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Limiting forms for capillary-gravity waves

Published online by Cambridge University Press:  21 April 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK and Institute of Oceanographie Sciences. Wormley, Surrey, UK
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Abstract

The form of steep capillary waves is of interest as a possible initial condition for the formation of air bubbles at a free surface. In this paper the limiting forms of pure capillary waves and of quasi-capillary waves are studied analytically. Crapper's finite-amplitude solution is expressed in a simple form, and is shown to be one of several exact elementary solutions to the pure-capillary free-surface condition. Among others are the solution z = w+sinh w, where w is the velocity potential, and also z = w3. The latter solution, though it represents a self-intersecting flow, can be used as the first in a sequence of approximations to the form of the steepest wave. Hence it is shown that the influence of gravity on the shape of the limiting ‘bubble’ is very small. The result is confirmed by an examination of Hogan's numerical calculations of limiting capillary-gravity waves.

In the crest of a limiting wave the particle velocity is almost constant and equal to the phase speed. This property makes it possible to apply a quasi-static approximation so as to determine the form of the crest, and hence to find an expression for the complete profile of a capillary-gravity wave of limiting steepness. It appears that there exists a solitary wave of capillary-gravity type on deep water.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 194 , September 1988 , pp. 351 - 375
Copyright
© 1988 Cambridge University Press

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