# Dynamic modes at supersonic speed

Our teacher gave us an assignment where we have to study the longitudinal stability of an X-15. I have to study it at 353.9 m/s which is Mach 1.2. Another group has to study it at 235.9 m/s so Mach 0.8.

I made a Matlab code computing the eigenvalues of the A matrix because I have to plot the phugoid and the short period modes. With my speed I get :

Eigenvalues [at 353.9 m/s]:
-0.6094 + 3.1782i
-0.6094 - 3.1782i
-0.0775 + 0.0000i
0.0473 + 0.0000i


While with the other speed I get:

Eigenvalues [at 235.9 m/s]:
-0.4093 + 2.1175i
-0.4093 - 2.1175i
-0.0070 + 0.0649i
-0.0070 - 0.0649i


In the second case, it is clear that the phugoid is the -0.0070 ± 0.0649i and the short mode is -0.4093 ± 2.1175i. While in the first case, what can I deduce ?

Is there a dynamic mode which is cancelled by the supersonic speed?

EDIT: The A matrix is the coefficients of the longitudinal motion equations: