  Here, we investigate the linear spatial stability of a parallel
two-dimensional compressible boundary layer on an adiabatic plate by
considering 2D and 3D disturbances. We employ the Compound Matrix Method for
the first time for compressible flows, which, unlike other conventional
techniques, can efficiently eliminate the stiffness of the original equation.
Our study explores flow Mach numbers ranging from low subsonic to supersonic
cases, to investigate the effects of flow compressibility and spanwise
variation of disturbances. We get some interesting results depending on the
flow Mach number. Mack (AGARD Report No. 709, 1984) reported the existence of
two unstable modes for Mach number greater than 3 from viscous calculations
(the so-called second mode) that subsequently fuse to create only one unstable
zone when Mach number increases. Our calculations show a series of unstable
modes for a Mach number greater than 3. The number of such modes is much more
than two (unlike what Mack reports). The number and the frequency extent of the
corresponding unstable zones increase with an increase in M, which is
significantly higher than subsonic or low-supersonic cases. While the shape of
the neutral curves for the second unstable mode for a Mach number greater than
4 is similar to the fused neutral curve shown by Mack for a Mach number of 4.8,
the characteristics of higher-order spatially unstable modes considering the
viscous stability of supersonic boundary layers remain unreported to the best
of our knowledge. The last one is the most novel element in the reported
results.
