Inverse non-linear problem of the long wave run-up on coast

TL;DR

This paper introduces a rigorous method to recover initial wave displacement from shoreline data using a transformation that simplifies non-linear shallow-water equations, verified through numerical examples.

Contribution

It presents a novel inverse problem approach for determining initial wave conditions from shoreline measurements using the generalized Carrier-Greenspan transformation.

Findings

Method successfully recovers initial wave displacement in simulated scenarios.

Numerical verification confirms the approach's accuracy for different coastal geometries.

The approach simplifies complex non-linear wave equations to linear form for easier analysis.

Abstract

The study of the process of catastrophic tsunami-type waves on the coast makes it possible to determine the destructive force of waves on the coast. In hydrodynamics, the one-dimensional theory of the run-up of non-linear waves on a flat slope has gained great popularity, within which rigorous analytical results have been obtained in the class of non-breaking waves. In general, the result depends on the characteristics of the wave approaching (or generated on) the slope, which is usually not known in the measurements. Here we describe a rigorous method for recovering the initial displacement in a source localised in an inclined power-shaped channel from the characteristics of a moving shoreline. The method uses the generalised Carrier-Greenspan transformation, which allows one-dimensional non-linear shallow-water equations to be reduced to linear ones. The solution is found in terms of Erd\'elyi-Kober integral operator. Numerical verification of our results is presented for the cases of a parabolic bay and an infinite plane beach.