Comparison of Solution Options for Line-Source Generated Short-Waves in a Wedge with Neumann and Robin Conditions on Respective Faces: Application to Waves on a Plane Beach

Abstract

High-frequency waves generated by a moving oscillatory line source over a plane beach are examined in the context of a linear non-hydrostatic theory. The work complements a recent study by the author of low-frequency wave generation in a similar environment (Q. Jl. Mech. Appl. Math.68 (2015) 421–460) and provides a special case of a wider class of problems to which belong also a number of problems on electromagnetic diffraction by impedance wedges currently of considerable interest. The latter are generally treatable only by numerical techniques and the opportunity is seized here to examine a comparison between such numerically generated solutions and analytic solutions which can be and are derived and evaluated for the water wave problem. In common with the electromagnetic problems, the water-wave solution is formulated in terms of an inverse Kontorovich–Lebedev transform the inversion of which is considered both by direct quadrature and by residue composition while the spectral function of the transform is also determined in two different ways namely as (i) (for special wedge angles) a semi-closed form entire function and (ii) a numerical solution of a singular integral equation (as commonly encountered in the electromagnetic problems) valid for more general wedge angles. It is concluded that the residue composition technique can be fraught with difficulties and, where spectral function structure allows, should be replaced by the strategy of direct numerical inversion. For the water-wave problem, the anticipated resonances at edge wave frequencies are shown to exist and their strengths prove to be consistent with the well-known findings of Ursell though the process of successfully detecting these purely from the numerical model is shown to be somewhat error prone.