Enter the Mach number to calculate the Mach angle, or enter a Mach angle to find the corresponding Mach number.
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Mach Angle Formula
MA = ASIN ( 1/ M) *57.2958
Variables:
- MA is the Mach Angle (degrees)
- M is the Mach number (dimensionless; must be โฅ 1)
The formula is a direct result of Mach cone geometry. When an object moves faster than sound, the pressure waves it emits form a cone. The sine of the half-angle equals the speed of sound divided by the object’s speed: sin(MA) = 1/M.
Mach Angle Reference Table
| Mach Number | Mach Angle | Real-World Reference |
|---|---|---|
| 1.0 | 90.0° | Sound barrier threshold (no cone) |
| 1.2 | 56.4° | Low supersonic fighters in afterburner |
| 1.5 | 41.8° | F/A-18 Hornet max speed (~Mach 1.8) |
| 2.0 | 30.0° | Common supersonic design point |
| 2.04 | 29.4° | Concorde cruise speed |
| 2.25 | 26.4° | F-22 Raptor maximum speed |
| 3.2 | 18.2° | SR-71 Blackbird cruise altitude speed |
| 5.0 | 11.5° | X-51A Waverider scramjet (hypersonic) |
| 25 | 2.3° | Space Shuttle peak re-entry speed |
Mach Cone Geometry
The Mach angle is the half-angle of the Mach cone, the conical shock front that forms around any object exceeding the speed of sound. At exactly Mach 1, the object keeps pace with the pressure waves it generates, producing a flat wave (90ยฐ). As Mach number increases, the object pulls further ahead with each passing moment, drawing those waves into an increasingly narrow cone. The geometry is purely kinematic: in one second, sound travels a distance c, while the object travels v. The resulting right triangle gives sin(MA) = c/v = 1/M.
Altitude Effect on Mach Angle
The speed of sound depends on air temperature, not pressure. Temperature drops with altitude up to the tropopause (~36,000 ft), where it stabilizes at -56.5ยฐC. This means the same physical airspeed corresponds to a higher Mach number (and narrower Mach angle) at altitude than at sea level. An aircraft flying at 3,400 km/h has M = 2.78 at sea level (ฮผ = 21.1ยฐ), but M = 3.20 at 36,000+ ft (ฮผ = 18.2ยฐ) because the speed of sound is 13% slower at altitude.
| Altitude | Temperature | Speed of Sound |
|---|---|---|
| Sea level | 15°C (59°F) | 340 m/s (661 kt / 761 mph) |
| 10,000 ft | -5°C (23°F) | 328 m/s (638 kt / 734 mph) |
| 36,000 ft (tropopause) | -56.5°C (-70°F) | 295 m/s (573 kt / 660 mph) |
| 80,000+ ft (SR-71 cruise) | ~-56.5°C to -50°C | ~295 to 299 m/s |
Mach Angle and Wing Sweep Design
Supersonic aircraft designers use the Mach angle to determine optimal wing sweep. When a wing’s leading edge is swept aft of the Mach line, the leading edge operates in a subsonic flow regime, reducing wave drag significantly. The minimum leading-edge sweep angle (measured from the spanwise axis) must exceed 90ยฐ minus the Mach angle. At Mach 2 (ฮผ = 30ยฐ), the minimum sweep is 60ยฐ. At SR-71 cruise of Mach 3.2 (ฮผ = 18.2ยฐ), it rises to 71.8ยฐ. This relationship drove the 58ยฐ ogival delta of the Concorde and the sharply swept chines of the SR-71.
FAQs
What is the Mach angle at Mach 2?
At Mach 2, the Mach angle is exactly 30°. This follows directly from the formula: arcsin(1/2) = 30°. At this speed, the Mach cone has a 30° half-angle, producing a 60° full apex angle.
Why does the Mach angle decrease as speed increases?
As the object moves faster, it outruns the pressure waves it emits by a greater margin each second. The cone of waves around it becomes narrower. Mathematically, as M increases, 1/M approaches zero, so arcsin(1/M) approaches 0°.
Can a subsonic object have a Mach angle?
No. The formula requires M ≥ 1. At subsonic speeds, the object moves slower than the pressure disturbances it creates, so those waves propagate in all directions ahead of the object and no cone forms.
What is the difference between the Mach angle and the oblique shock angle?
The Mach angle describes an infinitesimally weak disturbance (Mach wave) in uniform supersonic flow. An oblique shock angle depends on both Mach number and the flow deflection angle caused by the body. The Mach angle represents the minimum possible oblique shock angle at a given Mach number. For finite-strength shocks, the actual shock angle is always greater than the Mach angle.
How does altitude affect the Mach angle for a given airspeed?
Higher altitude means cooler air, which lowers the speed of sound, which raises the Mach number for the same indicated airspeed. A higher Mach number produces a smaller Mach angle. An aircraft climbing at constant true airspeed experiences a continuously decreasing Mach angle as it ascends through the troposphere.
What is the Mach number if the Mach angle is 30 degrees?
A 30° Mach angle corresponds to Mach 2. Since sin(30°) = 0.5, and M = 1/sin(MA), we get M = 1/0.5 = 2.0.
