  We test the validity of Isaacson's formula which states that high frequency
and low amplitude gravitational waves behave as a radiation fluid on average.
For this purpose, we numerically construct a solution of the vacuum Einstein
equations which contains nonlinear standing gravitational waves. The solution
is constructed in a cubic box with periodic boundary conditions. The time
evolution is solved in a gauge in which the trace of the extrinsic curvature
$K$ of the time slice becomes spatially uniform. Then, the Hubble expansion
rate $H$ is defined by $H=-K/3$ and compared with the effective scale factor
$L$ defined by the proper volume, area and length of the cubic box. We find
that, even when the wave length of the gravitational waves is comparable to the
Hubble scale, the deviation from Isaacson's formula $H\propto L^{-2}$ is at
most 3\% without taking a temporal average and is below 0.1\% with a temporal
average.
