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Interaction of capillary waves with longer waves. Part 1. General theory and specific applications to waves in one dimension

Published online by Cambridge University Press:  26 April 2006

Kenneth M. Watson  and
Steven B. Buchsbaum
Kenneth M. Watson
Affiliation:
Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego, CA 92093–0213, USA
Steven B. Buchsbaum
Affiliation:
Science Applications International Corporation, 10260 Campus Point Drive, San Diego, CA 92121, USA
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Abstract

A Hamiltonian formulation is used to investigate irrotational capillary wave dynamics. Dissipation is accounted for by putting the wave system in contact with a ‘heat bath’. The generation of short waves by longer waves is studied. It is found that millimetre-wavelength waves tend to be created on the forward face of a steep longer wave, while centimetre waves tend to form near the crest. Generation of capillary waves by wind waves is investigated. The results are compared with predictions of the Hasselmann transport equation. It is found that off-resonance interactions lead to significant corrections to the transport theory. The relative importance of three-wave and four-wave interactions is studied, as well as the role of triad resonances. For the capillary phenomena studied here, the four-wave terms in most cases lead to quantitative, but not qualitative, corrections to the three-wave only calculations. However, restricting interactions to the neighbourhood of triad resonances can give quite erroneous results. Use of a canonical transformation to pseudo-wave variables can greatly reduce numerical computation times.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 321 , 25 August 1996 , pp. 87 - 120
Copyright
© 1996 Cambridge University Press

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