Scattering of ultrasonic spherical waves by a pair of elastic shells immersed in a fluid

We have been investigating sonic and ultrasonic scattering of waves by either pairs of penetrable objects in boundless fluid media, or by single penetrable objects near boundaries of fluid half-spaces. These are exact benchmark solutions for configurations that will ultimately lead to more complex scattering situations. One case that we have not yet treated, and that we exactly analyze here, is that of a spherical ultrasonic wave scattered by two elastic, fluid-filled, spherical shells separated an arbitrary distance d, and immersed in a boundless fluid medium. Such a wave emerges from a point source that is arbitrarily close to the two shells, and the resulting scattered field is then determined anywhere in space. The exact solution presented here accounts for all orders of multiple scattering and the deformations of the two shells are described by the general 3-dimensional equations of elastodynamics. The pair of shells may or may not be of equal sizes or wall-thicknesses, they may or may not contain interior fluids, and they can be made of any substance. As in the related cases cited above, the determination of the resulting scattered pressure field requires the use of: (a) the addition theorem for spherical wavefunctions to reduce all fields to a common origin, and, (b) the summation of an infinite number of terms, each containing products of pairs of Wigner 3-j symbols. Many calculated results obtained in relatively broad frequency bands, illustrate the details of this analytical solution. As the point-source recedes to a very large distance away from the shells-relative to their radii and separation-we recover our earlier solution for plane-wave incidences.