  Some time ago it was pointed out that the presence of cosmological components
could affect the propagation of gravitational waves (GW) beyond the usual
cosmological redshift and that such effects might be observable in pulsar
timing arrays. These analyses were done at leading order in the Hubble constant
$H_0$, which is proportional to $\Lambda^{1/2}$ and $\rho_i^{1/2}$ ($\rho_i$
being the various cosmological fluid densities). In this work, we study in
detail the propagation of metric perturbations on a Schwarzschild-de Sitter
(SdS) background, close to the place where GW are produced, and obtain
solutions that incorporate corrections linear in $\rho_i$ and $\Lambda$. At the
next-to-leading order the corrections do not appear in the form of $H_0$ thus
lifting the degeneracy among the various cosmological components. We also
determine the leading corrections proportional to the mass of the final object;
they are very small for the distances considered in pulsar timing arrays but
may be of relevance in other cases. When transformed into comoving coordinates,
the ones used in cosmological measurements, this SdS solution does satisfy the
perturbation equations in a Friedmann-Lema\^itre-Robertson-Walker metric up to
and including $\Lambda^{3/2}$ terms. This analysis is then extended to the
other cosmological fluids, allowing us to consider GW sources in the Gpc range.
Finally, we investigate the influence of these corrections in pulsar timing
arrays observations.
