Mathematics of Wave Propagation

Abstract

9R3. Mathematics of Wave Propagation. - JL Davis (consultant). Princeton UP, Princeton. 2000. 395 pp. ISBN 0-691-02643-2. $49.50.Reviewed by LN Sankar (Sch of Aerospace Eng, Georgia Inst of Tech, Atlanta GA 30332-0150).There are a number of textbooks and references that have been published on wave propagation in fluids. One may also find numerous books that individually address the propagation of waves in solids, surface waves in water, and tidal waves. There are very few books that treat all of these related phenomena in a clear, unified manner. This book does this challenging job admirably well. The first chapter deals with the physics of propagating waves. Many classical problems such as the transverse oscillations of a string, acoustic waves in a duct, and compression waves in a bar are discussed. Topics such as the Doppler effect, dispersion, and group velocity are also covered. The mathematical aspects of these problems are not emphasized in this chapter. The second chapter deals with the theory behind hyperbolic partial differential equations. The method of characteristics is discussed and illustrated with increasingly complex model problems: 1D advection equation, 2D wave equation, and a system of first order linear and nonlinear partial differential equations, etc. Chapter 3 deals with how the partial differential equations developed in Chapter 2 may be solved. Classical mathematical techniques such as the separation of variables and Laplace transforms, as well as numerical methods that solve ODEs or algebraic equations along characteristic lines are covered. Numerous applications ranging from the vibration of a rectangular membrane to the propagation of current in an LC circuit are discussed. Having established the common mathematical foundation and the solution techniques for these problems, this book moves on to individual topics: wave propagation in viscous and inviscid fluids, elastic, viscoelastic and thermoelastic solids, and water waves. Each of these chapters may be read individually without reference to others. All the governing equations are derived from first principles, making it easier for the reader to understand and appreciate the physics behind these problems. This book ends with variational calculus based approaches (eg, Hamilton’s variational principle, Hamilton-Jacobi theory) to deriving the governing equations. Asymptotic approaches that rely on expansion of the solution for large wave numbers or frequency are also briefly discussed. There are numerous worked-out examples throughout the book. There are also exercises at the end of most of the chapters. These features make this work a suitable textbook for a senior-level or a graduate-level course on the theory of waves. In summary, Mathematics of Wave Propagation is an excellent book that covers seemingly diverse wave phenomena in a unified, coherent manner. Students and practicing engineers and physicists will find this book a useful addition to their collections.