Water waves over a rough bottom in the shallow water regime

TL;DR

This paper derives a new model for shallow water waves over a rapidly varying periodic bottom, revealing a resonance phenomenon that causes secular growth in surface wave patterns, supported by rigorous mathematical analysis.

Contribution

It introduces a coupled shallow water and nonlocal homogenization model with a novel resonance effect, extending understanding of wave-bottom interactions.

Findings

Discovery of a nonlinear Bragg resonance phenomenon.

Derivation of a coupled model with error estimates.

Validation of the model in 2D and 3D settings.

Abstract

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.