Attenuation of short surface waves by the sea floor via nonlinear sub-harmonic interaction

We consider the indirect mechanism for dissipation of short surface waves through their near-resonant interactions with long sub-harmonic waves that are dissipated by the bottom. Using direct perturbation analysis and an energy argument, we obtain analytic predictions of the evolution of the amplitudes of two short primary waves and the long sub-harmonic wave which form a near-resonant triad, elucidating the energy transfer, from the short waves to the long wave, which may be significant over time. We obtain expressions for the rate of total energy loss of the system and show that this rate has an extremum corresponding to a specific value of the (bottom) damping coefficient (for a given pair of short wavelengths relative to water depth). These analytic results agree very well with direct numerical simulations developed for the general nonlinear wave-wave and wave-bottom interaction problem.