In this paper, we investigate propagation of axial gravitational waves in the background of 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 discontinuity at the wave front.