Nonlinear multiphase obliquely propagating and standing gravity waves on arbitrary water depth

In this paper, we apply the method of multiple scales to derive a system of coupled nonlinear Schrödinger (NLS) equations for slowly varying standing and obliquely propagating gravity waves, which describe the evolution of the three-dimensional wavetrains on arbitrary water depth. We have studied analytically the influence of a second wave propagating in a different direction from the first wave. The NLS equations developed herein are valid for any arbitrary water depth except for the shallow water depth limit. These equations are then employed to investigate the modulational instability of the uniform Stokes wavetrains to small disturbances. The propagation of multiphase modes is taken into account for the case when group velocity projections overlap. The condition is obtained for wave instabilities and for the existence of a multiphase solitary envelope solution. Results exhibit that the instability growth rates are higher when two wavetrains are taken into account. Particular attention is paid to weakly nonlinear standing gravity waves on deep water. Moreover, a linear instability analysis of oblique perturbations has been made for both finite depth and deep water.