Acoustic scattering of a plane wave by two spherical elastic shells above the coincidence frequency

The acoustic scattering by two fluid-filled spherical elastic shells in close proximity to each other and insonified by plane waves at arbitrary angles of incidence is analyzed exactly in the frequency range that includes the midfrequency or coincidence enhancement region of the backscattered echoes. The incident and scattered wave fields are expanded in terms of the classical modal series and the addition theorem for the spherical wave functions is used to determine the exact expression for 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 found by simultaneously solving the Helmholtz equations governing the wave motion in the fluid medium in which the two shells are submerged as well as in the fluid media contained in the shells, together with the two sets of equations of motion of the two elastic shells obtained from the complete three-dimensional elasticity theory after satisfying the boundary conditions at all fluid-shell interfaces as well as the far-field radiation condition. Again, the numerical computation of the scattered pressure wave involves the solution of a truncated 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. Backscattered and bistatic echoes from two identical spherical elastic shells are extensively calculated. The result also exhibits the large enhancement present in the backscattered echoes for the endfire situation after the midfrequency or coincidence enhancement has taken place. This can be attributed to the effects of focusing by the front elastic shell and to the reflection and refocusing by the back elastic shell of the a(0) Lamb wave reradiation in the observer's direction.