Resonances of elastic scatterers in fluid half-spaces

A methodology is presented for the spectral analysis of the echo returns from elastic spheres submerged in a fluid half-space. The influence of a target depth on its resonances as manifested in the scattered field is assessed. Results are expressed in terms of Debye potentials in the fluid and in the solid. The unknown coefficients arising in these potentials are determined by the application of appropriate boundary conditions at the surface of the elastic sphere and at the free boundary of the fluid. These boundary conditions are satisfied exactly through the use of some known transformations of the basic wave functions. As the depth increases, our results degenerate to earlier results for spheres in unbounded media. In spite of the spherical symmetry of the scatterer, the field is found to be azimuthally dependent because of the influence of the plane free-surface of the fluid. The saddle-point method is used in the asymptotic evaluation of certain coefficients which exhibit the influence of depth on the complex eigenfrequencies of the scatterer. We demonstrate that as the sphere's depth increases, the influence of the boundary decreases. The rate of decrease increases as the azimuthal wavenumber
m
increases. Thus the influence of the boundary is greatest for either
m = 0
, or at shallow depths, or both.