Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Sonic boom — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. One signature, aged four times. The pressure signature of a slender body at four distances, computed by the exact Lax formula for the nonlinear propagation. Each point of the waveform moves forward in proportion to its own overpressure, so the compression at the front catches the undisturbed air and a shock forms there, while the expansion at the rear falls behind and forms a second one. What is left is an N-wave: two discontinuities and a straight line between them, spreading and weakening.

    The signature that forgets the shape

    The pressure field near a supersonic aeroplane depends on every part of it. What reaches the ground has two parameters. Two bodies whose near-field signatures differ by fifty-five per cent in peak and by their whole shape age into the same N-wave, to two and a half per cent.

    part 1 · compressible
  2. Below Mach 1.153 the boom turns back before it reaches the ground. The ray leaving the Mach cone straight down from an aeroplane at 11 km, at Mach 1.1, 1.15, 1.2, 1.5, 2, traced through a standard atmosphere whose sound speed rises from 295.1 m/s at the aeroplane to 340.3 m/s at the ground. A ray bends back upward where the local sound speed equals the aeroplane's speed, so Mach 1.1: turns at 4.00 km; Mach 1.15: turns at 0.25 km; Mach 1.2: lands 24.0 km on; Mach 1.5: lands 11.4 km on; Mach 2: lands 7.1 km on. The dividing speed, Mach 1.1533, is the ratio of the two sound speeds.

    The boom that turns back before the ground

    Sound is faster in the warm air near the ground, so a sonic boom's rays bend back upward on the way down. Whether any of them arrive is one comparison — the aeroplane's speed against the fastest sound beneath it — and the ray that just grazes the ground sets the edge of the carpet, which the uniform air of the ageing calculation cannot give it.

    part 2 · compressible
  3. The arrival map, and the place where it goes backwards. Where each boom ray lands on the ground, against the Mach number the aeroplane was doing when it launched it, for four accelerations from 15 km. In level flight this would be a straight line rising at the aeroplane's own speed. Here it falls before it rises: a ray launched at a higher Mach number is shorter and steeper, and near the cut-off it shortens faster than the aeroplane advances. Every minimum in these curves is a fold — two emission times delivering to one place — and on the fold itself neighbouring rays converge onto a single line.

    The carpet an accelerating aeroplane folds

    Level flight launches every boom ray with the same invariant, so they run parallel and each place hears one boom. Accelerate, and each successive ray is shorter than the last — shorter, near the cut-off, than the aeroplane's own advance — so later rays overtake earlier ones and the arrival map folds onto a line.

    part 3 · compressible
  4. What the rays say the edge is: nearly as loud, and then nothing. The overpressure across the carpet relative to the value under the track, from ray-tube spreading alone, at three Mach numbers from 15 km. It falls gently and then stops: at Mach 1.8 it is 0.92 ten kilometres out, 0.77 at twenty-three and 0.65 on the last ray, 36 km out, with silence beyond. Geometrical acoustics says the carpet ends at a cliff, and the boom at the edge of a real carpet fades as a rumble. The cliff is what the model says; it is also where the model stops.

    The edge is a rumble, not a quieter bang

    The rays that reach the outer half of a sonic-boom carpet arrive nearly horizontally, having travelled almost three times as far as the one under the track. Ray theory says they still carry two-thirds of the overpressure, right up to a line beyond which there is nothing. Neither half of that is what is heard — which is the useful result, because it says the edge's loudness is not a ray quantity at all.

    part 4 · compressible
  5. A boom gathers most of its age in the thin air near the aeroplane. The share of the total age gathered above each height, for the ray under the track and the last ray computed near the carpet's edge, with the share of the path length travelled above each height for comparison. Under the track 66 per cent of the age is gathered above the tropopause in 26 per cent of the path. The edge ray spends most of its path in the lowest few kilometres and gathers only 11 per cent of its age below 3 km, because the same pressure distorts dense air far more slowly than thin air.

    A boom is aged in the thin air it starts in

    The rays that reach the edge of a sonic-boom carpet travel two and a half times as far as the one under the track, and it is natural to expect their signatures to have aged accordingly. They have not. A pressure wave distorts thin air far faster than dense air, so two-thirds of a boom's ageing is done in the stratosphere near the aeroplane, and the extra kilometres near the ground add little — which decides how far out a boom shaped to be quiet stays quiet.

    part 5 · compressible

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