The acoustic scattering by a submerged, spherical shell. I: The bifurcation of the dispersion curve for the spherical antisymmetric Lamb wave

Abstract

The acoustic scattering by thin-walled, evacuated, elastic spherical shells immersed in water is studied. The analytic structure of the scattering amplitude in the complex-k plane is directly analyzed using Cauchy's residue theorem, and dispersion curves are presented for the lowest elastic modes of the fluid-loaded shell. It is found that fluid loading has a profound effect on the vacuum dynamical characteristics of the shell; the spherical equivalent of the first antisymmetric, fiat-plate Lamb wave for the fluid-loaded shell bifurcates into two distinct modes near the frequency that the vacuum dispersion curve transitions from a subsonic to a supersonic phase velocity. By way of contrast, the spherical equivalent of the first symmetric Lamb wave is essentially unaffected. The salient features of the free-field scattering process are also analyzed in terms of the resonance excitation of these modes. , PACS numbers: 43.30. Gv, 43.20.Fn, 43.40.Ey INTRODUCTIONThe study of the acoustic scattering from submerged, elastic targets has become increasingly fashionable in the last decade. Much of the more recent work has been concerned with resonance scattering theory and its implications for the inverse scattering problem. However, advances in this area will also have an impact on the more general question of fluid-elastic structure interactions, which have much more diverse applications. In this context, we note that the fluidloaded, elastic spherical shell is the simplest, nontrivial, three-dimensional example of such a dynamical system amenable to a numerical/analytical treatment over an extended frequency range. A solid physical understanding of the scattering from this target will surely yield dividends when the subtleties associated with more complicated structures are encountered. This acoustic scattering problem, which is the subject of the present article, thus takes on special significance.The earliest investigation of the acoustic scattering from an elastic spherical shell was apparently by Junger. • This article was quite modern in perspective in that it touched on a number of issues of current interest. For example, it was in this article that the concept of the separation of the scattering solution for shells into two terms, a "rigid body scattering" (background) contribution and a "radiation scattering" (resonant) contribution, was first introduced. While Junger's analysis also underscored the profound effects of fluid loading on the dynamical characteristics of the shell in vacuum, 2 this analysis must be regarded as incomplete; the thin shell theory employed by this author was later found to be inadequate for the bending modes of a spherical shell. 3 A scattering formalism utilizing the full linearized, elastic equations of motion for the spherical shell was later given by Goodman and Stern 4 and subsequently employed by Hickling • in a theoretical study of steady state and transient solutions. Hickling's analysis of the scattering of short transient pulses led him to propose that the component of the sc...

Fluid-structure effects of cloaking a submerged spherical shell

“…They include the works of Marston and co-workers, 10-14 € Uberall and co-workers, [15][16][17][18][19][20][21] and the excellent three papers by Sammelmann and Hackman. [22][23][24] The present work specifically addresses scattering from heavy fluid-loaded, cloaked spherical shells. The backscattering "form function" is employed to quantify the effectiveness of cloaks over a large frequency spectrum.…”

“…The backscattering "form function" is employed to quantify the effectiveness of cloaks over a large frequency spectrum. To provide as full as possible accounting, the authors adopt a treatment similar to that used in the earlier paper by Sammelmann and Hackman 22 by finding residues of the scattering amplitudes of various cloaks on a specific spherical shell of prescribed thickness. This allows the present authors to develop dispersion curves and corresponding phase/group velocity curves for the lowest symmetric and anti-symmetric modes excited by the shell and coupled fluid.…”

“…Because the full equations of elasticity are employed, use of the term thin-shell as applied by Junger and Feit, 25 to qualify for adopting approximate shell theories in the fluid-structure interaction is not appropriate, and would not be valid in the mid-and high-frequency regimes. The thin-shell designation is taken from Sammelmann and Hackman, 22 who label "very thin," "thin," and "thick" shells using thickness to radius ratios of h/a 0.005, 0.01 h/a 0.1, and h/a ! 0.1; for very thin, thin, and thick shells respectively.…”

A modified block FTF adaptive algorithm with applications to underwater target detection

“…the physical nature of the object, are discussed in [11]- [13]. The latter property would enable the adaptive filter to generate a "good" estimate of the specular component prior to the appearance of the resonant, and the former property guarantees that the system continues to provide a reasonable estimate of the specular part, even after the resonance has appeared.…”

“…Xi(X;*X i )-l be the LS solution for the ith block; then, (12) can be rearranged so that (13) or equivalently…”