Reflection of a shallow-water soliton. Part 2. Numerical evaluation.

Abstract

Reflection of a shallow-water soliton at a plane beach is investigated numerically based on the edge-layer theory developed in Part 1 of this series. The offshore behaviour in the shallow-water region is first obtained by solving the Boussinesq equation under the ‘reduced’ boundary condition. The spatial and temporal variations of the surface elevation are displayed for two typical values of the inclination angle of the beach. Using these solutions, the nearshore behaviour is then evaluated to obtain the surface elevation and the velocity distribution in the edge layer. Both offshore and nearshore behaviours furnish a full knowledge of the reflection problem of a shallow-water soliton. To check the applicability of the edge-layer theory, a ‘computational experiment’ is carried out based on the boundary-element method, in which the Laplace equation is solved numerically under the full nonlinear boundary conditions at the free surface without introducing the edge-layer concept. Both results show a fairly good agreement for the overall reflection behaviour of a shallow-water soliton except for the surging movement at the shoreline.