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The evolution of a weakly nonlinear, weakly damped, capillary-gravity wave packet

Published online by Cambridge University Press:  21 April 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

Longuet-Higgins's (1976) analysis of energy transfer within a narrow spectrum of gravity waves with approximately uncorrelated phases is generalized to accommodate capillarity and weak damping. The analysis is based on the corresponding generalization of Zakharov's (1968) evolution equation for weakly nonlinear, deep-water gravity-wave packets. The results for a symmetric normal spectrum are expressed in terms of elliptic integrals and depend, after appropriate scaling, on a single similarity parameter and on the sign of the curvature of the linear dispersion relation. Energy transfer is away from the peak of that spectrum if kl* < 0.393, where k is the wavenumber and l* is the capillary length (2.8 mm for water), but may be towards the peak if 0.343 < kl* < 0.707 (4.5 cm > 2π/k > 2.5 cm for water). The formulation is based on energy exchange through resonant quartets and is not valid in the neighbourhood of kl* = 0.707; at which the second harmonic of a capillary-gravity wave resonates with its fundamental (Wilton's ripples). The modulational instability of a weakly damped capillary-gravity wave is examined in an Appendix.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 187 , February 1988 , pp. 141 - 154
Copyright
© 1988 Cambridge University Press

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