  We study the behavior of gravitational waves and their backreaction on the
background in cosmological solutions of the five-dimensional Ho\v{r}ava-Witten
theory. As a dynamical background, we consider two cosmological solutions with
spatially flat expanding FRW branes, called $(\uparrow)$- and
$(\downarrow)$-solutions, in which the orbifold size increases and decreases in
time, respectively. % For these background solutions, the wave equation for the
tensor perturbation can be solved by the method of separation of variables, and
the mode functions are classified by a separation constant which can be
regarded as a graviton mass. We show that the spatial behavior of the mode
functions are the same for both background solutions, but the temporal behavior
is significantly different. We further show that for the $(\uparrow)$ solution,
the background bulk geometry is unstable against the backreaction of the
perturbation, while for the $(\downarrow)$ solution, the backreaction on the
bulk geometry can be neglected. We also show that, in contrast to the effect to
the bulk geometry, the backreaction of the perturbation significantly alters
the intrinsic geometry of the brane for the $(\downarrow)$ solution.
