Three-dimensional instabilities of periodic gravity waves in shallow water

A linear stability analysis of finite-amplitude periodic progressive gravity waves on water of finite depth has extended existing results to steeper waves and shallower water. Some new types of instability are found for shallow water. When the water depth decreases, higher-order resonances lead to the dominant instabilities. In contrast with the deep water case, we have found that in shallow water the dominant instabilities are usually associated with resonant interactions between five, six, seven and eight waves. For small steepness, dominant instabilities are quasi two-dimensional. For moderate and large steepness, the dominant instabilities are three-dimensional and phased-locked with the unperturbed nonlinear wave. At the margin of instability diagrams, these results suggest the existence of new bifurcated three-dimensional steady waves.