It is well known that, for a plane sinusoidal body wave of small amplitude in an elastic medium which is anisotropic in relation to a natural reference state, the kinetic and strain energy densities are universally equal and that, furthermore, the ray (or group) velocity of the wave coincides with the velocity of energy propagation and is co-directional with the normal to the slowness surface of the material at the point representing the wave. In this paper it is shown how an analogous set of relationships may be established for an acceleration wave of arbitrary form advancing into a stationary, stress-free region of a body composed of a non-heat conducting material. The counterpart of the equipartition of energy is found to hold regardless of the constitution of the body. For the remaining results the transmitting material is taken to be elastic, but subject, possibly, to one or more internal constraints. Some particular examples of constraints are discussed, namely incompressibility, inextensibility and combined incompressibility and inextensibility.