Sound scattering by resonantly excited, fluid-loaded, elastic spherical shells

Abstract

The scattering of sound waves by an air-filled, elastic, spherical shell in deep waters, in the frequency domain is analyzed. This exact analysis is based on the classical formulation of three-dimensional elastodynamics, for fluid-loaded spherical shells of arbitrary thickness. Form functions, residual responses, and the partial-wave expansions of both, are determined. Results are displayed in relatively wide frequency bands for various increasing shell thicknesses. The resonance features in the sonar cross sections (SCS) are isolated by means of a new hybrid modal background that substantially improves the results found with the earlier (rigid/soft) backgrounds of the resonance scattering theory (RST). The resonance features in the SCSs that correspond to each mode, and also to each of the various shell waves that propagate around its periphery are isolated. There seem to be over half-a-dozen shell (generalized Lamb and Stoneley) waves manifesting their influence in the SCSs within the examined band. Three large-amplitude features are most noticeable. There is a (slow) wave due to the double curvature of the shell which is responsible for a large, spiky feature at low frequencies. There is the spherical A0 wave, caused by the coincidence effect, responsible for a broad ‘‘bump’’ with superimposed spikes, which appears in the midfrequency region, near x∼(h′)−1, where h′ is the relative shell thickness. There is also a thickness resonance feature at high frequencies, which is due to a wave that propagates through the shell thickness at the incidence point. There is a spherical Stoneley-type wave that resides mostly in the water in the frequency region below coincidence, xc, at which the A0 wave is activated. Finally, there are two resonance families caused by spherical analogs of the symmetric, Sn, and the antisymmetric, An, Lamb waves originally studied in flat plates. These spherical generalizations are denoted by the same symbols Sn, An, with the understanding that they now refer to shells. This model’s results for phase velocities, cutoff frequencies, etc..., are compared to those produced by simpler approaches in order to size the approximations introduced by those simpler models. This serves to establish the benchmark nature of the present model and of the calculations it produces, which are displayed in many instances. Dispersion plots are generated for the phase velocities cp(x) of all the above-mentioned shell waves. These cp(x) are proportional to the (real parts of the) roots of determinantal conditions that are also analyzed. Critical angles, coincidence phenomena, and reflection and transmission properties of the shell are examined and all the required physical interpretations are given.