Water waves generated by instantaneous disturbances at the bed of a sloping beach

The two-dimensional problem of the generation of water waves due to instantaneous disturbances prescribed at the bed of a beach sloping at an arbitrary angle is studied here. It is formulated in terms of an initial-boundary-value problem for the velocity potential describing the motion in the fluid region assuming the linear theory. Using the Laplace transform in time and the Mellin transform in distance, the problem is reduced to solving a difference equation whose method of solution is of considerable importance in the literature. The form of the free surface is obtained in terms of a multiple infinite integral that is evaluated by the method of steepest-descent. For some prescribed forms of the disturbance at the bed of the beach, the free surface is depicted in a number of figures for different beach angles. It is observed that as the beach angle decreases, the maximum wave height increases, which is plausible.