Abstract

I will discuss the flow of water waves over bathymetry that varies periodically along one direction. Starting from the dispersionless linearized shallow water model, perturbation analysis shows that the periodic bathymetry induces an effective dispersion, distinct from the dispersion inherently present in water waves. This dispersion can be related to the effective dispersion introduced by changes in the bathymetry in non-rectangular channels. Numerical simulations using the (nonlinear) shallow water equations reveal that a balance between this effective dispersion and nonlinearity can create solitary waves. We derive a KdV-type equation that approximates the behavior of these waves in the weakly- nonlinear regime. We show that, depending on geometry, dispersion due to bathymetry can be much stronger than traditional water wave dispersion and can prevent wave breaking in strongly nonlinear regimes. Computational experiments using depth-averaged water wave models as well as a full two-fluid Navier-Stokes model confirm the analysis and suggest that experimental observation of these solitary waves is possible.