The cone the source leaves behind

The Doppler construction has one moving picture in it: a source emitting spherical fronts at regular intervals, each front expanding at the speed the medium imposes from wherever the source happened to be when it was emitted. Ahead of the source the fronts crowd together and the note rises; behind, they spread and it falls. The rung this ladder starts from computes both shifts and shows they are different formulas for a reason.

The construction — which is Huygens’ rule applied to a source rather than a front — says nothing about what happens as the source speed approaches the wave speed, and it does not stop working there. It simply has a new feature.

The angle, and what is not in it #

A front emitted at time $t_i$ has radius $c(T-t_i)$ at the moment $T$ of the drawing, and its centre is a distance $v(T-t_i)$ behind the source. The tangent line from the source to that circle therefore makes an angle whose sine is

$$\sin\theta = \frac{c(T-t_i)}{v(T-t_i)} = \frac{c}{v} = \frac{1}{M},$$

and the $(T-t_i)$ cancels — which is why the same line is tangent to every circle and there is an envelope at all. If the ratio depended on which front was chosen there would be no cone.

The cancellation deserves one more look, because it is what the whole figure rests on. Suppose the wave speed depended on the distance from the source, or the emission rate varied, or the source accelerated. Then the ratio r/d would differ from front to front, no single line would be tangent to all of them, and the envelope would be a curve rather than a straight cone — which is exactly what an accelerating source produces. The straight cone is the signature of steady motion in a uniform medium, and both conditions are visible in the picture as the equality of six angles.

That is the whole reason a photograph of a shock wave is a speedometer, and it is a rather unusual one: it needs nothing on board the object, no clock, no calibration and no knowledge of what the object is. A ballistics laboratory, needing no clock and no frame of its own, measures the speed of a bullet by photographing its cone and reading an angle with a protractor.

Why the fronts pile up rather than merely crowding #

Below Mach 1 the fronts crowd ahead of the source and the crowding is what the pitch shift measures: the wavelength ahead is $(c-v)/f$ rather than $c/f$, and the received frequency is $f\,c/(c-v)$, which is the shift for a moving source. At $v = c$ that denominator is zero and the received wavelength is zero — every front the source has ever emitted arrives at the same instant.

What happens past that point is not a catastrophe but a change of geometry. The fronts do not stop existing and they do not overlap: the source is now outside all of them, and the region they fill has a boundary. A point ahead of the cone has received nothing at all, a point inside it has received everything the source emitted before a certain moment, and the surface between the two is the envelope. Silence ahead, sound behind, and a discontinuity in between — which is what a shock is, described without any thermodynamics.

The barrier that was a divergence #

The phrase “sound barrier” was in use for years before anything flew through one, and it is worth saying where it came from, because the construction above shows no barrier at all — the fronts pile onto a plane at Mach 1 and the picture on either side is continuous.

What was in the way was a calculation. Subsonic aerodynamics is done by linearising the flow equations about the free stream, and every quantity that comes out of that linearisation carries a factor of $1/\sqrt{1-M^2}$: the lift on a wing, the pressure on a surface, the size of the disturbance the aircraft makes. At $M = 0.9$ that factor is 2.3; at 0.99 it is 7.1; at 1 it is infinite. Read literally, the theory predicts unbounded forces at the speed of sound.

It is not a physical prediction, and the tell is that the same factor appears in front of every quantity, which is the signature of an approximation failing rather than of a phenomenon. What actually happens near Mach 1 is that the disturbance stops being small — the linearisation’s own hypothesis — because local pockets of flow over a wing go supersonic while the aircraft as a whole does not, and shocks form on the wing. Those shocks are real, they separate the boundary layer behind them, and the drag they cause is real. So there was something in the way; what there was not was a wall.

The distinction matters for reading the rest of this essay. The Mach angle is exact for a weak disturbance at any supersonic speed, including 1.001, and it says nothing about how hard the air pushes back. Geometry and dynamics are separable here, and the “barrier” was entirely in the dynamics.

Where the cone meets the ground #

The consequence that most contradicts the popular account is geometric rather than acoustic.

So a listener on the ground hears the boom once, and the time at which they hear it has nothing to do with when the aircraft went supersonic. It has to do with when the hyperbola reaches them. An aircraft that goes supersonic over the Atlantic and stays there for two thousand kilometres lays down a strip of boom two thousand kilometres long, and every point in that strip hears exactly one.

There is a second consequence in that figure that is easy to miss. The hyperbola is open: it never closes, so the boom’s lateral reach is unbounded in this model, and the further from the track a listener stands the further behind the aircraft the boom arrives. At forty kilometres off the track and Mach 1.6 the trace is fifty-two kilometres behind, so the boom arrives nearly two minutes after the aircraft was abeam — by which time the aircraft is a hundred kilometres away and out of sight.

And the delay rises with speed rather than falling. The along-track delay is $h\sqrt{M^2-1}/(Mc)$, which increases with $M$ and approaches $h/c$ — the plain time for sound to fall straight down, forty-one seconds from twelve kilometres. That is the opposite of the intuition that faster means sooner, and the reason is that a narrower cone has to be dragged further before its surface reaches the ground.

One thing the angle is not. It is not the angle of the shock at the nose of the object. A real body makes an attached shock whose angle depends on the body’s own half-angle and on the Mach number, steeper than the Mach angle and merging with it far from the body. The Mach angle is the asymptotic angle of a weak disturbance, which is why it is a clean function of one variable while the shock at the nose is not.

What the construction cannot show #

The wavefront picture is exact about geometry and silent about everything else, and three of its silences matter.

It has no amplitude. The envelope is where infinitely many fronts arrive together; the construction says they coincide and does not say what the sum is. In reality the disturbance from an aircraft is not a point emission but a distribution of pressure over a body, and the pressure signature at the ground — a large-amplitude disturbance, so the linear wave equation no longer governs it — is the characteristic N-wave — a sharp rise, a linear fall, a second sharp rise — lasting a couple of hundred milliseconds. The two bangs a listener hears are its two edges.

The construction underneath is Huygens’, and it has no dynamics in it either. Every point of a front is a source of the next, and the new front is the envelope of those wavelets — so the cone is a statement about geometry and arrival times, and about nothing else. That is why the angle depends on a ratio of speeds and on no property of the air, the source, or how hard anything was pushed.

It has no nonlinearity. The Mach angle is derived for disturbances travelling at exactly the small-signal sound speed. A finite-strength shock travels faster than that, so its angle is a little steeper than arcsin(1/M), and the excess is a measure of the strength. The relation the figures draw is the weak-disturbance limit, and it is approached from above as the disturbance gets weaker.

The speed the whole construction divides by is worth one remark, because getting it right took a century. Newton computed it from the isothermal compressibility and came out sixteen per cent low; Laplace computed it from the adiabatic one and came out right. Compressions in a sound wave happen too fast for heat to leave them, and that single observation fixes the number every angle on this page is measured against.

And it has no refraction. That last figure is why a real boom carpet has an edge. The sound speed varies with height, so the rays that leave the cone are bent; those heading obliquely downward from the outer parts of the cone turn upward before reaching the ground, and beyond a lateral distance of some tens of kilometres the boom simply never arrives. The hyperbolas drawn above extend for ever and the real carpet does not, and the difference is entirely a gradient the model has none of.

A cone is a surface, so the energy in it spreads differently from a sphere’s. The energy per unit area falls as the distance from the axis rather than as its square — cylindrical rather than spherical spreading — which is why a sonic boom weakens with lateral offset more slowly than intuition suggests, and why the carpet it lays on the ground is as wide as it is.

What a listener actually experiences #

Putting the geometry and the acoustics together gives an account with no mystery in it.

An aircraft passes overhead at twelve kilometres, supersonic. Nothing is heard, because the cone has not reached the ground; the aircraft is ahead of its own sound in the strong sense that no sound it has made has arrived anywhere near the observer. Thirty seconds later the hyperbola sweeps across, and the pressure signature arrives: a rise of about a hundred pascals in a few milliseconds, a linear fall over two hundred milliseconds, and a second sharp rise back. The ear hears the two rises as two bangs a fifth of a second apart, which is why a boom is usually described as a double crack rather than a single one.

Then nothing. The cone has passed and there is no second cone. Whatever the aircraft does after that — accelerates, turns, decelerates through Mach 1 — produces changes in the cone’s angle and position rather than new booms, and those changes reach the ground as distortions of the same single signature.

The exception is a manoeuvre. A turn concentrates the cone on the inside of the curve, exactly as a curved mirror concentrates light, and a sufficiently tight one produces a focus at the ground where several parts of the cone arrive together. That is a superboom, it can be several times the ordinary overpressure, and it is a purely geometric effect — the envelope of an envelope.

The angle has no shape in it, and the loudness has nothing else #

The second refutation above says the cone’s angle does not depend on the aircraft. The pressure signature that arrives at the ground depends on almost nothing else, and holding the two statements together is the useful thing.

An aircraft is not a point. It is a distribution of volume and lift along its length, and each element of that distribution sends its own weak disturbance out along the same family of directions. Near the aircraft those disturbances are separate: a nose shock, a canopy shock, a wing shock, a tail shock, each with its own rise. What turns them into two bangs is the propagation itself. A region of higher pressure travels slightly faster than the undisturbed air, so the front of a compression catches up on what is ahead of it, and over ten kilometres of descent the whole signature coalesces into the single N shape — one shock at the front, one at the back, a linear expansion between them.

Which is where the design lever is. Nothing an engineer can do changes $\arcsin(1/M)$, and the coalescence is not inevitable if the distribution along the aircraft is arranged so that the disturbances never quite merge. Stretching the nose, spreading the lift along the length and controlling how cross-sectional area grows can deliver a signature at the ground that is a staircase of small steps rather than two large ones — the same total pressure change, delivered slowly enough that the ear reads it as a rumble instead of a crack. Aircraft built to test that idea are aiming at a shape rather than at a level.

So the two statements do not conflict, and they divide the subject. The geometry of where and when belongs to $M$ and the altitude alone. The amplitude and the waveform belong to the body, and they are the only things about a supersonic aircraft that its designer can negotiate.

Why the crack is a crack #

One number decides whether a signature is heard as a bang or as a rumble, and it is not the overpressure. It is the rise time.

A hundred pascals arriving over a second is a change in the weather. The same hundred pascals arriving over a few milliseconds is a sound with energy across the whole audible band, because a step has a spectrum and a short step has a wide one. So what the ear responds to is the sharpness of the two edges of the N, and the loudness of a boom is much more nearly a statement about that than about its height.

Left to itself the front would be extremely sharp indeed — the thickness of a shock in a gas is a few mean free paths, so a rise measured in fractions of a microsecond. Real booms rise over one to ten milliseconds, four orders of magnitude slower, and the reason is that air is not a simple fluid on that timescale: energy going into a compression partly goes into the vibration and rotation of the molecules, which take time to take it up and give it back, so the leading edge is smeared. The smearing depends on humidity, which is why the same aircraft on the same track is heard differently on different days.

And the number a microphone at the ground reads is about twice what arrived. The surface reflects, the incident and reflected waves add at the ground, and the pressure there is close to double the free-field value — so a figure quoted for a boom is a figure at a boundary rather than a property of the wave that carried it.

The same envelope elsewhere #

Nothing in the derivation was about air, and the same construction turns up wherever a source outruns a wave.

A boat’s wake is not this. The V behind a boat has a half-angle of about 19.5° whatever the speed, which looks like a Mach cone at Mach 3 and is a different phenomenon: water waves are dispersive, so the relevant construction involves the group velocity of a whole spectrum rather than a single speed, and the constant angle comes from an optimisation over wavelengths rather than from a ratio of speeds.

Deep-water waves differ in the one way that matters, and it changes the answer completely. Their speed depends on wavelength, so there is no single $c$ to divide into $v$ — and the wake’s angle comes out independent of the boat’s speed, at 19.47° for every boat that has ever been rowed. Same construction, same envelope, and a different result because the medium is dispersive.

A ship in shallow water does. Where the depth is small compared with the wavelength, water waves stop being dispersive and travel at √(gh) whatever their length — one speed, exactly the condition the Mach construction needs — and a vessel exceeding it makes a genuine cone with sin θ = 1/M, narrowing as it goes faster. Shallow-water wakes therefore look nothing like deep-water ones, and the difference is a dispersion relation rather than a boat.

Charged particles do it in a transparent medium. A particle moving faster than the phase speed of light in glass or water emits a cone with $\cos\theta = 1/(n\beta)$ — the same construction with the wave speed being $c/n$ — and the angle measures the particle’s speed for exactly the reason the Mach angle measures the aircraft’s — a ratio against a phase speed the medium fixes. Detectors that identify particles by measuring that angle are built on this figure.

And the same construction runs backwards. Given a photograph of a cone and nothing else, the speed follows from the angle; given the speed and the altitude, the delay follows; given the delay measured on the ground and the altitude known from radar, the speed follows again, by a completely independent route. Three quantities, two relations, and any two of them fix the third — which is the shape of a measurement rather than of a derivation, and it is what makes the geometry worth setting out so carefully.

Where this ladder goes next #

Four rungs of this ladder have now treated the source and the medium as the only two things in the problem. The next asks what happens when there are two media — a shift measured through a boundary, where the wave changes speed on the way — and finds that the Doppler shift and the refraction do not commute: the frequency is conserved across a boundary and the wavelength is not, so a moving source seen through moving water gives an answer that depends on which frame the boundary is at rest in.