# Shallow Water Dynamics on Linear Shear Flows and Plane Beaches

## TL;DR

This paper extends shallow-water equations to include shear flows and beach slopes, transforming them into a linear system solvable exactly, aiding the analysis of long wave run-up on beaches.

## Contribution

It introduces a novel approach to linearize complex shallow-water models with shear and slope, enabling exact solutions for wave run-up analysis.

## Key findings

- Exact solutions for wave propagation over shear flows and slopes.
- Analytical description of shoreline development during wave run-up.
- Method applicable to various beach and flow configurations.

## Abstract

Long waves in shallow water propagating over a background shear flow towards a sloping beach are being investigated. The classical shallow-water equations are extended to incorporate both a background shear flow and a linear beach profile, resulting in a non-reducible hyperbolic system. Nevertheless, it is shown how several changes of variables based on the hodograph transform may be used to transform the system into a linear equation which may be solved exactly using the method of separation of variables. This method can be used to investigate the run-up of a long wave on a planar beach including the development of the shoreline.