On two-dimensional packets of capillary-gravity waves

Abstract

The motion of a two-dimensional packet of capillary–gravity waves on water of finite depth is studied. The evolution of a packet is described by two partial differential equations: the nonlinear Schrödinger equation with a forcing term and a linear equation, which is of either elliptic or hyperbolic type depending on whether the group velocity of the capillary–gravity wave is less than or greater than the velocity of long gravity waves. These equations are used to examine the stability of the Stokes capillary–gravity wave train. The analysis reveals the existence of a resonant interaction between a capillary–gravity wave and a long gravity wave. The interaction requires that the liquid depth be small in comparison with the wavelength of the (long) gravity waves and the evolution equations describing the dynamics of this interaction are derived.