SOUND-SCATTERING BY A SPHERICAL OBJECT NEAR A HARD FLAT BOTTOM

We consider the scattering of plane acoustic waves by spherical object s 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 th e method of images. The final result for the sound field [cf. (21)] co nsists of four parts: the incident field and its reflection from the b oundary, 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 ca n be accomplished in an exact fashion by means of the addition theorem s for spherical wave-functions. These theorems are taken from the atom ic physics literature, where they are more frequently used. The requir ed coupling coefficients, b(mn), are obtained from the solution of an infinite linear complex system of transcendental equations with coeffi cients given by series. The system is suitably truncated to obtain num erical predictions for the form-functions by means of the Gauss-Seidel iteration method. Many calculations are displayed exhibiting the dist ortion that the proximity of the hard boundary causes on the free-spac e solution. The form-functions are graphed versus ka, for various valu es of the normalized separation D equivalent to d/a of the sphere from its image. They are also plotted versus the angle of observation, for fixed values of Omega = ka 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 t he 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 hat bottom.