  We derive the effects of a non-zero cosmological constant $\Lambda$ on
gravitational wave propagation in the linearized approximation of general
relativity. In this approximation we consider the situation where the metric
can be written as $g_{\mu\nu}= \eta_{\mu\nu}+ h_{\mu\nu}^\Lambda +
h_{\mu\nu}^W$, $h_{\mu\nu}^{\Lambda,W}<< 1$, where $h_{\mu\nu}^{\Lambda}$ is
the background perturbation and $h_{\mu\nu}^{W}$ is a modification
interpretable as a gravitational wave. For $\Lambda \neq 0$ this linearization
of Einstein equations is self-consistent only in certain coordinate systems.
The cosmological Friedmann-Robertson-Walker coordinates do not belong to this
class and the derived linearized solutions have to be reinterpreted in a
coordinate system that is homogeneous and isotropic to make contact with
observations. Plane waves in the linear theory acquire modifications of order
$\sqrt{\Lambda}$, both in the amplitude and the phase, when considered in FRW
coordinates. In the linearization process for $h_{\mu\nu}$, we have also
included terms of order $\mathcal{O}(\Lambda h_{\mu\nu})$. For the background
perturbation $h_{\mu\nu}^\Lambda$ the difference is very small but when the
term $h_{\mu\nu}^{W}\Lambda$ is retained the equations of motion can be
interpreted as describing massive spin-2 particles. However, the extra degrees
of freedom can be approximately gauged away, coupling to matter sources with a
strength proportional to the cosmological constant itself. Finally we discuss
the viability of detecting the modifications caused by the cosmological
constant on the amplitude and phase of gravitational waves. In some cases the
distortion with respect to gravitational waves propagating in Minkowski
space-time is considerable. The effect of $\Lambda$ could have a detectable
impact on pulsar timing arrays.
