With respect to the propagation through a cold, unmagnetized, electron plasma of nonlinear, highly superluminous, plane waves of fixed profile, with electric vector in a fixed plane parallel to the direction of propagation, it is known that, in addition to the familiar longitudinal and quasi-transverse waves, there can also be a third periodic wave. The perturbation method by which this third wave has previously been analysed is of restricted validity, and fails to describe how the wave disappears in the approach to the small-amplitude limit, where the longitudinal and transverse waves alone survive.