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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.
Keywords: Water waves, Shallow water, Rough bathymetry
@article{AIHPC_2012__29_2_233_0,
author = {Craig, Walter and Lannes, David and Sulem, Catherine},
title = {Water waves over a rough bottom in the shallow water regime},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {233--259},
publisher = {Elsevier},
volume = {29},
number = {2},
year = {2012},
doi = {10.1016/j.anihpc.2011.10.004},
mrnumber = {2901196},
zbl = {1329.76069},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2011.10.004/}
}
TY - JOUR AU - Craig, Walter AU - Lannes, David AU - Sulem, Catherine TI - Water waves over a rough bottom in the shallow water regime JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 233 EP - 259 VL - 29 IS - 2 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2011.10.004/ DO - 10.1016/j.anihpc.2011.10.004 LA - en ID - AIHPC_2012__29_2_233_0 ER -
%0 Journal Article %A Craig, Walter %A Lannes, David %A Sulem, Catherine %T Water waves over a rough bottom in the shallow water regime %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 233-259 %V 29 %N 2 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2011.10.004/ %R 10.1016/j.anihpc.2011.10.004 %G en %F AIHPC_2012__29_2_233_0
Craig, Walter; Lannes, David; Sulem, Catherine. Water waves over a rough bottom in the shallow water regime. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 233-259. doi: 10.1016/j.anihpc.2011.10.004
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