  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).
