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A non-local formulation of rotational water waves

Published online by Cambridge University Press:  08 November 2011

A. C. L. Ashton  and
A. S. Fokas
A. C. L. Ashton*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: acla2@damtp.cam.ac.uk
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Abstract

The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free surface. Here, in the two-dimensional case: (a) we generalize the relevant formulation to the case of constant vorticity, as well as to the case where the free surface is described by a multivalued function; (b) in the case of travelling waves we derive an upper bound for the free surface; (c) in the case of constant vorticity we construct a sequence of nearly Hamiltonian systems which provide an approximation in the asymptotic limit of certain physical small parameters. In particular, the explicit dependence of the vorticity on the coefficients of the Korteweg–de Vries equation is clarified.

JFM classification

Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 689 , 25 December 2011 , pp. 129 - 148
Copyright
Copyright © Cambridge University Press 2011

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