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A critique for shallow water theory

Published online by Cambridge University Press:  20 January 2009

A.G. Mackie
A.G. Mackie
Affiliation:
University of Edinburgh
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There is now a huge volume of literature dealing with exact and approximate solutions of the equations governing the motion of an inviscid, incompressible fluid. Major difficulties in finding exact solutions arise when the fluid is bounded by a socalled "free" surface on which the pressure is constant because the equation of such a surface is not known in advance. A boundary condition of this type arises when a mathematical analysis of water wave theory is attempted. Different approximations are, of course, appropriate to different circumstances and here we deal with a well known approximation for waves in shallow water which will be described in detail in the next section.The main ideas underlying this particular approximation have been known for some time but it is probably true to say that they were first brought into real prominence by Stoker [3] in a long article in the Communications of Pure and Applied Mathematics. The main ideas in this article were later developed in considerable detail in Stoker's book “Water Waves” [4]. More recently this approximation has been used by Sewell and Porter [2] as one of several examples in fluid and solid mechanics which are given a geometrical interpretation in terms of “constitutive surfaces” which has close links with catastrophe theory. An excellent account of this interesting approach, which lists many associated references, is given by Sewell [1].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 27 , Issue 2 , June 1984 , pp. 153 - 163
Copyright
Copyright © Edinburgh Mathematical Society 1984

References

1.Sewell, M. J., Legendre transformations and extremum principles, The Rodney Hill 60th Anniversary Volume (Pergamon Press, Oxford, 1981).Google Scholar
2.Sewell, M. J. and Porter, J., Constitutive surfaces in fluid mechanics, Math. Proc. Camb. Phil. Soc. 88 (1981), 517–546.CrossRefGoogle Scholar
3.Stoker, J. J., The formation of breakers and bores, Comm. Pure Appl. Math. 1 (1948), 1–87.CrossRefGoogle Scholar
4.Stoker, J. J., Water Waves (Interscience, New York, 1957).Google Scholar
5.Wehausen, J. V. and Laitone, E. V., Surface waves, Handbuch der Physik 9 (1960), 446–778.Google Scholar