Water waves over a rough bottom in the shallow water regime

Details Water waves over a rough bottom in the shallow water regime Water waves over a rough bottom in the shallow water regime

2011-10-24

arxiv.org/abs/1110.5155v1

Mathematics

Analysis of PDEs

Revised version, to appear in Annales de l'Institut Henri Poincar\'e

Scientific paper

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.

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