1 - Towards the Analytic Description of Overturning Waves

Summary

INTRODUCTION.

Till recently, one notable hiatus in the theory of surface waves was the absence of any satisfactory analysis to describe an overturning wave. In this category we include both the well-known “plunging breaker” and also any standing or partially reflected wave which produces a symmetric or an asymmetric jet, with particle velocities sometimes much exceeding the linear phase-speed.

A first attempt to describe the jet from a two-dimensional standing wave was made by Longuet-Higgins (1972), who introduced the “Dirichlet hyperbola”, a flow in which any cross-section of the free surface takes the form of a hyperbola with varying angle between the asymptotes. Numerical experiments by Mclver and Peregrine (1981) have shown this solution to fit their calculations quite well. The solution was further analysed in a second paper (Longuet-Higgins, 1976) where a limiting form, the “Dirichlet parabola”, was shown to be a member of a wider class of selfsimilar flows in two and three dimensions. Using a formalism introduced by John (1953) for irrotational flows in two dimensions, the author also showed the Dirichlet parabola to be one of a more general class of selfsimilar flows having a time-dependent free surface.

All the above flows were gravity-free, that is to say they did not involve g explicitly; they are essentially descriptions of a rapidly evolving flow seen in a frame of reference which itself is in free-fall.

A useful advance came with the development of a numerical time-stepping technique for unsteady gravity waves by Longuet-Higgins and Cokelet (1976, 1978). As later refined and modified by Vinje and Brevig (1981), Mclver and Peregrine (1981) and others, this has given accurate and reproducible results for overturning waves, with which analytic expressions can be compared.

A further advance on the analytic front came with the introduction by Longuet-Higgins (1980a) of a general technique for describing free-surface flows, that is flows satisfying the two boundary conditions

at a free surface. Particular attention was paid to the parametric representation of the flow in a form

where both the complex coordinate z = x + iy and the velocity potential x are expressed as functions of the intermediate complex variable u and the time t.