Plane incident p-waves which propagate through continuous (possibly absorptive) media are scattered by a fluid-filled spherical cavity contained in it. Using an approach familiar in nuclear scattering theory but novel to acoustics and elastodynamics, it is possible to express the scattering amplitudes (or the partial waves contained in them) for the scattered p-and s-waves in a form which clearly exhibits their dependence on two interacting contributions, one being the broad background of an ideally soft cavity, and the other the superimposed narrow spikes due to resonances excited in the cavity fluid. The cavity appears as a perfectly soft obstacle to the incident waves at all frequencies except in the near vicinity of the cavity eigenfrequencies where there is wave penetration into the filler fluid. When this happens, the interference of this wave with the ’’potential scattering’’ of the background is seen to cause the fluctuating character of the amplitudes (or cross section). We have numerically computed these isolated contributions for a variety of material combinations. The isolated resonances are traced (as Regge pole trajectories) as they reappear at the higher frequencies in subsequent partial waves. Due to its application in the analysis of acoustic-coating performance we have further studied the case of air-filled cavities in lossy rubber. The following findings have emerged: (1) The resonances are comparatively narrow, (2) their locations are apparently independent of the amount of absorption present, (3) absorption only affects the background, (4) shear absorption F1 only affects the mode-converted amplitude fps, and (5) the dilatational absorption, controlled by the parameter F, only influences the nonmode converted amplitude fpp.
We readdress the basic problem of the scattering of acoustic waves by an elastic sphere, now under the dissecting knife of the Resonance Scattering Theory (RST), with the purpose of illustrating the power of the method, and the versatile options and information it offers. These options include ease of understanding, novel physical interpretation of the phenomenon, and striking calculational simplicity. The principal findings presented here include: (a) the actual modal resonances, quantitatively separated from the corresponding modal backgrounds in the frequency (the "acoustic spectrogram"--already a target-identification tool) domain, and in the combined frequency and mode-order domain (the "response surface"). (b) The bistatic plots of the scattering cross section, to illustrate the point that if the cross s_ection is redetermined at certain selected observation angles, the resonance contributions from each individual mode can actually be isolated from those of all other modes. (c) A study of the nulls or dips present in the partial waves (i.e., modes), and in the summed cross section. We show the cause and physical meanings of these dips analytically and computationally, in both instances. (d) A derivation of the analytic conditions predicting the nulls and also the influence of the elastic resonance (SEM) poles in the scattered echoes. These conditions, which emerge from our scattering approach, are shown to be in agreement with early results of Love [•4 Treatise on the Mathematical Theory of Elasticty (Dover, New York, 1944)], obtained on a purely vibrational basis. A tungsten carbide sphere is used in all the examples, since this is a favorite target for experimental calibrations. Our future work will underline the intimate connection between his direct approach, and that (leading to the solution) of the inverse scattering problem for sonar target identification.
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.
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