The possibility of the damping of plane gravitational waves while propagating in a plasma medium is considered. The gravitational plasma frequency, is for a neutron star medium ~ 103Hz, which is the same as the frequency of the gravitational waves emitted by a collapsing star. So resonant damping of such waves within a collapsing star is probable. Estimates are made for the damping length for dense and dilute plasmas (also in the presence of magnetic fields). Analogies with Landau damping are made. Applications to other astrophysical situations are outlined.