Instabilities of Weakly Nonlinear Standing Gravity Waves

Instabilities of two-dimensional weakly nonlinear standing gravity waves on the surface of deep water are investigated on the basis of the Zakharov equation. It is found that the instability due to the resonant interaction among waves concerning the wave numbers k0 and -k0 gives the growth rate larger than that due to the resonant interaction among waves concerning k0. The restabilization of the entire system for large steepness does not occur for the standing wave. Furthermore, a coupled set of nonlinear Schrodinger type equations is derived from the Zakharov equation for long wave perturbations and a criterion for the Benjamin-Feir type instability is obtained.