The edge is a rumble, not a quieter bang

The boom that turns back before the ground found where a sonic boom lands and said, in its own list of limits, what it had not computed: “A ray says where energy goes, not how much arrives.” It ended by asking what the edge of a carpet sounds like, and noting that the edge ray travels nearly three times as far as the one under the track without being able to say what that does.

This is the amplitude calculation, and it has an unusual shape. It arrives at a definite answer — ray-tube spreading keeps the overpressure near the carpet’s edge at about two-thirds of the value under the track, and then, one ray further out, there is no ray and no boom. That answer is wrong at the edge, and knowing exactly how it is wrong is the point. A real carpet does not end at a cliff, and what is heard near its edge is not two-thirds of the bang under the track but a long low rumble. So the calculation’s value is in establishing that the edge’s loudness is not a ray quantity, and in supplying the numbers that say why.

What a ray tube carries #

Between two neighbouring azimuths, a tube of rays carries a fixed acoustic power — that is the content of the Blokhintzev invariant, and it is the acoustic version of the statement that a beam of light carries its energy along itself. The pressure amplitude is therefore set by the tube’s cross-section: squeeze it and the pressure rises, spread it and the pressure falls.

Two things set it at the ground, and both can be computed from the ray tracer the cut-off calculation wrote.

The first is how fast the landing point moves with azimuth. Near the track a degree of azimuth moves the landing point a modest distance; near the edge it moves it enormously, because the ray is bending through nearly horizontal and a small change in launch angle turns into a large change in where it comes down. The same power spread over more ground is less pressure.

The second is how obliquely the ray arrives, and it works the other way. Under the track at Mach 1.8 the ray meets the ground at 50 degrees to the horizontal; at the edge it grazes. The pressure is set by the tube’s cross-section normal to the ray, and a footprint on the ground is that cross-section seen at a slant: the footprint is longer than the tube by one over the cosine. So a given width of footprint corresponds to a narrower tube when the arrival is oblique, and a narrower tube is more pressure.

The tube’s area is therefore the footprint width times the cosine, and the amplitude goes as one over the square root of that product. An earlier version of this calculation had the cosine upside down — dividing the footprint by it rather than multiplying — which made the two effects compound instead of cancel and put the edge at two and a half per cent of the under-track boom. The check that catches it is simple and is now in the ledger: in air of one temperature the rays are straight, the tube is a cone, and the amplitude must fall as one over the square root of the distance from the track. The corrected calculation does that to four figures; the old one did not.

The wave being carried is not a sound wave in the ordinary sense, which matters for what follows. Every compression becomes a shock in the end is the statement that a finite-amplitude wave steepens until it has a discontinuity in it, and the discontinuity has a thickness set by viscosity — a few mean free paths in principle, and far more in a real signature because the rise is spread by atmospheric relaxation. The rise time is what makes a boom a bang.

Two effects that almost cancel #

It helps to see the two factors as one statement about geometry rather than as two corrections.

Think of the cone of rays leaving the aeroplane as a fan, and follow a wedge of it between two azimuths. Straight down, the wedge descends steeply and its footprint on the ground is close to its own cross-section. Out to the side, three things happen at once. The ray bends more, so the wedge is stretched laterally, which dilutes it. It arrives more obliquely, so a given cross-section lands on a longer footprint — which means the long footprint the stretching produced is less tube than it looks. And it has travelled further, so whatever the wave loses on the way it has lost more of.

Only the first two are in this calculation, and they are the ones a linear theory can supply. At the edge they become singular together: the lateral spreading diverges because the landing point runs away with azimuth, and the cosine vanishes because the ray comes in flat. Their product does neither. It stays finite right up to the grazing ray, so the amplitude does too.

That is the reason ray theory is qualitatively wrong at the edge, and it is a different reason from the one it might have been. The rays do not say the edge is quiet. They say it is loud right up to a line, and silent one step past it — a discontinuity in overpressure across a line on the ground, with nothing in the physics to make one. An error that is a factor of two somewhere is a modelling limitation; a cliff in a field of sound is a statement that the quantity being computed is not the quantity that matters there.

The answer: a gentle fall and then a cliff #

At Mach 1.8 from 15 km the carpet’s half-width is 37.8 km, and the overpressure is 0.93 of the under-track value a quarter of the way out, 0.81 halfway, 0.73 three-quarters of the way and 0.65 at the last ray computed, 95 per cent of the way to the edge. At Mach 1.3 the half-width is 22.7 km and the corresponding figures are 0.98, 0.93, 0.89 and 0.86. At Mach 2.5 the half-width is 43.7 km and they are 0.91, 0.77, 0.68 and 0.61.

The practically useful reading is that the boom does not halve anywhere on the carpet. Spreading alone takes off a third at most, and less at low Mach number, where the carpet is narrow and its rays are nearly alike. And the Mach 1.8 and Mach 2.5 curves lie almost on top of each other against distance: the faster aeroplane’s carpet is wider, but at a given distance from the track the two booms have spread by the same amount.

What a carpet’s width is worth knowing for #

A boom’s nuisance is not a binary. What is regulated, argued about and measured is the overpressure — pounds per square foot in the older literature, pascals in the newer — and a threshold somewhere in the middle of the range decides whether a flight is acceptable over a given place. A carpet width tells a planner where the boom stops existing, and on the ray calculation it is also very nearly where the boom stops mattering. The overpressure falls to 80 per cent of the under-track value about 21 km from the track at every Mach number from 1.4 to 2.6 — 64 per cent of the way out at Mach 1.6, half-way at Mach 2 — and not at all below Mach 1.4, where the whole carpet stays above it.

That is the correct way to read the mile-per-thousand-feet rule the cut-off calculation tested. The rule estimates the width at which the boom stops, and as far as spreading goes the carpet is close to a band of nearly uniform boom with a sharp edge — a plateau along the track with gently sloping shoulders, and most of its area at more than two-thirds of the peak. If the edge is quieter than that in practice, spreading is not the reason. Something the rays leave out is doing it, and the rest of this essay is about what.

Plotted against the launch azimuth instead, the three Mach numbers lie on top of one another for the first thirty degrees — 0.99 at ten degrees, 0.97 at twenty, 0.93 at thirty for all three — and part only near their own grazing angles. That says the falling-off is a property of the launch angle rather than of the flight, and it explains what a faster aeroplane actually buys: not a quieter edge, but a larger grazing azimuth, so more of the cone reaches the ground at all and the rays that do reach it have fallen further down the common curve.

The edge is not a cliff, and that is the finding #

Everything above is linear acoustics on a family of rays, and at the carpet’s edge it says the amplitude is about two-thirds of the under-track value on the last ray, with silence one step beyond.

Real carpets do not end like that. The boom near the edge is audible and identifiable — a long low rumble rather than a crack — it fades over a distance rather than stopping at a line, and it is audible some way outside where the last ray lands. Three separate things are missing, and each is worth naming because they fail differently.

Diffraction. Geometrical acoustics draws a sharp shadow boundary at the grazing ray, and a real wave does not have one — least of all a wave the rays have left at two-thirds strength right up to the boundary. Low frequencies bend into the shadow, so the field there decays rather than vanishing, and the decay is a function of frequency: the high-frequency content of the signature — which is what makes it a bang — is cut off hardest, and the low-frequency content survives. That is the mechanism of the rumble. The shadow is not quiet; it is filtered.

Nonlinear ageing along a long path. The rays that reach the edge have come a long way.

At Mach 1.8 the edge ray covers 2.68 times the under-track path, at Mach 2.5 2.99 times, and at Mach 1.3 1.93 — and almost all of the extra distance is through the lowest, warmest and densest air, where the ray is travelling nearly horizontally. Spreading has taken a third off the edge ray; the distance is a separate charge. The signature that forgets the shape is the essay about what distance does to an N-wave: the amplitude falls and the rise time lengthens, both as powers of the distance aged over. So the edge signature is not merely smaller. It is slower, and a slow rise is heard as a rumble whatever its amplitude.

The ground itself. A boom arriving a few degrees above the horizontal interacts with the surface quite differently from one arriving at fifty. Grazing incidence on a soft or irregular surface is the case where ground impedance, terrain and the surface boundary layer matter most, and none of that is a free-field calculation at all.

The speed the rays travel at is worth naming too, since it is the quantity the whole tracing rests on. What a signal travels at is the local sound speed, which depends on the temperature and on nothing else — so a ray’s path through the atmosphere is decided by a thermometer at every height and by no property of the aeroplane at all.

Why quiet-supersonic shaping does not reach the edge #

The practical consequence brings together the two halves of this argument, and it is the reason the calculation is worth making even though its answer at the edge is wrong.

Quiet supersonic design — the family of shapes the least-drag body is related to — works by arranging the aeroplane’s area distribution so that the signature arriving at the ground has not yet coalesced into a clean N-wave. The target is the rise time: a signature that rises over ten milliseconds instead of one is heard as a thump rather than a bang, at the same overpressure.

That is a statement about a particular propagation distance. The shaping is designed for the path under the track, and the edge ray’s path is two to three times longer — long enough that the shaped signature has more time to coalesce, and long enough that whatever margin the design held has been spent.

So the shaping is expected to work under the track and to be progressively less effective outward, and by the edge to have nothing left. The calculation here cannot confirm that, because it prices distance and not ageing; what it establishes is the size of the distance ratio, which is the input such a calculation would need. A design assessed only under the track has been assessed on the one ray whose path is shortest, and the carpet it lays is judged everywhere else.

There is one more reason the edge is a different acoustic object rather than a smaller one. A signature arriving a few degrees above the horizontal has been refracted through the whole depth of the troposphere at a shallow angle, and that is the path along which a jet’s noise is turned away from its own axis — refraction by a gradient does not merely attenuate a field, it reshapes its directivity, and what arrives at a grazing angle has been filtered on the way.

What this calculation is and is not #

It is linear acoustics. The Blokhintzev invariant holds for a weak wave whose amplitude does not affect its own propagation. A sonic boom at the ground is weak in that sense — tenths of a per cent of atmospheric pressure — but its shock is not, and the ageing that reshapes it is precisely the nonlinearity this invariant neglects.

It computes a ratio and not a pressure. Every number here is relative to the value under the track, and the constant that would turn it into pascals needs the aeroplane’s own near-field signature, its length and its lift. Nothing here supplies one.

The tube’s cross-section is computed in two directions and one of them is assumed uniform. The footprint width per azimuth is differenced from the tracer; the along-track extent is taken to be the same for every ray, which is exact for level flight at constant speed and is exactly what fails in the accelerating case — where the along-track spacing collapses to zero at the fold and the amplitude is a caustic problem rather than a spreading one.

And the atmosphere is the cut-off calculation’s. One lapse rate, no inversion, no wind, flat ground. A real lateral cut-off is set by the wind shear across the track as much as by the temperature, and a crosswind makes the carpet asymmetric — louder on one side and narrower on the other — which this symmetric calculation cannot show at all.

The asymmetry that a crosswind makes #

One omission deserves more than a line in the list below, because it is the difference between a symmetric calculation and a real day.

That calculation showed that a wind shear along the track moves the cut-off, and that only the difference between the wind aloft and the wind below counts. The same is true across the track, and it breaks the symmetry this essay has assumed throughout: a crosswind makes the effective sound speed larger on one side of the aeroplane and smaller on the other, so the grazing azimuth is not the same to port and to starboard. The carpet is wider on the downwind side and narrower on the upwind one, and the amplitude distribution across it is not the symmetric ridge drawn here.

The size of that asymmetry follows from the cut-off calculation’s arithmetic: a 40 m/s shear moves the along-track cut-off by 0.136 in Mach number, which is more than the whole margin a Mach 1.3 flight has above it. Applied across the track it would move the two edges by comparable amounts in opposite directions, so a crosswind of ordinary tropopause strength can make one edge of a carpet several kilometres further out than the other.

Nothing here computes it, and the tracer could: the invariant already carries a wind term, and the only change needed is to let the wind have a component across the track rather than along it. It is the first thing a forecast for a real flight would need, and it is a small extension of what already exists.

The checks are narrow because the claims are. In air of one temperature the amplitude follows the cone’s one over the square root of distance to within a sixth of a per cent, which is the check on the tube’s geometry. Across a real carpet it falls monotonically outward at every Mach number — it is required to, or the model is not saying what it is being read as saying — and at Mach 1.8 it is still 0.65 on the last ray. The path ratio at the edge exceeds 1.8 at all three Mach numbers. And the calculation refuses a carpet below the cut-off, where there is no edge to compute.

Still open: the rise time across the carpet #

The one number this essay wants and cannot compute is the rise time at the edge, and the pieces to get it are nearly all here.

The ageing calculation of the signature that forgets the shape propagates an N-wave through a stated distance of uniform air and returns its amplitude and rise time. The ray tracer here returns, for each azimuth, the path length and the density and sound speed all along it. Carrying the first along the second — a nonlinear propagation with a varying coefficient, which is a quadrature rather than a new theory — would give the signature at every station across the carpet, and with it the rise time, which is what the ear responds to and what every quiet-supersonic design targets.

Two things would come out of it. The first is whether the edge’s rumble is mostly ageing or mostly diffraction, which is the question the last section could not settle and which decides whether the effect scales with the aeroplane or with the atmosphere. The second is the shaped-signature question: run the same propagation on a deliberately shaped near-field signature and see how far out the shaping survives. If the answer is that it survives to a quarter of the half-width, a quiet supersonic aeroplane is quiet over a strip and ordinary over the rest of its carpet, and the width that gets quoted for it is the wrong width.