Oblique wave incidence on a plane beach: the classical problem revisited

Abstract

The non-hydrostatic description of three-dimensional waves incident over a plane beach at a long straight coastline is considered in terms of the inverse Kontorovich–Lebedev integral transform. This is seen as a natural extension to earlier work by the author where the two-dimensional (normal incidence) flow is expressed as an inverse Mellin transform, and similar simplifications in the description here are encountered. In particular computations are undertaken for a variety of beach slopes of the form α=π/2m where m is an integer and for a range of incidence angles. These computations have previously only been practical for beaches whose slope α is regarded as asymptotically small thereby allowing versions of the mild-slope equation to be used. For the chosen slope angles, the solution is established with rigour and methods of estimating near- and far-field asymptotics arise naturally in this discussion. For the case of perfect reflection, a previously known solution is recovered in closed form as a finite sum of exponential terms, and a shoreline ‘amplification factor’ a_γ is considered for these waves and is computed for a range of beach slopes through the entire spectrum of incidence angles. It is shown analytically that, in the limit of normal incidence, the value of a_γ approaches the well-known classical result a₀=m^1/2 and, for glancing incidence, Whitham's (1979) result is confirmed where the value approaches either 1 or 0 depending on whether the beach angle is or is not an angle at which a new Ursell edge wave mode appears (m odd).

As applications of the new development, comprehensive near-field expansions for arbitrary reflection are written and verified by computation. These permit the construction of refracted wavefronts and wave rays for arbitrary beach slope without the usual phase velocity assumptions. Instability is indicated at very oblique incidence where nonlinear modelling (Peregrine & Ryrie 1983) predicts ‘anomalous refraction’. Results are presented graphically and computation of derivatives of the potential enables estimation of the (second order) set-down seaward of the breaker zone. This is found to decrease as wave attack becomes increasingly oblique.