A shock solution for the nonlinear shallow water equations

A global shock solution for the nonlinear shallow water equations (NSWEs) is found by assigning proper seaward boundary data that preserve a constant incoming Riemann invariant during the shock wave evolution. The correct shock relations, entropy conditions and asymptotic behaviour near the shoreline are provided along with an in-depth analysis of the main quantities along and behind the bore. The theoretical analysis is then applied to the specific case in which the water at the front of the shock wave is still. A comparison with the Shen & Meyer (J. Fluid Mech., vol. 16, 1963, p. 113) solution reveals that such a solution can be regarded as a specific case of the more general solution proposed here. The results obtained can be regarded as a useful benchmark for numerical solvers based on the NSWEs.