Calculate Mach number from velocity and altitude using the International Standard Atmosphere model. Includes flow regime classification, isentropic relations, and altitude comparison table.
The Mach number is the ratio of an object's speed to the local speed of sound. Named after Austrian physicist Ernst Mach, it is the single most important parameter in compressible aerodynamics, determining whether a flow is subsonic (M < 1), transonic (0.8 < M < 1.2), supersonic (1.2 < M < 5), or hypersonic (M > 5). Each regime brings qualitatively different physics: subsonic flows behave smoothly, while supersonic flows generate shock waves and expansion fans that fundamentally alter drag, lift, and heat transfer characteristics.
The speed of sound depends on the medium's temperature, not altitude directly. In air, a = √(γRT) where γ = 1.4 for diatomic gases, R = 287.058 J/(kg·K), and T is the absolute temperature. Because temperature decreases with altitude in the troposphere (about 6.5°C per kilometer), the speed of sound drops from 340 m/s at sea level to about 295 m/s at the tropopause (11 km). In the lower stratosphere (11–20 km), temperature is nearly constant at −56.5°C, so the speed of sound remains at about 295 m/s.
This Mach number calculator uses the International Standard Atmosphere (ISA) to compute the local speed of sound from altitude, then determines the Mach number and flow regime. It also calculates isentropic stagnation relations — the temperature, pressure, and density ratios that a fluid element would experience if brought to rest — which are critical for designing supersonic inlets, nozzles, and wind tunnels.
Whether you are an aerospace engineering student, a pilot planning flight levels, or a physics researcher, accurately computing the Mach number requires the correct local speed of sound. This calculator removes guesswork by using the ISA model and provides the isentropic relations needed for compressible flow analysis. Keep these notes focused on your operational context. Tie the context to the calculator’s intended domain.
Mach Number: M = v / a. Speed of sound: a = √(γRT). ISA temperature: T = 288.15 − 0.0065·h (troposphere, h ≤ 11 km). Mach cone half-angle: μ = sin⁻¹(1/M) for M > 1. Stagnation temperature ratio: T₀/T = 1 + (γ−1)/2 · M². Stagnation pressure: p₀/p = (T₀/T)^(γ/(γ−1)).
Result: Mach 0.8376 — Transonic regime
At 10,000 m, ISA temp = 288.15 − 65 = 223.15 K. Speed of sound = √(1.4 × 287.058 × 223.15) = 298.5 m/s. Mach = 250/298.5 = 0.8376. This is in the transonic regime where local supersonic pockets begin forming over wing surfaces.
The ISA defines standard temperature, pressure, and density profiles from sea level to 86 km. Key layers:
| Layer | Altitude | Lapse Rate | Base Temp | |---|---|---|---| | Troposphere | 0–11 km | −6.5°C/km | 15°C | | Tropopause | 11–20 km | 0°C/km | −56.5°C | | Stratosphere | 20–32 km | +1°C/km | −56.5°C | | Upper strato. | 32–47 km | +2.8°C/km | −44.5°C |
| Vehicle | Year | Mach | Notes | |---|---|---|---| | Bell X-1 (Chuck Yeager) | 1947 | 1.06 | First piloted supersonic flight | | Concorde | 1976 | 2.04 | Supersonic commercial service | | SR-71 Blackbird | 1976 | 3.32 | Fastest air-breathing jet | | X-15 | 1967 | 6.7 | Hypersonic research aircraft | | Space Shuttle | 1981 | ~25 | Orbital reentry |
Below Mach 0.3, air behaves as essentially incompressible and density changes are negligible. Between 0.3 and 0.8, compressibility corrections (Prandtl-Glauert) become needed. Above Mach 0.8, shock waves begin forming on wing surfaces. Above Mach 1, the entire flow field reorganizes around oblique shocks and expansion fans, requiring supersonic aerodynamic theory.
The speed of sound depends on air temperature, not altitude or pressure directly. Because temperature decreases with altitude in the troposphere (−6.5°C/km), the speed of sound decreases. In the stratosphere, temperature stabilizes, and so does the speed of sound.
When an object moves faster than sound (M > 1), it outruns its own pressure waves. These accumulate along a cone whose half-angle is μ = sin⁻¹(1/M). At Mach 2, the cone half-angle is 30°. This cone boundary is experienced as a sonic boom on the ground.
Stagnation (total) conditions are the temperature, pressure, and density a fluid element would reach if brought isentropically to rest. They are always higher than static conditions and increase rapidly with Mach number — at Mach 3, stagnation temperature is 2.8× static temperature.
Even when the aircraft flies below Mach 1, air accelerates over curved surfaces (especially wings) to locally supersonic speeds. This mixed-flow region creates shock waves on the wing surface, causing wave drag and buffeting that require special transonic airfoil designs.
At hypersonic speeds, stagnation temperature is so high that air molecules dissociate and ionize. Aerodynamic heating becomes the dominant design challenge, requiring ablative heat shields or active cooling. The Space Shuttle reentered at about Mach 25.
Yes. Sound travels at ~1,480 m/s in water and ~5,960 m/s in steel. The Mach number is always defined relative to the local speed of sound in the medium.