The relativistic differential equations for the axial motion of an electron in a wave guide, excited by a progressive, sinusoidal, longitudinal, electric wave of constant speed less than that of light in vacuum, have been integrated. The first integral yields the results of J. C. Slater and gives the energy‐phase relationship, which is periodic. A second integration yields the phase distance relationship and thus completes the determination of the orbits. All separatrices are exactly integrable in terms of trigonometric and hyperbolic functions, while the general case requires elliptic integrals of the first and third kinds. The former are obtainable from tables, while the latter are evaluated by series expansions, or in the case of particles near the self‐crossing point of the separatrices where the convergence of the series expansion is slow, by the use of the addition theorem for elliptic integrals of the third kind.These results are applied to the case of the proposed new Purdue Linear Accelerator. Energy distribution of the particles coming out of the end of the first two sections of this accelerator is in this way obtained.