Scattering and diffraction of elastic P- and S-waves by a spherical obstacle: A review of the classical solution

Abstract

We re-derive the solution for scattering and diffraction of elastic waves by a single spherical obstacle. A complete catalog for the coefficients in the series’ expansions of scattered waves is presented. The classical solution is a superposition of incident and diffracted fields. Plane P- or S-waves are assumed. They are expressed as expansions of spherical wave functions which are tested against exact results. The diffracted field is calculated from the analytical enforcing of boundary conditions at the scatterer-matrix interface. The spherical obstacle is a cavity, an elastic inclusion or a fluid-filled zone. A complete set of wave functions is given in terms of spherical Bessel and Hankel radial functions. Legendre and trigonometric functions are used for the angular coordinates. Results are shown in time and frequency domains. Diffracted displacement amplitudes versus normalized frequency and radiation patterns at low, intermediate and high frequencies are reported. Synthetic seismograms for some relevant cases are computed.

Keywords

• Scattering
• diffraction
• spherical obstacle
• elastic waves
• Bessel and Hankel functions
• seismic response