On the Instability of Weakly Nonlinear Three-Dimensional Standing Waves

Stability Characteristics of weakly nonlinear three-dimensional standing gravity waves on the surface of deep water are investigated on the basis of the Zakharov equation. It is found that the resonance curves and the instability domains in the three-dimensional standing waves are larger than those in two-dimensional standing waves, and generally the instability due to two-dimensional disturbances predominates over that due to three-dimensional ones. The maximum growth rate for the two-dimensional standing waves is shown to be about 30 percent larger than that for the two-dimensional limiting waves.