring Classes End 4:00 p.m. — 5/3 (Fri) 5/2 (Fri) 5/1 (Fri) Spring Reading Days Begin — 5/4 (Sat) 5/3 (Sat) 5/2 (Sat) Spring Reading Days End — 5/5 (Sun) 5/4 (Sun) 5/3 (Sun) Examination Days Begin* — 5/6 (Mon) 5/5 (Mon) 5/4 (Mon) Examination Days End — 5/9 (Thurs) 5/8 (Thurs) 5/7 (Thurs) Baccalaureate — 5/12 (Sun) 5/11 (Sun) 5/10 (Sun) Commencement — 5/13 (Mon) 5/12 (Mon) 5/11 (Mon) Summer Session Begins — 5/20 (Mon) 5/19 (Mon) 5/18 (Mon) Summer Session Ends — 6/28 (Fri) 6/27 (Fri) 6/26 (Fri) *Classes/examinations must be held at scheduled times. Plan travel arrangements accordingly. 362 TRAVEL DIRECTIONS BY CAR: From Canton: • Rte 30 W for 30 miles into Wooster • Exit onto Madison Ave. • Turn right off exit and proceed into Wooster on Bever St. (Campus is on the right) From Akron: • I-76 W to Rte 21 S • Rte 21 S to Rte 585 • Rte 585 W for 21 miles into Wooster to Wayne Ave. • Right on Wayne Ave. to stop sign at Bever St. • Left on Bever St. (Campus is on the left) From Cleveland: • I-71 S to Rte 83 (Wooster Exit) • South on Rte 83 for 14 miles to Wooster • Once you enter Wooster, turn right at Friendsville Road • Go through 4 stoplights until Burbank becomes Bever St. (Campus is on the left) From Columbus: • I-71 N to Rte 30 E • Rte 30 E for 28 miles into Wooster • Exit onto Madison Ave. • Turn right off exit and proceed into Wooster on Bever St. (Campus is on the right) Beall Avenue Mechanicsburg Road Oak Hill Road Highland Avenue Oldman Road Cleveland Road Gasche Street Milltown Road Portage Road Bowman Street Friendsville Road From Columbus (I-71 North) From Akron (I-76 West and Route 21) From Cleveland (I-71 South) From Canton Street Wayne Avenue Liberty Street Burbank Road Bever Madison Avenue Exit ★The College of Wooster 30 30 585 83 BY AIR: Wooster is served by Cleveland Hopkins International Airport and Akron-Canton Airport. A shuttle service is available from the College for service to and from the airport. 1 Andrews Library 2 Ebert Art Center Art History Studio Art Art Museum Publications 3 Freedlander Theatre 4 Galpin Hall Academic Affairs Business Office Deans’ Offices Finance and Business President’s Office Development Offices 5 Gault Admissions Center 6 Gault Alumni Center Alumni Relations Development Offices 7 Gault Library for Independent Study Financial Aid Registrar 8 Grounds Crew Garages 9 Human Resources 10 Kauke Hall Humanities Social Sciences 11 Lilly House Educational Planning and Advising Center 12 Longbrake Student Wellness Center 13 Lowry Student Center Dining Hall Mom’s (restaurant) Post Office Scot Lanes Wilson Bookstore 14 Mateer Hall Biology 15 McGaw Chapel 16 Burton D. Morgan Hall Business Economics Economics Education Information Technology Psychology 17 Olderman House Public Information 18 Overholt House Interfaith Campus Ministries 19 Pearl House 20 President’s Home 21 Rubbermaid Student Development Center Career Services Learning Center 22 Culbertson/Slater Complex Keys and IDs Residence Life Security and Protective Services 23 Scheide Music Center 24 Scot Center 25 Scovel Hall Geology Philosophy 26 Service Center 27 Severance Hall Chemistry 28 Taylor Hall Computer Science Mathematical Science Physics 29 Timken Library Science Library 30 Westinghouse Power Plant 31 Westminster Church House 32 Wishart Hall Communication Freedlander Speech and Hearing Clinic Theatre and Dance WCWS-FM 33 The Wooster Inn 34 Armington Physical Education Center 35 L.C. Boles Golf Course 36 Cindy Barr Field 37 Carl W. Dale Soccer Field 38 D.J. Hard Tennis Courts 39 Murray Baseball Field 40 Papp Stadium 41 Softball Diamond 42 Timken Gymnasium 43 Andrews Hall 44 Armington Hall 45 Babcock Hall Ambassadors Program Center for Diversity and Global Engagement Off-Campus Study 46 Bissman Hall 47 Bornhuetter Hall 48 Compton Hall 49 Douglass Hall 50 Gault Manor 51 Holden Hall 52 Kenarden Lodge 53 Kittredge Hall 54 Luce Residence Hall 55 Stevenson Hall 56 Wagner Hall 48 42 15 24 25 26 27 30 35 36 37 38 46 43 44 45 47 29 8 4 5 7 6 16 17 18 52 51 49 50 19 20 21 23 28 22 32 31 34 41 9 11 3 2 39 40 1 33 12 13 14 53 54 55 56 10 The Wooster Campus INDEX A AH (arts and humanities) course, 219, 223 abbreviations, 21 about the College, 5-12 academic honors, 268 Academic Integrity, Code of, 239 applications of, 213, 228, 261 (sec. 4), 262 academic probation, 236-237 academic standing, 237 accreditation and memberships, 8 adding a course, 230 (sec. 4) audit, 230 (sec. 7), 233 late registration fee, 230 (sec. 8) administration, 338-344 admission, 253-262 appeal, 260 application procedure, 254-255 application timetable, 253 deferred, 255 implications of, 260-262 international students, 256-257 readmission, 238 transfer students, 257-260 advanced placement, 258-260 class standing and, 236 international, 256, 258 See also individual departments advising, academic, 18-19, 242 pre-professional, 206 Africana studies, 22-26 alcohol policies, 262 Ambassadors Program, 241 Ancient Mediterranean Studies. See Classical Studies Anthropology. See Sociology and Anthropology appeal of policies, 238 Archaeology, 27-29 architecture, pre-professional program in, 207-208 Art and Art History, 29-37 Art Museum, 19, 242, 244 arts program in New York, 200, 217 athletic facilities, 328, 329, 335 athletics, intercollegiate and intramural, 244-245 auditing courses, 233, 263 appeal, 238 deadline, 230 (sec. 7), 233 grading, 231 B Bachelor of Arts, 218-219, 226 Bachelor of Music, 138-140, 219-222, 226 Bachelor of Music Education double degree, 226 music therapy, 140, 224-226 public school teaching, 140, 222-224 Besançon, Wooster in, 109-110 Biochemistry and Molecular Biology, 38-40 Biology, 40-47 Board of Trustees, 336-338 British Advanced-Level Examination, 258 buildings and facilities academic, 327-330 administrative, 330-331 athletic, 328, 329, 335 other, 334-336 residence halls and houses, 331-334 business, pre-professional program in, 208 Business Economics, 47-50 internship, 216 C C (studies in cultural difference) course, 218, 220, 222, 224 calendar, academic, 360-361 Campus Council, 244 campus activities, 244-249 campus map, 363 career services, 240-241 Caribbean Advance Proficiency Examinations (CAPE), 258 Center for Diversity and Global Engagement, 241 Center for Entrepreneurship, 242 chairs, endowed, 278-283 Chemical Physics, 50-51 Chemistry, 51-56 364 365 China, Wooster in, 214 Chinese Studies, 56-59 Christian groups on campus, 247 class attendance, 230 (sec. 3) class hours, 235 class standing, 236 Classical Studies, 59-65 clubs and sections, 248 codes. See academic integrity; social responsibility College Writing course, 125, 232 (sec. D) commencement, 228 double degrees and, 226 See also graduation Communication, 65-72 workshop credits, 234 Comparative Literature, 73-76 comprehensive fee. See fees Computer Science, 77-80 CONSORT, 20 contact information, inside back cover core values, 10-11 counseling career, 240-241 health and personal, 252-253 course abbreviations, 21 course load, 230-231 course numbering, 20-21 credits, 230-231 advanced placement, 259-260 international, 256, 258 amount per course, 231 Bachelor of Arts, 218-219 Bachelor of Music, 219-222 Bachelor of Music Education music therapy, 224-226 public school teaching, 222-224 communication workshops, 234 fractional. See fractional credits international, 256-257 internships, 128, 214, 234 maximum, in major or department, 219 maximum, per semester, 227, 231 military courses, 235 minimum, 227 music instruction, 146-147, 234 music performance, 146, 150-151, 234 off-campus study, 212-217 overload. See overload physical education, 158-159, 234 theatre workshops and performance 234 transfer, 227, 257-260 cultural difference. See C (studies in cultural difference) course cultural life, 244-249 curriculum, 14-17 D dance. See Theatre and Dance deadlines adding a course, 230 (sec. 4) admissions applications, 253 audit, 230 (sec. 7) deferred admission, 255 double degree, 226 double major, 229 dropping a course, 230 (sec. 5) grade change, 233 (sec. L) in absentia graduation, 207, 228 incomplete, grade of, 232 (sec. G) major, declaration of, 229 medical withdrawal, 232 (sec. F) minor, declaration of, 230 off-campus study, 212-213 payment of fees, 263-264 readmission applications, 238 S/NC declaration, 230 (sec. 6) student-designed major, 229 transfer credits, 258 Dean’s List, 268 deferred admission, 255 degree requirements, 218-227 Bachelor of Arts, 218-219 Bachelor of Music, 219-222 Bachelor of Music Education, 222-226 double degree, 226 general requirements, 227 repeated courses and, 232 (sec. H) transfer credit and, 251-252 degrees offered, 218 departmental honors, 269 dining, 249-250 agreement, 262 directions to the College, 362 directories, 336-359 disability services, 243 distribution requirements, 218-220, 222-223, 224-225 See also degree requirements 366 Diversity and Global Engagement, Center for, 241 diversity groups, 248 double counting Bachelor of Arts, 218 Bachelor of Music, 219, 220 Bachelor of Music Education music therapy, 224, 225 public school teaching, 222, 223 double majors, 229 minor, 229-230 repeated courses, 232 (sec. H) double degree, 226 double major, 229 dropping a course, 230 (sec. 5) dual degree programs, 207-212 E early decision and early action, 255 East Asian Studies, 80-82 Ebert Art Center, 242 Economics, 83-86 Education, 86-91 Educational Planning and Academic Advising Center, 18-19, 242 employment, student, 266 endowed resources chairs, 278-283 funds, 283-292 scholarships, 292-327 engineering, pre-professional program in, 208-209 English, 91-101 English language proficiency, 256-257 enrollment and security deposit, 251, 255, 262, 263 Entrepreneurship, Center for, 242 Environmental Studies, 102-104 Ethics and Society internship, 215 expulsion, 261 extracurricular activities, 244-249 F faculty, 18 active, 347-355 adjunct, 355 committees, 356-358 emeritus, 344-347 on leave, 356 student evaluations of, 235 Family Educational Rights and Privacy Act (FERPA), 233 fees, 262-265 application, 254, 255 billing and payment, 263-264 comprehensive fee, 262-264 course change, late, 230 (sec. 8) enrollment and security deposit, 251, 255, 262, 263 late payment fee, 263 medical, 263 Monthly Payment Plan, 263-264 music lessons, 262-263 off-campus study, 263 overload, 262 refund insurance plan, 265 registration late, 230 (sec. 8) re-registration, 263 withdrawal from College, 264-265 Film Studies, 104-105 final examinations, 233 (sec. J), 235 financial aid, 265-268 application, 254 (sec. 7), 266 international students, 256 standing, 237 withdrawal from College, 264-265 First-Year Seminar, 125 degree requirement, 218, 219, 222, 224 grading, 232 (sec. D) transfer students, 257-258 food services. See dining foreign language requirement Bachelor of Arts, 218 Bachelor of Music, 219 international students, 257 forestry and environmental studies, pre-professional program in, 209 fractional credits, 231, 234 final examinations and, 233 music, 146, 234 physical education, 158-159, 234 fraternities. See clubs and sections French, 106-110 full-time status, 231 academic probation and, 236-237 funds, endowed, 283-292 G G, grade of, 231 (sec. A) general education requirements abbreviations, 21 Bachelor of Arts, 218-219 367 Bachelor of Music, 219-220 Bachelor of Music Education music therapy, 224-225 public school teaching, 222-223 transfer credits and, 257-258 gender studies. See Women’s, Gender, and Sexuality Studies Geology, 110-112 German Studies, 113-117 GLCA. See Great Lakes Colleges Association Goliard, The, 248 Gospel Choir, 150, 245 grade point average, 231-232 (sec. C) academic standing and, 237 Dean’s List and, 268 departmental honors and, 269 graduation requirement, 227-228 Latin Honors and, 269 off-campus study and, 213, 231-232 (sec. C) repeating a course, 232 (sec. H) transfer credit and, 257 grades, 230-234 A-F, grades of, 231-232 (secs. A, C) changing, 233 (sec. L) faculty obligations, 232-233 (sec. I) final, 233 (sec. J) G, grade of, 231-232 (secs. A, C) H, grade of, 231-232 (secs. A, C) I (incomplete), grade of, 232 (sec. G) Independent Study, 231 (sec. A), 232 (sec. D), 233 (sec. K) L, grade of, 231 (sec. A) pass/fail. See S/NC grade release of, 233 (sec. M) W, grade of, 232 (sec. E) See also grade point average; S/NC grade graduate qualities, 11-12 graduation, 228 double degrees and, 226 honors, 269 in absentia, 207, 228 See also degree requirements grants, 13, 243, 265, 267 Great Lakes Colleges Association (GLCA), 8, 213 Greek. See Classical Studies Greek life. See clubs and sections H H, grade of, 231-232 (secs. A, C) HSS (history and social sciences) course, 219, 223, 225 health, pre-professional programs in, 210-211 health services and counseling, 252-253 Hebrew. See Religious Studies Hillel group, 247 History, 118-125 history of the College, 5-8 home-schooled students, 255 honor societies, 269 honors and prizes, academic, 268-277 housing, 250-251 agreement, 251, 262 I I (incomplete), grade of, 232 (sec. G) Independent Study, 12-14 Bachelor of Arts, 219 Bachelor of Music, 220-222 Bachelor of Music Education music therapy, 225 public school teaching, 223 course load, 227 double major, 229 evaluation of, 233 (sec. K) final examinations and, 233 (sec. J) grades, 231-232 (secs. A, D), 233 (sec. K) residency, 227 Index yearbook, 248 Information Technology, 19 Institute for the International Education of Students (IES) French offerings, 109 German offerings, 117 interdepartmental courses, 125-128 interdisciplinary studies, 16-17, 126-127 Interfaith Campus Ministries, 246-247 International Baccalaureate (IB), 256, 258 International Relations, 128-130 international students admission, 256-257 advanced placement, 256, 258 Office of International Student Affairs, 241 organizations, 248 internships, 128, 214-217, 234 See also individual departments intramural sports, 244-245 368 J Jazz Combo, 246 Jazz Ensemble, 150, 246 Jewish life on campus, 247 junior class standing, 236 L L, grade of, 231 (sec. A.1) Latin. See Classical Studies Latin American Studies, 130-132 Latin honors, 269 law, pre-professional program in, 211 leadership and liberal learning, 127 Learning Across the Disciplines (LAD), 219, 259 Learning Center, 242-243 leave of absence, 250, 264 libraries, 19-20, 327-328, 330 Lilly Project, 243, 245 loans, 266 Longbrake Student Wellness Center, 252-253, 334 Lowry Center, 251-252, 334 M MNS (mathematical and natural sciences) course, 219, 223 major credits, maximum in, 219 declaration of, 229 double, 229 grades, minimum, 227, 236 pass/fail (S/NC), 232 (sec. D) student-designed, 229 transfer credits, 256-258 map of campus, 363 Math Center, 243 Mathematics, 132-137 meal plan, 249 medical plan, student accident and sickness, 253, 263 merit scholarships, 267-268 military course credit, 235 minor, declaration of, 230 pass/fail (S/NC), 232 (sec. D) mission statement, 10 molecular biology. See Biochemistry and Molecular Biology Monthly Payment Plan, 263-264 music Department of, 137-151 lesson fee, 262-263 minor, 141 performance credits, 146, 234 performance groups, 150-151, 245-246 private instruction, 146, 262-263 scholarships, 267 See also degree requirements music therapy, 140, 148-149, 224-226 See also music Muslim life on campus, 241 N NC, grade of, 231, 232 (secs. E, G) grade point average, 231-232 (sec. C) network, campus, 19 Neuroscience, 151-153 New York Arts Program, 200, 217 Noor, 247 O off-campus living, 250-251 Off-Campus Studies (OCS) office, 212-213 off-campus study, 212-217 appeal after deadline, 238 art, 29 biology, 43 Chinese Studies, 57 domestic programs, 214-217 East Asian Studies, 81 endorsed programs, 213 fee, administrative, 263 French, 106, 107, 109-110 German, 114, 117 grade point average, 231-232 (sec. C) international relations, 125 Latin American Studies, 131 mathematics, 133-134 prerequisites, 213 Russian, 180 South Asian Studies, 189 study abroad programs, 214 theatre and dance, 199-200, 216, 217 tuition and fees, 213 urban studies, 201 withdrawal, 261 OhioLINK, 20 organizations and clubs, student, 248-249 overload, 230-231 369 appeal, 238 music education, 222-224 P pass/fail. See S/NC grade payment plans, 263-264 Peace by Peace, 247 petitions for exceptions to academic policies, 238 Phi Beta Kappa, 269 Philadelphia Center, 217 Philosophy, 154-158 physical disabilities services, 243 Physical Education, 158-160 maximum credits, 234 Physics, 160-164 placement tests, 256, 258 See also individual departments Political Science, 164-170 pre-professional and dual degree programs, 206-212 forestry and environmental studies, 209 health professions, 210-211 music, 138. See also Bachelor of Music; Bachelor of Music Education pre-architecture, 207-208 pre-business, 208 pre-engineering, 208-209 pre-law, 211 pre-seminary studies, 211-212 pre-social work, 212 Presidents of the College, 336 prizes. See honors and prizes, academic probation. See academic probation Program in Writing, 16 Psychology, 170-174 practicum, 216 public school teaching (music), 140, 222-226 See also music publications, student, 248 Q Q (quantitative reasoning) course advanced placement and, 258 degree requirement, 218, 220, 223, 225 residence requirement, 257-258 R R (religious perspectives) course, 218, 220, 223, 224-225 radio, 69, 246 readmission, 238, 261 refunds, 263, 264-265 registration, 230-235 implications of, 260-262 late, 230 (sec. 8) medical data, 261 re-registration fee, 263 termination or denial of, 260-261 religious life on campus, 246-247 religious perspectives. See R (religious perspectives) course Religious Studies, 175-179 repeated courses, 228, 232 (sec. H), 237, 256, 258 requirements for graduation. See degree requirements residence halls and houses, 331-334 policies, 250-251, 262 residence requirements, 227, 258 resident assistants, 251 Russian Studies, 179-182 S S/NC grade, 232 (sec. D) appeal regarding status, 239 College Writing course, 232 (sec. D) deadline to declare, 230 (sec. 6) First-Year Seminar, 232 (sec. D) limit, 232 (sec. D) major and the, 232 (sec. D) minor and the, 232 (sec. D) NC, grade of, 232 (secs. E, G) grade point average, 231-232 (sec. C) S, minimum equivalent grade, 232 (sec. D) transfer students, 232 (sec. D) Sapere Aude, 248 satisfactory academic progress, 236 scholarships, 267-268, 292-327 Scot Band, 150, 245 Scot’s Key, 262 security deposit, 251, 255, 262, 263 security and protective services, 252 semester, 235 Seminary Semester, 215 370 seminary studies pre-professional program, 211-212 senior class standing, 236 Social Responsibility, Code of, 239-240 applications of, 213, 228, 234, 239-240, 262, 262 social work, pre-professional program in, 212 Sociology and Anthropology, 182-189 sophomore class standing, 236 sororities. See clubs and sections South Asian Studies, 189-190 Spanish, 190-195 sports. See athletics Student Accident and Sickness Medical Plan, 253, 263 student-designed major, 229 Student Government Association (SGA), 247 student life, 244-249 student services, 249-253 student teaching. See teaching studio art. See Art and Art History study abroad. See off-campus study summer session, 217 international Wooster programs, 214 theatre and dance summer study, 199 transfer credits, 258 T teacher licensure, 86-88 French, 107 German, 114 mathematics, 134 political science, 166 sociology, 183 Spanish, 192 teaching apprenticeship, 127 teaching, student, 90-91 termination of enrollment, 261 Thailand, Wooster in, 214 Theatre and Dance, 195-200 campus opportunities, 248, 249 internships, 216 scholarships, 268 workshop credits, 234 times, of classes, 235 TOEFL (Test of English as a Foreign Language), 256-257 transcript, 228, 231-232 (secs. A, C), 256 transfer credits, 257-260 advanced placement (AP), 258-260 deadline to submit, 258 equivalence, 258 French, 107 maximum, 257, 258 repeated courses, 232 (sec. H), 258 Spanish, 191 summer school, 258 transfer students admission, 257-260 class standing, 236 First-Year Seminar, 257, 258 general education courses, 257, 258 grade point average, 231 (sec. C) Independent Study, 257 major courses, 257, 258 quantitative reasoning (Q) course, 257, 258 S/NC credits, 232 (sec. D) writing intensive (W) course, 257, 258 Trinidad and Tobago, Wooster in, 214 Trustees, Board of, 336-338 tuition. See fees Tuition Management Systems Monthly Payment Plan, 263 Tuscany, Wooster Summer in, 214 U United Nations Semester, 215 Urban Studies, 200-202 V veterans’ education, 235 Voice, The, 248 W W, grade of, 232 (sec. E) W (writing intensive) course, 218, 219, 222, 224 W† (not all sections writing intensive) course, 21 warning, academic, 236 Washington Semester, 170, 214-215 WCWS radio station, 69, 246 371 withdrawal from the College, 236, 261, 264 fees, 264 housing and, 250 medical, 232 (sec. F) readmission, 238 Women’s Advisory Board, 359 Women’s, Gender, and Sexuality Studies, 202-206 practicum, 216-217 Wooster Activities Crew (WAC), 248-249 Wooster Chorus, 150, 245 Wooster Dance Company, 249 Wooster Ethic, 239 Wooster in programs, 214 Wooster Inn, 336 Wooster Jazz Ensemble, 150, 246 Wooster Presbyterian Church, 247 Wooster Singers, 150, 245 Wooster Symphony Orchestra, 150, 246 Wooster Symphony Chamber Orchestra, 246 Wooster Volunteer Network, 249 Worthy Questions, 246 writing College Writing course, 125, 232 (sec. D) Program in Writing, 16 Writing Center, 243-244 writing intensive. See W (writing intensive) course Y Year One Journal, 248 yearbook, 248
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