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Crest instabilities of gravity waves. Part 2. Matching and asymptotic analysis

Published online by Cambridge University Press:  26 April 2006

Michael S. Longuet-Higgins ,
R. P. Cleaver  and
M. J. H. Fox
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
M. J. H. Fox
Affiliation:
Berkeley Nuclear Laboratories, Nuclear Electricity plc, Berkeley, Gloucestershire GL13 9PB, UK
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Abstract

In a previous study (Longuet-Higgins & Cleaver 1994) we calculated the stability of the flow near the crest of a steep, irrotational wave, the ‘almost-highest’ wave, considered as an isolated wave crest. In the present paper we consider the modification of this inner flow when it is matched to the flow in the rest of the wave, and obtain the normal-mode perturbations of the modified inner flow. It is found that there is just one exponentially growing mode. Its rate of growth β is a decreasing function of the matching parameter ε and hence a decreasing function of the wave steepness ak. When compared numerically to the rates of growth of the lowest superharmonic instability in a deep-water wave as calculated by Tanaka (1983) it is found that the present theory provides a satisfactory asymptote to the previously calculated values of the growth rate. This suggests that the instability of the lowest superharmonic is essentially due to the flow near the crest of the wave.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 259 , 25 January 1994 , pp. 333 - 344
Copyright
© 1994 Cambridge University Press

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References

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