Generation of long waves by an oscillating line source above a flat beach of angle ${\alpha=\pi =/2M, M \epsilon z}$: the 3-d Case

A problem of long-wave generation by an oscillatory line source is discussed in the context of 3-d flow over a plane beach of slope angle ${\alpha=\pi =/2M, M \epsilon z}$ extending previous 2-d work by the author with the use of a Kontorovich–Lebedev transform. The placing of the source at a zero of the classical long-wave scattering potential is known to lead to a system of trapped waves and provides a robust check on this generalised theory. The development is based on a solution to a second order source-induced inhomogeneous difference equation and leads to an extension and correction of previous work by Morris (Proc. Camb. Phil. Soc. (1976), 79, 573) where a Green's function (for a reduced problem) was utilised for the case of a beach of unit gradient only. The solution provided in that work was recently deemed incomplete by the author (Appl. Math. Letters, 48, (2015), 135–142) following a parallel investigation on a discrepancy in numerical results from the well-known bench mark models of Peters and Roseau for the classical beach problem. While a full water column response requires a trigonometric factorisation, the complexity of which increases with ${M}$, surface response can be more readily calculated using an Abel-summable form of the inverse transform. Full water column analysis is restricted to the simple cases ${M = 2, M = 3}$. It is found, through the asymptotic theory in the transform space that, contrary to Morris' conjecture, there is a resonance at cut-off only for odd values of ${M}$ and this is verified through computation as wave-number approaches its critical value. Moreover, for cut-off resonance it is found that these amplitudes are directly proportional to the profile of the shortest edge wave possible (in source coordinates) for the chosen value of ${M}$. As a result, it therefore follows that cut-off resonance disappears if (for ${m\geq 3}$) the source is placed at a zero of this highest order edge wave. It is also found that the flow regime is generally of standing wave type landward of the source position, an observation which might impact on any subsequent mass transport discussions. An alternative method is also outlined in brief, based on recent work on electromagnetic scattering by a wedge shaped dielectric, here formulated to provide a numerical solution from a singular integral equation but designed to give results valid for arbitrary beach slope. The spectral function thus defined is shown to be consistent with that found in the analytic modelling for the case of unit slope. It remains to establish a robust numerical procedure which will combine appropriately discretised radiation conditions in the spectral space with an efficient means to invert the transform numerically.