Non-linear capillary waves generated by steep gravity waves

Abstract

The train of capillary waves which appears on the forward face of a steep gravity wave is discussed by considering the capillary waves as stationary waves on a slowly varying running stream, using the non-linear capillary wave solution of Crapper (1957) and the recent method due to Whitham (1965 a, b ). Then energy input and damping are introduced into an energy equation which becomes a non-linear ordinary differential equation for the capillary wave steepness. Numerical solutions for various gravity wavelengths and steepnesses are discussed, and some light is thrown on the problem of the breaking of gravity waves.