Shock trains on a planar beach

Following on from the author's previous work, the propagation of trains of shock waves on a planar beach is studied in the framework of the nonlinear shallow water equations. The analysis is based on the use of a quasi-analytical solution valid for a shock wave which is fed by a constant Riemann invariant. The asymptotic behaviour of a train of such shock waves is inspected and novel approximate analytical solutions are provided. These are useful both for representing fundamental physical scenarios (e.g. propagation of saw-tooth spilling breakers in the surf zone) and for benchmarking wave-resolving and wave-averaged theoretical/numerical solutions. Finally, a study of the energy dissipation induced by the shock train is provided.