As an extension of the classic water wave problem, a non-local formulation describing the evolution of the interface between two density stratified fluids with a rigid lid at the surface and underwater topography along the bottom is presented. We derive a system of non-local equations which describe the evolution of the interface and its associated velocity potentials. From these, we derive a bi-directional shallow water Benney-Luke (BL) equation, and from this a unidirectional Kadomtsev-Petviashvili (KP) equation at the interface is derived in 2+1 dimensions. To conclude, numerical solutions of the KP equation are presented which show the impact of various bathymetric variations on nonlinear wave formation along the interface.
