Water waves over a rough bottom in the shallow water regime

  • Walter Craig

    Department of Mathematics, McMaster University, Hamilton, ON L8S 4K1, Canada
  • David Lannes

    Département de Mathématiques et Applications, Ecole Normale Supérieure, 45 rue dʼUlm, F-75230 Paris Cedex 05, France
  • Catherine Sulem

    Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada


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.

Cite this article

Walter Craig, David Lannes, Catherine Sulem, Water waves over a rough bottom in the shallow water regime. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 2, pp. 233–259

DOI 10.1016/J.ANIHPC.2011.10.004