Modulational instability in crossing sea states over finite depth water

Abstract

Nonlinear evolution equations are derived in a situation of crossing sea states characterized by water waves having two different spectral peaks. The nonlinear evolution equations derived here are valid for any water depth except for shallow water depth case. These evolution equations are then employed to study the instability properties of two Stokes wave trains considering both unidirectional and bidirectional perturbations. Figures have been plotted showing the growth rate of instability for various depths of water and for different values of the angle of interaction of the two wave systems. All the figures serve as an evidence to the fact that freak waves can be formed as a result of modulational instability in crossing sea states over finite depth water. It is observed that the growth rate of instability in crossing sea states situation over finite depth water is much higher than that for infinite depth case and it increases with the decrease of the depth of water.