When the warning cannot arrive

There is a temptation to think of supersonic flow as subsonic flow with the numbers turned up. It is not. Something categorical happens at Mach one, and it is easier to state in terms of information than in terms of speed: above Mach one, part of the fluid has not been told that the body is coming, and never will be.

The cone, and why it is straight #

The construction is the whole argument, and it is elementary enough to do by hand.

A pulse emitted a time $t$ ago has grown to radius $at$. The source has meanwhile travelled $Ut$, so the pulse’s centre is $Ut$ behind. A line from the source tangent to that circle makes an angle $\mu$ with the path where

$$\sin \mu = \frac{at}{Ut} = \frac{1}{M}$$

and the time $t$ has cancelled. That cancellation is the reason the envelope is a straight line rather than some curve: every pulse, of every age, is tangent to the same pair of lines, because each has grown in exactly the proportion its centre has receded.

Below Mach one the construction has no solution — $\sin\mu$ would have to exceed one — and the geometric statement of that is the one in the figure on the previous rung: every circle still contains the source. At Mach one exactly, $\mu = 90°$, and all the circles touch at a single plane through the source. Above it the angle narrows: 30° at Mach 2, 19.5° at Mach 3, 11.5° at Mach 5. Faster aircraft drag narrower cones.

Measured off the circles rather than quoted #

The formula $\mu = \arcsin(1/M)$ is one line long and every textbook has it, which is exactly why this site does not put it in a caption and leave it there.

The figure and the formula are two different objects. A caption stating an angle that the drawing does not actually have is the characteristic failure of illustrated physics, and it is invisible: nobody measures a picture with a protractor. So the check here measures the tangent angle independently off each circle in the family — a circle of radius $r$ whose centre is $d$ behind the source subtends $\arcsin(r/d)$ — and requires every one of them to agree with $\arcsin(1/M)$ to within $10^{-9}$ radians. If the circles were drawn from a different rule than the caption claims, the family would not share a tangent and the build would stop.

That check is also what makes the figure worth drawing rather than replacing with the formula. The straightness of the envelope is a consequence being demonstrated, not an assumption being drawn.

The zone of silence, and what it is not #

Outside the cone the fluid is undisturbed. Not “slightly disturbed”; undisturbed, in the linearised description, because no signal has reached it.

This has a consequence that overturns the intuition built up in the ideal-flow essays. There, a body placed anywhere in the fluid changed the pressure everywhere at once, and the flow began deflecting round an obstacle long before reaching it. That behaviour is not a property of fluids; it is a property of incompressible fluids, in which the signal speed is infinite. Supersonically, the flow runs straight into the front of the body with no advance warning whatever, and has to do all of its turning abruptly. That is where shocks come from.

Two things the zone of silence is not, both worth stating because both are common.

It is not silence in the ordinary sense. An observer standing outside the cone hears nothing from this source yet. The cone sweeps past, and then they are inside it, and then they hear everything at once — which is the sonic boom.

It is not a wake. A wake is fluid that has been disturbed and has not recovered. The zone outside the cone is fluid that has not been disturbed at all, which is the opposite condition. The region inside the cone contains both the disturbance and, further back, the viscous wake the body also leaves, and those are separate objects with separate causes.

The boom is a carpet, not an event #

The common belief that the boom happens at the moment of transition is worth stating carefully, because it is false in an interesting way rather than a careless one.

The cone goes where the aircraft goes. Its intersection with the ground is a hyperbola that sweeps along beneath the flight path for as long as the aircraft is supersonic, so every point on that track receives the pressure jump as the cone passes over it. The boom is a carpet dragged across the landscape, tens of kilometres wide, not a bang emitted at Mach one.

The geometry is easy to make concrete. At Mach 2 the half-angle is 30°, so the cone from an aircraft at eleven kilometres meets the ground at a horizontal distance of $11/\tan 30° = 19.1$ km behind it. Anyone hearing the boom is hearing an aircraft that has been past for the best part of a minute, and is now nineteen kilometres away.

The angle is set by a ratio, so the air’s temperature is in it #

The cone angle depends on the Mach number and on nothing else, which sounds like a statement that it depends only on the aircraft. It is not, because the Mach number has the air in it.

An aircraft holding 600 metres per second true airspeed is at Mach 1.76 at sea level, where the speed of sound is 340.3, and its cone half-angle is 34.6°. The same aircraft at the same true airspeed at the tropopause, where the speed of sound is 295.1, is at Mach 2.03 with a half-angle of 29.4°. It has not changed speed and its cone has narrowed by five degrees.

This is the same point the previous rung made about the speed of sound, arriving in a form that has geometric consequences. Nothing about compressible flow is a property of the vehicle alone.

A Mach line is the zero-strength limit of a shock #

The lines in these figures carry no pressure jump. That is what makes them Mach lines rather than shocks, and the relationship between the two is not a matter of terminology.

A shock that turns the flow through no angle at all is a Mach line, and the chart says so quantitatively: the weak branch of the θ–β–M relation approaches $\arcsin(1/M)$ as the deflection goes to zero. Turn the flow a little and the wave leans further forward and acquires a small pressure jump; turn it more and it leans further still. The Mach cone is therefore not a different kind of object from the shock on a wedge — it is the same object at zero strength.

That relationship is why the cone drawn from wavefronts is a good picture of a real aircraft’s disturbance and a bad picture of its strength. The geometry is nearly right; the physics of what crosses the line is not there at all.

The equations change type, which is a real change #

The information argument has a precise mathematical statement, and it is the reason compressible aerodynamics needed new mathematics rather than new coefficients.

The linearised potential equation for steady compressible flow is

$$(1 - M^2)\,\phi_{xx} + \phi_{yy} = 0$$

For $M < 1$ the bracket is positive and this is Laplace’s equation with a stretched $x$ — an elliptic equation. Elliptic equations have no preferred direction of influence, no characteristics, and a solution at every point that depends on the boundary data everywhere. That is the mathematics of a fluid in which the news arrives instantly.

For $M > 1$ the bracket is negative and the equation is the wave equation with $x$ playing the part usually taken by time — a hyperbolic equation. Hyperbolic equations have characteristics, along which information travels, and a solution at each point that depends only on a bounded region of the data upstream. The characteristics are the Mach lines, at $\pm\mu$ to the flow, and the bounded region is the interior of the upstream cone.

So the cone in the figure is not merely a picture of some sound waves. It is the domain of dependence of the differential equation, drawn.

The change of type is worth taking seriously as a change in what a solution even is. An elliptic problem is a boundary-value problem: the answer everywhere depends on conditions all round the outside, and it has to be solved as one object, which is why panel methods and conformal maps invert a matrix over the whole body at once. A hyperbolic problem is a marching problem: the answer at a station depends on the station before it, and can be stepped forward face by face with no inversion at all. That is why a supersonic aerofoil can be solved exactly by hand while a subsonic one of the same shape cannot. The mathematics is not harder above Mach one. It is easier, and differently shaped.

It also explains a lasting confusion about what “supersonic” describes. A flow at Mach 0.99 and a flow at Mach 1.01 differ by two per cent in speed and by everything in structure, and no amount of care with the numbers will make one behave like the other. The threshold at Mach 0.3 discussed on the previous rung is a convention about tolerable error. The threshold at Mach 1 is not a convention about anything.

The other cone, and the design freedom it buys #

The upstream cone is the region a point can be affected by. There is a second one pointing the other way — the region a point can affect — and it has consequences a subsonic designer has no analogue for.

Everything a body disturbs lies inside its own downstream Mach cone, and nothing outside it is touched at all. So two objects that lie outside each other’s cones do not interact, however close they are: a nacelle beside a fuselage, a fin beside a wing, one part of a configuration beside another. In subsonic flow that is unthinkable — the elliptic equation makes everything feel everything — and above Mach one it is exact within the linearised description.

Which is a real design freedom, and an unusual one: parts of a supersonic aircraft can be shaped independently and then assembled, provided the geometry keeps them out of one another’s cones. And since the cone narrows as the Mach number rises, more of a configuration becomes independent the faster it flies.

The same geometry decides how a wing’s edges behave, and this is where it earns its place in a design office. Sweep a leading edge back far enough that it lies behind the Mach cone from the apex, and the component of the free stream normal to that edge is subsonic: the edge can be rounded, it can carry leading-edge suction, and it behaves in the way subsonic intuition expects. Leave it ahead of the cone and the normal component is supersonic; the edge must be sharp, it carries a shock, and it pays wave drag.

That single distinction — subsonic or supersonic leading edge — is why supersonic wings are swept so sharply, and why the sweep required rises with the design Mach number. It is the cone in this essay, laid over a planform.

The one place both types coexist #

The awkward range is not supersonic flight. It is the transonic band, where the free stream is subsonic and pockets of the flow over the body are not.

An equation that changes type inside its own domain, at a location that has to be determined as part of the solution, is a genuinely hard problem — and it is the reason transonic aerodynamics resisted theory for so long and yielded to computation rather than to analysis. What the flow over a wing does there is a rung of its own.

Characteristics are not a metaphor #

The claim that information travels along Mach lines can be seen in a picture, because there is a flow in which the Mach lines are individually visible as separate objects rather than as an envelope.

Two features of that figure are the hyperbolic structure made visible. The fan is bounded by the two Mach lines and the flow between them is being turned continuously — so a point inside the fan is influenced only by what is upstream along its own characteristic, and a point just outside the leading wave has not been influenced at all. And the fan opens, which is what stops it becoming a discontinuity; the corresponding compression does the opposite, and that asymmetry is the whole difference between a fan and a shock.

What the solver computes, and how it is checked #

Everything drawn here comes from wavefronts, which is a handful of circles, and assertMachAngleFromWavefronts, which measures them.

The assertion has two branches because the physics does. At or below Mach one it requires every front to contain the source — with equality permitted exactly at Mach one, since that is the sonic case rather than a failure of it. Above Mach one it computes the tangent half-angle off each circle separately and requires them all to agree.

The rejection test hands it a family of circles that is nearly right: radii and centres that grow at slightly different rates, so no common tangent exists. That family draws a picture which looks entirely convincing — a spray of circles with something cone-shaped about them — and the check refuses it, which is the point. The failure mode being guarded against is a figure that illustrates the caption rather than deriving it.

What the picture cannot show #

Three things, and the third is the one that matters.

The circles are drawn as a discrete sample of a continuous emission, and their spacing carries no meaning; drawing twice as many changes nothing about the envelope.

Amplitude is not drawn. Every circle has the same stroke, so the figure is silent about how strong the disturbance is when it arrives, and in reality it falls off with distance.

And the cone drawn here is a Mach cone, not a shock. A Mach line is the path of an infinitesimal disturbance and carries no jump in pressure at all: crossing one changes the state by an infinitesimal amount. A real aircraft is a finite disturbance, its compression waves coalesce, and what actually sweeps the ground is a shock — a finite pressure jump, travelling slightly faster than the Mach angle would suggest, with entropy behind it. The figure draws the linear limit of an object that is not linear, and the difference is exactly the difference between a Mach wave and a shock.

Who found it, and when #

Ernst Mach photographed a supersonic bullet in 1887, using a spark shadowgraph triggered by the bullet itself, and the plate showed the cone. That photograph is the reason the ratio bears his name, and it is a useful reminder about the order in which things happened: the cone was seen before the mathematics that explains it was assembled.

The angle itself had been derived earlier — Doppler’s work on moving sources in 1842 contains the construction — and the hyperbolic character of supersonic flow was understood by Prandtl and his students in the 1900s and 1910s. The engineering consequences waited until the 1940s, when aircraft first started encountering them, at which point a body of mathematics that had been sitting unused for thirty years became urgent.

Where the ladder goes next #

The fluid ahead has no warning, so it cannot turn gradually. What happens instead is a discontinuity, and the surprising part is that the conservation laws permit it: mass, momentum and energy are all satisfied across a jump, which is not obvious and is why shocks exist at all.

Before that, though, there is a question this rung has skipped. If density is now free to change, what is the quantity that stays constant along a streamline, given that Bernoulli’s constant was built on a density that does not? That is the next essay in this field, and its answer is an energy rather than a pressure.