Overconvergence of the Near-Field Expansion for Linearized Waves Normally Incident on a Sloping Beach

Abstract

Small amplitude irrotational waves over a uniformly sloping beach are described by an integral equivalent to the conventional inverse Mellin transform. Solutions are established in terms of an inversion integral that is absolutely convergent and uniformly so for all angles $\theta $ in$ - \alpha \leqq \theta \leqq 0$, where $\theta = \alpha ,\,( \alpha < \pi/ 2)$ is the bed, and $\theta = 0$ is the surface. It is established that the near-field asymptotic expansion is overconvergent for all slope angles, and this is shown to hold for bounded as well as unbounded solutions.