  We establish the possibility of Landau damping for gravitational scalar waves
which propagate in a non-collisional gas of particles. In particular, under the
hypothesis of homogeneity and isotropy, we describe the medium at the
equilibrium with a J\"uttner-Maxwell distribution, and we analytically
determine the damping rate from the Vlasov equation. We find that damping
occurs only if the phase velocity of the wave is subluminal throughout the
propagation within the medium. Finally, we investigate relativistic media in
cosmological settings by adopting numerical techniques.
