Acoustic scattering of a plane wave by two spherical elastic shells

Abstract

The acoustic scattering by two spherical elastic shells in close proximity insonified by plane waves at arbitrary angles of incidence is analyzed exactly in the low- and intermediate-frequency ranges. The incident and scattering wave fields are expanded in terms of the classical modal series and the addition theorem for the spherical wave functions facilitates the exact expression of the sound fields scattered by each spherical elastic shell in the presence of the other, referred to coordinate systems at the centers of either spherical shell. The solution to the scattering problem is obtained by simultaneously solving the Helmholtz equation governing the wave motion in the fluid medium in which the two shells are submerged and the two sets of equations of motion of the two elastic shells satisfying the boundary conditions at all fluid–shell interfaces and the far-field radiation condition. Numerical computation of the scattered wave pressure involves the solution of the truncation of an ill-conditioned complex matrix system the size of which depends on how many terms of the modal series are required for convergence. This in turn depends on the value of the frequency, and on the proximity of the two spherical elastic shells. The ill-conditioned matrix equation is solved using the Gauss–Seidel iteration method and Twersky’s method of successive iteration double checking each other. Backscattered echoes from two identical spherical elastic shells are extensively calculated. The result also demonstrates that the large amplitude low-frequency resonances of the echoes of the neighboring elastic shells shift downward with proximity to each other. This can be attributed to the increase of added mass for the vibration of the shells.