Pugilistic Animus, BOAC,
Thanks gents for providing the link to some of my old 'work', that's going back a bit now.
Hurricane6, I regret that there's not going to be any simplification of the formula used for Machmeter calibration, the essentials of a complete and correct calculation are there. The only 'relief' that I can provide is the reduction of the (Y-1)/Y exponent to 2/7.
I note that in one of your intermediate posts you refer to supersonic flight. The formulae for Subsonic (including transonic) and Supersonic flight are -
SUBSONIC (Bernoulli's equation for M<1)
M = SQR (5 X (Qc/Ps + 1)^((Y-1)/Y) - 1)
SUPERSONIC (Rayleigh Supersonic Pitot equation)
M= .88128485 SQR ((Qc/Ps +1) X (1-1/(7 X M^2)^2.5)
(For Qc measured behind the normal shock).
(M appears on both sides of the supersonic equation. The most practical solution of the supersonic equation is to make initial Mach calculation from the subsonic formua, and then by several iterations apply successive Mach number results to the supersonic algorithm as differential between successive results approaches zero).