3R15. Linear Water Waves: A Mathematical Approach. - N Kuznetsov (Russian Acad of Sci, St Petersburg, Russia), V Maz’ya (Univ of Linkoping, Sweden), B Vainberg (Univ of N Carolina, Charlotte, NC, Sweden). Cambridge UP, Cambridge, UK. 2002. 513 pp. ISBN 0-521-80853-7. $100.00.
Reviewed by J Miles (Inst of Geophysics and Planetary Phys, UCSD, La Jolla CA 92093-0225).
As its subtitle suggests and its preface proclaims, “The aim of the present book is to give a self-contained and up-to-date account of mathematical results in the linear theory of water waves.” An almost identical aim is expressed by Stoker in the opening sentence of his 1957 monograph: “The purpose of this book is to present a connected account of the mathematical theory of wave motion in liquids with a free surface….” But Stoker is now dated, and the authors opine that “there is no monograph on the progress achieved in the more mathematical approach to the linear water-wave theory during the last few decades.”
The book is divided into three parts covering, respectively, monochromatic waves associated with the uniform translation of a body, ship waves, and forced unsteady waves. There is also an introductory chapter in which the linear equations governing gravity-wave motion are derived from the Euler and continuity equations and the kinematic boundary conditions. There are good subject and author indexes and a 370-entry bibliography that supplements the earlier bibliographies of Stoker  and Wehausen and Laitone  and is particularly valuable in its coverage of the post-1960 Russian literature.
Specific problems are attacked using integral-equation and Green’s-function techniques at the level of Morse and Feshbach . These techniques are standard in mathematical physics and facilitate a uniform approach to a variety of problems, but at least in the present case, they may obscure the physics. This is especially evident in the authors’ treatment of group velocity. After calculating the energy flow across a geometric surface, they remark that, “The velocity of energy propagation is known as the group velocity. However, it does not play any significant role in the considerations presented in this book, and we restrict ourselves to references to… Stoker… and Wehausen and Laitone…” But, above all, it is dispersion, and hence the distinction between wave speed and group velocity that separates water waves from such simpler phenomena as sound waves. It plays a major role in the classical presentations of Lamb , Stoker (, §3.4), and Wehausen and Laitone (1960, §15), and (in this reviewer’s view) any discussion of water waves that omits dispersion (which, as far as this reviewer could determine, does not appear in either the text or the index of the present monograph) is idiosyncratic in the extreme.
This reviewer concludes that Linear Water Waves: A Mathematical Approach is indeed “A Mathematical Approach.” It may be of interest to applied mathematicians with a secure understanding of the physics of dispersive waves, and it deserves a place in the fluid-mechanics section of any large, technical library, but it is not for the novice.