The motion of water on a uniformly sloping beach, just after a bore reaches the shore, is studied. The shllow-water equations are used to fomulate a singular hyperbolic problem, which is solved by the help of RIEMANN'S Method and of the trutural theory of quasilinear hyperbolic equations developed in gas dynmics. The shore line is found to advance suddenly with non-zero velocity and then to move up and down the beach with constant, negative acceleration. The solution s shown to contain limit lines indiating a rather unexpeted, secondary bore in the back-wsh. (Author).