Fourth-order nonlinear evolution equations for a capillary-gravity wave packet in the presence of another wave packet in deep water

Abstract

Starting from the Zakharov integral equation, two coupled fourth-order nonlinear equations have been derived for the evolution of the amplitudes of two capillary-gravity wave packets propagating in the same direction. The two evolution equations are used to investigate the stability of a uniform capillary-gravity wave train in the presence of another having the same group velocity. The relative changes in phase velocity of each uniform wave train due to the presence of the other one have been shown in figures for different wave numbers. The condition of instability of a wave of greater wavelength in the presence of a wave of shorter wavelength is obtained. It is observed that the instability region for a surface gravity wave train in the presence of a capillary-gravity wave train expands with the increase of wave steepness of the capillary-gravity wave train. It is found that the presence of a uniform capillary-gravity wave train causes an increase in the growth rate of instability of a uniform surface gravity wave train.