We consider the spectral stability problem
for Floquet-type systems such as the wave equation
with periodic forcing ψ.
Our approach is based on a comparison with finite-dimensional approximations.
Specific results are obtained for a system where the forcing
is due to a coupling between the wave equation
and a time-period solution of a nonlinear beam equation.
We prove (spectral) stability for some period and instability for another.
The finite-dimensional approximations are
controlled via computer-assisted estimates.