On the normal incidence of linear waves over a plane incline partially covered by a rigid lid

The effect is examined on infinitesimal standing waves over a plane beach when restricted by the arbitrary placing of a finite rigid (or permeable) lid of length ℓ on the undisturbed surface. A uniformly bounded solution for the potential function is obtained by a Green's function method. The Green's function is derived and manipulated, for subsequent computational expedience, from a previously known solution for the problem of an oscillating line source placed at an arbitrary location in the sector. Applications are made to both the case of plate anchored at the origin and the case of plate anchored some distance at sea (drifted plate problem). In both cases water column potentials and equipotentials are constructed from the numerical solution of a Fredholm equation of the second kind by finite difference discretization. Solutions are further extended to include the logarithmically singular standing wave, combination with which allows the construction of progressing waves. Computation of initially incoming progressing wave envelopes demonstrates the emergence of a partially standing wave pattern shoreward of the plate. There is no difficulty, in principle, to extend the theory to any number of plates, and this is verified by computation for the case of two plates. A new shoreline radiation condition is constructed to allow formulation, in the usual way, of the reflection/transmission problem for the plate, and results are in good qualitative agreement with a similar model on a horizontal plane bed. It is argued that the Green's function constructed here could be used in a number of diverse problems, of this linear nature, where all, or part, of the submerged boundary is that of a plane incline.