This paper considers the propagation of a plane elastic pulse through a nonhomogeneous, elastic, and isotropic medium. The inhomogeneity is represented as a constant velocity gradient (caused by changes in the elastic properties) which is assumed to be parallel to the direction of propagation of the pulse. It is shown that it is impossible to transmit a “clean” pulse through such a medium—the disturbance will have a definite onset, but will be followed by a train of slowly decaying oscillations. The period of these oscillations is related to the existing velocity gradient. The case of a reflection from a free boundary normal to the velocity gradient is considered in detail and it is shown that the motion imparted to the boundary will not be a replica of the motion in the incident pulse.