# John Wehausen and his in-depth look at surface waves

John V. Wehausen was the fifth Executive Editor at Mathematical Reviews, from 1950 to 1956.  Wehausen has connections to the University of Michigan (BS and PhD) and to Brown (instructor), both of which institutions were hosts to Mathematical Reviews at one time or another.  The November 2015 issue of the AMS Notices has a short article about Wehausen, focusing on his long survey article (333 pages!) on surface waves, but also with a bit of information about Wehausen.  The author, Nikolay Kuznetsov, points out how Wehausen was able to provide a bridge between mathematicians in the Soviet Union and those in the West.  From early on, Mathematical Reviews, has made a concerted effort to cover mathematics published in Russian.  This was particularly important in the Cold War era, but remains important today.  For more information about him, there is an entry for Wehausen on Wikipedia and an informative obituary at UC Berkeley from 2005.

Here is the review of Wehausen’s  s u r v e y  on water waves.  There is an online copy of the survey hosted by UC Berkeley.

MR0119656 (22 #10417)
Wehausen, John V.; Laitone, Edmund V.
Surface waves. 1960 Handbuch der Physik, Vol. 9, Part 3 pp. 446–778 Springer-Verlag, Berlin
76.00

This long article is concerned with motions in water subject to a gravitational force, when free surfaces or interfaces are present. (Not all such flows are treated here. Volume 48 contains articles by A. Defant on tidal motion and by H. U. Roll on ocean waves, and the present volume contains another related article on free-surface flows by D. Gilbarg.) The scope of the work is further explained in the introduction in the following words:

“The subject of water waves engaged many of the mathematicians and mathematical physicists of the last century. Moreover, the last several years have brought a renewed interest in the theory of water waves. In addition to this extensive literature on theoretical aspects of the subject, there have also been many experimental investigations, usually carried out by hydraulic engineers. Hydraulic engineers have also produced an extensive literature, both theoretical and experimental, on open channel flow, flow over weirs and through sluice-gates, etc.; included is a considerable literature on numerical and graphical methods of solving the equations involved. Oceanographers have produced their own literature, usually emphasizing different aspects of the subject. The theory of ship waves has produced its own literature.

“All this material is pertinent to this article. Clearly some selection has to be made. We have followed roughly the following rules: Fundamental results are derived in full. The treatments of various special problems are selected so as to exemplify particular methods, other methods being mentioned only by literature citation. Experimental results are not usually reproduced, but references are given. Numerical methods of solving equations are not treated at all. The more special problems of hydraulic engineering are also not treated. Geophysical aspects which are omitted have already been mentioned.”

The article is divided into seven chapters, as follows: (A) Introduction (446–447); (B) Mathematical formulation (447–455); (C) Preliminary remarks and developments (455–469); (D) Theory of infinitesimal waves (469–667); (E) Shallow-water waves (667–714); (F) Exact solutions (714–757); (G) Bibliography (757–778). Chapter E is by Laitone, the other chapters are by Wehausen, and it is convenient to discuss their contributions separately.

Wehausen’s contribution is a review article, and a most valuable and thorough one. Can it be that Wehausen has read critically all the 700 works listed in the bibliography? From sample tests the reviewer is inclined to think that he has. It is impossible to give more than a brief discussion of the contents. Chapter A outlines the scope of the work. Chapter B gives the exact equations and boundary conditions for viscous and inviscid fluids. Chapter C discusses with care the schemes of approximation leading to the classical infinitesimal-wave and shallow-water theories, and also contains certain exact considerations on wave velocity momentum and energy. Chapter D, which forms the greater part of the review, is devoted almost entirely to the potential theory of infinitesimal waves. The velocity potential$\phi(x,y,z,t)$ satisfies Laplace’s equation with simple linear boundary conditions. The general theory of this system is not yet understood. Thus for the important case of time-periodic motion the general uniqueness problem is still unsolved, and only a few partial results are known. A picture of surface-wave behaviour is, however, beginning to emerge from the solutions of a variety of boundary-value problems which are described in the review. Chapter F describes the known theory relating to the exact non-linear equation, a part of the subject which will be less familiar to readers than the theory of infinitesimal waves.

Among omissions (probably due to lack of space) is the comparison with experiments which have shown that much of the theoretical work on inviscid fluids is directly applicable to real fluids. It has also been noted that several sections contain no reference to any other author and are presumably due to Wehausen himself. One of the most noteworthy is section 15 on group velocity and the propagation of disturbances and of energy. That energy propagates with the group velocity has appeared to many students as an unexpected coincidence. It will appear less so after Wehausen’s discussion.

Wehausen deserves to be congratulated on a scholarly and well-written review. We now turn briefly to Chapter E on shallow-water waves, by Laitone. This author has preferred to concentrate on a few aspects rather than to give a survey of all that is known. (Particularly on some non-linear aspects our knowledge at present is slight.) The treatment is thorough and interesting.

To sum up, this article on surface waves is a most worthy contribution to the Encyclopaedia of physics which many workers in fluid mechanics would be glad to possess. Unfortunately the price of the complete volume is too high for a wide distribution. The publishers would perform a service by separate publication of this article, perhaps after the lapse of some time.

Reviewed by F. Ursell