Exact non-linear wave solutions are obtained for waves stationary in a uniformly moving frame for the case of a Maxwellian electron plasma in a uniform ion background. It is found that there are no waves below a minimum wave velocity; anharmonic waves develop above the critical velocity, but only up to a maximum amplitude, depending on the wave velocity; frequency amplitude relations indicate that small amplitude linearized theory is not a good approximation, except for large wave velocities; and in the limit of zero amplitude, twice the frequency of oscillation given by linearized theory for any wave velocity is obtained. (Author).