Obliquely incident surface waves in shallow water of variable depth

The propagation properties of obliquely incident, weakly nonlinear surface waves in shallow water of varying depth are studied analytically. The depth changes slowly in a direction that makes a constant angle with the propagation direction of the incident wave, initially travelling in a region of uniform depth. In the adjacent inhomogeneous region, depth variations are relatively slow. On the other hand, it is assumed that these occur on a scale shorter than that on which the wave evolves. As a consequence, the problem can be reduced to an evolution equation with constant coefficients. Since weak three-dimensional effects are also taken into account, this equation is related to the KP equation (Kadomtsev & Petviashvili 1970). Based on these results, the mechanism of mass transfer is studied. In a subsequent analysis, devoted to the case of a normally incident wave, the problem of describing the leading-order mass balance is solved. A normally incident solitary wave breaks up into a finite number of separate solitons if certain specific conditions are satisfied, such as the condition that this wave enters a region of smaller depth. In the more general case of an obliquely incident solitary wave, it is shown that this phenomenon can also occur, although the conditions are different.