An analysis is presented of the forced torsional vibrations of a cylindrical rod connected to an elastic half-space under the condition that the circumferential displacement at the free end of the rod, where the disturbing moment is applied, varies proportionally with the distance from the rod axis. Both the Pochhammer-Chree and elementary theory are utilized. The response curve of the rotational amplitude at the free end of the rod, obtained from the Pochhammer-Chree theory, and that obtained from the elementary theory almost coincide with each other except in the case where the rod and the half-space are of the same material. The amplitude attenuation is largest when the rod and the half-space are of the same material. The maximum values of half-space displacement distribution at the interface lie within the rod edge.