Edge-Layer Theory for Shallow-Water Waves over a Step–Reflection and Transmission of a Soliton

Abstract

This paper develops an `edge-layer' theory for weakly nonlinear shallow-water waves propagating over a step bottom. Introducing an `edge-layer' in the vicinity of the step, the matched-asymptotic expansion method is applied to establish `reduced' boundary conditions relevant to Boussinesq equations in the uniform shallow-water regions extending on the outside of the `edge-layer'. From the matching conditions, two `reduced' boundary conditions are derived which incorporate the higher-order corrections to the well-known Lamb's conditions. On applying the conditions thus derived, a problem of reflection and transmission of a single soliton incident upon the step is discussed. It is found that for a moderate depth-ratio, Lamb's conditions still provide correct boundary-values except the phase shift at the step.