Solitary water waves created by variations in bathymetry

Abstract

We study the flow of water waves over bathymetry that varies periodically along one direction. We derive a linearized, homogenized model and show that the periodic bathymetry induces an effective dispersion, distinct from the dispersion inherently present in water waves. We relate this dispersion to the well-known effective dispersion introduced by changes in the bathymetry in non-rectangular channels. Numerical simulations using the (non-dispersive) 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 confirm the analysis and suggest that experimental observation of these solitary waves is possible.

References in corpus (1)

• Effective Rankine-Hugoniot conditions for shock waves in periodic media

Cited by in corpus (3)

• A dispersive effective equation for transverse propagation of planar shallow water waves over periodic bathymetry
• On the Rate of Error Growth in Time for Numerical Solutions of Nonlinear Dispersive Wave Equations
• An efficient implicit scheme for the multimaterial Euler equations in Lagrangian coordinates