Complete integrated solutions have been found in terms of elementary functions for the motion of an electron on the axis of a progressive wave linear accelerator with a constant wave speed, &bgr;=1. The character of the solutions has been discussed and asymptotic relations found.Calculations have been made for the third section of the Purdue linear accelerator, in which the curves for the first two sections where &bgr;<1 have been extended. The energy distribution of the emergent beam was computed. It was found possible to design a linear accelerator with constant wave speed sections which, without preliminary bunching, would produce a beam of greater homogeneity in energy than might have been expected.