Different aspects of the hodograph transform and Riemann invariants for the shallow water equations
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The aim of this work is to visualize irrotational long wave on a sloping beach by following the approach of Carrier and Greenspan . We first derive the non-linear shallow-water equations for sloping beach and then find the Riemann invariants. The Riemann invariants are then used to implement a proper hodograph transformation in order to transform the equations into linear form. By using separation of variables the exact solutions of the linear equations are found and the results are plotted for different values of runup and run-down time. Furthermore, in this study we obtain shallow-water equations for shear flow which are also called Benney equations . These equations are written in a vector form  to find the characteristic form and the Riemann invariants of the shallow-water equations for shear flow over a flat bed.