Acoustic scattering by a spherical body near a plane boundary

Abstract

Acoustic scattering is studied by considering spherical bodies near plane boundaries when they are insonified by plane waves at arbitrary incidence angles. Using the method of images, there is an image sphere on the side of the boundary opposite to the side containing the real sphere, and both the sphere and its image contribute to the resulting sound field. The plane boundary is introduced and the scattering problem is solved. The resulting sound field consists of four parts: the incident field, its reflection from the boundary, the scattered field from the sphere, and that scattered by its image. The whole solution is referred to the center of the real sphere, and any depth below the boundary is possible. The treatment is analytic and exact, and it uses the addition theorems for spherical wave functions and the whole (atomic-physics-like) machinery of the coupling of various angular momentum vectors. The required coupling coefficients bmn emerge from the solution of an infinite linear system of algebraic equations, which is appropriately truncated to obtain numerical predictions for the ‘‘form functions.’’ Form functions that account for either one or two of the four components of the total acoustic field, as well as for the overall effect of all four contributions together, are considered. The two individual components are shown so that the distortions they induce in the field can be separately assessed. All the form functions are displayed versus ka, for various values of the normalized sphere depth d/a. These plots also serve to predict quantitatively the critical depths beyond which the effect of the boundary becomes negligible. They also provide exact benchmark curves against which the accuracy of some approximate techniques, based on the numerical evaluation of certain integral equations can be assessed.