7R10. Nonlinear Hyperbolic Waves in Multi-Dimensions.Monographs and Surveys in Pure and Applied Mathematics, Vol 121.- P Prasad (Dept of Math, Indian Inst of Sci, Bangalore, India). Chapman and Hall/CRC, Boca Raton FL. 2001. 338 pp. ISBN 1-58488-072-4. \$89.95.

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

In this book, the authors have discussed the propagation of a curved nonlinear wavefront for the ordinary fluid dynamics (ie without considering magneto-fluids) according to two fundamental physical processes: $i$) at different points of the wavefront, it travels with different speeds depending on the local amplitude leading to a longitudinal stretching of rays; and $ii$) a lateral deviation of rays is produced due to non-uniform distribution of the amplitude of the wave front.

This book consists of ten chapters. The first three chapters introduce all necessary mathematical concepts and some basic results. For example, the fundamentals of nonlinear hyperbolic waves are given in Chapter 1. In Chapter 2, the basic results of hyperbolic systems are presented. These results include $i$) discussion of hyperbolic system of first order equations in two independent variables with a canonical form of a system of linear and semi-linear equations, and $ii$) the hyperbolic system in more than two variables together with propagation of discontinuities of first order derivatives along rays is given. In Chapter 3, the results for simple wave high frequency approximation and ray theory are given. Specifically, the Huygen’s method of wavefront construction, Huygen’s method and ray theory, and Fermat’s principle are discussed in detail. The next seven chapters deal with specifics of the wave propagation of nonlinear hyperbolic systems; the derivation of weakly nonlinear ray theory (WNLRT) is given in Chapter 4.

The stability of solutions for one-dimensional weakly nonlinear wave propagation as well as waves in a multi-dimensional steady transonic flow near a sonic singularity are discussed in Chapter 5. In Chapter 6, a special case of WNLRT for polytropic gas was presented in detail for two spatial dimensions. In Chapter 7, the authors presented a single conservation law approach to derive the compatibility conditions on a shock which is claimed as a New Theory of Shock Dynamics (NTSC). To establish this NTSC, the authors present a proof of the existence and uniqueness of the solution and made comparison of the numerical results deduced from NTSD with the exact solution which shows good agreement. Then, this NTSD is applied to a one-dimensional piston problem to illustrate its capabilities in Chapter 8. In Chapter 9, the authors presented a derivation of the compatibility conditions on a shock in multi-dimensions using shock ray theory. The final Chapter 10 includes the application of NTSD to the propagation of a curved weak shock.

Overall, the book has three parts: $i$) The first three chapters have given all the necessary fundamental mathematical concepts, models and methods to calculate complete evolutionary history of a curved nonlinear wave front consisting of shock front. $ii$) The detailed discussions of WNLRT and NTSD are presented in Chapters 4-8; and $iii$) the specific applications of shock ray theory and NTSD are given in Chapters 9 and 10, respectively.

In summary, Nonlinear Hyperbolic Waves in Multi-Dimensions includes numerical examples to illustrate theory, the quality of the graphic is good and an adequate subject index is provided. The author has achieved his goal as a textbook for advanced undergraduate applied mathematics. However, it could be used as a textbook for first year graduate students in engineering and physics if the author had presented some problems at the end of each chapter. This should be able to be remedied by the instructor. This reviewer strongly recommends that scientists and engineers who are interested in these topics read this book; it would be a useful reference book.