Mach Number Formula
Calculate Mach number from speed and altitude.
Classify flight as subsonic, transonic, supersonic, or hypersonic using the speed of sound at altitude.
Need to calculate, not just reference? Use the interactive version.
Open Mach Number Calculator →
The Formula
M = v / a
The Mach number compares an object's speed to the local speed of sound. Mach 1 is exactly the speed of sound. Above Mach 1 is supersonic flight.
Variables
| Symbol | Meaning |
|---|---|
| M | Mach number (unitless) |
| v | Speed of the object (m/s) |
| a | Speed of sound in the surrounding medium (m/s) |
Flight regimes: Subsonic (M < 0.8), Transonic (0.8-1.2), Supersonic (1.2-5.0), Hypersonic (M > 5.0)
Example 1
A jet flies at 680 m/s at altitude where speed of sound is 295 m/s
M = 680 / 295
M ≈ 2.31 (supersonic)
Example 2
A commercial airplane flies at 250 m/s at sea level (a = 343 m/s)
M = 250 / 343
M ≈ 0.73 (subsonic)
When to Use It
Use the Mach number when:
- Classifying the speed regime of aircraft or projectiles
- Designing aircraft and wind tunnels
- Predicting shock wave formation at high speeds
- Calculating compressibility effects on aerodynamic forces
Limitations
- The speed of sound changes with altitude and temperature — a plane flying at Mach 0.85 is moving slower in km/h at cruise altitude than at sea level
- Above Mach 5 (hypersonic), real gas effects (dissociation, ionization) mean the simple M = v/a formula is insufficient for aerodynamic design
- Mach number alone does not determine drag or shock wave behavior — shape, angle of attack, and boundary layer conditions all play major roles
Key Notes
- Formula: Ma = v / a: v is the object's speed; a is the local speed of sound. Sound speed in air ≈ 343 m/s (1,235 km/h) at 20°C at sea level. At altitude, colder temperatures reduce the speed of sound, changing the Mach number for the same airspeed.
- Flow regimes: Subsonic: Ma < 0.8 (smooth flow); transonic: 0.8–1.2 (mixed regions of sub/supersonic flow, high wave drag); supersonic: 1–5 (shock waves form); hypersonic: Ma > 5 (extreme heating, dissociation of air molecules).
- Sonic boom and shock waves: When Ma ≥ 1, the aircraft outruns the pressure waves it creates. These stack into a shock wave — a sudden pressure discontinuity heard on the ground as a loud boom. The boom is continuous as long as the aircraft is supersonic overhead.
- Temperature effect on sound speed: a = √(γRT/M) where γ ≈ 1.4 for air, R is the gas constant, T is absolute temperature, and M is molar mass. At 10 km altitude (−50°C), a ≈ 299 m/s — significantly lower than at sea level.
- Applications: Mach number governs aircraft aerodynamic design, jet engine performance (inlet/nozzle shaping), wind tunnel testing, missile guidance, and re-entry vehicle thermal protection design.