  Linearised relativistic elasticity equations of motion are considered for a
rod and a spinning ring encountering a gravitational wave. In the case of the
rod, the equations reduce to a wave equation with appropriate boundary
conditions. Using Fourier transforms, the resonant frequencies are found and an
explicit distributional solution is given, both for a plus- and a
cross-polarised gravitational wave. In the case of the spinning ring, the
equations are coupled wave equations with periodic boundary conditions. Using a
Fourier series expansion, the system of wave equations is recast as a family of
ordinary differential equations for the Fourier coefficients, which are then
solved via Fourier transforms. The resonant frequencies are found, including
simple approximate expressions for slowly rotating rings, and an explicit
distributional solution is obtained in the case of the non-spinning ring.
Interestingly, it is possible to tune the resonant frequencies by adjusting the
angular velocity of the spinning ring.
