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

TL;DR

This paper derives an effective dispersive equation to model the transverse propagation of shallow water waves over periodic bathymetry, capturing their two-dimensional shape and behavior with high accuracy.

Contribution

It introduces a homogenized Boussinesq-type equation for shallow water waves over periodic bathymetry, extending previous one-dimensional models to two dimensions.

Findings

Effective equations accurately predict wave shape and evolution.

Numerical results show strong agreement with full shallow water equations.

Homogenized model captures transverse wave behavior over periodic bathymetry.

Abstract

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 equations, showing that the transversely-averaged wave approximately satisfies a Boussinesq-type equation, while the lateral variation in the wave is related to certain integral functions of the bathymetry. Thus the homogenized equations 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.