Reflection of interface solitary waves at a slope

Abstract

This paper deals with the problem of the reflection of interface solitary waves at a slope in a two-layer fluid system with rigid upper and lower boundaries. An edge-layer theory is developed to treat the region near to the slope in which shallow-water theory is not valid. We work to O (ϵ 1 2 ) so that the effects of the surging movement at the shoreline are included. After deriving the ‘reduced boundary conditions’ relevant to the shallow-water equations, a perturbation procedure involving methods of strained coordinates and inner-outer expansions is employed to solve this problem. Analytical results, such as maximum run-up at the slope and the time when it occurs, are presented. For a plane slope with an angle of inclination ν=π/4, we present some graphical results. It is found that, for the reflection problem, the properties of the depression mode differ from those of the elevation mode in many ways. In Appendix A, we also present an alternative way of obtaining the second order solutions for interface solitary waves in an infinite region.