We study eigenfrequencies and propagator expansions for damped wave
equations on
compact manifolds. Under the assumption of geometric control, the
propagator is shown to admit an
expansion in terms of finitely many eigenmodes near the real axis,
with an error term exponentially
decaying in time. In the presence of a nondegenerate elliptic closed
geodesic not meeting the support
of the damping coefficient, we show that there exists a sequence of
eigenfrequencies converging rapidly
to the real axis. In the case of Zoll manifolds, we show that the
propagator can be expanded in terms
of the clusters of the eigenfrequencies in the entire spectral band.