The Mach number (M or Ma), often only Mach, (/mɑːk/; German: [max]) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound.^[1][2] It is named after the Czech physicist and philosopher Ernst Mach.

${\displaystyle \mathrm {M} ={\frac {u}{c}},}$

where:

M is the local Mach number,

u is the local flow velocity with respect to the boundaries (either internal, such as an object immersed in the flow, or external, like a channel), and

c is the speed of sound in the medium, which in air varies with the square root of the thermodynamic temperature.

By definition, at Mach 1, the local flow velocity u is equal to the speed of sound. At Mach 0.65, u is 65% of the speed of sound (subsonic), and, at Mach 1.35, u is 35% faster than the speed of sound (supersonic). Pilots of high-altitude aerospace vehicles use flight Mach number to express a vehicle's true airspeed, but the flow field around a vehicle varies in three dimensions, with corresponding variations in local Mach number.

The local speed of sound, and hence the Mach number, depends on the temperature of the surrounding gas. The Mach number is primarily used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a gas or a liquid. The boundary can be travelling in the medium, or it can be stationary while the medium flows along it, or they can both be moving, with different velocities: what matters is their relative velocity with respect to each other. The boundary can be the boundary of an object immersed in the medium, or of a channel such as a nozzle, diffuser or wind tunnel channelling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless quantity. If M < 0.2–0.3 and the flow is quasi-steady and isothermal, compressibility effects will be small and simplified incompressible flow equations can be used.^[1][2]

Etymology

The Mach number is named after the physicist and philosopher Ernst Mach^[3] according to a proposal by the aeronautical engineer Jakob Ackeret in 1929.^[4] The word Mach is always capitalized since it derives from a proper name, and since the Mach number is a dimensionless quantity rather than a unit of measure, the number comes after the word Mach; the second Mach number is Mach 2 instead of 2 Mach (or Machs). This is somewhat reminiscent of the early modern ocean-sounding unit mark (a synonym for fathom), which was also unit-first, and may have influenced the use of the term Mach. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's number, never Mach 1.^[5]

Overview

Mach number is a measure of the compressibility characteristics of fluid flow: the fluid (air) behaves under the influence of compressibility in a similar manner at a given Mach number, regardless of other variables.^[6] As modeled in the International Standard Atmosphere, dry air at mean sea level, standard temperature of 15 °C (59 °F), the speed of sound is 340.3 meters per second (1,116.5 ft/s; 761.23 mph; 1,225.1 km/h; 661.49 kn).^[7] The speed of sound is not a constant; in a gas, it increases proportionally to the square root of the absolute temperature, and since atmospheric temperature generally decreases with increasing altitude between sea level and 11,000 meters (36,089 ft), the speed of sound also decreases. For example, the standard atmosphere model lapses temperature to −56.5 °C (−69.7 °F) at 11,000 meters (36,089 ft) altitude, with a corresponding speed of sound (Mach 1) of 295.0 meters per second (967.8 ft/s; 659.9 mph; 1,062 km/h; 573.4 kn), 86.7% of the sea level value.

Appearance in the continuity equation

As a measure of flow compressibility, the Mach number can be derived from an appropriate scaling of the continuity equation.^[8] The full continuity equation for a general fluid flow is:

where ${\displaystyle D/Dt}$ is the material derivative, ${\displaystyle \rho }$ is the density, and ${\displaystyle {\bf {u}}}$ is the flow velocity. For isentropic pressure-induced density changes, ${\displaystyle dp=c^{2}d\rho }$ where ${\displaystyle c}$ is the speed of sound. Then the continuity equation may be slightly modified to account for this relation:The next step is to nondimensionalize the variables as such:where ${\displaystyle L}$ is the characteristic length scale, ${\displaystyle U}$ is the characteristic velocity scale, ${\displaystyle p_{\infty }}$ is the reference pressure, and ${\displaystyle \rho _{0}}$ is the reference density. Then the nondimensionalized form of the continuity equation may be written as:where the Mach number ${\displaystyle {\text{M}}=U/c}$. In the limit that ${\displaystyle {\text{M}}\rightarrow 0}$, the continuity equation reduces to ${\displaystyle \nabla \cdot {\bf {u}}=0}$ — this is the standard requirement for incompressible flow.

Classification of Mach regimes

While the terms subsonic and supersonic, in the purest 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 flight (free stream) M = 1 where approximations of the Navier-Stokes equations used for subsonic design no longer apply; the simplest explanation is that the flow around an airframe locally begins to exceed M = 1 even though the free stream Mach number is below this value.

Meanwhile, the supersonic regime is usually used to talk about the set of Mach numbers for which linearised theory may be used, where for example the (air) flow is not chemically reacting, and where heat-transfer between air and vehicle may be reasonably neglected in calculations.

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.

Generally, NASA defines high hypersonic as any Mach number from 10 to 25, and re-entry speeds as anything greater than Mach 25. Aircraft operating in this regime include the Space Shuttle and various space planes in development.

Regime	Flight speed					General plane characteristics
	(Mach)	(knots)	(mph)	(km/h)	(m/s)
Subsonic	<0.8	<530	<609	<980	<273	Most often propeller-driven and commercial turbofan aircraft with high aspect-ratio (slender) wings, and rounded features like the nose and leading edges. The subsonic speed range is that range of speeds within which, all of the airflow over an aircraft is less than Mach 1. The critical Mach number (Mcrit) is lowest free stream Mach number at which airflow over any part of the aircraft first reaches Mach 1. So the subsonic speed range includes all speeds that are less than Mcrit.
Transonic	0.8–1.2	530–794	609–914	980–1,470	273–409	Transonic aircraft nearly always have swept wings, causing the delay of drag-divergence, and often feature a design that adheres to the principles of the Whitcomb Area rule. The transonic speed range is that range of speeds within which the airflow over different parts of an aircraft is between subsonic and supersonic. So the regime of flight from Mcrit up to Mach 1.3 is called the transonic range.
Supersonic	1.2–5.0	794–3,308	915–3,806	1,470–6,126	410–1,702	The supersonic speed range is that range of speeds within which all of the airflow over an aircraft is supersonic (more than Mach 1). But airflow meeting the leading edges is initially decelerated, so the free stream speed must be slightly greater than Mach 1 to ensure that all of the flow over the aircraft is supersonic. It is commonly accepted that the supersonic speed range starts at a free stream speed greater than Mach 1.3. Aircraft designed to fly at supersonic speeds show large differences in their aerodynamic design because of the radical differences in the behavior 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, MiG-31, North American XB-70 Valkyrie, SR-71 Blackbird, and BAC/Aérospatiale Concorde.
Hypersonic	5.0–10.0	3,308–6,615	3,806–7,680	6,126–12,251	1,702–3,403	The X-15, at Mach 6.72 is one of the fastest manned aircraft. Also, 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 Mach 5 X-51A Waverider.
High-hypersonic	10.0–25.0	6,615–16,537	7,680–19,031	12,251–30,626	3,403–8,508	The NASA X-43, at Mach 9.6 is one of the fastest aircraft. Thermal control becomes a dominant design consideration. Structure must either be designed to operate hot, or be protected by special silicate tiles or similar. Chemically reacting flow can also cause corrosion of the vehicle's skin, with free-atomic oxygen featuring in very high-speed flows. Hypersonic designs are often forced into blunt configurations because of the aerodynamic heating rising with a reduced radius of curvature.
Re-entry speeds	>25.0	>16,537	>19,031	>30,626	>8,508	Ablative heat shield; small or no wings; blunt shape. Russia's Avangard (hypersonic glide vehicle) is claimed to reach up to Mach 27.

High-speed flow around objects

Flight can be roughly classified in six categories:

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