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Book DetailsStudent Mathematical LibraryIAS/Park City Mathematics SubseriesVolume: 3; 2000; 196 ppMSC: 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 onedimensional waves using functions of two variables, provides an introduction to partial differential equations, and discusses computeraided visualization techniques.
The second part of the book discusses traveling waves, leading to a description of solitary waves and soliton solutions of the KleinGordon and KortewegdeVries 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 selfcontained, 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.ReadershipAdvanced undergraduates, graduate students, and research mathematicians interested in nonlinear PDEs.

Table of Contents

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 Kortewegde Vries equation

Chapter 6. The SineGordon equation

Chapter 7. The wave equation

Chapter 8. D’Alembert’s solution of the wave equation

Chapter 9. Vibrations of a semiinfinite 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


Additional Material

Reviews

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 "theoremproof" 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 wellwritten and muchneeded book!
MAA Online


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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 onedimensional waves using functions of two variables, provides an introduction to partial differential equations, and discusses computeraided visualization techniques.
The second part of the book discusses traveling waves, leading to a description of solitary waves and soliton solutions of the KleinGordon and KortewegdeVries 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 selfcontained, 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.
Advanced undergraduates, graduate students, and research mathematicians interested in nonlinear PDEs.

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 Kortewegde Vries equation

Chapter 6. The SineGordon equation

Chapter 7. The wave equation

Chapter 8. D’Alembert’s solution of the wave equation

Chapter 9. Vibrations of a semiinfinite 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.
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 "theoremproof" 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 wellwritten and muchneeded book!
MAA Online