Mach Number
The Mach number is the ratio of an object's speed to the local speed of sound. Below Mach 1 the flow is subsonic, at Mach 1 it is transonic, above Mach 1 supersonic, and above Mach 5 hypersonic.
M = v / a
Velocity from Mach Number
Rearranges the Mach equation to find the actual velocity when you know the Mach number and the local speed of sound. Useful for converting between Mach and absolute speed units.
v = M × a
Speed of Sound from Mach Number
Determines the local speed of sound when you know the object's velocity and Mach number. The speed of sound depends on the fluid and its temperature.
a = v / M
How It Works
The Mach number expresses how fast something moves compared with the speed of sound in the surrounding medium. When Ma is less than 1 the flow is subsonic; at Ma near 1 it enters the transonic zone where shock waves first appear; above Ma 1 it is supersonic; and beyond Ma 5 it is hypersonic, where extreme heating dominates the aerodynamics. Because the speed of sound depends on temperature rather than pressure, the same vehicle speed corresponds to a higher Mach number at colder, higher altitudes. In standard sea-level air at 20 °C the speed of sound is about 343 m/s (767 mph).
Example Problem
A fighter jet flies at 680 m/s at an altitude where the speed of sound is 310 m/s. Determine the aircraft's Mach number and classify its flow regime.
- Identify the known quantities: object velocity v = 680 m/s, local speed of sound a = 310 m/s.
- Write the Mach number definition: M = v / a.
- Substitute the values: M = 680 / 310.
- Compute the division: M = 2.193 (rounded to three decimal places).
- Compare to regime thresholds: M > 1 means the jet is supersonic.
- Interpret the result: at Mach 2.19 oblique shock waves form on the aircraft's leading edges, and compressibility effects dominate the aerodynamics.
At Mach 2.19 the jet is well into the supersonic regime. Oblique shock waves form on its leading edges, and aerodynamic drag is significantly higher than at subsonic speeds.
When to Use Each Variable
- Solve for Mach Number — when you know the velocity and speed of sound and want to determine the flow regime (subsonic, supersonic, etc.).
- Solve for Velocity — when you know the Mach number and speed of sound and want to find the actual speed in standard units.
- Solve for Speed of Sound — when you know the velocity and Mach number and want to determine the local acoustic conditions.
Key Concepts
The Mach number classifies flow into four regimes: subsonic (M < 1), transonic (M near 1), supersonic (M > 1), and hypersonic (M > 5). Each regime exhibits fundamentally different aerodynamic behavior. The speed of sound depends on temperature, not pressure — in an ideal gas, a = sqrt(γ × R × T). At altitude, lower temperatures reduce the speed of sound, so the same airspeed corresponds to a higher Mach number.
Applications
- Aircraft design: determining drag, lift, and heating characteristics at different flight regimes
- Wind tunnel testing: setting test conditions to match operational Mach numbers for scale models
- Rocket propulsion: analyzing nozzle flow where exhaust gases accelerate through Mach 1 at the throat
- Ballistics: classifying projectile speeds and predicting shock wave patterns
Common Mistakes
- Assuming Mach 1 is a fixed speed — it varies with temperature; Mach 1 is about 767 mph at sea level but only 660 mph at cruising altitude
- Using sea-level speed of sound at altitude — the speed of sound drops with temperature, so the same airspeed yields a higher Mach number at altitude
- Neglecting compressibility effects near Mach 1 — aerodynamic coefficients change drastically in the transonic region and cannot be extrapolated from subsonic data
Frequently Asked Questions
What does the Mach number tell you about an object's speed?
The Mach number expresses speed as a multiple of the local speed of sound. Ma = 0.5 means the object moves at half the speed of sound; Ma = 2 means twice the speed of sound. This ratio determines whether shock waves, compressibility drag, and aerodynamic heating are significant.
How fast is Mach 1 in miles per hour?
At standard sea-level conditions (15 °C), Mach 1 is about 767 mph (1,235 km/h or 343 m/s). At jet cruising altitude where the temperature is roughly −57 °C, Mach 1 drops to about 660 mph because colder air carries sound more slowly.
Why does the speed of sound change with altitude?
Sound propagates through molecular collisions, and temperature controls how fast those molecules move. In the troposphere, temperature drops about 6.5 °C per kilometer, so sound slows down. In the stratosphere, temperature levels off and the speed of sound stabilizes near 295 m/s.
What is the difference between subsonic, transonic, and supersonic?
Subsonic flow (Ma < 0.8) behaves like an incompressible fluid with smooth streamlines. Transonic flow (Ma 0.8–1.2) is a mixed regime where pockets of supersonic flow create local shock waves and drag rises sharply. Supersonic flow (Ma > 1.2) features well-defined oblique shocks and expansion fans.
What happens to drag at the sound barrier?
As an aircraft approaches Mach 1, normal shock waves form on its surfaces and the drag coefficient can double or triple — the so-called transonic drag rise. With sufficient thrust the aircraft punches through, and drag decreases somewhat in the fully supersonic regime beyond Mach 1.2.
How hot does an aircraft get at hypersonic speeds?
At Mach 5 and above, air friction heats leading edges to 1,000–2,000 °C or more. Spacecraft re-entering at Mach 25 can exceed 3,000 °C. Materials such as reinforced carbon-carbon or ceramic tiles are needed to survive these temperatures.
Can the Mach number apply to fluids other than air?
Yes. The Mach number works in any compressible medium — helium, steam, water, even stellar plasma. The only change is the local speed of sound, which depends on the medium's bulk modulus and density. In water, sound travels at roughly 1,480 m/s, so a torpedo at 100 m/s is only Ma ≈ 0.068.
Mach Number Formula
The Mach number quantifies how fast an object travels relative to the surrounding speed of sound:
Ma = v / a
Where:
- Ma — Mach number (dimensionless ratio)
- v — velocity of the object (m/s, mph, km/h, or knots)
- a — local speed of sound in the medium (m/s)
Because the speed of sound varies with temperature and medium, the same vehicle speed can correspond to different Mach numbers at different altitudes or in different gases. In dry air at 20 °C, the speed of sound is approximately 343 m/s (767 mph).
Worked Examples
Aviation
What is the Mach number of a commercial jet at cruise speed?
A Boeing 787 cruises at about 252 m/s (565 mph) at 35,000 ft, where the speed of sound is roughly 295 m/s due to the colder air.
- Ma = v / a = 252 / 295
Ma = 0.854
At Mach 0.85 the jet is high subsonic. Commercial airliners stay below Mach 1 to avoid the transonic drag rise and shock-wave buffeting.
Aerospace
How fast does a re-entering spacecraft travel in Mach?
A capsule re-enters the atmosphere at roughly 7,800 m/s. At 80 km altitude the speed of sound is about 310 m/s.
- Ma = v / a = 7800 / 310
Ma = 25.16
Re-entry at Mach 25 is deep in the hypersonic regime. Extreme aerodynamic heating demands ablative heat shields or thermal-protection tiles.
Wind Tunnel Testing
What tunnel flow speed reproduces Mach 2 for a scale model?
A supersonic wind tunnel operates with air at 20 °C (speed of sound 343 m/s). The test requires Mach 2 flow over a 1:10 scale wing model.
- v = Ma × a = 2 × 343 m/s
v = 686 m/s
The tunnel must accelerate air to 686 m/s. Engineers often cool or pressurize the working gas to reach the target Mach number at lower absolute speeds, reducing structural loads.
Speed of Sound in Common Media
The Mach number always compares an object's speed to the speed of sound in the medium it is moving through, so the reference speed changes with the gas, liquid, or solid. Mach number is most often quoted relative to air (~343 m/s at 20 °C), but sound travels far faster through stiff, dense solids. Click a row to load that speed of sound into the calculator, keeping your current velocity or Mach number (if you are currently solving for the speed of sound, the calculator switches to solving for the Mach number so the looked-up speed becomes an input).
| Medium | Speed of sound (m/s) |
|---|---|
| Rubber | 60 |
| Air (20 °C) | 343 |
| Helium (20 °C) | 1,007 |
| Lead | 1,210 |
| Water (25 °C) | 1,493 |
| Seawater | 1,531 |
| Ice | 3,200 |
| Gold | 3,240 |
| Oak (wood) | 3,850 |
| Copper | 4,600 |
| Glass | 5,640 |
| Steel | 5,960 |
| Aluminum | 6,420 |
| Diamond | 12,000 |
Values at typical conditions (gases ~20 °C). The speed of sound rises with temperature in gases and varies with composition, density, and stiffness in liquids and solids, so treat these as representative figures. Because Mach number is most often referenced to air, aviation and aerospace speeds usually assume the local speed of sound in the atmosphere. Synthesized from standard references (CRC Handbook of Chemistry and Physics; NIST; Engineering ToolBox). See the sound wave speed calculator to compute the speed of sound from a medium's properties.
Related Calculators
- Cauchy Number Calculator — compressibility ratio equal to Mach² for an ideal gas
- Eckert Number Calculator — assess viscous dissipation heating in high-speed flows
- Knudsen Number Calculator — determine rarefied-gas flow regime at high altitudes
- Reynolds Number Calculator — characterize whether flow is laminar or turbulent
- Sound Wave Calculator — compute the speed of sound used in the Mach number formula
- Speed Unit Converter — convert between m/s, mph, km/h, and knots
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