  We study the general infrared behavior of the power spectrum of a stochastic
gravitational wave background produced by stress tensor in the form bilinear in
certain dynamical degrees of freedom. We find $\Omega_{\text{GW}}\propto k^3$
for a very wide class of the sources which satisfy a set of reasonable
conditions. Namely, the $k^3$ scaling is universally valid when the source term
is bounded in both frequency and time, is effective in a radiation-dominated
stage, and for $k$ smaller than all the physical scales associated with the
source, like the peak frequency, peak width, and time duration, etc. We also
discuss possible violations of these conditions and their physical
implications.
