On the stability of steep gravity waves

Abstract

Previous calculations of the normal mode perturbations of steep gravity waves have suggested that the lowest superharmonic mode n = 2 becomes unstable at around ak = 0.436, where 2a is the crest-to-trough height of the unperturbed wave and k is the wavenumber. This would correspond to the wave steepness at which the phase speed c is a maximum (considered as a function of ak). However, numerical calculations at such high wave steepnesses can become inaccurate. The present paper studies analytically the conditions for the existence of a normal mode at zero limiting frequency. It is proved that for superharmonic perturbations such conditions will occur only for a pure phase-shift (corresponding to n = 1) or when the speed c is stationary with respect to the wave steepness, that is when dc = 0. Hence the limiting form of the instability found by Tanaka (J. phys. Soc. Japan 52, 3047-3055 (1983)) near the value ak = 0.429 must be a pure phase-shift.