Travelling waves on a membrane: Reflection and transmission at a corner of arbitrary angle .1.

This article is the first part of a study into the reflection, transmission and scattering of waves on the semi-infinite faces of a compressible fluid wedge of arbitrary angle. This class of problems, in which the wedge surfaces are described by high order impedance conditions (that is, containing derivatives with respect to variables both normal and tangential to the boundary), is of great interest in structural acoustics and electromagnetism. Here, for mathematical convenience, the canonical problem of a fluid wedge with two plane membrane surfaces is examined. Forcing is taken as an unattenuated fluid coupled surface wave incident from infinity on one of the wedge faces. Explicit application of the edge-constraints allows the boundary value problem to be formulated as an inhomogeneous difference equation which is then solved in terms of Maliuzhinets special functions. An analytical solution is thus obtained for arbitrary wedge angle and the membrane wave reflection and transmission coefficients are deduced. The solution method is straightforward to apply and can easily be generalized to any boundary or edge conditions. Also in Part I, the solution obtained for the case of a wedge of angle 2 pi is compared with that determined by the Wiener-Hopf technique. The two methods are in complete agreement. In the second half of this work the reflection coefficients calculated here will be shown to confirm those given previously in the literature for certain specific wedge angles. A full numerical study, for a range of fluid-membrane parameter values, will also be presented in Part II.