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An Introduction to the Mathematical Theory of Wave

Roger Knobel
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 3
[Reviewed by
Mark McKibben
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An undergraduate math or science major typically pursues a study of differential equations through two courses: ordinary differential equations and "elementary" partial differential equations. There is a tremendous gap between the levels of these two courses. While most books used for a first course in ordinary differential equations are written at a level amenable to an undergraduate audience without requiring much in the way of physical intuition (or much math beyond integral calculus), the same cannot be said about the vast majority of partial differential equations books. Often, the exposition is unintuitive and a careless omission of critical details gives rise to a magical development of a topic rather than a mathematical one. Further, it seems very common for authors to gleefully assume the reader has a working knowledge of physics. But yet, the titles relentlessly contain the word "elementary" and the prerequisite is only "a good course in multivariable calculus." Hence, the conception that the study of partial differential equations is so much more difficult than that of ordinary differential equations is perpetuated by such exposition, often leaving the average student with the uneasy feeling that his journey has ended before it began. This is why I am ecstatic to have found such a brilliantly written book as Roger Knobel's An Introduction to the Mathematical Theory of Waves.

The goal of this book is to make accessible the basic theory of partial differential equations to undergraduates who are assumed to have only completed the basic calculus sequence and preferably a course in ordinary differential equations. The author proposes to do this via an in-depth discussion of the theory of waves. The text is more focused on depth of understanding than on superficially exposing the reader to a wide variety of topics. Specifically, the heat equation and Laplace equation are both omitted from discussion, whereas many other books devote one chapter of twenty to each of the topics and leave many of the details to the reader. The primary goal of the book, as I see it, is to carefully investigate only one class of equations in this one-semester course, and to do so slowly, methodically, and thoroughly. Although the book is divided into three main parts (Introduction, Traveling and Standing Waves, and Conservation Laws) the text is really a unified, well-thought out compilation of 56 very short lessons.

The journey begins with a very quick introduction to what partial differential equations are and a brief hint at what lies ahead. Then, very elegantly and smoothly, it works through various topics so that the final lesson is focused on the rather abstract notion of a weak solution of a partial differential equation (a topic which it is not common even to attempt at the undergraduate level). Rather than providing an exhaustive list of topics, I simply summarize the main points addressed in the text. Following a well-motivated, easy-to-read preliminary chapter, the notions of traveling and standing waves are discussed (both analytically and graphically) and examples are provided. Armed with these two concepts, the next leg of the journey is focused on an extremely clear derivation of the wave equation, as well as methods used to "solve it" under various conditions (e.g., D'Alembert's formula and Fourier Series). The more advanced topics in the final part are not common to an undergraduate course, but are made palatable by investigating them in the context of a mathematical model of traffic flow (adapted from Haberman's book Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow). These topics include a discussion of the method of characteristics, shocks, gradient catastrophes, and the notion of a weak solution.

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 in 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, it seems, 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.

Appropriate use of computer-algebra systems (primarily Matlab) is integrated throughout the text rather than being relegated to the end of the chapters as a "Lab Activity." The technology is used as the need for it naturally arises, for example when it helps to visualize the concept being presented. As a reader, at no time did I feel that the usage of the computer was contrived. (The necessary code is provided within the context of the material for the reader's convenience.)

The exposition of the material is very clear. Particularly close attention is devoted to bridging the gap between the physical and mathematical elements of the theory, as well as providing all crucial computational details. This aids in the development of the reader's intuition of the subject. The book practically invites the reader to take this journey and it is written in such a way that the reader is not likely to get lost along the way.

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.

So, what's wrong with the book? Drawing from my experience with partial differential equations texts, I found myself asking this very question as I turned each page, only to be pleasantly surprised that nothing was wrong with the next page. In fact, as I began this review I was so pessimistic that despite the fact that my expectation of poor exposition had been nullified time and time again, I still "knew" that the very next topic would contain some typical flaw or oversight exhibited in many other texts. Happily, each time I was proven wrong. The book is extremely well written. The only slight criticism I have (and it's a selfish one) is that I wish the code for MAPLE-V was given more often (or even in place of Matlab). But that's it! Kudos to Roger Knobel on having produced such a well-written and much-needed book!

Mark McKibben ( is assistant professor of mathematics at Goucher College in Baltimore, Maryland. His research areas are nonlinear analysis, abstract evolution equations, and integral equations. His most recent work deals with abstract nonlinear nonlocal Cauchy problems in abstract spaces and controllability of solutions to such problems. He is the co-author of the book Algebra (with Dave Keck and Shane Rosanbalm, Wiley, 1998, 2nd edition) and is currently writing texts in real analysis and differential equations.


  • Introduction to waves
  • A mathematical representation of waves
  • Partial differential equation

Traveling and standing waves

  • Traveling waves
  • The Korteweg-de Vries equation
  • The Sine-Gordon equation
  • The wave equation
  • D'Alembert's solution of the wave equation
  • Vibrations of a semi-infinite string
  • Characteristic lines of the wave equation
  • Standing wave solutions of the wave equation
  • Standing waves of a nonhomogeneous string
  • Superposition of standing waves
  • Fourier series and the wave equation

Waves in conservation laws

  • Conservation laws
  • Examples of conservation laws
  • The method of characteristics
  • Gradient catastrophes and breaking times
  • Shock waves
  • Shock wave example: Traffic at a red light
  • Shock waves and the viscosity method
  • Rarefaction waves
  • An example with rarefaction and shock waves
  • Nonunique solutions and the entropy condition
  • Weak solutions of conservation laws
  • Bibliography
  • Index