Ray synthesis of the form function for backscattering from an elastic spherical shell: Leaky Lamb waves and longitudinal resonances

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An acoustic ray analysis is employed in synthesizing the form function for backscattering, f(θ=π,ka), from a fluid-loaded evacuated elastic spherical shell where k is the wave number of the incident plane wave and a is the outer radius of the shell. The synthesis contains a component associated with a specular reflection, fsp, and contributions from leaky Lamb waves. The contribution fl of the lth leaky Lamb wave is expressible in a Fabry–Pérot resonator form [P. L. Marston, J. Acoust. Soc. Am. 83, 25–37 (1988)]. A comparison of the ray synthesis for f(ka) with the exact partial-wave series representation for a 440c stainless-steel shell verifies the usefulness of the ray synthesis for the present case of a shell. The present synthesis is also new in that it includes the effects of longitudinal resonances on fsp. A novel ray synthesis of fsp indicates a significant resonance effect near the condition kLh=nπ (n=1,2,…). The thickness of the shell is h, and kL=ω/cL is the longitudinal wave number where cL is the longitudinal speed of sound in the elastic material. The ray synthesis demonstrates that the curvature of the shell is essential to the modeling of longitudinal resonances. Although acoustic ray modeling is generally a high-frequency technique, the ray synthesis of f(ka) for a 440c stainless-steel shell appears to be applicable for ka as small as 7. Certain anomalies in the synthesis are investigated to better understand the limitations of the present ray model.