A formulation of multiple scattering by many bounded obstacles using a multicentered, T supermatrix

Abstract

The acoustic scattering from many interacting, bounded, three-dimensional obstacles has been treated by several authors [see, for example, V. Twersky, J. Math. Phys. 8, 589 (1967) or B. Peterson and S. Ström, J. Acoust. Soc. Am. 56, 771 (1974)]. In particular, Peterson and Ström extended the single-obstacle, transition (T) matrix formalism to several obstacles (including all orders of multiple scattering) by using the translation properties of the spherical basis functions to translate the multiply scattered fields of each obstacle to a common origin. Later, their formalism was extended to treat elastic wave scattering by using spherical vector basis functions [A. Boström, J. Acoust. Soc. Am. 67, 399 (1980)]. However, for numerical results, the present state of development is cumbersome to apply to more than two obstacles and convergence of the rescattering matrices is sensitive to obstacle separation. In this paper, a multicentered, T-matrix formalism for acoustic and elastic wave scattering is given, based on field expansions centered on each obstacle. Like previous approaches, it also incorporates the single-obstacle T matrix and sums the multiple scattering series exactly. The result is simpler and less numerically sensitive to the separation of the obstacles. The extension of the formalism to treat the scattering from infinite systems that periodically repeat an arbitrary group of obstacles is also given. Numerical calculations based on the exact formalism for scattering from a linear array of as many as ten spherical obstacles are presented. The validity of approximating the scattered field from larger finite arrays by the field scattered from finite sections of an infinite array is considered.