Mapped spheroidal wave-envelope elements for unbounded wave problems

This paper describes a family of axisymmetric, spheroidal Ôwave envelopeÕ elements for modelling exterior wave problems. They are of variable radial order and can be used to represent steady and transient wave Þelds. The formulation is presented for the axisymmetric case using elements which are based on oblate and prolate spheroidal geometries. These o¤er the prospect of reduced dimensionalityÐin comparison to conventional, spherically formulated elementsÐwhen used to represent wave Þelds in the vicinity of slender or ßat objects. Conjugated weighting functions are used to give frequency-independent acoustic ÔmassÕ, Ôsti¤nessÕ and ÔdampingÕ matrices. This facilitates a simple extension of the method to transient problems. The e¤ectiveness and accuracy of the method is demonstrated by a comparison of computed and analytic solutions for sound Þelds 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 ba§e.