Appendix F: Derivation of Air Speed Equations

Overview

Nomenclature:

p	air pressure
?	air density
v	air speed
?	specific heat ratio
T _s	static air temperature (free stream)
T _t	total air temperature (at zero air velocity)
R	ideal gas constant
a	speed of sound
M	Mach number

The energy conservation equation, given by Bernoulli's theorem is [1-2]

Assuming that air expands adiabatically implies

Eliminating ? from (F.1) and (F.2) gives

Integrating (F.3) yields

Substituting for c from (F.2) in the above gives

Using the ideal gas law p = pRT _s in Eq. (F.4) yields

If we denote by T _t the temperature at which the air speed v= 0, then the above equation becomes

From sound wave theory

Since

Then (F.5) becomes

From the ideal gas law and Eq. (F.2) we get

Substituting for the temperature ratio in (F.8) gives

1. John D. Anderson, Fundamentals of Aerodynamics, McGraw Hills, NY, New York, 1991

2. L.M. Milne-Thompson, Theoretical Aerodynamics, Macmillan & Co LTD, 1958

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