Preprint A555/2007
An optimal Boussinesq model for shallow water wave-microstructure interaction

AndrĂ© Nachbin | Garnier, Josselin | Kraenkel, Roberto

**Keywords: **
Korteweg-de Vries equation | effective media | water waves

In this paper we consider the propagation of water waves
in a long-wave asymptotic regime, when the bottom topography is periodic
on a short length scale.
We perform a multiscale
asymptotic analysis of the full potential theory model
and of a family of reduced Boussinesq systems parameterized by a free parameter
that is the depth at which the velocity is evaluated.
We obtain explicit expressions for the coefficients
of the resulting effective KdV equations.
We show that it is possible to choose the free parameter
of the reduced model so as to match the KdV
limits of the full and reduced models. Hence the reduced model is optimal regarding
the embedded linear weakly dispersive and weakly nonlinear characteristics of the
underlying physical problem, which has a microstructure.
We also discuss the impact of the rough bottom on the effective wave propagation.
In particular nonlinearity is enhanced and we can distinguish two regimes
depending on the period of the bottom where the dispersion is either enhanced or reduced
compared to the flat bottom case.