  We study the damping of $f(R)$ gravitational waves by matter in flat
spacetime and in expanding universe. In the former case, we find that the
Landau damping of scalar mode in $f(R)$ theory exists, while that of the tensor
mode in general relativity does not; we also present the viscosity coefficients
and dispersion relations of the two modes. In the later case, we investigate
the evolution of tensor and scalar modes in Friedmann-Robertson-Walker (FRW)
cosmology with a matter distribution; by considering the case of $f(R)=R+\al
R^2$, we analysis the influence of parameter $\al$ on wave damping,and put
restrictions on its magnitude.
