Nonlinear ripples of Kelvin–Helmholtz type which arise from an interfacial mode interaction

Abstract

An analysis is made of the small-amplitude capillary–gravity waves which occur on the interface of two fluids and which arise out of the interaction between the Mth and Nth harmonics of the fundamental mode. The method employed is that of multiple scales in both space and time and a pair of coupled nonlinear partial differential equations for the slowly varying wave amplitudes is derived. These equations describe, correct up to third order, the progression of a wavetrain and are generalizations of the nonlinear Schrödinger-type equations used by many authors to model wave propagation. The equations are solved and formal power series expansions of the corresponding wave profiles obtained. Many different wave configurations can arise, some symmetric others asymmetric. It is found that an important influence on the type of waves which can occur is whether the ratio of the interacting wave modes is greater or less than two. Finally, an examination of the stability of the waves to plane wave perturbations is carried out.