An Introduction to the Mathematical Theory of Waves

by 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.

Genres: Mathematics

196 pages, Paperback

First published January 1, 1999

Reviews

[Tomáš Ševček]

I am currently teaching an undergrad course based on this book. Overall, it is a nice introductory book on PDEs/waves that covers way more than just the basic separation of variables that most introductory classes on PDEs focus on. Namely, it also covers topics such as traveling waves for linear and nonlinear PDEs (and the 1-soliton solution to the KdV equation is derived in the book this way), dispersion relations, d'Alembert's formula and characteristics for the wave equation, linear and nonlinear conservation laws, the method of characteristics for conservation laws, gradient catastrophes, shock waves, the viscosity method, rarefaction waves, the entropy condition and weak forms of conservation laws. Moreover, multiple PDEs are derived in the text, including the inhomogeneous wave equation, the Gordon-Sine equation, the heat equation via conservation laws and others. In addition, the reader is introduced to other useful equations and models, such as the viscous/inviscid Burgers equation, the Gordon-Klein equation and various traffic flow models. The prerequisites really are just Calculus 1 to 3 and an introductory course on ODEs, which makes this book quite accessible.
However, as this book mainly focuses on waves, there are certain PDE concepts which are omitted. There is no mention of the Laplace equation as the book focuses primarily on wave phenomena. Separation of variables is only applied to the wave equation, not the heat equation. All of the equations are one-dimensional in space. Also, when discussing traveling waves and dispersion relations, one is just a few steps away from discussing linearization and pattern formation, which I have also included in the class, but the concepts are omitted in the book.
Overall, this is a strong and accessible introductory book on waves and I recommend it to anyone interested in wave phenomena, but if you are using this book as your first exposure to PDEs, I also recommend supplementing it with another standard PDE textbook (such as Strauss, Olver or maybe Evans if you have the background).