ACOUSTIC SCATTERING BY A SPHERE IN A CIRCULAR CYLINDRICAL WAVE-GUIDE

Multipole expansions are used to solve the problem of the acoustic scattering of an arbitrary mode in a circular cylindrical waveguide by a sphere situated on its axis. Both hard and soft boundary conditions for both the guide and the sphere are considered. The multipoie method considered here, although only being appropriate to the above geometry, provides fairly simple equations from which accurate results can be obtained without much computational effort. These results, whilst being interesting in their own right, can be used as a check for more complicated numerical schemes that solve problems for arbitrary geometries. The multipole formulation also enables simple approximate formulae for the reflection and transmission coefficients to be obtained in the low-frequency and small-sphere limits. The same method, with a different set of multipoles, can be used to solve the problem of potential flow past a sphere in a circular tube, and this is done in an Appendix.