Mach number

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In fluid mechanics, Mach number (M or Ma) /ˈmɑːx/ is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound.[1][2]

\mathrm{M} = \frac {{v}}{{v_\text{sound}}}


M is the Mach number,
vsound is the speed of sound in the medium.



In French, the Mach number is sometimes called the "nombre de Sarrau" ("Sarrau number") after Émile Sarrau who researched explosions in the 1870s and 1880s.[4]


The Mach number is commonly used both with objects traveling at high speed in a fluid, and with high-speed fluid flows inside channels such as nozzles, diffusers or wind tunnels. As it is defined as a ratio of two speeds, it is a dimensionless number. At Standard Sea Level conditions (corresponding to a temperature of 15 degrees Celsius), the speed of sound is 340.3 m/s[5] (1225 km/h, or 761.2 mph, or 661.5 knots, or 1116 ft/s) in the Earth's atmosphere. The speed represented by Mach 1 is not a constant; for example, it is mostly dependent on temperature and atmospheric composition and largely independent of pressure. Since the speed of sound increases as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on the fluid temperature around it. Mach number is useful because the fluid behaves in a similar way at the same Mach number. So, an aircraft traveling at Mach 1 at 20°C or 68°F, at sea level, will experience shock waves in much the same manner as when it is traveling at Mach 1 at 11,000 m (36,000 ft) at −50°C or −58F, even though it is traveling at only 86% of its speed at higher temperature like 20°C or 68°F.[6]

Classification of Mach regimes[edit]

While the terms "subsonic" and "supersonic" in the purest verbal sense refer to speeds below and above the local speed of sound respectively, aerodynamicists often use the same terms to talk about particular ranges of Mach values. This occurs because of the presence of a "transonic regime" around M = 1 where approximations of the Navier-Stokes equations used for subsonic design actually no longer apply, the simplest of many reasons being that the flow locally begins to exceed M = 1 even when the freestream Mach number is below this value.


In the following table, the "regimes" or "ranges of Mach values" are referred to, and not the "pure" meanings of the words "subsonic" and "supersonic".


Regime Mach mph km/h m/s General plane characteristics
Transonic 0.8-1.2 610-915 980-1,470 270-410 Transonic aircraft nearly always have swept wings, delaying drag-divergence, and often feature design adhering to the principles of the Whitcomb Area rule.
Supersonic 1.2–5.0 915-3,840 1,470–6,150 410–1,710 Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behaviour of flows above Mach 1. Sharp edges, thin aerofoil-sections, and all-moving tailplane/canards are common. Modern combat aircraft must compromise in order to maintain low-speed handling; "true" supersonic designs include the F-104 Starfighter, SR-71 Blackbird and BAC/Aérospatiale Concorde.
Hypersonic 5.0–10.0 3,840–7,680 6,150–12,300 1,710–3,415 Cooled nickel-titanium skin; highly integrated (due to domination of interference effects: non-linear behaviour means that superposition of results for separate components is invalid), small wings, such as those on the X-51A Waverider

High-speed flow around objects[edit]

Flight can be roughly classified in six categories:

Regime Subsonic Transonic Sonic Supersonic Hypersonic High-hypersonic
Mach <0.8 0.8–1.2 1.0 1.2–5.0 5.0–10.0 >10.0

For comparison: the required speed for low Earth orbit is approximately 7.5 km/s = Mach 25.4 in air at high altitudes. The speed of light in a vacuum corresponds to a Mach number of approximately 881,000 (relative to air at sea level).



Transsonic flow over airfoil 1.svg Transsonic flow over airfoil 2.svg
(a) (b)






High-speed flow in a channel[edit]


The obvious result is that in order to accelerate a flow to supersonic, one needs a convergent-divergent nozzle, where the converging section accelerates the flow to sonic speeds, and the diverging section continues the acceleration. Such nozzles are called de Laval nozzles and in extreme cases they are able to reach hypersonic speeds (Mach 13 (9,896 mph; 15,926 km/h) at 20°C).




\mathrm{M} = \frac{v}{v_\text{sound}}


M is the Mach number
v is velocity of the moving aircraft and
vsound is the speed of sound at the given altitude

Note that the dynamic pressure can be found as:

q = \frac{\gamma}{2} p\, \mathrm{M}^2




qc is impact pressure (dynamic pressure) and
p is static pressure
\ \gamma\, is the ratio of specific heat of a gas at a constant pressure to heat at a constant volume (1.4 for air).


 \frac{q_c}{p} = \left[\frac{\gamma+1}{2}\mathrm{M}^2\right]^\left(\frac{\gamma}{\gamma-1}\right)\cdot \left[  \frac{\gamma+1}{\left(1-\gamma+2 \gamma\, \mathrm{M}^2\right)} \right]^\left(\frac{1}{ \gamma-1 }\right)

Calculating Mach Number from Pitot Tube Pressure[edit]



\mathrm{M} = \sqrt{5\left[\left(\frac{q_c}{p}+1\right)^\frac{2}{7}-1\right]}\,

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:

\mathrm{M} = 0.88128485 \sqrt{\left(\frac{q_c}{p} + 1\right)\left(1 - \frac{1}{7\,\mathrm{M}^2}\right)^{2.5}}


qc is dynamic pressure measured behind a normal shock

As can be seen, M appears on both sides of the equation. The easiest method to solve the supersonic M calculation is to enter both the subsonic and supersonic equations into a computer spreadsheet such as Microsoft Excel, Calc, or some equivalent program to solve it numerically. 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 in the supersonic equation. Then a simple iteration of the supersonic equation is performed, each time using the last computed value of M, until M converges to a value—usually in just a few iterations.[7] Alternatively, Newton's method can also be used.

See also[edit]


  1. ^ a b Young, Donald F.; Bruce R. Munson, Theodore H. Okiishi, Wade W. Huebsch (2010). A Brief Introduction to Fluid Mechanics (5 ed.). John Wiley & Sons. p. 95. ISBN 978-0-470-59679-1. 
  4. ^ Blackmore, John T. (1972). Ernst Mach: His Life, Work, and Influence. University of California Press. p. 112. ISBN 978-0-520-01849-5. 
  6. ^ ([1]). National Aeronautics and Space Administration website page "Mach Number", NASA.

External links[edit]