  In this paper, we investigate the propagation of axial gravitational waves in
the background of the flat FRW universe in $f(R,T)$ theory. The field equations
are obtained for unperturbed as well as axially perturbed FRW metric. These
field equations are solved simultaneously to obtain the unknown perturbation
parameters. We find that the assumed perturbations can affect matter as well as
four-velocity. Moreover, ignoring the material perturbations we explicitly
obtain an expression for four-velocity. It is concluded that axial
gravitational waves in the curvature-matter coupling background can produce
cosmological rotation or have memory effect if the wave profile has a
discontinuity at the wavefront.
