Longitudinal instability of an induction linac with acceleration

The question arises as to what effect acceleration, which so far has been ignored, has on the longitudinal instability of an induction linac. The answer is not much for the anticipated acceleration rate (1-2 MeV/m) and minimum e-folding distance for the instability (50-500 meters). However, the total unstable growth is significantly reduced over distances which are long enough for appreciable acceleration to occur. The purpose of this work is to give a calculation for the instability, including a constant acceleration rate. Some interesting features emerge-for example, the velocity of the head is a more convenient independent variable than axial position and for an initially sinusoidal perturbation of the velocity in time, the number of oscillations along the pulse is constant; as the pulse shortens in time the frequency increases