Editing Mach number (section)

==Calculation==

When the speed of sound is known, the Mach number at which an aircraft is flying can be calculated by
<math display="block">
\mathrm{M} = \frac{u}{c}
</math>

where:
* M is the Mach number
* ''u'' is [[velocity]] of the moving aircraft and
* ''c'' is the [[speed of sound]] at the given altitude (more properly temperature)

and the speed of sound varies with the [[thermodynamic temperature]] as:

<math display="block">c = \sqrt{\gamma \cdot R_* \cdot T},</math>

where:
* <math>\gamma\,</math> is the [[Heat capacity ratio|ratio of specific heat]] of a gas at a constant pressure to heat at a constant volume (1.4 for air)
* <math> R_*</math> is the [[specific gas constant]] for air.
* <math> T, </math> is the static air temperature.

If the speed of sound is not known, Mach number may be determined by measuring the various air pressures (static and dynamic) and using the following formula that is derived from [[Bernoulli's principle|Bernoulli's equation]] for Mach numbers less than 1.0. Assuming air to be an [[ideal gas]], the formula to compute Mach number in a subsonic compressible flow is:<ref name="Olson">Olson, Wayne M. (2002). [http://www.aviation.org.uk/pdf/Aircraft_Performance_Flight_Testing.pdf "AFFTC-TIH-99-02, ''Aircraft Performance Flight Testing''"]. Air Force Flight Test Center, Edwards Air Force Base, California: United States Air Force. {{webarchive |url=https://web.archive.org/web/20110904190008/http://www.aviation.org.uk/pdf/Aircraft_Performance_Flight_Testing.pdf |date=September 4, 2011 }}</ref>

<math display="block">\mathrm{M} = \sqrt{\frac{2}{\gamma- 1 }\left[\left(\frac{q_c}{p} + 1\right)^\frac{\gamma - 1}{\gamma} - 1\right]}\,</math>

where:
* ''q<sub>c</sub>'' is [[impact pressure]] (dynamic pressure) and
* ''p'' is [[static pressure]]
* <math>\gamma\,</math> is the [[Heat capacity ratio|ratio of specific heat]] of a gas at a constant pressure to heat at a constant volume (1.4 for air)

The formula to compute Mach number in a supersonic compressible flow is derived from the [[Rayleigh number|Rayleigh]] supersonic pitot equation:

<math display="block">\frac{p_t}{p} =
  \left[\frac{\gamma + 1}{2}\mathrm{M}^2\right]^\frac{\gamma}{\gamma-1} \cdot
  \left[\frac{\gamma + 1}{1 - \gamma + 2\gamma\, \mathrm{M}^2}\right]^\frac{1}{\gamma - 1}
</math>

===Calculating Mach number from pitot tube pressure===
Mach number is a function of temperature and true airspeed.
Aircraft [[flight instruments]], however, operate using pressure differential to compute Mach number, not temperature.

Assuming air to be an [[ideal gas]], the formula to compute Mach number in a subsonic compressible flow is found from Bernoulli's equation for {{nowrap|M < 1}} (above):<ref name="Olson" />
<math display="block">\mathrm{M} = \sqrt{5\left[\left(\frac{q_c}{p} + 1\right)^\frac{2}{7} - 1\right]}\,</math>

The formula to compute Mach number in a supersonic compressible flow can be found from the Rayleigh supersonic pitot equation (above) using parameters for air:

<math display="block">\mathrm{M} \approx 0.88128485 \sqrt{\left(\frac{q_c}{p} + 1\right)\left(1 - \frac{1}{7\,\mathrm{M}^2}\right)^{2.5}}</math>

where:
* ''q<sub>c</sub>'' is the dynamic pressure measured behind a normal shock.

As can be seen, M appears on both sides of the equation, and for practical purposes a [[root-finding algorithm]] must be used for a numerical solution (the equation is a [[septic equation]] in M<sup>2</sup> and, though some of these may be solved explicitly, the [[Abel–Ruffini theorem]] guarantees that there exists no general form for the roots of these polynomials). It is first determined whether M is indeed greater than 1.0 by calculating M from the subsonic equation. If M is greater than 1.0 at that point, then the value of M from the subsonic equation is used as the initial condition for [[fixed point iteration]] of the supersonic equation, which usually converges very rapidly.<ref name=Olson/> Alternatively, [[Newton's method]] can also be used.