Mach Number Calculator
Mach Number Explained: A Complete Guide to Subsonic, Transonic, and Supersonic Flow
Mach number is the ratio of an object’s speed to the speed of sound in the surrounding medium. It is the key non-dimensional parameter in compressible flow and aerodynamics because many physical behaviors— shock waves, drag rise, aero heating, and noise—change dramatically as a vehicle crosses Mach 1. Formally,
where \(v\) is the flow or vehicle speed and \(a\) is the local speed of sound. If \(M=0.70\), the flow is subsonic; if \(M=1\), it is traveling at the speed of sound; if \(M=2\), it is supersonic, and so on. Because the speed of sound depends on the gas and its temperature, the same ground speed can correspond to different Mach numbers at different altitudes or in different gases (air vs. helium).
Speed of Sound Depends on Temperature, Gas Properties, and Altitude
For a calorically perfect gas, the local speed of sound is
where \(\gamma\) is the ratio of specific heats (≈1.4 for dry air near standard conditions), \(R\) is the specific gas constant (287.05 J/kg·K for air), and \(T\) is the absolute temperature (K). Because temperature decreases with altitude in the lower atmosphere, \(a\) typically decreases with altitude—so the same aircraft true airspeed can correspond to a higher Mach number at cruising altitude than at sea level.
Typical “sea level standard day” values yield \(a \approx 340\ \mathrm{m/s}\) (about 761 mph). At 11 km (≈36,000 ft), the stratosphere’s lower temperature can drop \(a\) to roughly \(295\ \mathrm{m/s}\), increasing Mach number at the same true airspeed.
Flow Regimes by Mach Number
- Subsonic: \(M \lesssim 0.8\). Compressibility effects are modest; shocks do not form.
- Transonic: \(0.8 \lesssim M \lesssim 1.2\). Mixed subsonic/supersonic pockets appear; shocks form and move; drag rises sharply.
- Supersonic: \(1.2 \lesssim M \lesssim 5\). Well-defined shock waves; strong compressibility effects dominate design.
- Hypersonic: \(5 \lesssim M \lesssim 10\). Intense aerodynamic heating; thin shock layers; high-temperature chemistry may begin.
- High-hypersonic: \(M \gtrsim 10\). Real-gas effects and ablation, radiation, and dissociation become important.
Unit Conversions & Typical Values
Mach number is dimensionless, but you’ll often convert speeds for context. Useful conversions:
- 1 m/s = 3.6 km/h ≈ 2.237 mph ≈ 1.944 knots
- At sea level on a standard day, \(M=1\) ≈ 340 m/s ≈ 1,225 km/h ≈ 661 knots ≈ 761 mph
Example: A jet cruising at 250 m/s. At sea level \(a \approx 340\ \mathrm{m/s}\), so \(M \approx 0.74\). At 11 km altitude \(a \approx 295\ \mathrm{m/s}\), so \(M \approx 0.85\). Same speed, different Mach numbers due to temperature.
How to Calculate Mach Number (Step by Step)
Case A — Speed and Temperature Known
1) Convert temperature to kelvin if needed (K, °C, °F → K). 2) Compute \(a\). 3) Divide your speed by \(a\) to get \(M\).
Case B — Speed and Speed of Sound Known
If your instrumentation already provides \(a\) (or “local sonic speed”), just divide.
Case C — Estimating Temperature
If you don’t have temperature, use a standard-atmosphere estimate for your altitude or a measured outside air temperature (OAT). Remember Mach number is based on the static temperature, not stagnation temperature; onboard avionics handle this for pilots.
Why Mach Number Matters for Design and Safety
- Drag Rise: Near \(M\approx0.8\)–1.0, wave drag increases sharply; wings, nacelles, and inlets must be shaped to delay/soften shocks.
- Shock Waves & Buffet: Shocks can cause boundary-layer separation and buffet; modern wings use supercritical airfoils to manage this.
- Thermal Effects: At supersonic and especially hypersonic speeds, aerodynamic heating dictates materials and leading-edge design.
- Noise: Sonic booms (overpressure waves) are a direct consequence of shock formation at supersonic speed.
- Engine Performance: Compressors, turbines, and inlets are sensitive to Mach number for efficiency and stall margin.
Common Mistakes & Best Practices
- Confusing ground speed with airspeed: Mach number uses speed relative to the air, not the ground.
- Ignoring temperature: Assuming a constant “speed of sound” leads to wrong Mach estimates at altitude or in different gases.
- Using stagnation temperature by accident: Ensure you’re using static temperature for \(a=\sqrt{\gamma R T}\) unless your instrument converts appropriately.
- Mixing units: Convert all quantities consistently (m/s with SI \(R\) and \(T\) in kelvin).
Applications: Aviation, Space, Turbomachinery, Automotive, and Sports
Aviation: Cruise schedules and certification often specify Mach number (e.g., Mach 0.85). Space & reentry: Hypersonic vehicles encounter extreme heating and shock interactions. Turbomachinery: Blade tip Mach affects efficiency and noise. Automotive & cycling: Even at subsonic speeds, compressibility can matter for very fast prototypes, while wind-tunnel corrections reference local Mach.
Mach Number FAQ
Is Mach 1 always 343 m/s?
No. 343 m/s is a common textbook value for 20°C air at sea level. The true value varies with temperature and gas composition. Use \(a=\sqrt{\gamma R T}\).
What is “transonic” speed?
Transonic refers to the range where parts of the flow are subsonic and other parts are supersonic, typically from about Mach 0.8 to 1.2. Shock waves appear and move over surfaces, creating drag rise and buffeting.
Can Mach number be less than zero?
Mach number is a magnitude ratio and is non-negative. Direction is handled by velocity vectors; Mach itself is conventionally ≥ 0.
Why do pilots fly by Mach instead of knots at altitude?
Because the speed of sound changes with temperature, maintaining a target Mach number better preserves aerodynamic and compressibility conditions across altitude and temperature changes than holding a fixed indicated airspeed.
Does humidity change Mach 1?
Slightly. Humid air has a different effective gas constant and can marginally alter \(a\), but temperature is the dominant factor for most calculations.