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Crest instabilities of gravity waves. Part 1. The almost-highest wave

Published online by Cambridge University Press:  26 April 2006

Michael S. Longuet-Higgins  and
R. P. Cleaver
Michael S. Longuet-Higgins
Affiliation:
Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA 92093-0402, USA
R. P. Cleaver
Affiliation:
Gas Research Centre, British Gas, Ashby Road, Loughborough, LE11 3QU, UK
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Abstract

It is shown theoretically that the crest of a steep, irrotational gravity wave, considered in isolation, is unstable. There exists just one basic mode of instability, whose exponential rate of growth β equals 0.123(g / R)½, where g denotes gravity and R is the radius of curvature at the undisturbed crest. A volume of water near the crest is shifted towards the forward face of the wave; the ‘toe’ of the instability is at a horizontal distance 0.45R ahead of the crest. The instability may represent the initial stage of a spilling breaker. On small scales, the ‘toe’ may be a source of parasitic capillary waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 258 , 10 January 1994 , pp. 115 - 129
Copyright
© 1994 Cambridge University Press

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