**Student Mathematical Library**

Volume: 3;
2000;
196 pp;
Softcover

MSC: Primary 35;
Secondary 00

**Print ISBN: 978-0-8218-2039-1
Product Code: STML/3**

List Price: $30.00

Individual Price: $24.00

**Electronic ISBN: 978-1-4704-2231-8
Product Code: STML/3.E**

List Price: $28.00

AMS Member Price: $22.40

MAA Member Price: $25.20

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# An Introduction to the Mathematical Theory of Waves

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*Roger Knobel*

This book is based on an undergraduate course taught at the
IAS/Park City Mathematics Institute (Utah) on linear and nonlinear waves. The
first part
of the text overviews the concept of a wave, describes one-dimensional
waves using functions of two variables, provides an introduction to
partial differential equations, and discusses computer-aided
visualization techniques.

The second part of the book discusses traveling waves, leading to a
description of solitary waves and soliton solutions of the
Klein-Gordon and Korteweg-deVries equations. The wave equation is
derived to model the small vibrations of a taut string, and solutions
are constructed via d'Alembert's formula and Fourier series.

The last part of the book discusses waves arising from conservation
laws. After deriving and discussing the scalar conservation law, its
solution is described using the method of characteristics, leading to
the formation of shock and rarefaction waves. Applications of these
concepts are then given for models of traffic flow.

The intent of this book is to create a text suitable for
independent study by undergraduate students in mathematics,
engineering, and science. The content of the book is meant to be
self-contained, requiring no special reference material. Access to
computer software such as Mathematica®, MATLAB®, or Maple®
is recommended, but not necessary. Scripts for MATLAB applications will
be available via the Web. Exercises are given within the text to
allow further practice with selected topics.

This book is published in cooperation with IAS/Park City Mathematics Institute

#### Readership

Advanced undergraduates, graduate students, and research mathematicians interested in nonlinear PDEs.

#### Reviews & Endorsements

An interesting first reading on high analysis at an elementary level.

-- European Mathematical Society Newsletter

The book offers a student an excellent introduction to some of the most interesting wave phenomena that have physical significance, and at the same time it also serves to explain some of the deeper mathematical issues that are involved. It can be recommended to all undergraduates who wish to learn something about physics wave phenomena of various types.

-- Mathematical Reviews

The style of this book is not that of a typical textbook. For one, the very short sections (few exceed five pages in length) have a more interactive, conversational flavor rather than the usual "theorem-proof" style of most texts. This is not to say that it lacks in precision; far from it, in fact. Very carefully constructed short exercise lists occur frequently throughout the book and often times, immediately following a discussion of a difficult topic: they are not all collected and placed, out of context, at the end of the chapter. It is the intention that every exercise be completed as part of the journey through the material, and not simply to practice a technique. The problems are all very relevant to the material presented and many challenge the student to extend the theory he or she just learned in a slightly tangential direction. Also, a common theme in the text is to revisit the same problem at several different points in the book and each time investigate it more carefully using the theory just developed. This spiraling approach is very clever, and it instills in the reader a sense of what is going on.

The exposition of the material is very clear. All in all, this book provides a sturdy bridge from a course on ordinary differential equations, and so I would recommend it, without batting an eyelash, to any of my differential equations students who wish to continue their study independently. Further, I feel that it could be very useable as a text for a first course in partial differential equations. Kudos to Roger Knobel on having produced such a well-written and much-needed book!

-- MAA Online

#### Table of Contents

# Table of Contents

## An Introduction to the Mathematical Theory of Waves

- Cover Cover11
- Chapter 1. Introduction to waves 318
- Chapter 2. A mathematical representation of waves 722
- Chapter 3. Partial differential equations 1328
- Chapter 4. Traveling waves 2338
- Chapter 5. The Korteweg-de Vries equation 3146
- Chapter 6. The Sine-Gordon equation 3752
- Chapter 7. The wave equation 4560
- Chapter 8. D’Alembert’s solution of the wave equation 5368
- Chapter 9. Vibrations of a semi-infinite string 5974
- Chapter 10. Characteristic lines of the wave equation 6782
- Chapter 11. Standing wave solutions of the wave equation 7792
- Chapter 12. Standing waves of a nonhomogeneous string 87102
- Chapter 13. Superposition of standing waves 95110
- Chapter 14. Fourier series and the wave equation 101116
- Chapter 15. Conservation laws 113128
- Chapter 16. Examples of conservation laws 119134
- Chapter 17. The method of characteristics 127142
- Chapter 18. Gradient catastrophes and breaking times 137152
- Chapter 19. Shock waves 145160
- Chapter 20. Shock wave example: Traffic at a red light 153168
- Chapter 21. Shock waves and the viscosity method 159174
- Chapter 22. Rarefaction waves 165180
- Chapter 23. An example with rarefaction and shock waves 173188
- Chapter 24. Nonunique solutions and the entropy condition 181196
- Chapter 25. Weak solutions of conservation laws 187202
- Back Cover Back Cover1218