Lamb and creeping waves around submerged spherical shells resonantly excited by sound scattering

Abstract

Acoustic wave scattering by spherical shells in water in the resonance region is studied. The interaction is studied classically and by the resonance scattering theory (RST). The connection between internal resonances and the Lamb waves excited in the shell is analyzed. Conditions are derived that govern the propagation of Lamb waves in a spherical shell when it is fluid-loaded or in vacuo. These general conditions reduce to the usual conditions for Lamb waves in plates, in the large radii limit. The connection is established between the outer scattering problem and the internal vibrational problem that excites the shell resonances, and it is demonstrated that the creeping-wave series that synthesizes the Franz waves around the shell, can be obtained quite simply without the use of the Watson–Sommerfeld Method (WSM). All that is required is suitable one-term expansions of the denominators of the scattering amplitudes. This eliminates the need for the cumbersome excursions through the complex angular momentum plane ν of the WSM. Calculations are presented for the summed form functions of seven shells of three compositions and thicknesses over quite broad frequency bands. Subtraction of rigid or soft backgrounds permits one to draw many conclusions as to the behavior of the shell response in various spectral regions. The relative phase between the form function and the background is also displayed. This phase is helpful for the identification of actual resonances and for the choice of proper background. It is shown how the poles of the scattering amplitude in the x plane split into two subsets. One set, the Franz set, depends only on shell shape, and is related to the external Franz (creeping) waves in the water. This set does not change with varying shell thickness or composition. The other set, the Lamb set, depends only on material composition, and is related to the Lamb waves in the shell. These poles are displayed in various cases, and it is shown that for thin shells, only one family of Lamb poles (the zeroth-order symmetric family, s0) is dominant, which explains the relatively simple structure of the form function of a thin shell. For thicker shells, many modes interact, and one is forced to go to the usual partial-wave analysis of the RST. A very accurate picture of the scattering process taking place around and inside the shell emerges from this analysis.