Two-Dimensional Surface Wave Propagation over Arbitrary Ridge-Like Topographies

We extend the conformal mapping technique for water waves over topography, from its natural two-dimensional (2D) setting to three dimensions (3D) where the Laplacian, from potential theory, is no longer invariant. Nontrivial 3D flows of interest consider 2D surface waves propagating over large amplitude, nonsmooth, ridge-like topographies. The conformal mapping extension allows the derivation of asymptotic long wave models without the mild-slope assumption on the topography. The present reduced model generalizes the terrain-following Boussinesq system [A. Nachbin, SIAM J. Appl. Math., 63 (2003), pp. 905--922] to 3D. We present numerical simulations involving different topographical regimes. Our model agrees very well with predictions based on the linear fully dispersive potential theory equations. Fully 2D, weakly nonlinear wave simulations are presented in a regime usually not accessible to a reduced Boussinesq-type model: a tsunami interacting with the continental shelf and a solitary wave having an oblique incidence upon a highly irregular topography.