Mean Radiation Force of Shear Plane Waves on a Sphere in an Elastic Medium

Abstract

The mean (time-averaged) longitudinal force component (i.e. acting along the direction of wave propagation) arising from the interaction of linearly-polarized plane progressive shear elastic waves, incident upon a sphere embedded in an elastic medium, is considered. Exact partial-wave series expansions are derived based on the integration of the radial component of the time-averaged elastodynamic Poynting vector in spherical coordinates. The method is verified stemming from the law of energy conservation applied to elastic scattering. The analytical modeling is useful and provides improved physical understanding of shear-to-compressional (S $\to $ P) mode conversion, as well as shear-to-shear (S $\to $ S) and transverse-to-transverse (T $\to $ T) mode preservation in the context of the mean elastic force. The elastic wave scattering formulation based on Debye’s shear and transverse potentials is solved first, and used subsequently to derive the mathematical expression of the mean force efficiency. Numerical computations illustrate the analysis with particular emphasis on the components related to mode preservation, coupling and conversion separately. It is shown here that the total force originates from individual interactions of scattering terms between the scattered pure shear (S $\to $ S) and transverse (T $\to $ T) waves, in addition to shear-to-transverse (S $\rightleftarrows $ T) coupling, and a shear-to-compression (S $\to $ P) mode conversion that contributes negligibly to the total mean force. The benchmark solution presented in this analysis for the time-averaged elastic force of shear plane progressive waves can be utilized to validate numerical methods (such as the FEM, BEM, FDTD or other). The results can provide a priori information for the optimization and design of experimental setups in various applications in biomedical ultrasound, elastography and elasticity imaging, shear-wave activation of implantable devices, characterization of biological tissue, seismology and other related applications in elastic wave scattering and radiation force.