$T$-matrix methods in acoustic scattering

Some further refinements are described for the $T$-matrix approach to acoustic scattering. From the structure of the matrices involved, one can infer the Rayleigh limit explicitly for objects having no density contrast. One finds $TRay=iR−R2$⁠, where the $R$-matrix involves integrals of the regular spherical wave functions over the object’s surface. The index of refraction and loss factor can be chosen as desired, and energy balance and reciprocity requirements are found to be met. The derivation can be extended to obtain the Rayleigh expansion, effectively describing $T$ as a series in ascending powers of the ratio of object size to wavelength. In trial cases, the series converges throughout the Rayleigh region and somewhat beyond. Bodies of high aspect ratio are also considered, where difficulties arise due to precision loss during numerical integration. Loss ranges from 4 or 5 significant figures (2:1 spheroid) to 22 figures (40:1 spheroid) or more. A class of surfaces has been found for which this problem can be avoided, however, enabling one to treat a variety of body shapes up to aspect ratios of 100:1 with no difficulty.