Abstract The dynamic response of a linearly elastic rod with material properties which are described by random functions of position along the rod is studied. A general formulation to be satisfied by the ensemble averaged field quantities is derived. This formulation is considered in some detail for those situations in which the magnitude of the random variations is vanishingly small and those in which the length scale on which these variations are observable is small compared to the length scale on which the variations in the averaged field quantities are observable. The infinite rod dispersion spectrum is obtained for the former situation. A dynamic “effective modulus” theory is obtained for the latter situation. The physical implications of the results are discussed.