⋙ Sonic Boom and Mach Cone
The source moves faster than sound: the waves cannot spread ahead and merge into a shock-wave cone with angle μ = asin(1/M).
Mach number M1,40
Emission rate8 /s
Animation speed×1,0
View:
Examples:
Motion parameters
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Mach number M = v / c
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Mach angle μ = asin(1/M)
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Speed, m/s (for c = 343 m/s)
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Speed, km/h
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Flight regime
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Boom lag h·√(M² − 1)
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Booms heard by the observer
Mach angle: sin μ = c / v = 1 / M Cone condition: M > 1
Boom lag behind the aircraft at altitude h: d = h · cot μ = h · √(M² − 1) Delay: Δt = d / v
Boom lag behind the aircraft at altitude h: d = h · cot μ = h · √(M² − 1) Delay: Δt = d / v
- Every time the source emits a wavefront, it spreads out as a circle at the speed of sound c (75 px/s in the simulation). The source travels at M·c, so for M > 1 it outruns its own wavefronts, and their envelope becomes a cone, marked by the orange lines.
- For M < 1 the wavefronts are always ahead of the source (Doppler effect), and there is no cone. At M = 1 the wavefronts pile up at a single point in front of the source, which corresponds to the “sound barrier”.
- The “With ground and observer” view shows a vertical cross-section of the flight. The orange dot is the place where the cone touches the ground. When it reaches the observer, the boom is heard, and the window on the right shows the N-wave: a sharp pressure jump, a gradual drop below atmospheric pressure, and a return.
- The boom is heard not at the moment the aircraft passes overhead, but later, when the aircraft has already moved away by the distance h·√(M² − 1). The “Boom sound” button plays the double boom heard on the ground. The sound is simulated, so this is only an illustration.
Enter the aircraft’s speed, the air temperature at its altitude, and the flight altitude. The speed of sound depends on temperature, so the Mach number of the same aircraft changes with altitude.
Speed v, km/h
Air temperature, °C
Flight altitude h, m
Examples:
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Speed of sound c, m/s
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Mach number M
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Mach angle μ, °
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Boom lag d, m
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Boom delay after the flyover, s
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Flight regime
c ≈ 331.3 · √(1 + t / 273.15) m/s, M = v / c, μ = asin(1 / M)
d = h · √(M² − 1), Δt = d / v
d = h · √(M² − 1), Δt = d / v
Mach angle and boom lag as a function of Mach number
| M | Angle μ | Lag per 1 km of altitude | Regime |
|---|---|---|---|
| 1,0 | 90,0° | 0 m | sound barrier |
| 1,1 | 65,4° | 458 m | transonic |
| 1,2 | 56,4° | 663 m | supersonic |
| 1,4 | 45,6° | 980 m | supersonic |
| 1,5 | 41,8° | 1118 m | supersonic |
| 2,0 | 30,0° | 1732 m | supersonic |
| 3,0 | 19,5° | 2828 m | supersonic |
| 5,0 | 11,5° | 4899 m | hypersonic (from M = 5) |
Sonic Boom and Mach Cone Online: Simulation and Calculator
A free interactive tool that shows what happens when a sound source moves faster than sound. Change the Mach number and the emission rate and watch the circular wavefronts merge into a shock-wave cone with angle μ = asin(1/M). The ground view shows when an observer hears the sonic boom, along with the pressure N-wave. The calculator computes the speed of sound for a given temperature, the Mach number, the cone angle and the boom delay for an aircraft or a bullet.
Key concepts
Mach number
The ratio of a body’s speed to the speed of sound in the medium. M < 1 is subsonic, M > 1 is supersonic speed.
M = v / c
Mach cone
The envelope of the circular wavefronts from a source moving faster than sound. Sound does not reach ahead inside the cone, and the shock wave is concentrated on its surface.
sin μ = 1 / M
Sonic boom
A sharp pressure jump as the shock wave passes an observer. Usually two booms are heard, separated by about a tenth of a second (N-wave).
Δt = h · √(M² − 1) / v
Where it is used
- Aviation: designing supersonic aircraft and routes that minimize the boom over populated areas
- Ballistics: the shock wave of a bullet or a projectile
- Rocketry and spacecraft re-entry
- Aerodynamics: analyzing shock-wave angles on wings and in nozzles
- Measuring speed from the cone angle in gas-dynamics experiments
- Teaching wave physics and acoustics
How to use
- Look at M = 0.6: the wavefronts spread around the source, there is no cone
- Increase the Mach number to 1 and beyond: a cone appears and narrows as M grows
- Change the emission rate to see more or fewer wavefronts
- Switch to the ground view, place the observer and wait for the boom
- In the “Calculator” tab, enter the speed, temperature and altitude of a real aircraft
Frequently asked questions
How is the Mach cone formed?
Every wavefront emitted by the source spreads out as a circle at the speed of sound c. If the source moves faster (v > c), it outruns its own wavefronts, and all the circles are left behind it. The tangent to these circles forms a cone with its apex at the source, and the shock wave builds up along it.
What formula gives the cone angle?
The half-angle at the apex μ is given by sin μ = c/v = 1/M. So at M = 1.4 the angle is μ ≈ 45.6°, at M = 2 it is 30°, and at M = 5 it is ≈ 11.5°. The faster the source, the narrower the cone.
Why is the sonic boom heard not under the aircraft but later?
The shock wave propagates along the cone trailing behind the aircraft. The cone reaches the ground at a point shifted back by d = h·√(M² − 1), where h is the altitude. The observer hears the boom when this point reaches them, that is, after the aircraft has already moved away by the distance d.
Why does a sonic boom consist of two bangs?
The pressure profile near a supersonic body has an N shape: a sharp pressure rise at the nose, a slow decline, and a sharp recovery at the tail. The two sharp changes are perceived as two bangs, separated by roughly a tenth of a second for a large aircraft.
Is it true that a herdsman hears the “sound barrier” in the crack of a whip?
Yes, the tip of a whip accelerates during the swing to a speed above the speed of sound, and the crack is a small sonic boom. The same happens with a bullet flying faster than sound: its shock wave is a weak Mach cone.