A numerical study of one stage of the transition region from laminar to turbulent flow of the boundary layer over a flat plate was performed. The theoretical development follows that of Benney and Lin in which the interaction of a two−dimensional and a restricted three−dimensional periodic disturbance was considered up to second order terms in a series expansion. Several laminar profiles were studied, all belonging to the Falkner−Skan family. The assumption of equal wave speeds for the two−dimensional and three−dimensional disturbances was relaxed, giving rise to a slow modulation of the mean secondary flow. The resulting mean secondary flow compared favorably with both the existing experimental work and the restricted analytical solution of Benney. The flow depended principally upon the disturbance spanwise wavenumber, &bgr;, whose variation resulted, in some instances, in a system of counter−rotating vortices, one on top of the other, for the mean secondary flow.