Journal of Fluid Mechanics

Resonant interactions between two trains of gravity waves

M. S.  Longuet-Higgins a1
a1 National Institute of Oceanography, Wormley, Surrey

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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 [doteqdot /Doteq R: eq, even dots] 17°; the rate of growth with horizontal distance is a maximum when θ [doteqdot /Doteq R: eq, even dots] 24°. The calculations show that it should be possible to detect the tertiary wave in the laboratory.

(Published Online March 28 2006)
(Received June 20 1961)