Long wave interaction over varying topography

The propagation of long waves on the surface of a three-dimensional fluid domain bounded below by slowly varying topography is considered. There are two important limits: If the initial data can be written in terms of a discrete set of one-dimensional wavefronts, the resulting wave field is described by a set of variable coefficient Korteweg-de Vries (KdV) equations for each wave along its characteristic curve. Waves along different characteristics interact with each other yielding phase shifts that depend on the wave amplitudes, the angle between the rays and the local depth. If the initial data is modulated slowly in the direction parallel to the wavefronts, the wave field is described by variable coefficient Kadomtsev-Petviashvili (KP) equations along rays. For topography varying only in one direction, we calculate explicit results for the interaction between two sets of periodic or solitary waves and show the equivalence of a single nearly normally incident KdV wave and a normally incident KP wave.