Nonlinear wave focusing on water of finite depth

The problem of freak wave formation on water of finite depth is discussed. Dispersive focusing in a nonlinear medium is suggested as a possible mechanism of giant wave generation. This effect is considered within the framework of the nonlinear Schrödinger equation and the Davey–Stewartson system, describing 2+1-dimensional surface wave groups on water of finite depth. In the 2+1-dimensional case, the dispersive grouping is accompanied with a geometrical focusing. Necessary wave conditions for the occurrence of such a phenomenon are discussed. Influence of non-optimal phase modulation and presence of strong random wave component are found to be weak: they do not cancel the mechanism of wave amplification. The mechanism of dispersive focusing is compared with the wave enhancement due to the Benjamin–Feir instability, which is found to be extremely sensitive with respect to weak random perturbations.