A moving body sends out sound in every direction, all the time, and each burst spreads as a circle at the speed of sound from wherever the body was. Slower than sound, the body stays inside its own circles: they crowd together ahead of it and spread out behind — the Doppler effect, and the reason a passing car's note drops. At exactly the speed of sound every circle touches the same point, the nose, and the pressure piles up into a wall. Faster than that, the body leaves its circles behind, and all of them are tangent to one cone.
In a time t the body travels vt and its first circle grows to a radius of at. The cone's edge is the tangent from the body to that circle, so sin μ = at/vt = 1/M. At Mach 2 the half-angle is 30°; at Mach 1.1 it is 65°, nearly flat; a rifle bullet at Mach 2.7 drags a cone 22° either side of its path.
A sonic boom is not an event that happens when something "breaks the barrier". It is that cone sweeping over you, and it sweeps over everyone along the whole route for as long as the flight stays supersonic. What you hear is two cracks a fraction of a second apart — one from the bow shock, one from the tail — because the pressure jumps up, falls through ambient, and jumps back: an N-wave.
It is tempting to think sound is slower in thin air. It is slower up there, but not because the air is thin. In an ideal gas the speed of sound is √(γRT/Mmol), where Mmol is the gas's molar mass (not the Mach number): temperature, and what the gas is made of. Pressure cancels out entirely, because thinner air is also lighter air, and the two effects are exactly equal. The stratosphere is slow because it is −56.5 °C, and above 20 km, where it warms again, sound speeds up again.
Mach number is a ratio, not a speed. Mach 2 at sea level on a standard day is 681 m/s; at 11 km it is 590 m/s, thirteen per cent slower through the air for the same number. Aircraft limits are quoted in Mach high up precisely because it is the ratio, not the speed, that decides how the air behaves around the wing.
Because sound bends towards the slower air, and the air near the ground is warmer and faster. Follow the shock downward: its angle to the horizon shrinks as the sound around it speeds up, and if it reaches a height where the local speed of sound matches the body's speed, it is travelling horizontally — and then it bends back up. That is Mach cutoff. At 11 km on a standard day it happens below about Mach 1.15, which is the whole idea behind quiet supersonic flight over land: go fast, but not so fast that the ground can hear.
No. The cone exists the whole time the body is supersonic, so a boom is heard along the entire route, by everyone the cone sweeps across. The barrier crossing itself is not special on the ground; what is special is that the cone first exists then.
Ahead of a moving source the wavefronts are squeezed together and behind it they are stretched out, so you hear the note raised as it approaches, by a factor of 1/(1−M), and lowered as it leaves, by 1/(1+M). The drop between the two is what you hear. A car at 100 km/h is Mach 0.08, and the note falls by nearly three semitones — enough to hear clearly, as everyone has.
An ideal gas for the speed of sound, √(γRT/Mmol), with a constant ratio of specific heats. Real air departs from it by well under a tenth of a per cent in the range here; humidity raises the speed slightly (about half a per cent in warm, saturated air), and that is not modelled.
The International Standard Atmosphere for temperature with height: 15 °C at sea level falling 6.5 °C per kilometre to −56.5 °C at 11 km, constant to 20 km, then warming 1 °C per kilometre — shifted up or down by the day-temperature offset. Real days have inversions and weather this does not.
Still air. Wind shifts where a boom lands and can cause or prevent cutoff on its own; it is not modelled. The boom's path to the ground is the acoustic ray through the layered temperature profile, and only straight down the flight path — the width of the boom carpet either side is not computed.
Choosing another gas fills the whole atmosphere with it at the same temperatures, which no atmosphere does; it is there to show what γ and molar mass do. The flyby sound is a synthesis of the Doppler shift and the N-wave, not a recording.
A shock heading down into warmer air bends, because sound bends towards where it travels slower. If it meets air where sound is as fast as the aircraft, it runs level and then curves back up: the boom is refracted away before it lands. That gives a speed limit for silence at every height and temperature. It is the fastest sound speed anywhere between the aircraft and the ground, divided by the sound speed at the aircraft. On a standard day at 11 km that is 340.3 ÷ 295.1, about Mach 1.15.
1. Stay under the cutoff speed (“Mach cutoff”). Boom Supersonic reported that its XB-1 demonstrator flew at up to Mach 1.12 on 28 January 2025, and at Mach 1.18 at about 36,500 ft on 10 February, over the Mojave, with ground microphones recording no boom. The company calls this “Boomless Cruise” and says its planned Overture airliner could fly overland this way at up to about Mach 1.3, depending on the day. The Mach 1.18 flight is above this page's standard-day cutoff at that height. Real temperatures and winds set the limit, not the textbook atmosphere.
2. Let the weather help. What bends the shock is the sound speed plus the wind along the flight path. A day that is hot at the surface, or a wind that grows along the direction of flight near the ground, raises the cutoff speed. The same flight can be boomless heading one way and audible heading back. This page shifts the whole atmosphere by one day-temperature offset and has no wind, so it shows the principle, not a flight plan.
3. Reshape the boom instead of bending it. NASA's X-59 takes a different route: the boom does reach the ground, but a long, carefully shaped nose and body keep its shocks from merging into the two sharp jumps of an N-wave, so the ground should hear a soft thump, not a crack. It first flew in October 2025 and first went supersonic in June 2026. NASA plans to fly it over communities to measure how people respond.
Not quite. Under a cutoff flight a weak, low rumble can still leak through to the ground, but it has nothing like the double crack of a boom that lands.
The United States has prohibited civil supersonic flight over land since 1973 (14 CFR 91.817), by speed rather than by noise, so a flight with no boom on the ground was still not allowed. An executive order in June 2025 directed the FAA to replace the ban with a noise-based standard. The FAA published that proposal in July 2026. Until the final rule takes effect, the old limit still applies.
A shock reaches the ground only if the aircraft is faster than the speed of sound at the ground, not merely at its own altitude: 340 m/s there against 295 at 11 km, the air some 70 °C warmer. Follow the shock down and the sound speeds up, the margin shrinks and the front flattens; where the local speed of sound equals the aircraft’s it runs horizontally and turns back up. That is Mach cutoff, why a jet at Mach 1.1 in the stratosphere can be supersonic and silent below, the basis of quiet supersonic flight over land.
The cone reaches you long after the aircraft was overhead: by the time you hear the double crack the aircraft is twenty kilometres further on and you are looking in the wrong place.
The tip is the thinnest, lightest part and the throw’s energy runs out along it into less and less mass, so it speeds up and passes the speed of sound; the crack is its Mach cone, the first man-made object to break the barrier, thousands of years before aircraft.
Near Mach 1 the air accelerating over the aircraft expands and cools, and in humid air the water condenses into a cone of cloud, then re-evaporates at the shock behind it. It is photographed constantly and can appear at Mach 0.9.
Two aircraft at the same true airspeed can be at different Mach numbers because the air around them is at different temperatures. That is why airliners cruise at, and have ceilings in, a Mach number: it is the ratio that decides where shocks form on the wing.
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