Resonant interfacial capillary-gravity waves in the presence of damping effects

An investigation is made into the waves of small and moderate amplitude which may occur at the interface of two inviscid fluids of different densities. The external forces are those of gravity and surface tension and the waves are due to the resonant interaction between the M th and N th harmonics of the fundamental mode. In contrast to previous studies, damping effects are taken into account. Important parameters in the problem are the velocity and density ratios. A pair of coupled nonlinear Schrodinger-type partial differential equations for the wave amplitudes is derived which model the evolution of the waves, correct up to third order. A wide variety of sinusoidal solutions to the equations is shown to exist, irrespective of the values assigned to the parameters. The stability of these solutions to small modulational perturbations is considered. It is found that when the damping is due to dissipation then the waves are stabilised.