A MULTISCALE MODEL FOR WEAKLY NONLINEAR SHALLOW WATER WAVES OVER PERIODIC BATHYMETRY

We study the behavior of shallow water waves over periodically varying bathymetry,based on the first-order hyperbolic Saint-Venant equations. Although solutions of this system areknown to generally exhibit wave breaking, numerical experiments suggest a different behavior inthe presence of periodic bathymetry. Starting from the first-order variable-coefficient hyperbolicsystem, we apply a multiple-scale perturbation approach in order to derive a system of constant-coefficient high-order partial differential equations whose solution approximates that of the originalsystem. The high-order system turns out to be dispersive and exhibits solitary-wave formation, inclose agreement with direct numerical simulations of the original system. We show that the constant-coefficient homogenized system can be used to study the properties of solitary waves and to conductefficient numerical simulations.