Last updated: 2026-06-16

Mach number

The Mach number is a dimensionless ratio comparing the speed of an object (or flow) to the local speed of sound in the surrounding medium. Named after the Austrian physicist Ernst Mach, who photographed supersonic projectiles and documented shock waves in the 1870s, it is the standard quantity for characterising compressibility effects in aerodynamics and gas dynamics.

Formula

The Mach number $M$ is defined as

where $v$ is the speed of the object and $c$ is the local speed of sound. Both quantities must be in the same units; the result is dimensionless.

Worked example

An aircraft travels at 500 m/s through air where the speed of sound is 343 m/s (dry air, 20 °C).

The aircraft is flying at roughly Mach 1.46, well within the supersonic regime. A Mach cone trails the aircraft, and an observer on the ground hears a sonic boom after the aircraft passes.

Flight regimes

The Mach number is used to define four flow regimes, each with distinct physics:

Subsonic (M below 0.8). The flow behaves as nearly incompressible everywhere around the body. Conventional thin-airfoil and lifting-line theories are accurate, and drag is dominated by skin friction and induced drag from lift.

Transonic (M from 0.8 to 1.2). Regions of both subsonic and supersonic flow coexist simultaneously around the aircraft. Local shock waves form on wing surfaces even before the aircraft itself reaches Mach 1, causing a sharp rise in wave drag — historically known as the "sound barrier" — and control-surface effectiveness changes sign. Transonic aircraft design requires area-ruling and careful profile shaping.

Supersonic (M from 1.2 to 5). The flow is supersonic everywhere. An oblique shock attaches to any sharp leading edge, or a detached bow shock stands ahead of a blunt body. Wave drag is a dominant contributor to total drag. Aerodynamic heating begins to matter for sustained flight, and intake design must decelerate the incoming air to subsonic conditions before the compressor or combustion chamber.

Hypersonic (M at 5 or above). Aerodynamic heating becomes the primary engineering challenge. The kinetic energy of the impinging air is large enough to dissociate diatomic oxygen and nitrogen and, at very high Mach numbers, to ionise the gas. The resulting plasma can blackout radio communications, as experienced during atmospheric re-entry. Thermal protection systems — ablative shields or reusable ceramic tiles — are essential.

Why the speed of sound varies

In an ideal gas the speed of sound is

where $\gamma$ is the ratio of specific heats, $R$ is the specific gas constant for the gas, and $T$ is the absolute temperature in kelvin. In dry air, $\gamma \approx 1.4$ and $R \approx 287\,\text{J kg}^{-1}\text{K}^{-1}$, giving approximately $c \approx 331\sqrt{T/273}$ m/s.

As altitude increases through the troposphere, temperature falls at roughly 6.5 °C per kilometre, so the speed of sound also decreases. A fighter jet flying at 800 km/h at sea level is at about Mach 0.65; the same true airspeed at 11 km altitude, where the temperature has dropped to roughly −57 °C and the speed of sound is about 295 m/s, corresponds to Mach 0.75. For this reason, pilots track both indicated airspeed (for aerodynamic loads) and Mach number (for compressibility margins).

The sonic boom and Mach cone

When an object moves faster than sound, the pressure waves it generates cannot propagate forward. They pile up into a conical shock surface — the Mach cone — with a half-angle given by

The intersection of this cone with the ground creates a hyperbolic ground track that sweeps along as the aircraft moves. Observers hear a loud sonic boom as the high-pressure region of the shock passes over them, not just at the moment the aircraft breaks Mach 1.

Applications

• Aviation: aircraft are certified for specific Mach envelopes, and autopilots enforce Mach-number limits to prevent buffet and flutter.
• Ballistics: rifle bullets typically travel at Mach 2 to Mach 4; the Mach number governs the shape of the shock and the drag coefficient.
• Space re-entry: vehicles returning from orbit enter the atmosphere at Mach 20 to Mach 25, requiring extensive thermal protection.
• Industrial gas flows: nozzles in gas turbines and wind tunnels are designed using isentropic Mach-number relations to achieve desired pressures and velocities.

Limits of this calculator

This calculator assumes the speed of sound is given as a fixed input. In practice, the speed varies continuously with altitude and temperature along the flight path. For transient or climbing flight, the local value of $c$ should be updated for each altitude increment. The calculator also assumes the object moves through a homogeneous medium at rest; for flows in moving air masses, the Mach number should be referenced to the air speed relative to the object, not the ground speed.