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Resonant interactions between two trains of gravity waves

Published online by Cambridge University Press:  28 March 2006

M. S. Longuet-Higgins
M. S. Longuet-Higgins
Affiliation:
National Institute of Oceanography, Wormley, Surrey
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Abstract

In a previous paper, Phillips (1960) showed that two or three trains of gravity waves may interact so as to produce a fourth (tertiary) wave whose wave-number is different from any of three primary wave-numbers k1, k2, k3, and whose amplitude grows in time. Such resonant interactions may produce an appreciable modification of the spectrum of ocean waves within a few hours. In this paper, by a slightly different method, the interaction is calculated in detail for the simplest possible case: when two of the three primary wave-numbers are equal (k3 = k1).

It is found that, when k1 and k2 are parallel or antiparallel, the interaction vanishes unless k1 = k2. Generally, if θ denotes the angle between k1 and k2, the rate of growth of the tertiary wave with time is a maximum when θ [eDot ] 17°; the rate of growth with horizontal distance is a maximum when θ [eDot ] 24°. The calculations show that it should be possible to detect the tertiary wave in the laboratory.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 12 , Issue 3 , March 1962 , pp. 321 - 332
Copyright
© 1962 Cambridge University Press

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References

Hasselmann, K. 1961 On the non-linear energy transfer in a wave spectrum. Conf. on Ocean Wave Spectra, Easton, Md, 1–4 May 1961. U.S. Nat. Acad. Sci. Publ. (in the Press).Google Scholar
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. A, 245, 535–81.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar