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Aerospace
Mach angle of a weak disturbance
Angle between the flow direction and a weak Mach wave in uniform supersonic flow at Mach number M > 1.
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- Result
- μ [°]
A small disturbance in a supersonic stream cannot travel upstream; it spreads as a Mach wave at the angle μ = arcsin(1/M) to the flow. At Mach 2 this is 30°, and the angle narrows as M rises. The region ahead of the wave is the zone of silence. A shock of finite strength is stronger and steeper, at an angle greater than μ.
Angle between the flow direction and a weak Mach wave in uniform supersonic flow at Mach number M > 1. The page gives it in degrees. A finite-strength shock is steeper than this wave, with a larger angle.
Calculator
- Sine of the angle, 1/M0.5
- Tangent of the angle, 1/√(M² − 1)0.5774
- Mach angle in radians0.5236 rad
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The Mach waves lie at 30° to the flow, so a source is heard only inside a cone of half-angle 30° behind it.
- A body moving at this speed travels 1.732 times faster than a disturbance spreads sideways, which is why the wave angle is set by that ratio.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| M Mach number | 1.01 to 10 | 5.739 | 81.93 | 76.19 | +10% | −2.96 ° |
Sweep of Mach number
| Mach number | Mach angle (°) |
|---|---|
| 1.01 | 81.93 |
| 2.134 | 27.95 |
| 3.258 | 17.88 |
| 4.381 | 13.19 |
| 5.505 | 10.47 |
| 6.629 | 8.677 |
| 7.752 | 7.411 |
| 8.876 | 6.469 |
| 10 | 5.739 |
Sweep one variable
Variables and how each one contributes
Mach angle
What the equation gives you. Everything else feeds it.
Mach number
The upstream Mach number, greater than 1. A higher Mach number gives a narrower Mach cone. Raising it by 10 % (to 2.2) moves the result by −9.88 %, from 30 to 27.04 °.
Worked example with the values above
- Ratio: 1/M = 1/2 = 0.5.
- Arcsine: asin(0.5) = 0.523599 rad.
- μ = 30°.
Assumptions and limits
- Calorically perfect gas with a constant heat-capacity ratio, and isentropic deceleration to the pitot tube.
- Subsonic flow for the pitot and calibrated-airspeed formulas; a shock forms ahead of the probe above Mach 1.
Common mistakes
- Using the incompressible ½ρV² formula at high Mach number, where it underestimates impact pressure.
- Confusing true, equivalent and calibrated airspeed.
- Treating a recovery temperature as a static temperature.
Statistics and uncertainty
Air-data conversions are exact for the stated models, so errors come from the instrument: pitot and static ports carry position errors of 1 % to 2 % in speed, and a poorly designed temperature probe can have a recovery factor that differs from 0.9 by several per cent.
Where it is used
- Converting instrument readings to speed, Mach number and static temperature.
- Understanding what the airspeed indicator and temperature probes really measure.
Maths you need
- Stagnation point: where the flow is brought to rest
- Dynamic pressure: q = ½ρV²
- Speed of sound: a = √(γRT)