In this paper, the instability criterion of steady uniform flows in open channels with any section is derived by considering the time growth or decay of an infinitesimal disturbed motion of fluid surface. The mathematical analysis shows that the criterion by this approach, based upon the momentum equation neglecting the curvature of fluid surface, is identical with the expression of V. V. Vedernikov's criterion for instability of free surface in real fluids, using certain approximations of Saint Venant. It is well interesting to note that this expression of the criterion for instability becomes the same condition to maintain final patterns of roll-waves in steep inclined channels, obtained by argumenting their hydraulic characteristics
