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/M): 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/M), 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.
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