Longitudinal axial resonances in the form function for backscattering from a fluid spherical shell

Abstract

Reverberations of longitudinal waves between the inner and outer surfaces of a hollow shell can strongly affect the backscattering amplitudes at frequencies associated with a thickness resonance. The phenomena are studied for the idealized case of vanishing shear stresses by letting the material of the shell be an inviscid fluid; the surrounding fluid is water. An exact partial-wave series (PWS) gives the form function f (θ,ka) and plots of |f(θ = π,ka)| display a structure from resonances. Here, ka = 2πa/λ is the size parameter, where a is the outer radius of the shell and λ is the wavelength in water of the incident acoustic wave. The resonant structure is also recovered in a geometrical calculation of f (π,ka), which sums the amplitudes associated with rays multiply reflected within the curved shell. The geometric synthesis demonstrates that the effects of curvature are essential to modeling f (π,ka). The analysis gives the geometric divergence factors of successive internal reflections. In addition to numerical comparisons with the PWS, the geometrical synthesis is tested by considering several limiting cases. These limiting cases correctly give results anticipated from elementary considerations. [Worked supported by ONR.]