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Capillary–gravity waves of solitary type and envelope solitons on deep water

Published online by Cambridge University Press:  26 April 2006

Michael S. Longuet-Higgins
Michael S. Longuet-Higgins
Affiliation:
Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA 92093-0402, USA
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Abstract

The existence of steady solitary waves on deep water was suggested on physical grounds by Longuet-Higgins (1988) and later confirmed by numerical computation (Longuet-Higgins 1989; Vanden-Broeck & Dias 1992). Their numerical methods are accurate only for waves of finite amplitude. In this paper we show that solitary capillary-gravity waves of small amplitude are in fact a special case of envelope solitons, namely those having a carrier wave of length 2π(T/ρg)1½2 (g = gravity, T = surface tension, ρ = density). The dispersion relation $c^2 = 2(1-\frac{11}{32}\alpha^2_{\max)$ between the speed c and the maximum surface slope αmax is derived from the nonlinear Schrödinger equation for deep-water solitons (Djordjevik & Redekopp 1977) and is found to provide a good asymptote for the numerical calculations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 703 - 711
Copyright
© 1993 Cambridge University Press

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