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On the crest instabilities of steep surface waves

Published online by Cambridge University Press:  10 April 1997

MICHAEL LONGUET-HIGGINS  and
MITSUHIRO TANAKA
MICHAEL LONGUET-HIGGINS
Affiliation:
Institute of Nonlinear Science, University of California, San Diego, La Jolla, CA 92093-0402, USA
MITSUHIRO TANAKA
Affiliation:
Department of Applied Mathematics, Faculty of Engineering, Gifu University, 1-1 Yanagido, Gifu 501-11, Japan
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Abstract

The forms of the superharmonic instabilities of irrotational surface waves on deep water are calculated for wave steepnesses up to 99.9% of the limiting value. It is found that as the limiting wave steepness is approached the rates of growth of the lowest two unstable modes (n=1 and 2) increase according to the asymptotic law suggested by the theory of the almost-highest wave (Longuet-Higgins & Cleaver 1994; Longuet-Higgins, Cleaver & Fox 1994; Longuet-Higgins & Dommermuth 1997). Moreover, each eigenfunction becomes concentrated near the wave crest, with a horizontal scale proportional to the local radius of curvature at the crest. These are therefore ‘crest instabilities’ in the original sense.

Similar calculations are carried out for the normal-mode instabilities of solitary waves in shallow water, at steepnesses up to 99.99% of the limiting steepness. Similar conclusions are found to apply, though with greater accuracy.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 336 , 10 April 1997 , pp. 51 - 68
Copyright
© 1997 Cambridge University Press

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