Coupling coefficient G for the Fabry‐Perot representation of backscattering from spheres and cylinders: Quantitative GTD for elastic objects in water

Abstract

For each class of elastic surface wave, the contribution fsw to the total form function may be written in a Fabry‐Perot representation [K. L. Williams and P. L. Marston, J. Acoust. Soc. Am. 79, 1702–1708 (1986)]. For backscattering |fsw| = |G exp[− 2β(π − θsw)]/[j + exp(− 2πβ + i2πxc/csw)]|, where x = ka and for spheres j = + 1. The surface wave's phase velocity and angular attenuation coefficients are csw and β while sin θsw = c/csw. The exact form function was synthesized by summing various fsw with a specular term. An expression for G, which had to be evaluated by numerical differentiation, was derived for solid spheres via a Sommerfeld‐Watson transformation. In the present research, a simple approximation for |G| is derived by matching fsw with the RST result near a resonance when β is small. This gives |G| ≈ 8πβc/csw for spheres while for circular cylinders j = − 1 and |G| ≈ 8πβ/(πx)1/2. Comparison with the exact |G(x)| for Rayleigh waves on solid tungsten carbide and aluminum spheres gives agreement even when 2πβ is not ≪ 1. These approximations of |G| should also apply to the scattering of short tone bursts and to the fsw for hollow elastic shells. [Work supported by ONR.]