Analytical solution for a vibrating rigid sphere with an elastic shell in an infinite linear elastic medium

TL;DR

This paper provides an analytical solution for a vibrating rigid sphere with an elastic shell in an infinite elastic medium, revealing complex wave phenomena and potential wave filtering applications based on frequency and material properties.

Contribution

It introduces a novel analytical model for a vibrating core-shell sphere in an elastic medium, enabling analysis of wave separation and filtering effects.

Findings

Frequency-dependent wave phenomena observed

Ability to filter out longitudinal or transverse waves

Application of FFT shows wave separation in pulsed vibrations

Abstract

The analytical solution is given for a vibrating rigid core sphere, oscillating up and down without volume change, situated at the center of an elastic material spherical shell, which in turn is situated inside an infinite (possible different) elastic medium. The solution is based on symmetry considerations and the continuity of the displacement both at the core and the shell - outer medium boundaries as well as the continuity of the stress at the outer edge of the shell. Furthermore, a separation into longitudinal and transverse waves is used. Analysis of the solution shows that a surprisingly complex range of physical phenomena can be observed when the frequency is changed while keeping the material parameters the same, especially when compared to the case of a core without any shell. With a careful choice of materials, shell thickness and vibration frequency, it is possible to filter out most of the longitudinal waves and generate pure tangential waves in the infinite domain (and vice-versa, we can filter out the tangential waves and generate longitudinal waves). When the solution is applied to different frequencies and with the help of a fast Fourier transform (FFT), a pulsed vibration is shown to exhibit the separation of the longitudinal (L) and transverse (T) waves (often called P- and S-waves in earthquake terminology).