The new mode of instability found by Tunney et al.  is studied with
viscous stability theory in this article. When the high-speed boundary layer is subject
to certain values of favorable pressure gradient and wall heating, a new mode becomes
unstable due to the appearance of the streamwise velocity overshoot (U(y) > U∞) in
the base flow. The present study shows that under practical Reynolds numbers, the
new mode can hardly co-exist with the conventional first mode and Mack’s second
mode. Due to the requirement for additional wall heating, the new mode may only
lead to laminar-turbulent transition under experimental (artificial) conditions.