Long Wave Run-Up Resonance in a Multi-Reflection System

Abstract

Wave reflection and wave trapping can lead to long wave run-up resonance. After reviewing the theory of run-up resonance in the framework of the linear shallow water equations, we perform numerical simulations of periodic waves incident on a linearly sloping beach in the framework of the nonlinear shallow water equations. Three different types of boundary conditions are tested: fully reflective boundary, relaxation zone, and influx transparent boundary. The effect of the boundary condition on wave run-up is investigated. For the fully reflective boundary condition, it is found that resonant regimes do exist for certain values of the frequency of the incoming wave, which is consistent with theoretical results. The influx transparent boundary condition does not lead to run-up resonance. Finally, by decomposing the left- and right-going waves into a multi-reflection system, we find that the relaxation zone can lead to run-up resonance depending on the length of the relaxation zone.