The two-dimensional stability of periodic and solitary wave trains on a water surface over arbitrary depth

The two-surface dimensional dynamics of a periodically or soliton-shaped modulated wave train on a water surface is investigated, using the Davey–Stewartson model. The depth h is taken to be uniform. The results obtained are twofold: the general behavior of the Benjamin–Feir (BF) instability is found for an arbitrary stationary envelope profile, thus generalizing Hayes’ analysis for uniform wave trains, and a new ‘‘K degeneracy’’ instability is found. The instability is always limited to ∼45° around the direction of propagation of the basic wave train and this critical angle decreases as the modulation becomes stronger. The new instability covers a narrow range of acute angles for 2πh/λ≤1.363, where λ is the wavelength of the carrier wave, and all angles for 2πh/λ≥1.363. Thus our results add new significance to the famous critical number 1.363. A simple explanation of how the new instability comes about concludes the paper. Agreement with previous work is demonstrated.