  We study in a Brill-Hartle type of approximation the back-reaction of a
superposition of linear gravitational waves in an Einstein-de Sitter background
up to the second order in the small wave amplitudes $h_{ik}$. The wave
amplitudes are assumed to form a homogeneous and isotropic stochastic process.
No restriction for the wavelengths is assumed. The effective stress-energy
tensor $T^{e}_{\mu\nu}$ is calculated in terms of the correlation functions of
the process. We discuss in particular a situation where $T^{e}_{\mu\nu}$ is the
dominant excitation of the background metric. Apart from the Tolman radiation
universe, a solution with the scale factor of the de Sitter universe exists
with $p = -\rho$ as effective equation of state.
