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An Introduction to the Mathematical Theory of Waves

Roger Knobel University of Texas-Pan American, Edinburg, TX
Available Formats:
Softcover 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 MAA Member Price:$25.20
AMS Member Price: $22.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$45.00
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An Introduction to the Mathematical Theory of Waves
Roger Knobel University of Texas-Pan American, Edinburg, TX
Available Formats:
 Softcover 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 MAA Member Price:$25.20 AMS Member Price: $22.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$45.00
• Book Details

Student Mathematical Library
IAS/Park City Mathematics Subseries
Volume: 32000; 196 pp
MSC: Primary 35; Secondary 00;

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.

• Part 1. Introduction
• Chapter 1. Introduction to waves
• Chapter 2. A mathematical representation of waves
• Chapter 3. Partial differential equation
• Part 2. Traveling and standing waves
• Chapter 4. Traveling waves
• Chapter 5. The Korteweg-de Vries equation
• Chapter 6. The Sine-Gordon equation
• Chapter 7. The wave equation
• Chapter 8. D’Alembert’s solution of the wave equation
• Chapter 9. Vibrations of a semi-infinite string
• Chapter 10. Characteristic lines of the wave equation
• Chapter 11. Standing wave solutions of the wave equation
• Chapter 12. Standing waves of a nonhomogeneous string
• Chapter 13. Superposition of standing waves
• Chapter 14. Fourier series and the wave equation
• Part 3. Waves in conservation laws
• Chapter 15. Conservation laws
• Chapter 16. Examples of conservation laws
• Chapter 17. The method of characteristics
• Chapter 18. Gradient catastrophes and breaking times
• Chapter 19. Shock waves
• Chapter 20. Shock wave example: Traffic at a red light
• Chapter 21. Shock waves and the viscosity method
• Chapter 22. Rarefaction waves
• Chapter 23. An example with rarefaction and shock waves
• Chapter 24. Nonunique solutions and the entropy condition
• Chapter 25. Weak solutions of conservation laws

• Reviews

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

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
IAS/Park City Mathematics Subseries
Volume: 32000; 196 pp
MSC: Primary 35; Secondary 00;

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.

• Part 1. Introduction
• Chapter 1. Introduction to waves
• Chapter 2. A mathematical representation of waves
• Chapter 3. Partial differential equation
• Part 2. Traveling and standing waves
• Chapter 4. Traveling waves
• Chapter 5. The Korteweg-de Vries equation
• Chapter 6. The Sine-Gordon equation
• Chapter 7. The wave equation
• Chapter 8. D’Alembert’s solution of the wave equation
• Chapter 9. Vibrations of a semi-infinite string
• Chapter 10. Characteristic lines of the wave equation
• Chapter 11. Standing wave solutions of the wave equation
• Chapter 12. Standing waves of a nonhomogeneous string
• Chapter 13. Superposition of standing waves
• Chapter 14. Fourier series and the wave equation
• Part 3. Waves in conservation laws
• Chapter 15. Conservation laws
• Chapter 16. Examples of conservation laws
• Chapter 17. The method of characteristics
• Chapter 18. Gradient catastrophes and breaking times
• Chapter 19. Shock waves
• Chapter 20. Shock wave example: Traffic at a red light
• Chapter 21. Shock waves and the viscosity method
• Chapter 22. Rarefaction waves
• Chapter 23. An example with rarefaction and shock waves
• Chapter 24. Nonunique solutions and the entropy condition
• Chapter 25. Weak solutions of conservation laws
• An interesting first reading on high analysis at an elementary level.