Wave Generation on a Plane Beach by a Moving Pressure Disturbance or by a Stationary Bottom Protrusion

The initial-value problem (IVP) on a plane beach is considered where the leading-order motion is a uniform longshore flow. We consider waves created by surface pressure disturbances or bottom protrusions. Comparison is made with work by Evans (J. Fluid Mech. 1988) wherein the specific pressure chosen is here shown to be orthogonal to the space of continuous-spectrum solutions. The solution to the IVP is shown, in the limit of large time, to approach Evans's steady-state solution. Special consideration is given to a travelling pressure wave both with and without the longshore current and asymptotics are considered for observers leading, trailing and moving with the wave. Generalization of the work shows that all disturbances generated by a pressure distribution will ultimately approach edge-wave profiles and thus decay exponentially in the cross-shore coordinate. The classical result is generalized, for unidirectional flow in infinitely deep water, that disturbances are 'trapped' when the pressure applied has compact support on a certain length L. It is shown that, for beaches no shallower than 30 per cent, the equivalent result holds for pressures having longshore compact support on a length L/ sin a, where a is the beach angle.