Sound scattering by a spherical object near a hard flat bottom

We consider the scattering of plane acoustic waves by spherical objects near a plane hard surface. The angles of incidence are arbitrary and so are the distances of the objects from the hard boundary. We use the method of images. The final result for the sound field consists of four parts: the incident field and its reflection from the boundary, which are shown combined; the scattered field from the sphere, and that scattered by its image. These last two appear coupled since both sphere and image are repeatedly interacting with each other. The entire solution is referred to the center of the real sphere. This can be accomplished in an exact fashion by means of the addition theorems for spherical wave-functions. These theorems are taken from the atomic physics literature, where they are more frequently used. The required coupling coefficients, b/sub mn/, are obtained from the solution of an infinite linear complex system of transcendental equations with coefficients given by series. The system is suitably truncated to obtain numerical predictions for the form-functions by means of the Gauss-Seidel iteration method. Many calculations are displayed exhibiting the distortion that the proximity of the hard boundary causes on the free-space solution. The form-functions are graphed versus ka, for various values of the normalized separation D/spl equiv/d/a of the sphere from its image. They are also plotted versus the angle of observation, for fixed values of /spl Omega/=La and D. These plots are the exact benchmark curves against which the accuracy of approximate solutions, found by other methods, could be assessed. They could also serve to determine the distances above the bottom, beyond which the bottom effect could be neglected. This is an idealized model to predict the distorted sonar cross section of a hard spherical object near a hard flat bottom.