Dispersion relations for gravity waves in a deep fluid: second sound in a stormy sea
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We consider the nonlinear processes of interaction between a random field of short deep gravity waves and a deep long gravity wave, as well as the effects of nonlinear interactions among the driven short waves. The interactions are characterized by two types of invariants. First, the short waves posses an adiabatic invariant as regards the interaction with the long waves. Second, a collision between short waves conserved frequency and wave vector. As a consequence of these invariants, extra degrees of freedom appear. This results in a two-fluid description at the surface of the liquid for the mean flow of the long wave plus a distribution of waves. The two-fluid theory implies a spectrum of oscillations of the surface where, besides the usual gravity waves, there exists a surface mode with longitudinal oscillation. The regimes of validity of this hydrodynamic description are discussed, and the solutions for the dispersion relation are also presented. The power spectrum of a stormy sea is derived within the framework on nonlinear wave interactions. Finally, we comment on the relevance of the results to phase-velocity measurement is a wind-wave laboratory experiment, as well as possible experiments to observe the theoretically predicted new mode.
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