On nonlinearly detuned third harmonic ripples between two stratified fluids

Abstract

The problem of two semi-infinite fluids in uniform horizontal motion parallel to their interface is studied. Attention is focused on the interfacial disturbances that are caused by the interaction between a fundamental mode and its third harmonic. A series expansion for the disturbance profile is obtained in which the leading-order amplitudes are assumed to be slowly varying functions in time and space. By use of this expression we are able to derive a pair of coupled nonlinear Schrödinger-type equations which model the evolution of the interface. Solutions to this system are found and thus we are able to describe the possible wave profiles, which turn out to be tripleor quintuple-dimpled. We also find that at perfect resonance three profiles are always possible but that at near-resonance there may be one or three profiles depending on the values of the parameters present in the problem.