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1R8. Wave Motion. Cambridge Texts in Applied Mechanics. - J Billingham and A King (Univ of Birmingham, UK). Cambridge UP, Cambridge, UK. 2000. 468 pp. Softcover. ISBN 0-521- 63450-4. (Hardcover ISBN 0-521-63257-9 $110). $37.95.

Reviewed by Shi Tsan Wu (Dept of Mech and Aerospace Eng, Univ of Alabama, Sparkman Dr, Huntsville AL 35899).

Wave motions are most common phenomena which we encounter in daily life. For example, when someone signs or plays a musical instrument, the standing waves in their vocal cords and piano strings produce a pressure change as sound waves which is audible. On the other hand, the existence of waves can be seen in the universe in the electromagnetic waves that cover a spectrum from low frequency radio waves, through visible light to x-ray and gamma rays. This text is designed for use by advanced undergraduates in applied mathematics with potential to be used for physics and engineering students. The authors also state that this is not a book about numerical methods for wave propagation.

In this book, the authors have divided their presentation into three parts. Part I contains Chapters 1 through 6 which deal with linear wave theory. All of the analytical techniques of 19th century mathematics concerning linear wave theory are presented. These techniques include separation variables, Fourier series and Fourier transforms which are used to reveal the characteristics of linear waves on stretch string (Ch 2), linear sound waves (Ch 3), linear water waves (Ch 4), linear waves in solids (Ch 5), and electromagnetic waves (Ch 6).

Part II includes Chapters 7 through 9. The nonlinear wave theory with 20th century mathematics are presented. The authors begin in Chapter 7 by examining hyperbolic systems governed by propagation of information on characteristics. The fundamentals of method characteristics are introduced. Examples are used to illustrate the mathematical theory which are traffic flow and nonlinear gas dynamics (shocks). The authors then move on to study nonlinear water waves presented in Chapter 8, which includes the topics of the nonlinear shallow water waves, the effect of nonlinearity on deep water gravity waves (Stokes Expansion), the Korteweg-deVries equation for shallow water waves and nonlinear capillary waves. To conclude this part, the subject of chemical and electrochemical waves are presented in Chapter 9. Specific topics discussed are the law of mass action, molecular diffusion, reaction-diffusion system, auto-catalytic chemical waves with unequal diffusion coefficients, and the transmission of nerve impulses (the Fitzhugh-Nagumo equation).

Part III, the final part of the book, covers more advanced topics which include Chapters 10 through 12. In Chapter 10, the authors present various physical systems that can be modeled using Burger’s equation. These include more complicated analyses of the traffic flow and weakly nonlinear compression gasdynamics. The analysis of scattering and diffraction of both scalar and vector waves through apertures and past obstacles are presented in Chapter 11. In the final Chapter (12), the authors describe the use of the inverse scattering transform to solve the Korteweg-deVries (KdV) equation and the nonlinear Schro¨dinger (NLS) equation. The KdV equations are used to analyze the propagation of long waves on shallow water and the NLS equation governs the propagation of dispersive wavepackets in a nonlinear medium such as pulses of light in optical fibers. The book also includes an appendix for useful mathematical formulas and physical data which are very useful.

In summary, this is a very well written textbook. This reviewer agrees with the authors claim that this is a textbook for advanced undergraduate in applied mathematics. But this reviewer would suggest that the book could also be used as a textbook for first-year graduate students in physics and engineering disciplines as a reference for scientists and engineers who are interested in analytical methods and solutions for wave motions.