The signature that forgets the shape

The pressure field near a supersonic aeroplane is a complicated thing that depends on every part of it — the nose, the canopy, the wing, the inlets, the tail. Whitham’s theory reduces the whole of that to one function of one variable, the F-function, and then propagates it outwards.

The propagation is where the essay is. A point of the signature at overpressure $p$ travels faster than one at zero, so the characteristics are

$$y = \tau - \lambda F(\tau),\qquad \lambda\propto\sqrt r,$$

and after enough distance they cross. Where they cross, the multivalued profile is replaced by a discontinuity placed so that the area under the curve is preserved — the same equal-area rule the kinematic-wave essay uses, and the same one a gas shock obeys.

What the propagation makes #

The far-field limit of that process is an N-wave: a front shock, a linear fall, a rear shock. That is what a boom sounds like — two bangs, a few hundred milliseconds apart, close enough together to be heard as one.

No shock-fitting logic is used to produce those pictures. The Lax formula

$$p(y) = F(X^*),\qquad X^* = \arg\min_X\Big[\frac{(y-X)^2}{2\lambda} + \Phi(X)\Big],\qquad \Phi' = F$$

does the equal-area construction automatically and places every shock correctly, without numerical diffusion. It is the same construction the previous essay in this field uses in a different variable, and it is exact rather than a scheme.

The scaling #

Measured over a sweep of age: the peak overpressure falls as $\lambda^{-0.493}$ and the duration grows as $\lambda^{+0.491}$, against predictions of $\mp\tfrac12$.

In physical distance, with $\lambda\propto\sqrt r$ and the geometric $1/\sqrt r$ spreading already in $F$:

$$\Delta p\propto r^{-3/4},\qquad \text{duration}\propto r^{1/4}.$$

So a boom heard twice as far away is forty per cent weaker and nineteen per cent longer. The $-3/4$ is why altitude helps and helps slowly: doubling the cruise altitude buys a forty per cent reduction, which is not enough to make a difference and is expensive.

The information the ageing destroys #

Now the result the essay is for. The N-wave has exactly two parameters, and they depend on the F-function through a single number — the largest value of its running integral.

So two bodies with the same value of that number produce identical far-field signatures, however different they are.

Aged side by side and compared in the area norm, the difference between their signatures falls from seventy-three per cent to two and a half — a factor of twenty-eight.

Everything the near field knew about how the volume and the lift were distributed along the aeroplane is gone. What survives is a length, through the duration, and a single number combining the lift and the volume, through the amplitude — and both arrive mixed with the altitude and with the atmosphere in between.

Why the wave ends up as an N #

The shape is worth deriving rather than asserting, because it explains why almost every distant blast signature in nature is the same shape.

Between the two shocks the solution is $p = y/\lambda$: a straight line through the origin with slope $1/\lambda$. That comes straight from the characteristics — a point that has travelled a distance $\lambda p$ from its starting phase is at $y = \tau - \lambda p$, and for the fluid well behind the front the starting phase has become irrelevant, so $y \approx -\lambda p$.

The two shocks sit where that line meets the undisturbed gas on either side, and their positions are fixed by conservation: the area under the positive part of the wave cannot change, because the equal-area construction preserves it. So the half-length is $L = \sqrt{2\lambda A}$ with $A$ the positive area, and the amplitude is $L/\lambda = \sqrt{2A/\lambda}$.

Two consequences drop out. The slope between the shocks is $1/\lambda$ and depends on nothing about the source at all. And the amplitude and duration depend on the source only through $A$ — one number. That is the whole of the forgetting, in two lines, and it is why the shape is universal.

It is also why the same N-wave appears behind a bullet, an explosion, a lightning strike and an aeroplane. Different sources, different near fields, one asymptotic shape, and only the two parameters distinguishing them — and it is why a blast wave’s late-time signature is an N-wave too, arrived at from a completely different beginning.

How loud, and why that is the wrong question #

Some numbers, because the subject exists for them.

A supersonic transport at cruise produces about 100 pascals of overpressure at the ground — a hundredth of a per cent of atmospheric pressure, which sounds negligible and is not. What makes it intolerable is the rise time: the pressure goes from zero to 100 pascals in a few milliseconds, and the ear responds to the rate rather than to the amplitude.

That is why the design target is not a smaller boom but a slower one. Halving the overpressure requires either doubling the altitude, which is impossible, or a fourfold reduction in the equivalent area distribution, which is a different aeroplane. Stretching the rise time from two milliseconds to twenty is achievable by shaping and changes the sound from a bang to a distant thump.

So the useful metric is not the peak pressure at all. Modern low-boom work uses perceived-level measures that weight the spectrum, and a signature with the same peak and a slower front scores twenty to thirty decibels lower on them — a difference of the same order as the difference between a rifle shot and a car door.

What the far field would need to remember more #

It is worth asking what would have to be true for the signature to carry more information, because the answer says where the theory’s limits are.

The forgetting happens because characteristics cross and are absorbed into shocks. Before that, the propagation is exactly invertible: given a signature at any distance short of the first crossing, the F-function can be recovered exactly by running the map backwards.

So the information survives up to the first shock formation and not beyond. A microphone flown close enough to the aeroplane records a signature from which the aeroplane’s equivalent area distribution can be reconstructed, and that is how near-field boom measurements are made — a probe aircraft flying a few body lengths below.

At the ground, several body lengths have become several thousand, and the reconstruction is no longer possible in principle rather than in practice. There is no measurement technique that recovers it, because the information is not attenuated, it is gone.

Why that changes what a designer does #

If the far field forgets the shape, then shaping for a quieter boom is impossible — which is the conclusion the industry drew for thirty years and it is wrong, for a reason worth stating exactly.

The N-wave is the asymptotic signature. It is what the propagation produces after enough distance, and enough is set by the amplitude and by the length of the aeroplane. A long enough aeroplane, with a carefully shaped area and lift distribution, can arrange for the front shock to be still forming when the signature reaches the ground.

Then what arrives is not an N-wave. It is a shaped signature with a slow rise instead of a front shock, and a slow rise is heard as a thump rather than a bang — an enormous perceptual difference for the same overpressure.

So quiet supersonic design works on preventing the ageing rather than on reducing the amplitude, and the design variable is not the size of the disturbance but the shape of the F-function near its front. That is why demonstrator aircraft for low-boom flight are long, slender and oddly shaped at the nose: the nose is doing the work of stretching the front of the signature so that it has not finished coalescing by the time it arrives.

What the F-function is #

A word about the object, because it is the whole of the near field compressed into one curve.

For a slender body at supersonic speed the disturbance at a distance is a function of one variable — the distance along a Mach line — and $F(\tau)$ is that function. It is computed from the body’s cross-sectional area distribution and its lift distribution together, through an integral that weights the second derivative of the equivalent area.

Two things follow. It is the same area distribution the wave-drag integral uses, so a body designed for least wave drag has a smooth F-function, and the two design problems are related without being the same. And lift enters it too, through an equivalent area that includes the accumulated lift ahead of each station — so a boom cannot be designed for by shaping the geometry alone, and the lift distribution is part of the answer.

The boom and the drag are the same integral #

There is a connection between this essay and the least-drag body that is worth making explicit, because it is the one place where two apparently separate design problems share an object.

Both are computed from the equivalent area distribution — the cross-sectional area of the body plus a term accounting for the lift accumulated ahead of each station. Wave drag is a quadratic functional of its second derivative; the F-function is a linear functional of the same quantity.

That means a body optimised for one is not optimised for the other, and the two objectives are not even in conflict in a simple way. Minimum wave drag wants the area distribution smooth and symmetric — Sears–Haack. Minimum boom wants the front of the F-function stretched, which usually means a long slow nose ramp and an asymmetric distribution, and pays for it in drag.

The exchange rate between them is the central design trade of a quiet supersonic aeroplane, and it is why such an aeroplane looks nothing like a minimum-drag one. It is also why the area rule, which is the same integral applied at Mach one, produces a shape that is good for drag and says nothing about boom.

Where lift comes into it #

One point that is easy to miss: a sonic boom is not made by the aeroplane’s volume alone.

A body of revolution at zero lift makes a boom from its volume. An aeroplane makes lift, and lift is a momentum flux delivered to the air, which shows up in the far field exactly as an extra area distribution would. The two contributions add, and for a transport-sized aeroplane at cruise the lift contribution is the larger of the two.

That has a consequence worth carrying. The boom is proportional to the weight, at fixed length and altitude, because the lift equals the weight. So a heavier aeroplane makes a louder boom for reasons that no amount of shaping can remove, and the only levers are length — which stretches the signature — and altitude.

It also means that a boom carries a measurement of the aeroplane’s weight, mixed into the same single number as its volume. That is one of the two things the far field remembers, and it cannot be separated from the other.

Reading the ground signature #

Three habits, and they follow from the two-parameter result.

Do not read the shape. A recorded N-wave’s departures from a straight line between its two shocks are atmospheric — turbulence in the boundary layer distorts the rise times and can double or halve the peak from one aircraft pass to the next.

Do quote the impulse. The area under the positive phase is a more robust measurement than the peak, and it maps back onto the F-function’s integral, which is the quantity that survives. It is the same preference aeroacoustics has for integrated quantities over peak ones, and for the same reason: the peak is where the measurement is least reproducible.

And state the altitude. An overpressure without one is a number that cannot be compared with anything, because the $r^{-3/4}$ law is doing most of the work.

Where it lands, which the straight rays cannot say #

The propagation above is radial, so every signature reaches the ground and reaches it at the same strength. Put the stratification back and three facts appear, none of which the uniform model can contain, and all three decide who hears the boom.

A boom is a carpet, not an event. The N-wave is attached to the aeroplane and trails it as a cone, so what sweeps the ground is the cone’s intersection with it — a hyperbola travelling at the aeroplane’s ground speed, for as long as the aeroplane is supersonic. Every observer inside that swathe hears one boom, once, as the cone passes; the aeroplane has been making it continuously. The idea that the bang marks the moment of crossing the speed of sound is the same mistake as reading a bow wave as an event rather than a shape, and the carpet is wide: roughly a mile of width for every thousand feet of altitude, so a transport at cruise lays down a band some fifty miles across.

Below a cut-off Mach number, nothing lands at all. The speed of sound rises towards the warmer air near the ground, so rays refract upwards, and one launched at a shallow enough angle turns before it arrives. At about Mach one and a sixth in a standard atmosphere every ray does, and the boom is reflected back into the sky — which is a real operational technique rather than a curiosity, and it sets the edge of the carpet too. There the surviving ray grazes, and what is heard is a rounded rumble rather than a bang.

And rays can converge. Accelerating or turning, the aeroplane launches successive rays that cross, and at the caustic the pressure is two to five times the flat-carpet value. Those focused superbooms are why acceleration through the transonic is flown out over water.

Where the same mathematics has been before #

This is the third time in this collection that a smooth disturbance has become a discontinuity by the same mechanism, and the three are worth putting together.

A flood wave, where the flux law is imported and there is no momentum equation.

A sound wave, where the propagation speed comes from the gas and the coefficient of nonlinearity is $\tfrac12(\gamma+1)$.

And a sonic boom, which is the second case with a three-dimensional geometric spreading laid over it — the $1/\sqrt r$ that makes $\lambda$ grow as $\sqrt r$ rather than as $r$, and which is why the exponents are three-quarters and a quarter rather than one and zero.

All three produce a shock from smooth data at a definite distance, and all three place it by the same equal-area rule.

What is not in the flow #

The distance.

The near field contains the aeroplane. The far field contains two numbers. Nothing has been approximated away — the propagation is exact and reversible right up to the moment the first characteristic crosses another — and after that the equal-area construction discards information irreversibly.

This is the one essay of the group whose missing quantity is not something the model leaves out but something the propagation destroys. What the ground can measure is a projection of the aeroplane onto two numbers, and no amount of care with the microphone recovers the rest.

The model limit #

Three.

A uniform atmosphere. The real one is stratified, so rays curve and the propagation is not radial. The section above says what that does to where a boom lands; none of it is in the figures, which propagate every signature straight down.

No absorption and no turbulence. Molecular relaxation gives the front shock a finite rise time — a few milliseconds — and atmospheric turbulence distorts it unpredictably. Both are what a real recorded signature is dominated by at the front.

And a slender body. The F-function description needs the disturbance to be small and the body to be long compared with its diameter. An aeroplane at Mach 2 qualifies; a blunt vehicle does not, and neither does anything near the aeroplane itself, where the whole theory is being used outside the region it describes. The near field of a real wing has shocks and expansions attached to it that the F-function represents only in the large, which is why near-field probing is done several body lengths away rather than at one.