A dispersive effective equation for transverse propagation of planar shallow water waves over periodic bathymetry

We study the behavior of shallow water waves propagating over bathymetry that varies periodically in one direction and is constant in the other. Plane waves traveling along the constant direction are known to evolve into solitary waves, due to an effective dispersion. We apply multiple-scale perturbation theory to derive an effective constant-coefficient system of Boussinesq-type equations that not only accurately describe these waves but also predict their full two-dimensional shape in some detail. Numerical experiments confirm the good agreement between the effective equations and the variable-bathymetry shallow water equations.