UK Nonlinear News, May 1999

A Modern Introduction to the Mathematical Theory of Water Waves

By R. S. Johnson

Reviewed by J. D. Gibbon

Cambridge University Press, 1997.
Hardback: ISBN 0521 59172 4, pounds 55.00.
Paperback: ISBN 0521 59832 X, pounds 19.95

This is the seventeenth book in the CUP series ``Cambridge Texts in Applied Mathematics''. The subject of water waves is a tricky area because the combination of nonlinear equations of motion with a nonlinear moving free surface conspires to form a difficult set of obstacles for the student who wants to gain an understanding of the terrain of the subject. The universal complaint of older university teachers in the western world that students are ``not what they were in our day'' has its foundation in modern students' shaky manipulative competence and lack of confidence in handling the fundamental equations of their subject. Today such matters need to be spelt out line by line in a clear and careful way without making large jumps in the calculations. Books of this type are a valuable addition to the series because they are very good at performing this task in the sense that they take the student through a calculation from start to finish, thereby inspiring confidence.

The book consists of five chapters and has the remarkable total of 249 exercises with answers and solution hints at the back. It begins by laying out the governing equations of fluid dynamics and their scaling properties with a considerable discussion of the boundary conditions for water waves. Chapter 2 is split into two parts, linear and nonlinear. The linear part consists of a study of gravity and edge waves, ray theory and a considerable and very nice section on ship waves. The nonlinear part consists of studies in Stokes' waves, hydraulic jumps and bores (such as the Severn bore), and a section on waves on sloping beaches.

It is natural in such a study to spend some time on weakly nonlinear dispersive phenomena. Chapters 3 and 4 are devoted to this with chapter 3 concentrating mainly on the Korteweg-de Vries (KdV) equation and the various methods for finding solutions of it. Here there is some overlap with the book by Drazin and Johnson Solitons: an introduction in the same series. Chapter 4 looks at surface wave packets and is based around the Nonlinear Schrödinger (NLS) and Davey-Stewartson (DS) amplitude equations.

The problems studied in the first four chapters are based on inviscid flows; Chapter 5 looks at the effect of viscosity on some of these. One example is how solitary waves are attenuated by this effect. Another is a study of the evolution of an undular bore through two examples, the second being the KdV-Burgers equation.

Overall the book would make an excellent textbook for graduate or advanced undergraduate courses. My only (very mild) criticism is that it has a somewhat old fashioned air in the sense that out of 200 references, excluding four to the author's own work, there is only one that I could find that is post-1990 that is not a reference to a book. In this sense it reflects a style of work of the period up to the middle 1980's but not much beyond. Other styles of work on water waves tend to use a combination of nonlinear analysis and numerical methods to explore various phenomena but these get no mention. Of course, Johnson's book is textbook and not a research monograph but the reader should understand that it represents the distillation of one important part of the literature but not all of it.

UK Nonlinear News thanks Cambridge University Press for providing a copy of this book for review.

J.D. Gibbon,
Department of Mathematics,
Imperial College of Science, Technology and Medicine.

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Last Updated: 8th April 1999.