Transverse Wave Instability on the Surface of a Liquid of Finite Depth.

A study is made of the stability of the wave field on the surface of a liquid of arbitrary depth. The wave spectrum is assumed to be narrow with respect to transverse wavenumbers k and broad with respect to longitudinal wavenumbers. From the known Hamiltonian equations for the field dynamics of weakly nonlinear waves an equation for the Fourier amplitude of the wave field is obtained which generalizes the well-known nonlinear Schroedinger equation to the case of a broad spectrum with respect to k. On the basis of the linearized equation, an analysis is made of the transverse stability of the waves. It is shown that on the surface of a deep liquid an arbitrary wave spectrum is stable with respect to long-wave transverse modulation. On the surface of shallow water an instability occurs the spatial and time scales of which are determined by the energy and shape of the wave spectrum.