Engineering · Aerospace & Aeronautics
Mach Number Calculator
Calculate the Mach number of an object moving through a fluid by dividing its velocity by the local speed of sound.
Calculator
Formula
M is the Mach number (dimensionless). v is the velocity of the object (m/s). a is the local speed of sound (m/s). γ (gamma) is the adiabatic index of the gas (1.4 for dry air). R is the specific gas constant for air (287.05 J/kg·K). T is the absolute temperature of the fluid in Kelvin (K).
Source: Anderson, J.D. — Introduction to Flight, 8th Ed., McGraw-Hill; ISO 2533:1975 Standard Atmosphere.
How it works
The Mach number (M) is a dimensionless quantity that compares an object's speed to the speed of sound in the medium through which it travels. This ratio is critical because the behavior of a fluid around a body changes dramatically depending on whether the flow is subsonic (M < 1), transonic (M ≈ 0.8–1.2), supersonic (1 < M < 5), or hypersonic (M > 5). Below M = 1, pressure disturbances from the object can propagate ahead of it; above M = 1, shock waves form and the aerodynamic drag behavior changes fundamentally.
The formula is M = v / a, where v is the object's velocity and a is the local speed of sound. The speed of sound itself depends on the fluid's thermodynamic state: for an ideal gas, a = √(γRT), where γ is the adiabatic index (ratio of specific heats), R is the specific gas constant (287.05 J/kg·K for dry air), and T is the absolute temperature in Kelvin. Because temperature decreases with altitude in the troposphere, the speed of sound — and therefore the Mach number for a given true airspeed — varies significantly with altitude. At sea level on a standard day (15°C), the speed of sound in air is approximately 340.3 m/s (1225 km/h or 661.5 knots), while at cruising altitude (~11 km, −56.5°C) it drops to about 295 m/s.
Mach number is used throughout aerospace engineering in aircraft and spacecraft design, nozzle and inlet geometry optimization, wind tunnel testing, ballistic trajectory calculations, and gas dynamics research. Commercial airliners typically cruise at M 0.78–0.85, military fighters operate supersonically up to M 2–3, and experimental vehicles like the SR-71 Blackbird flew at M 3.3. Rocket re-entry vehicles must manage hypersonic conditions above M 5 where aerodynamic heating becomes extreme.
Worked example
Consider a fighter jet flying at a true airspeed of 600 m/s at an altitude where the ambient temperature is −30°C. We want to find its Mach number.
Step 1 — Convert temperature to Kelvin:
T = −30 + 273.15 = 243.15 K
Step 2 — Calculate the local speed of sound:
a = √(γRT) = √(1.4 × 287.05 × 243.15) = √(97,733.7) ≈ 312.6 m/s
Step 3 — Calculate the Mach number:
M = v / a = 600 / 312.6 ≈ M 1.919
This result places the aircraft firmly in the supersonic regime. At this Mach number, the jet is generating oblique shock waves, and its inlet and nozzle geometry must be designed to efficiently handle supersonic compression and expansion. The velocity in more familiar units is 600 × 3.6 = 2160 km/h or approximately 1166 knots.
Limitations & notes
This calculator assumes an ideal gas with constant specific heat ratios, which is accurate for dry air at moderate temperatures but breaks down at very high temperatures (above ~1000 K) where vibrational and dissociative effects become significant — conditions common in hypersonic flight and re-entry. The formula uses the specific gas constant for dry air (287.05 J/kg·K); for other gases or humid air, the gas constant must be adjusted accordingly. The adiabatic index γ also varies with temperature for real gases, so selecting a fixed value introduces approximation error at temperature extremes. This calculator computes the free-stream Mach number and does not account for local Mach variations at specific points on an airfoil or body surface. Additionally, the formula does not apply to liquids or non-ideal flow conditions such as two-phase flows. For flight planning and navigation, note the distinction between true airspeed (TAS) and indicated airspeed (IAS); this calculator uses TAS.
Frequently asked questions
What is Mach 1 in km/h at sea level?
At sea level on a standard day (15°C / 59°F), the speed of sound is approximately 340.3 m/s, which equals about 1225 km/h or 661.5 knots. This value decreases with altitude as temperature drops, so Mach 1 at 35,000 ft is approximately 1062 km/h (573 knots).
Why does the Mach number change with temperature but not pressure?
The speed of sound in an ideal gas depends only on temperature (a = √(γRT)), not directly on pressure, because changes in pressure at constant temperature are accompanied by proportional changes in density that cancel out. As a result, two aircraft at the same altitude but in air masses of different pressures will have the same Mach number for the same true airspeed, provided temperatures are equal.
What is the difference between subsonic, transonic, supersonic, and hypersonic?
These regimes describe distinct aerodynamic flow behaviors: subsonic (M < 0.8) has no shock waves; transonic (M ≈ 0.8–1.2) features mixed sub- and supersonic regions with local shocks forming on the airfoil; supersonic (1.2 < M < 5) produces well-defined shock and expansion waves; and hypersonic (M > 5) involves extreme aerodynamic heating, boundary layer interactions, and real-gas chemical effects.
What value of γ should I use for air?
For dry air at typical atmospheric temperatures (−50°C to +50°C), γ = 1.4 is the standard accepted value. This corresponds to a diatomic ideal gas where translational and rotational modes are fully active. At very high temperatures, vibrational excitation reduces γ below 1.4, and dissociation at extreme temperatures (above ~2500 K) further complicates the value.
Can this calculator be used for flow through a nozzle or duct?
Yes — as long as you input the local flow velocity and local static temperature at the point of interest inside the nozzle or duct, this calculator will correctly return the local Mach number. For nozzle design involving stagnation conditions and isentropic relations, you would also need to incorporate total temperature and pressure calculations alongside this result.
Last updated: 2025-01-15 · Formula verified against primary sources.