If acoustic scattering by a single sphere is the most basic problem of scalar scattering, then sound scattering by a pair of spheres is next in the hierarchy of complexity. The problem has been formulated by several approaches in the past, but no actual detailed studies have been openly published so far. Two spheres insonified by plane waves at arbitrary angles of incidence are considered. The solution of this simplest of multiple-scattering problems is generated by exactly accounting for the interaction between the two spheres, which can be strong or weak depending on their separation, compositions, frequency, and directions of observation. The tools to attack this type of problem are the (forward/backward) addition theorems for the spherical wave functions, which permit the field expansions—all referred to the center of one of the spheres—by means of Wigner (3-j) symbols. The fields scattered by each sphere are obtained as pairs of (double) sums in the spherical wave functions, with coefficients that are coupled through an infinite set of two linear, complex, algebraic equations. These are then solved (by truncation) and used to obtain (i) the scattered fields and (ii) the scattering cross section of the pair of spheres. These exact results are illustrated with many plots of the form functions at various relevant incidence angles, separations, frequencies, etc. Finally, some asymptotic approximations for this problem that are analytically simple are obtained. They are displayed and compared to the exact solutions found above, with quite satisfactory results, even for the simple approximations used here. Thus the phenomenon is described, explained, graphically displayed, physically interpreted, and reduced to a simple accurate approximation in some important cases.
A large set of dolphin-emitted acoustic pulses ("echolocation clicks") have been examined, which were reflected from various elastic shells that were suspended, underwater, 4.5 m in front of the animal in a large test site in Kaneohe Bay, Hawaii. A carefully instrumented analog-to-digital system continuously captured the emitted clicks and also the returned, backscattered echoes (A/D conversion at 500 kHz). Using standard conditioning techniques and food reinforces, the dolphin is taught to push an underwater paddle when the "correct" target-the one he has been trained to identify-is presented to him. He communicates his consistently correct identifying choices in this manner. Many echoes returned by three types of cylindrical shells in both the time and frequency domains as well as in the joint time-frequency (t-f) domain, by means of Wigner-type distributions have been examined. It will be shown exactly how specific features observable in these displays are directly related to the physical characteristics of the shells. This processing takes advantage of certain fundamental resonance principles to show which echo features contain information about the size, shape, wall thickness, and material composition of both the shell and its filler substance. In the same fashion that these resonance features give the identifying characteristics of each shell, it is believed they may also give them to the dolphin. These echo features may allow him to extract the target properties by inspection without any need for computations. It is claimed that this may be the fundamental physical explanation of the dolphin's amazing target ID feats, upon which they base their recognition choices. This claim may be substantiated by the detailed analysis of many typical echoes returned by various shells, when they are interrogated by several dolphins. Thus far, this analysis of many echoes from many shells has only been carried out for a single dolphin.
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