A study is made of the motion of the shock wave produced when a piston is impulsively set in motion with constant velocity into a diatomic gas at rest, conditions being such that behind the shock there is a zone of vibrational or dissociational relaxation. The dissociation case is treated by the method of linearized characteristics, using a form for the rate equation due to Freeman (1958), while the vibrational case is included in a discussion in which the third-order equation for acoustic disturbances derived by Chu (1958) is applied to the flow behind the shock. It is shown that in either case the initial speed of the shock is that corresponding to frozen flow between it and the piston, but that the lower speed calculated from equilibrium thermodynamics is approached at large times. Distributions of pressure and velocity between the piston and the shock are found: at long times after the start of the motion these are precisely those given by the Bethe-Teller (1941) theory for partly dispersed shock waves. Some applications to other shock motions are discussed.