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When two electromagnetic waves propagate along the fixed direction of an uniform and constant magnetic field in a homogeneous, cold, and collisionally damped plasma, the effects of their nonlinear interaction are large and so important in two special cases. One is the case in which the beat frequency equals the characteristic plasma frequency and moreover, one of the wave frequencies equals the gyration frequency of electrons (or ions). The other is the nonlinear evolution of the wave pattern formed by an appropriate superposition of the two waves in a manner which produces standing waves from them in the linear approximation. These two cases have been theoretically investigated in this paper. Moreover, the nonlinear evolution of the standing‐wave patterns has been discussed with a view to initiate the exploration of the application side of this effect.