Mapped spheroidal wave-envelope elements for unbounded wave problems

Abstract

This paper describes a family of axisymmetric, spheroidal ‘wave envelope’ elements for modelling exterior wave problems. They are of variable radial order and can be used to represent steady and transient wave fields. The formulation is presented for the axisymmetric case using elements which are based on oblate and prolate spheroidal geometries. These offer the prospect of reduced dimensionality—in comparison to conventional, spherically formulated elements—when used to represent wave fields in the vicinity of slender or flat objects. Conjugated weighting functions are used to give frequency-independent acoustic ‘mass’, ‘stiffness’ and ‘damping’ matrices. This facilitates a simple extension of the method to transient problems. The effectiveness and accuracy of the method is demonstrated by a comparison of computed and analytic solutions for sound fields generated by a rigid sphere in steady harmonic oscillation, by a rigid sphere excited from rest, and by a circular plate vibrating in a plane baffle.