A Perturbation Analysis of an Interaction between Long and Short Surface Waves

Abstract

The interaction of finite‐amplitude long gravity waves with a small‐amplitude packet of short capillary waves is studied by a multiple‐scale method based on the invariance of the perturbation expansion under certain translations. The result of the analysis is a set of equations coupling the complex amplitude of the packet of short waves with the long‐wave velocity potential and surface elevation. The short wave is described by a Ginzburg‐Landau equation with coefficients that depend on properties of the long wave. The long‐wave potential and surface elevation satisfy the usual free‐surface conditions augmented by forcing terms representing effects of the short waves. The derivation removes some of the restrictions imposed in earlier studies.