Resonance scattering of elastic waves by an elastic sphere in an elastic medium

Abstract

We study elastic wave scattering by an elastic sphere within an elastic matrix under the light of the Resonance Scattering Theory (RST). We consider shear (s) and compressional (p) plane wave incidence on a sphere of radius a, and the four cases of elastic wave scattering that they engender (i.e., pp, ps, sp, and ss). In all cases we isolate the resulting modal resonances by suitable background subtraction and generate the resonance spectrogram of the target in a wide frequency band that includes the resonance region where λ < a. We exhibit the SEM poles that appear in this general case in the complex-frequency plane. We study their connection with the surface waves that circumnavigate the spherical inclusion, both inside and outside it. This work contains four simple but important cases that we have extensively studied in the past [viz., (i) (visco) elastic sphere in a fluid, (ii) fluid-filled cavity, (iii) fluid sphere in a dissimilar fluid, and (iv) impenetrable (rigid/soft) inclusions in a (visco) elastic matrix] and they all emerge from it as particular cases. We show the specific form of the RST in the general case and the pertinent SEM poles with their interpretation and relationship to surface waves. We determine the phase velocities and attenuations from the pole positions, and also show the response surface, cross sections, modal resonances, etc. in wide frequency bands. This work considerably extends the 1960 analysis of Truell, Einspruch, and their associates on this classical problem.