  We examine the classical theory of stellar wind formation. The theory
requires that to form a supersonic stellar wind, a subsonic flow speed must
start at a specific initial speed from the coronal base, called eigenspeed, go
along a continuous eigenfunction, and reach the sonic point, which is where the
flow speed equals the sonic speed, while the critical condition, which is where
the effective driving force is zero, is satisfied. Any mismatch between the
sonic point and critical condition distances results in either subsonic winds
when the initial speed is below the eigenspeed, or no stellar wind when the
initial speed is above the eigenspeed. However, because the critical condition
is determined by the stellar wind temperature profile, which depends on
ionization process at the top of the chromosphere and the heating process
around the coronal base but not by the processes at the sonic point, the
requirement of sonic point satisfying the critical condition is generally not
met and hence the momentum equation in the conventional theory encounters
difficulty when the initial speed is above the eigenspeed. To resolve the
difficulty, we propose a discontinuity between the sonic point and the critical
condition to reach supersonic stellar wind solutions. As a result, supersonic
stellar winds can be produced when the initial speed in the coronal base is
greater than the eigenspeed. The critical solution or eigen function provided
by the conventional stellar wind model describes the condition that separates
the supersonic stellar winds from subsonic ones.
