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.

Nonlinear Evolution Equations of Co-propagating Waves over Finite Depth Fluid

“…In the limiting case when the carrier wave numbers of the two wave packets considered by Debsarma and Das [12] are equal and capillarity is ignored, the coefficients of the evolution equation (26) of Debsarma and Das [12] matches with the corresponding coefficients of the evolution equation 4of Onorato et al [5] and also with equation (1) of Roskes [2] for the case when the angle of interaction between the two wave packets is zero.…”

“…In the present work, we extend the investigation of Debsarma and Das [12] to the case of finite depth fluid. Following Zakharov [13] integral equation approach we derive a pair of coupled nonlinear evolution equations correct up to third order in wave steepness for a pair of co-propagating surface gravity waves over finite depth fluid.…”

“…They found that plane wave solutions of the system become modulationally unstable when higher order dispersive terms are retained in the system. Debsarma and Das [12] derived a pair of fourth-order nonlinear evolution equations for two capillary-gravity wave packets propagating in the same direction and having different carrier wave numbers for the case of infinite depth water. Using these two evolution equations they studied stability properties of a uniform surface gravity wave train in the presence of a uniform capillary-gravity wave train, when the group velocities of the two wave trains are nearly equal.…”

“…We also perform modulational instability analysis of two co-propagating uniform Stokes wavetrains. Unlike Debsarma and Das [12] the stability analysis carried out here is not restricted to the case when the group velocities of the two wave packets are nearly equal. The paper is organized as follows: in Sect.…”

Nonlinear Evolution Equations for Broader Bandwidth Wave Packets in Crossing Sea States

“…It is assumed that the spectral widths | ⃗ |/ , | ⃗ |/ are of order ( ) for narrow band wave packet and of order ( 1/2 ) for broad band wave packet, being the order of smallness of wave steepness. The process of derivation of envelope equations adapted here is similar to the process of derivation of evolution equations for a surface gravity wave packet in the presence of another wave packet described in Debsarma and Das [16].…”