Focusing of surface waves by variable bathymetry

Scattering of a monochromatic train of surface gravity waves incident on a finite region of arbitrary three-dimensional smoothly varying bathymetry is considered in this paper. The full three-dimensional linear water wave theory is approximated by the depth-averaged modified mild-slope equations and a Greens function approach is used to derive domain a integral equation for the function relating to the unknown surface over the varying bed. A simple but robust and effective numerical scheme is described to approximate solutions to the integral equation. The method is applied to bathymetries which exhibit focusing in the high-frequency ray-theory limit and used to illustrate that focusing occurs at finite wavelengths where both refractive and diffractive effects are included. Specifically, examples of elliptical and bi-convex lens bathymetries are considered.