Exact solutions for shoaling waves in shallow water
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The dynamics of shallow-water waves at the surface of an inviscid and incompressible fluid over a background shear flow approaching a sloping beach are investigated. First, we derive the nonlinear shallow-water equations in the presence of both background shear flow and a sloping beach. In this case, the hyperbolic shallow-water equations are not reducible and is it not straightforward to find the Riemann invariants. However, using intuition gained from the case of a shear flow over a flat bed, Riemann invariants can nevertheless be found. The Riemann invariants provide a proper hodograph transformation which is combined with several additional changes of variables to put the equations in linear form. This linear equation can be solved using the method of separation of variables. In this way, we are able to find exact solutions which give us a prediction of the shoaling process and of the development of the waterline (run-up). Our work is inspired by the method of Carrier and Greenspan . Therefore, a careful description of the Carrier-Greenspan method is presented first.