The carpet an accelerating aeroplane folds

The boom that turns back before the ground traced boom rays through a real atmosphere and found the two things level flight decides: a Mach number below which nothing lands, and an edge past which nothing lands. It ended by naming a third thing it could not compute — what happens when the aeroplane accelerates — and this is that calculation.

It needs no new physics. Every ray is the same ray the cut-off calculation traces, with the same Snell invariant and the same quadrature; the only change is that successive rays are launched at successive Mach numbers, because the aeroplane is speeding up between them. What comes out of that one change is a map that folds.

The arrival map, and why level flight’s cannot fold #

Write down where a ray lands. An aeroplane at height $h$ launches a ray at the moment it is at $x_e$ doing Mach $M$; the ray travels $X(M)$ horizontally on its way down and takes $\tau(M)$ to do it, so it arrives at

$$x_g = x_e + X(M), \qquad t_g = t_e + \tau(M).$$

In level flight $M$ never changes, so $X$ never changes, and $x_g$ is $x_e$ plus a constant. The map from emission time to arrival place is the aeroplane’s own motion shifted bodily downstream, it rises at the aeroplane’s speed, and it has no turning point of any kind.

At Mach 1.6 from 15 km, level flight’s map advances 1,416 metres for every three seconds of flight, forever. That is the control, and it is checked rather than taken on trust: the map is required to be strictly increasing, and it is.

One convention needs stating, and it is the one the airspeed indicator’s own reading makes awkward: the Mach number here is measured against the air at flight height, which is the only one an aeroplane’s instruments can report, and it is not the ratio of the aeroplane’s speed to the sound speed at the ground. The cut-off is exactly that second ratio, which is why it is a number the cockpit cannot compute.

Why a later ray can arrive first #

Now let $M$ rise. The quantity that matters is $X(M)$, the ray’s horizontal reach, and near the cut-off it does something violent.

At the cut-off the ray leaves the cone travelling almost horizontally and turns back before reaching the ground at all, so as the Mach number approaches it from above the reach grows without limit. Just above it, from 15 km, the ray lands 41.0 km downstream. By Mach 1.26 it lands 23.9 km on, by Mach 1.6 13.8 km, and by Mach 1.87 only 10.3 km.

Why the reach diverges where it does #

The divergence of the reach is worth a paragraph of mechanism, because it is the engine of everything here and it is a purely geometrical fact about a turning ray.

A ray’s descent is governed by its vertical slowness, $q(z) = \sqrt{1/a(z)^2 - 1/V^2}$, and the horizontal distance it covers is $\int s_x\,\mathrm dz/q$. At the cut-off Mach number, $q$ vanishes at the ground: the ray arrives travelling horizontally, and the integrand blows up over the last stretch of its descent. Below the cut-off $q$ vanishes above the ground and the ray turns; above it, $q$ at the ground is small but positive, and the integral is finite but large.

So the reach is an integral with a square-root singularity approaching it. It diverges as the inverse square root of the margin above the cut-off — which is why the first ray to land goes forty kilometres and the ray launched a Mach number later goes twenty-four. Nothing is delicate about that divergence and nothing numerical about it either: it is the same inverse-square-root that calculation removes with a substitution in order to integrate the ray at all.

It also explains why the fold exists for every acceleration rather than only for hard ones. However gently the aeroplane accelerates, there is a Mach number close enough to the cut-off at which the reach is falling arbitrarily fast — faster, eventually, than any fixed advance. The fold is not a threshold effect that switches on above some acceleration; it is a consequence of a divergence, and a divergence always wins eventually.

So the arrival position is a race. The aeroplane is moving forward, adding to $x_e$; the reach is shrinking, subtracting from $X$. Where the reach shrinks faster than the aeroplane advances, the arrival position moves backwards — and a ray launched later lands nearer.

Accelerating at one metre per second squared from 15 km, the first ray that lands at all leaves at Mach 1.1593 and lands 56.0 km past the point of Mach one. Rays launched over the next forty seconds land progressively nearer, down to 48.8 km at Mach 1.2266, and then the arrivals start moving downstream again. The minimum is the fold.

The two booms are not an echo #

It is worth separating the double arrival from the thing it is most often mistaken for, because a listener cannot tell them apart and the mechanisms are unrelated.

A boom heard twice from a reflection is an echo: one ray, one arrival, and a second arrival from a surface some distance away, delayed by twice the travel time to it. A boom heard twice from a fold is two separate rays, launched at different moments by an aeroplane at two different speeds, which have taken different paths through the atmosphere and arrived at one place. The delay between them is not a distance to anything — it is the difference between the two rays’ travel times, which at one metre per second squared from 15 km is of order thirty seconds near the fold and shrinks to nothing on it.

That shrinking is the signature. Standing well inside the doubled band, a listener hears two distinct events; standing nearer the fold, the two close up; and on the fold they coincide, which is what makes the pressure there large rather than merely doubled. An echo does not do that anywhere.

What a fold means on the ground #

Two consequences follow, and they are different from each other.

The first is arithmetic. Between the fold at 48.8 km and the first arrival at 56.0 km, every place is reached by two rays — one launched before the fold and one after, seconds apart, at different Mach numbers and arriving at different times. A listener there hears two booms.

The second is the one the phenomenon is named for. At the fold itself the arrival map’s derivative vanishes, which means neighbouring rays — a continuum of them, not two — converge onto one line. The acoustic energy carried between two neighbouring rays is spread over a ground footprint that has shrunk to nothing, and geometrical acoustics answers with an infinite pressure. That is a caustic, and the measured overpressure on one is two to five times the ordinary carpet’s rather than infinite, because the infinity is a failure of the ray approximation and not a prediction about air.

The line where it lands is a focus line, and it is what a “superboom” is. It is also why an accelerating supersonic transport was required to complete its acceleration over water: the ordinary carpet is a nuisance and the focus line is a different order of thing.

The mechanism should be put beside the two related statements already in hand. When the warning cannot arrive is the geometry of a cone behind a steadily moving source, and the sound now is the source then is the mapping from emission time to arrival time for a source that moves — which is exactly the map that folds here, with the ground added and the speed allowed to change. The fold is what that mapping does when the source is accelerating rather than merely supersonic.

A fold is a familiar thing seen from an unfamiliar side #

Nothing about this is peculiar to sound. A family of rays parameterised by one number, arriving at a surface parameterised by another, is a map between two lines, and the singularities of a map between two lines are folds. That is the whole classification: generically, a smooth map from a line to a line is either invertible or has a fold, and nothing else happens.

The optical version is the one everybody has seen. A rainbow is the fold of the map from raindrop impact parameter to exit angle — rays pile up at the minimum deviation, the brightness diverges there in geometrical optics, and what is actually seen is a bright band of finite width whose profile is an Airy function. The bright caustic on the bottom of a swimming pool is the same thing with a wavy surface for the family. And the inferior mirage used to explain the cut-off is its close relative: rays turning back from a hot surface, with a caustic at the turning envelope.

What is different here is which parameter indexes the family. In a rainbow it is where the ray enters the drop; in a mirage it is the launch angle. Here it is time — the family is indexed by when the aeroplane launched the ray, and the fold exists because the launch condition is changing while the source moves. That makes the boom focus a fold in a space-time map rather than in a purely spatial one, which is why it lands on a line across the ground rather than on a curve in the sky.

It also sets the expectation for the amplitude. On a fold, geometrical theory gives an infinite intensity and the true answer is finite, larger than the surroundings by a factor that depends on the wavelength — in optics by $\lambda^{-1/6}$, which is why a rainbow is bright but not blinding. For a shock there is no single wavelength; what sets the finite answer is the signature’s own rise time, and the measured factor of two to five is the result. That is a different calculation from this one and its scaling is not the optical one.

Where the fold lands, and it is never ahead #

The position of the fold is what an operating rule would need, and it depends on the acceleration in a way that is not obvious until it is computed.

From 15 km, a gentle acceleration of half a metre per second squared folds at Mach 1.1877 — barely above the 1.1533 cut-off — and lands the focus 67.9 km past the point where the aeroplane passed Mach one. One metre per second squared folds at Mach 1.2266 and 48.8 km; two at Mach 1.2945 and 36.5 km; four at Mach 1.4016 and 27.9 km.

The trend runs two ways at once and both are worth holding. A gentle acceleration spends longer near the cut-off, where the reach is changing fastest, so it folds at a Mach number barely above it — but by the time it does the aeroplane has travelled a long way, so the focus lands far downstream. A hard acceleration folds at a higher Mach number and much nearer.

The focus is never ahead of the aeroplane. Across this whole range it trails by between 10.8 and 24.0 kilometres, and the harder the acceleration the further behind it falls — because a hard acceleration has the aeroplane moving fast by the time its early rays arrive. The picture of a boom arriving in front of the aeroplane that made it does not come out of this calculation.

Height moves it as the cut-off moves. Below the tropopause the cut-off falls — there is less cold air between the aeroplane and the ground — so a lower flight folds at a lower Mach number, and the ray has less height to cross, so the focus lands much nearer. The two effects run the same way, which makes the focus distance a steep function of cruise height.

Nothing about the signature is in any of this. What travels along each ray is a wave that has already become a shock and has been ageing ever since, and what a signal actually travels at is the sound speed of the local air rather than anything about the aeroplane. The rays carry the geometry and nothing else.

What an operating rule gets from this #

The numbers above are the ones a route planner would need, and they turn a qualitative rule into an arithmetic one.

A supersonic transport accelerating from subsonic cruise lays an ordinary carpet from the moment it passes its cut-off Mach number, and lays a focus line once, at the fold. The carpet is a nuisance spread over tens of kilometres of width; the focus is a line of two to five times the overpressure. The rule that follows is that the acceleration has to be completed clear of anywhere that matters, and “clear” means by the focus distance rather than by the distance to the point of Mach one.

From 15 km those are very different numbers. At one metre per second squared, the aeroplane passes Mach one and its focus lands 48.8 kilometres further on — so a coast crossed at Mach one still has fifty kilometres of sea to spare before the focus is safe. At half a metre per second squared it is 67.9 kilometres, and the gentler acceleration is the worse case, which is the opposite of the intuition that a gentle manoeuvre is a quiet one.

The other reading is that the focus can be moved deliberately. A step change of acceleration during the transonic moves the fold along the track, and a sufficiently non-uniform acceleration — fast through the Mach numbers near the cut-off and gentle afterwards — shortens the distance at which the fold lands. Whether an acceleration profile exists that avoids a fold entirely is the interesting question, and the answer from the section above is that it does not: the reach diverges at the cut-off, so a fold forms whatever the profile. What can be chosen is where.

What the ray calculation cannot say about a caustic #

The pressure on the focus line is not computed and cannot be. A caustic is precisely where geometrical acoustics fails: the ray-tube area goes to zero, the amplitude the theory returns is infinite, and the real field there is decided by diffraction, by the finite width of the signature, and by the nonlinearity of a shock that is no longer weak. What the calculation locates is the line; how loud it is needs a different theory, and the measured factor of two to five is a borrowed observation.

The fold’s position is sensitive to the atmosphere in the way the cut-off itself is. The reach diverges at the cut-off, and the cut-off moves with the surface temperature and with a wind shear aloft — a 40 m/s headwind aloft raises it from 1.153 to 1.289. A fold located to within a kilometre here would move by tens of kilometres on a different day, and forecasting one is the same forecasting problem the cut-off calculation describes, with a derivative in it.

The acceleration is constant and straight. A real transonic acceleration is neither: thrust varies, drag peaks near Mach one, and the aeroplane is usually climbing at the same time. Each of those changes the map’s shape and none is here.

And a turn does the same thing sideways. An aeroplane in a turn launches successive rays at successive azimuths, which converges them laterally for the same reason acceleration converges them along the track. It is a genuinely different calculation — the invariant is no longer the same for neighbouring rays — and it is not attempted.

The checks are the ones the argument needs. Level flight’s map advances by the same amount between every pair of rays and never goes backwards. All four accelerations fold. Every fold forms above the cut-off, and a harder acceleration folds at a higher Mach number than the one below it. And the calculation refuses an aeroplane that is not accelerating, which is the case with no fold to find.

One last comparison, because the doubled band has a cousin in the collection. The sound that only leaves is the account of a jet’s own noise escaping in a cone its scaling law does not predict, and the mechanism is the same refraction by a gradient that bends these rays — there in the shear of the jet, here in the temperature of the atmosphere. A field bent by a gradient makes shadows and caustics whatever is doing the bending.

Still open: the second fold, and where deceleration puts it #

The map falls and then rises, and this essay has taken the minimum. What it has not asked is whether the map folds again.

An aeroplane that keeps accelerating eventually reaches Mach numbers where the reach changes slowly, so the map rises smoothly and there is no second fold from the same mechanism. But the reach has a second feature the cut-off calculation found and this one has not used: above the tropopause the rays are straight, and below it they curve, so $X(M)$ has a change of character at the Mach number whose ray bottoms out at the tropopause. Whether that produces a second, weaker fold — and where — is a calculation the tracer here can already do and has not been asked.

Beside it is deceleration, which has been ignored throughout and is the more common manoeuvre over land. A decelerating aeroplane launches successively longer rays, so its arrival map’s slope is steeper than the aeroplane’s own speed and it cannot fold forward — but the rays behind it can converge in the other direction, and near the cut-off the reach diverges just as fast on the way down as on the way up. Whether a deceleration through the cut-off makes a focus, and whether it lands ahead of the aeroplane where the acceleration’s never does, is the next thing the same map would answer.