Travelling waves on a membrane: reflection and transmission at a corner of arbitrary angle. II

Abstract

This is the second part of an investigation into the reflection and transmission of fluid coupled membrane waves at a corner of arbitrary angle. In part one of this work (Abrahams & Lawrie 1995) an exact solution was obtained for a model problem comprising of a fluid wedge of arbitrary angle 4 β bounded by two identical semi-infinite plane membranes and forced by a surface wave incident along one face of the wedge. The problem was decomposed into a symmetric and an antisymmetric sub-problem and closed form expressions for the reflection coefficients, R s and R a respectively, were derived. The solution method incorporates several fundamental advancements on the work of Maliuzhinets (1958) and offers a constructive approach by which wedge problems with higher order boundary conditions can be solved easily. In this part of the investigation it is demonstrated how, for rational wedge angles, the formulae of part I can be exploited to yield simple exact or asymptotic expressions for R s , R a and, therefore, the reflection and transmission coefficients for the full problem. Further, a numerical implementation of the analytic solution enables these coefficients to be determined for all wedge angles β (0 ≼ β ≼ π), both for heavy and moderate fluid loading. The results confirm the reflection coefficients known previously for a few special wedge angles, and highlight several interesting trends. In particular it is found that, in the heavy fluid loading limit, there is a remarkably simple relationship between the phases of R s , R a and β .