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

Using Zakharov integral equation approach, a pair of coupled non-linear evolution equations are derived for two co-propagating weakly nonlinear gravity wave packets over finite depth fluid. Equations obtained here are not valid for resonant or quasi-resonant three-wave interactions and also inapplicable for shallow water. The two evolution equations are then employed to perform modulational instability analysis of a pair of co-propagating uniform Stokes wavetrains. It is found that the relative change in phase speed of one uniform wavetrain increases with the increase in wave steepness of the other wavetrain, but it decreases with the increase in the depth of the medium. It is also observed that the growth rate of instability of one uniform wavetrain increases with the increase in wave steepness of the second wavetrain and also with the decrease in the depth of the medium.