Coupling coefficient for leaky waves on thick spherical shells from elasticity theory

Abstract

An approximation of leaky wave scattering amplitudes for thick spherical shells in water was previously demonstrated for backscattering and (through the optical theorem) the total cross section [S. G. Kargl and P. L. Marston, J. Acoust. Soc. Am. 88, 1103–1113 (1990); 89, 2545–2558 (1991)]. In the convention of that formulation, it was sufficient to approximate the complex leaky wave coupling coefficient Gl for the thick shell by 8πβlc/cl over the frequency range considered. In the present work, calculations are given pertaining to that approximation that are based on full elasticity theory and a generalization of Watson transformation results for solid spheres. The computations [S. G. Kargl, Ph.D. dissertation, Wash. State Univ. (1990)] show that while ‖Gl‖ is well approximated as noted, φl=arg(Gl) tends to negative values for low ka [roughly in proportion to (ka)−1] for both the s0 wave and the supersonic region of the a0 wave. At large ka, φl becomes small in agreement with the approximation. Over a range of ka above the coincidence frequency the φl(ka) curves for the a0 and s0 waves are similar and ‖φl‖<1 rad. [Work supported by ONR.]