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A variational approach to the problem of deep-water waves forming a circular caustic

Published online by Cambridge University Press:  21 April 2006

T. R. Marchant  and
A. J. Roberts
T. R. Marchant
Affiliation:
Department of Applied Mathematics, University of Adelaide, GPO Box 498, Adelaide, South Australia 5001
A. J. Roberts
Affiliation:
Department of Applied Mathematics, University of Adelaide, GPO Box 498, Adelaide, South Australia 5001
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Abstract

We consider the situation where a deep-water wavetrain approaching from infinity forms a circular caustic, is glancingly reflected at the caustic, and then propagates on out to infinity. At every point in the wavefield there are two wavetrains, the incident and reflected waves. Thus the wavefield can be treated as a slowly varying field of short-crested waves. This work generalizes that of Peregrine (1981) who considered a wavefield of incident waves only. The problem is formulated using the averaged-Lagrangian variational approach of Whitham (1974). Owing to the circular symmetry of the problem, the governing differential equations can be reduced to a set of algebraic equations at each radius. Results for the wave steepness and wavenumber are presented. These indicate that the nonlinear caustic occurs at a larger radius than does the linear caustic, and that the ray paths are no longer straight but curve away from the caustic. It is found that the slowly varying assumption is invalid at the caustic radius. To overcome this we derive, by the method of multiple scales, a modified nonlinear Schrödinger equation which is valid in this region. The solution of this equation, involving the second Painlevé transcendent, is then asymptotically matched to the slowly varying solution to provide a complete description of the wavefield.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 194 , September 1988 , pp. 581 - 597
Copyright
© 1988 Cambridge University Press

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