Abstract In classical theory of stellar wind formation, supersonic stellar wind starts with an initial speed of the eigenspeed at the inner boundary of the corona, goes along a continuous eigenfunction, reaches the sonic point while requiring a critical condition to be satisfied, and becomes supersonic across the critical point. If the initial speed is smaller than the eigenspeed, the wind is subsonic, and if the initial speed is greater than the eigenspeed, stellar wind cannot form. Since the initial flow speed is determined by ionization processes at the top boundary of the chromosphere and chromospheric and coronal heating processes, the initial speed of the wind can often be greater than the eigenspeed, posing a dilemma to the classical stellar wind theory which predicts no stellar wind under such conditions. We examine the classical stellar wind evolution equation and find that when the initial speed is greater than the eigenspeed, it cannot hold at the sonic point. In 1D steady state gasdynamics, the evolution equation can be rewritten with two expressions, one at the sonic point and one for everywhere else. The Rankine–Hugoniot relations are used to connect the solutions across the sonic point. A discontinuity standing in the flow that travels at the local sonic speed can facilitate the sonic transition. Supersonic winds can form when the inner boundary speed is greater than the eigenspeed. The critical solution separates the parameter regimes of supersonic from subsonic winds, and most supersonic stellar winds do not go through the critical point or critical points.