emil
John T is asking I think that you perform a little substitution in two thermodynamics equations.
The first equation should be familiar, that of temperature rise with Mach Number? You use it to calculate total air temperature,yes? Page 171 of Hill and Peterson on Mechanics and Thermodynamics of Propulsion is one reference.
T2/T1 = 1 +[(g-1)/2]*M^2 where g is the ratio of specific heats also called "gamma".
The second equation is on the same page and deals with pressure rise
P2/P1 = 1+ e[(T2/T1)-1)]^[g/(g-1)]
In this equation e is the adiabatic efficiency which we'll say is unity.
Re-arrange the first equation to obtain
[(T2/T1)-1] = [(g-1)/2]*M^2
Substitute that into equation two and you have
P2/P1 = 1 + {[(g-1)/2]*M^2}^(g/(g-1))
So provided that the process is adiabatic and 100% efficient if we know the pressure ratio P2/P1 and assume that the value of gamma is pretty much constant at 1.4 then the Mach Number pops out.
Some authors use a slightly different presentation [Saravannamuttoo, Rogers] but I'll stick to JT's version.
And he did say Mach Number not the speed of sound.
Hope this helps and I hope I have not mistyped.
The "E"