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Aymptotic analysis of interaction of a surface wave with two internal waves Jamali, Mirmosadegh


The motion of a surface wave in a two-layer fluid can lead to generation of two internal waves through a resonance mechanism under certain circumstances. Two subjects related to this interaction are studied theoretically here. These are the behavior of the waves in threedimensional interaction when the density difference between the two layers is small, and the effect of a diffuse interface on the interaction. In the first study, the three-dimensional interaction of a surface wave with two oblique internal waves is analyzed asymptotically in an attempt to obtain simple approximate expressions for the growth rate as well as the kinematic properties of the internal waves. The non-dimensional density difference 8 is taken as a perturbation parameter, and the first few terms in the expansions of the desired quantities are derived. The results indicate that the internal-wave numbers are O(δ⁻¹), one order larger than the surface-wave number. Also, at leading order the internal wave frequencies are equal to ω₀/2, and the directions of the two internal waves differ by 180 . An important finding is that an immediate consequence of taking δ as a small parameter is that the internal waves become deep-water waves in both layers. According to the asymptotic analysis, the interaction coefficients α₁ and α₂ are 0(l) and are equal at leading order. The second study concerns the generation of two internal waves by a surface wave on a thin diffuse interface. As in the first analysis, the non-dimensional density difference δ is taken as a small perturbation parameter. In addition, it is assumed that the diffuse interface is small compared to the internal wavelengths by taking it to be order δ². A three-layer system admits two modes of internal wave motion, and similarly two modes of interaction are found possible through the analysis. These are interaction between a surface wave and two first mode internal waves, and interaction between a surface wave, a first-mode and a secondmode internal wave. It is shown that, contrary to the first mode, in the second mode of interaction the waves are not sub-harmonic to the surface wave. An important finding is that the growth rate in the first mode is higher than in the second. This implies that in a real situation the interaction between a surface wave and two first-mode internal waves has more chance to occur.

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