  Standard sirens - GW sources with an EM counterpart - can be used to measure
H_0 directly which should help to ease the existing Hubble tension. However, if
the source has a relative velocity to the expanding universe on top of its
motion due to the Hubble flow, a relativistic redshift affects the redshift of
the EM counterpart and the apparent distance of the GW source, and thus it
needs to be corrected to obtain accurate measurements. We study the effect of
such a relative velocity on GWs for a source in an expanding universe showing
that the total redshift of the wave is equal to the product of the relativistic
redshift and the cosmological redshift. We, further, find that a relative
velocity of the source changes its apparent distance by a factor (1+z_rel)^2 in
contrast to a linear factor for the cosmological redshift. We discuss that the
additional factor for the relativistic redshift is a consequence of a
velocity-dependent amplitude for GWs. We consider the effect of the relative
velocity on the chirp mass and the apparent distance of the source an observer
would infer when ignoring this velocity. We find that for different
astrophysical scenarios the error can range between 0.1 % and 7 % for the chirp
mass while the error in the apparent distance can be between 0.25 % and 15 %.
Furthermore, we consider the error introduced in the measurement of H_0 using
standard sirens for two cases: (i) when the effect of velocity on the redshift
of the EM counterpart is considered but not on the apparent distance obtained
from GWs and (ii) when the effect of the relative velocity is ignored
completely. We find that in the first case the error can reach 1 % for a source
moving due to the peculiar velocity of its host galaxy and that in the second
case the error can be more than 5 % for a source at the distance of GW150914
with the same velocity.
