Impedance of a sphere oscillating in an elastic medium with and without slip

TL;DR

This paper analyzes the dynamic impedance of a sphere oscillating in an elastic medium, providing simplified formulas and modifications for slip conditions at the interface.

Contribution

It derives a simplified expression for the impedance of a sphere in an elastic medium and extends it to include slip conditions at the interface.

Findings

Impedance can be expressed using three lumped impedances.

Slip at the interface modifies the transverse impedance.

The modified formula accounts for interfacial impedance.

Abstract

The dynamic impedance of a sphere oscillating in an elastic medium is considered. Oestreicher's formula for the impedance of a sphere bonded to the surrounding medium can be expressed simply in terms of three lumped impedances associated with the displaced mass and the longitudinal and transverse waves. If the surface of the sphere slips while the normal velocity remains continuous, the impedance formula is modified by adjusting the definition of the transverse impedance to include the interfacial impedance.