Radiation force of compressional plane waves on a sphere embedded in an elastic medium†

The aim of this work is to derive exact partial-wave series expressions for the radiation force of plane compressional progressive waves propagating inside an elastic medium and incident upon an embedded elastic sphere. The analytical modeling is needed to provide fundamental physical understanding of the underlying phenomenon of mode conversion and its contribution to the acousto-elastic radiation force, and in experimental design. In the context of linear elasticity theory, a rigorous derivation for the acousto-elastic radiation force, based on the integration of the time-averaged radial component of the elastodynamic Poynting vector (or power flow density), is presented and discussed. Initially, the elastic scattering problem is determined and subsequently used to derive the mathematical expression for the acousto-elastic radiation force of progressive compressional waves. The method is also verified using the extended optical theorem for elastic compressional plane waves. Extension to the case of elastic plane standing wave is also provided. Particular importance is made on the contributions of elastic mode preservation (P → P) and mode conversion (P → S) to the acousto-elastic radiation force. Numerical computations for the dimensionless radiation force efficiency and its components demonstrate the importance of compressional-to-shear mode conversion in the scattering by the sphere encased in a linearly-elastic medium. The analytical formalism presented here can be used to validate numerical methods, and the results of the simulations can be utilized as a priori knowledge in optimizing and designing acousto-elastic radiation force experiments involving elastic compressional progressive waves on a sphere in acoustically-engineered materials applications, elasticity imaging methods, activation of implantable devices, characterization of biological tissue, and non-destructive evaluation to name some examples.