  We present novel analytical solutions for linear-order gravitational waves or
tensor perturbations in a flat Friedmann-Robertson-Walker universe containing
two perfect fluids, radiation and pressureless dust, and allowing for neutrino
anisotropic stress. One of the results is applicable to any sub-horizon
gravitational wave in such a universe. Another result is applicable to
gravitational waves of primordial origin, for example produced during
inflation, and works both before and after they cross the horizon. These
results improve on analytical approximations previously set out in the
literature. Comparison with numerical solutions shows that both these
approximations are accurate to within $1\%$, or better, for a wide range of
wave-numbers relevant for cosmology.
