Concept

Shock wave — where it appears

A discontinuity across which pressure, density and temperature jump within a few mean free paths. The jump is fixed entirely by conservation of mass, momentum and energy, and only the second law decides which of the two allowed jumps a flow may take.

Named by 25 essays across 3 fields — each of them below, with the objects they name alongside it.

A source at Mach 2.00, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

When the warning cannot arrive

Supersonic flow is not fast flow. It is flow in which the fluid ahead has been told nothing, because the body is outrunning its own pressure signals — and that single change turns an equation of one type into an equation of another.

compressible · Speed of sound

Mach number: one number, four different flows. Mach number is speed ÷ speed of sound. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.

When air stops being incompressible

Air is a gas and can obviously be squeezed, yet most of aerodynamics treats its density as fixed. The assumption holds until the flow approaches the speed at which pressure information travels — and then everything changes at once.

regimes · Mach

One nozzle, five back pressures, five different flows. The duct above, and the static pressure along it below, computed station by station from the local area. Where a shock stands inside the divergent section its position was solved for rather than placed: the shock spends total pressure, which sets the subsonic Mach number at the exit, which has to match the imposed back pressure.

One area, two answers

The area of a duct at a station fixes the Mach number there — twice over. A ratio of 2.5 is satisfied at Mach 0.24 and again at Mach 2.44, nothing local chooses between them, and where the choice cannot be made consistently a shock appears inside the duct to join the two.

compressible · Area mach

The shock in a river

Shallow water is a gas whose ratio of specific heats is two. The white water below a weir is a shock wave, momentum is conserved across it exactly, energy is not, and one direction is forbidden for the same reason an expansion shock is forbidden — which makes the analogy exact to first order and wrong at the second.

applied · Open-channel

A normal shock at Mach 2.00, and what crosses it unchanged. The state in front of the shock and the state behind it. Every ratio was computed from the standard jump relations and then substituted back into mass, momentum and energy, which is an independent route — a mistyped exponent in the total-pressure expression cannot survive a momentum balance it never appeared in. The residuals are printed below because a check nobody can see is a check nobody can audit.

The jump the equations allow

A shock is a discontinuity in a fluid, which sounds like a breakdown of the description rather than a solution of it. It is a solution: mass, momentum and energy can all be satisfied across a jump, and every ratio across one follows from that alone.

compressible · Shock

Where the flow over the section first reaches Mach one. Two curves running towards each other. The falling one is the section's peak suction, solved exactly in incompressible flow and then deepened by the Prandtl–Glauert factor as the free-stream Mach number rises. The rising one is the pressure coefficient at which the local flow would be exactly sonic. Where they cross is the critical Mach number, and above it there is a pocket of supersonic flow on a subsonic aeroplane.

The pocket on top of the wing

An airliner cruising at Mach 0.85 has subsonic flow almost everywhere and a patch of supersonic flow over its wing. Closing that patch takes a shock, the shock separates the layer beneath it, and the correction that predicts all of this also predicts, in the plainest terms, where it stops being true.

regimes · Mach

The second law is the only thing that forbids the other half of this curve. Entropy change across a normal shock, against the Mach number in front of it. The solid branch is the compression shock that exists. The dashed branch below Mach one is the expansion shock, and it satisfies mass, momentum and energy exactly — the residuals are zero to machine precision. It is refused by the second law alone, the one statement in the problem that no conservation residual can show.

The only law that forbids it

The jump conditions permit a discontinuity in either direction. An expansion shock conserves mass, momentum and energy exactly — the residuals are zero to machine precision — and it does not exist. Nothing that can be drawn rules it out.

compressible · Shock

Everything a normal shock does, against the Mach number in front of it. Four quantities across a normal shock, each scaled to fit one axis. Pressure and density rise without limit and without bound as the Mach number grows; the Mach number behind falls towards a floor it never passes; and the total pressure — the flow's ability to be turned back into speed — collapses. That last curve is why a supersonic intake is designed around avoiding a single strong shock.

What a shock costs

All the heat survives a shock and none of it is lost. What is lost is the ability to turn that heat back into speed — 27.9 per cent of it at Mach 2 and 93.8 per cent at Mach 5 — and every supersonic intake ever built is a scheme for paying less.

compressible · Shock

A 10° wedge at Mach 2.00 has two shocks that solve it. The wedge turns the flow through a fixed angle, and the θ–β–M relation offers two shock angles that achieve it. The weak solution, drawn steeply forward, leaves the flow supersonic and is what a wedge in a free stream produces. The strong solution leaves it subsonic and appears where downstream pressure forces it. Nothing local to the wedge chooses between them.

A shock that leans

Tilt a shock and only the velocity component across it is changed — the component along it passes through untouched. That single observation turns every oblique shock into a normal shock in disguise, and it is why a wedge at Mach 2 leaves the flow supersonic while a blunt nose does not.

compressible · Oblique shock

The same 10° turn, taken both ways. A supersonic stream turned away from itself expands through a fan of Mach waves and keeps every bit of its total pressure. Turned into itself through the same angle it shocks, and pays. Nothing in the equations distinguishes the two cases except the sign of the angle: compression waves converge and steepen into a front, expansion waves diverge and spread.

Turning the other way is free

Compression through ten degrees at Mach 2 costs 1.54 per cent of the total pressure. Expansion through the same ten degrees costs exactly nothing — not approximately nothing, nothing — and the two are the same equations with the sign of one angle changed.

compressible · Expansion fan

The Hugoniot of a gas that can burn, and the gap in the middle of it. Pressure against specific volume, both scaled on the unburnt gas. The lower curve is the ordinary shock Hugoniot, which passes through the initial state because a jump of zero strength satisfies mass, momentum and energy. Adding a heat release lifts it away, and the initial state now sits in a region no wave can reach: between the two branches a Rayleigh line would need a positive slope, and its slope is minus the square of the mass flux. A burning gas has no weak waves available to it at all.

The other branch of the same curve

Put heat into the jump conditions and the Hugoniot lifts away from the initial state, leaving a gap no wave can occupy. A burning gas has no weak waves: it must run supersonically or subsonically, and the conservation laws pick the first speed exactly and say nothing at all about the second.

compressible · Detonation

The two ways a shock can meet a wall. Left: the incident shock from the wedge reaches the wall and a second shock turns the flow back parallel to it, meeting at a point. Right: at a larger wedge angle no reflected shock can turn the flow that far, and the intersection lifts off the wall into a triple point with a nearly normal Mach stem standing on the surface and a slip line trailing downstream. The two configurations are drawn at the angles the solver returns, with every shock angle computed rather than sketched.

When a shock cannot bounce

A shock reflects off a wall until the reflected shock runs out of turning, which happens at a wedge angle well below the free stream's own limit. Between the two boundaries both configurations exist, both are stable, and which one appears depends on which direction the experiment came from.

compressible · Mach reflection

A curved shock, and the entropy each streamline picks up crossing it. A parabolic bow shock ahead of a blunt nose at Mach six, with the streamlines drawn arriving horizontally and a marker at each crossing whose size is the total pressure lost there. The streamline through the nose crosses a normal shock and keeps three per cent of its total pressure; one four nose radii out crosses at fourteen degrees and keeps ninety-four per cent. Every streamline gets a different entropy, and the stagnation enthalpy is the same on all of them.

The spin a shock leaves behind

A curved shock gives every streamline a different entropy rise and the same stagnation enthalpy. Crocco's theorem then forces vorticity into a flow with no viscosity anywhere — and it scales as the inverse of the shock's radius of curvature, exactly, so a straight shock makes none.

compressible · Crocco

The density ratio and the nose pressure coefficient, against Mach number. Two quantities across a normal shock, on a logarithmic Mach axis. Both approach limits that depend on γ and on nothing else: six for the density ratio and 1.8394 for the pressure coefficient at the stagnation point. By Mach five the second is within three per cent of its limit and by Mach twenty within a tenth of a per cent. Above that the flow round a blunt body has stopped depending on how fast it is going and started depending on what the gas is.

A shock that lies on the body

Above about Mach eight the flow round a blunt body stops depending on how fast it is going. The density ratio, the nose pressure coefficient and the shock standoff all reach limits set by γ alone — and going from a perfect gas to a dissociating one halves the standoff.

compressible · Shock layer

One signature, aged four times. The pressure signature of a slender body at four distances, computed by the exact Lax formula for the nonlinear propagation. Each point of the waveform moves forward in proportion to its own overpressure, so the compression at the front catches the undisturbed air and a shock forms there, while the expansion at the rear falls behind and forms a second one. What is left is an N-wave: two discontinuities and a straight line between them, spreading and weakening.

The signature that forgets the shape

The pressure field near a supersonic aeroplane depends on every part of it. What reaches the ground has two parameters. Two bodies whose near-field signatures differ by fifty-five per cent in peak and by their whole shape age into the same N-wave, to two and a half per cent.

compressible · Sonic boom

The shock radius against time, from an equation that was given no exponent. The thin-shell energy balance integrated forward from a small initial radius, on logarithmic axes. The two-fifths power is not put in: the ordinary differential equation is Ṙ = √(E/AρR³), and the straight line is what it does. The fitted slope is 0.39983 and the fitted prefactor is 0.90721 against a closed form of 0.90702 — the small residuals being the integration's memory of where it started, which the similarity solution has no equivalent of.

A radius that gives the energy away

Four quantities, three dimensions, one group. The radius of a strong blast must be a constant times (Et²/ρ)¹⁄⁵, and nothing about the device, the chemistry or the initial size can appear. The exponent is free and the constant is not.

compressible · Blast wave $A sinusoid, distorting on its way to a shock. A finite-amplitude sound wave at four fractions of the distance to shock formation, computed by inverting the implicit simple-wave solution. Each point of the waveform travels at its own speed, so the compressions catch up with the rarefactions ahead of them and the profile leans forward. At σ = 1 the front is vertical. The linear theory says the first panel is the answer at every distance, for ever.$

Every compression becomes a shock in the end

Linear acoustics has no time scale in it, which is the sign that something has been thrown away. A 120-decibel tone shocks after three hundred metres and a jet engine after twenty; the distance goes exactly as the reciprocal of the amplitude, and nothing is exempt.

compressible · Characteristics

So splitting a turn into N ramps costs one over N squared. A twelve-degree compression at Mach 3, done in one ramp and in up to sixty-four. The entropy is N times a cube of one Nth, so it falls as exactly the inverse square of the number of ramps — the measured exponent is −2.00 — and sixty-four ramps cost a two-hundred-and-fifty-sixth of what one costs.

A compression that costs nothing in the end

Turning a supersonic stream away from itself is free and turning it into itself is not. But the price of a compression is the cube of its strength, so splitting one turn into N turns costs one over N squared — and in the limit the compression is free too.

compressible · Expansion fan

A compression piston, and where its characteristics first cross. Sixty C+ characteristics from an accelerating piston, drawn in the distance-time plane. Each is a straight line, because the invariant makes the state along it constant; later ones are faster, because the gas ahead of them has been compressed; so they converge, and the first crossing is the shock. The envelope formula gives 2.7529 and the first actual crossing is at 2.7510.

Two numbers that do not change

One-dimensional unsteady gas flow carries two quantities that are exactly constant along two families of curves. That single fact turns a pair of coupled partial differential equations into a family of straight lines, and gives an exact speed at which a gas outruns its own expansion.

compressible · Characteristics

The convergence exponent depends on the gas, which a dimensional exponent cannot. R ∝ (−t)^α for a converging shock, against the ratio of specific heats, for cylindrical and spherical symmetry. Guderley's exact values are marked and the agreement is to four figures. The Sedov blast's two-fifths is drawn beside them: it is the same for every gas, because it comes from dimensions and a conserved energy, and γ is dimensionless.

An exponent dimensions cannot give

A blast wave's radius goes as the two-fifths power of time, and the two-fifths is arithmetic: count the dimensions and it falls out. A shock converging on a point goes as the 0.717 power, and no amount of counting will produce that number — because it depends on the gas, and γ is dimensionless.

regimes · Similarity

A 20° cone at Mach 2: the shock at 37.80°, and a flow still compressing behind it. The conical flow round a cone of 20° half-angle at Mach 2, from the Taylor–Maccoll equation. The shock sits at 37.80° from the axis and turns the flow crossing it through only 8.57°, leaving it at Mach 1.693. Between the shock and the surface every ray from the apex carries its own state: the Mach number falls from 1.693 just behind the shock to 1.568 at the surface and the pressure rises from 1.586 to 1.912 times the free stream's — the rest of the turn, made without a shock. Nothing depends on the distance from the apex, so the rays are lines of constant state.

A cone finishes its turn after the shock

A wedge turns a supersonic stream all at once, at its shock. A cone of the same angle does not: its shock turns the flow only part of the way and leaves the rest to a smooth compression between the shock and the surface. Solved from Taylor and Maccoll's equation, the cone's shock is weaker, keeps more of the total pressure, carries less than half the wedge's surface pressure, and stays attached to 40.7° at Mach 2 where the wedge gives up at 23°.

compressible · Oblique shock

Below Mach 1.153 the boom turns back before it reaches the ground. The ray leaving the Mach cone straight down from an aeroplane at 11 km, at Mach 1.1, 1.15, 1.2, 1.5, 2, traced through a standard atmosphere whose sound speed rises from 295.1 m/s at the aeroplane to 340.3 m/s at the ground. A ray bends back upward where the local sound speed equals the aeroplane's speed, so Mach 1.1: turns at 4.00 km; Mach 1.15: turns at 0.25 km; Mach 1.2: lands 24.0 km on; Mach 1.5: lands 11.4 km on; Mach 2: lands 7.1 km on. The dividing speed, Mach 1.1533, is the ratio of the two sound speeds.

The boom that turns back before the ground

Sound is faster in the warm air near the ground, so a sonic boom's rays bend back upward on the way down. Whether any of them arrive is one comparison — the aeroplane's speed against the fastest sound beneath it — and the ray that just grazes the ground sets the edge of the carpet, which the uniform air of the ageing calculation cannot give it.

compressible · Sonic boom

The arrival map, and the place where it goes backwards. Where each boom ray lands on the ground, against the Mach number the aeroplane was doing when it launched it, for four accelerations from 15 km. In level flight this would be a straight line rising at the aeroplane's own speed. Here it falls before it rises: a ray launched at a higher Mach number is shorter and steeper, and near the cut-off it shortens faster than the aeroplane advances. Every minimum in these curves is a fold — two emission times delivering to one place — and on the fold itself neighbouring rays converge onto a single line.

The carpet an accelerating aeroplane folds

Level flight launches every boom ray with the same invariant, so they run parallel and each place hears one boom. Accelerate, and each successive ray is shorter than the last — shorter, near the cut-off, than the aeroplane's own advance — so later rays overtake earlier ones and the arrival map folds onto a line.

compressible · Sonic boom

What the rays say the edge is: nearly as loud, and then nothing. The overpressure across the carpet relative to the value under the track, from ray-tube spreading alone, at three Mach numbers from 15 km. It falls gently and then stops: at Mach 1.8 it is 0.92 ten kilometres out, 0.77 at twenty-three and 0.65 on the last ray, 36 km out, with silence beyond. Geometrical acoustics says the carpet ends at a cliff, and the boom at the edge of a real carpet fades as a rumble. The cliff is what the model says; it is also where the model stops.

The edge is a rumble, not a quieter bang

The rays that reach the outer half of a sonic-boom carpet arrive nearly horizontally, having travelled almost three times as far as the one under the track. Ray theory says they still carry two-thirds of the overpressure, right up to a line beyond which there is nothing. Neither half of that is what is heard — which is the useful result, because it says the edge's loudness is not a ray quantity at all.

compressible · Sonic boom

A boom is aged in the thin air it starts in

The rays that reach the edge of a sonic-boom carpet travel two and a half times as far as the one under the track, and it is natural to expect their signatures to have aged accordingly. They have not. A pressure wave distorts thin air far faster than dense air, so two-thirds of a boom's ageing is done in the stratosphere near the aeroplane, and the extra kilometres near the ground add little — which decides how far out a boom shaped to be quiet stays quiet.

compressible · Sonic boom

Named alongside it

The objects these essays reach for when they reach for this one.

Model limit Mach number Discontinuity Entropy Measurement Speed of sound Total pressure Characteristics Signal speed Sonic boom Conservation Far field

All concepts