The governing system of differential equations for the linear problem of the transient interaction of plane acoustic waves and a submerged elastic cylindrical shell is transformed into a system of Volterra integral equations of the second kind. The integral equations are solved by a step-by-step integration scheme and numerical results to the problem are obtained exactly within the limit of series solution imposed by the Gibb’s phenomenon and within the limit of numerical truncation and roundoff errors. Detailed features of the transient response of the shell were revealed.
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.
Ahstract-We consider the scattering of plane acoustic waves by spherical objects near a plane hard surface. The angl£'$ 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 [ef. (21)] consists of four parts: the incident field and its refiection 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, bmn, 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 == d I a of the sphere from its image. They are also plotted versus the angle of observation, for fixed values of S! = 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 the distances above the bottom, beyond which the bottom effeet could be neglected. This is an idealized model to predict the distorted sonar cross section of a hard spherical object near a hard fiat bottom. L I NTRODUCTION To STUDY acoustic scattering bv an object near an interface or by many objects placed in various spatial configurations, the starting point is the addition theorems. Such theorems allow us to considcr thc scattered fields produced by the various bodics, or by the body and its image (or images), all referred to a common origin. A number of works lIJ-[7] have dealt with this subject. The use of these theorems in physical cases leads to Wigner (3 -j) symbols. In their quantum mechanical or atorn.ic physics context, these computational tools are discussed in many works [8]- [14]. Their use in classical physics is straightforward. Scattcring by two or more hodies which are relatively close to each other, and thcir interaction is significant, is essentially a multiple scattering problem. The literat ure on this related topic is also nontrivial Manuscript
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