Spatial wave dynamics of steady oblique wave interactions

In several models for waves on water of finite depth, a great variety of surface-elevation profiles for steadily traveling waves can be generated by a spatial evolution law in which the time-like variable is one of the horizontal space variables. This is the case for both linear and weakly nonlinear models, including the Kadomtsev–Petviashvili equation (KP-II). At the linear level, these waves correspond to steady oblique interactions of one-dimensional wave trains. The shape of the waves is determined in a similar way in all cases – by initial data that specifies the wave slope along a single line oriented either parallel or perpendicular to the direction of propagation. As one application we compute a steady two-dimensional wake due to a pressure disturbance moving at near-critical speed in a weakly nonlinear regime. We find that the disturbance can generate solitary waves traveling obliquely in the wake, a phenomenon recently realized experimentally and found to be relevant to the safe operation of high-speed ferries.