The wing the equation is really solving

Somewhere in every account of compressible flow there is a factor $\beta = \sqrt{1-M^2}$ dividing a pressure coefficient, presented as a correction: the answer for the incompressible flow, divided by that, and a note that it fails near Mach one.

It is not a correction. It is a change of shape, and treating it as a factor works in two dimensions for a reason that is a coincidence of dimension rather than of physics. In three dimensions the shape it changes is the aspect ratio, and everything about how compressibility affects a real wing follows from that.

The substitution, and what it does to the boundary #

Linearise the equation for a small disturbance and it is

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

Substitute $\bar{y} = \beta y$, $\bar{z} = \beta z$, and it becomes Laplace’s equation. The compressible problem is an incompressible problem — of a different body, in a different space, with a scale factor on the answer.

Nothing about the air has changed. What has changed is the shape the equation sees.

Two dimensions, where the transformation hides #

In two dimensions the stretched body is a section that has been made thicker and more cambered by $1/\beta$ — and a thin section’s linearised answer is proportional to its thickness and camber, so the stretch can be undone by scaling the answer instead. The two operations cancel and what is left is

$$C_p = \frac{C_{p,0}}{\beta},$$

evaluated on the original section, which is the rule everybody quotes.

It is a special case. It survives because a section has one geometric dimension to stretch and one scale to divide by, and the two are the same number. Add a third dimension and they stop being the same number.

Three dimensions, where it does not #

Stretch a wing. The span is a $y$ dimension and the chord is an $x$ dimension, so the span is multiplied by $\beta$ and the chord is not. The aspect ratio $b^2/S$ therefore becomes $\beta\,AR$.

The wing whose incompressible flow the equation is solving is a wing of lower aspect ratio. At Mach 0.7 an aspect ratio of eight is really an aspect ratio of 5.71; at Mach 0.85 it is 4.21.

Run that through the lifting line and the algebra tidies itself:

$$a(M) = \frac{a_{\text{inc}}(\beta AR)}{\beta} = \frac{1}{\beta}\cdot\frac{2\pi}{1 + 2/(\beta AR)} = \frac{2\pi}{\beta + 2/AR}.$$

The $\beta$ in the finite-wing term has cancelled against the $\beta$ in front. The Mach number now sits beside a term that has no Mach number in it, and the larger that term is — the shorter the wing — the less of the correction survives.

The check is that the site’s lifting-line solver reproduces that closed form to six figures at three aspect ratios and four Mach numbers, having never been given it. The solver is told to solve a wing of aspect ratio $\beta AR$ and the answer is divided by $\beta$; the formula is a consequence, not an input.

The numbers, and what they mean for an aeroplane #

At Mach 0.7:

• AR 20 — slope 5.712 at rest, 7.718 compressible. A gain of 35.1 per cent.
• AR 8 — 5.027 at rest, 6.517 compressible. A gain of 29.7 per cent.
• AR 2 — 3.142 at rest, 3.666 compressible. A gain of 16.7 per cent.
• The section rule promises 40.0 per cent to all three.

So the correction that everybody applies as though it were a property of the air is out by more than a factor of two between a glider’s wing and a fighter’s. Compressibility is a property of the wing.

The design consequence runs the other way from the way the rule is usually read. A high-aspect-ratio wing gains more lift-curve slope with Mach number, which sounds like an advantage and is a liability: a steeper slope means a larger lift change for a given gust or a given change of incidence, and it means the aircraft’s stability derivatives move faster with speed. Short wings are less sensitive to Mach number for the same reason they are less efficient — their answer is dominated by a term that compressibility does not touch.

What the stretched space looks like #

The transformation is easier to trust once it is read as a statement about the field rather than about the answer. Compressible flow over a body is the incompressible flow over the same body with every distance across the stream stretched — so the disturbance reaches further out at Mach 0.7 than it does at rest, by a factor of $1/\beta = 1.40$, and further still at Mach 0.85, by 1.90.

That is the physical content of the whole rule. Compressibility makes a body’s influence reach further sideways, because information about it travels at a finite speed and the body is closing on that speed. The pressure at a fixed point above the wing is therefore larger; the pressure on the wing itself is larger; and the ratio is $1/\beta$.

It also says which quantities are unaffected. Anything measured along the stream is untouched — a chordwise position, the location of the pressure peak, where the recovery starts. Anything measured across it is stretched. A section’s pressure distribution therefore keeps its shape in the chordwise coordinate and changes only its scale, which is why the rule can be applied to a whole $C_p$ curve at once rather than point by point.

What the transformation is not #

It is worth being precise about what is exact here and what is not, because the rule has a reputation for being unreliable that it half deserves.

The transformation is exact for the equation it is applied to. Given the linearised potential equation, the affine map is an identity, not an approximation, and this site’s two routes to it agree to machine precision.

The linearisation is what is approximate. Dropping the nonlinear terms requires the disturbance to be small compared with the free stream, which requires the body to be thin and the incidence small. So the accuracy of the Prandtl–Glauert rule is not a function of Mach number in the first instance — it is a function of thickness, and its Mach dependence enters because a thicker effective section is a larger disturbance.

And the rule dies at the critical Mach number, not at Mach one. Once a supersonic pocket appears on the section the equation has changed type in part of the domain and no transformation of an elliptic equation is going to help.

The stretched-section picture also explains the failure without any extra machinery. As the Mach number rises the effective section gets thicker; a thicker section has a stronger suction peak; the peak reaches sonic conditions sooner. So the rule’s own transformation predicts the conditions under which the rule stops applying — the critical Mach number is where $C_{p,0}/\beta$ first touches the sonic value, an equation with the correction factor on both sides of the problem.

What the picture cannot show #

No shocks, no wave drag, no thickness effects beyond the linear ones. Everything here is linearised potential flow, which has no mechanism for entropy and therefore none for wave drag.

The transformation of a swept wing is not the transformation of a straight one. Stretching $y$ changes the sweep angle as well as the aspect ratio — $\tan\Lambda$ becomes $\beta\tan\Lambda$ — so a swept wing’s compressible behaviour is a different wing in two respects at once, and the independence principle this site computes elsewhere is the cleaner way to reach that case.

And the lifting line is not a wing. It has no chordwise loading, no thickness and no tips beyond a mathematical point, so the answers above are the lifting line’s answers transformed rather than a wing’s.

What the rule does to a pressure distribution #

Since the transformation stretches only the direction across the stream, a section’s pressure distribution keeps its shape along the chord and changes only its scale. That has two consequences worth stating as facts a designer can use.

The suction peak stays where it is. A section whose peak sits at fifteen per cent of chord at low speed has it at fifteen per cent at Mach 0.7 as well, and the recovery that follows starts from the same station. What has changed is the depth of the peak, by the factor $1/\beta$, and therefore how steep the recovery behind it has to be — so a section that separates at a given peak suction reaches that peak at a lower lift coefficient as the Mach number rises.

And the whole curve scales together. A designer who has shaped a rooftop and a recovery at low speed does not need to reshape them for compressibility; the same section produces the same shape of curve with everything multiplied. That is why low-speed section design remained useful into the transonic era, and why the arrival of genuinely transonic sections — supercritical ones, with their flat rooftops and rear loading — required abandoning the linear rule rather than refining it.

The limit is the same in both statements. The rule multiplies a curve, so it can be trusted until the multiplied curve touches the sonic pressure coefficient somewhere, and after that the flow has a pocket in it and no factor applied to a subsonic answer describes anything.

What the rule does to the other two coefficients #

Lift is the coefficient everybody applies the rule to, and it is the one where the rule is least interesting, because the answer is simply the factor. The moment and the induced drag are where the affine map says something that could not have been guessed from a correction factor, and in both cases the answer is that nothing happens at all.

The aerodynamic centre does not move. The transformation leaves every chordwise station alone and multiplies every pressure ordinate by the same number. A distribution scaled uniformly has its centroid in the same place, so the centre of pressure sits at the same fraction of chord at Mach 0.7 as at rest, and the lift and the pitching moment about any fixed point are multiplied by an identical $1/\beta$. Their ratio — which is what an aerodynamic centre is — is therefore a Mach-number invariant, and linearised subsonic theory puts it at the quarter chord for every section at every subsonic Mach number whatever.

That is worth holding beside what aircraft actually do. The aerodynamic centre of a real wing moves back as it approaches its critical Mach number, by something like a quarter of the chord, and the nose-down pitching moment that results is one of the defining hazards of early transonic flight. The rule above says that shift is not compressibility in the sense the rule describes. Every part of it is nonlinear: the pocket, the shock that terminates it, the separation the shock provokes, and the loss of rear loading behind it. The linear theory’s silence here is a correct prediction, not a gap — it says that the pitch-up and the tuck cannot be understood as a scaling of a subsonic answer, and the history of the aircraft that discovered them the hard way is a demonstration that it is right.

And the induced drag of a given lift coefficient is untouched. This one falls out of the algebra in two lines and is easy to disbelieve. Take the elliptically loaded wing, whose induced drag coefficient is $C_L^2/(\pi AR)$. In the transformed problem the aspect ratio is $\beta AR$ and the lift coefficient is $\beta C_L$, so the incompressible induced drag is $\beta^2 C_L^2/(\pi \beta AR)$; divide by $\beta$ as the transformation requires and

$$C_{D_i} = \frac{C_L^2}{\pi AR},$$

with no $\beta$ anywhere. The compressible answer is the incompressible formula, at the wing’s own aspect ratio, at the wing’s own lift coefficient.

The reason is worth stating without the algebra. Induced drag is the price of the wing’s trailing vorticity, the loading that minimises it is elliptic, and stretching a span uniformly maps an ellipse to an ellipse — so the transformation takes the optimum to the optimum and cannot change what the optimum costs. Compressibility changes the incidence a wing needs for a given lift and does not change what that lift costs in induced drag. An airliner’s induced drag at cruise is what its aspect ratio and its lift coefficient say it is, and the Mach number enters only through the fact that flying faster means flying at a smaller $C_L$.

What the rule cannot correct is the drag it never had. Linearised subsonic potential flow gives every body zero profile drag and zero wave drag, exactly as ideal flow always does, so there is no subsonic answer to scale. Dividing a measured drag polar by $\beta$ is not a mild abuse of the rule; it is applying a transformation to a quantity the transformed equation does not contain. The drag rise that matters at high subsonic speeds is wave drag, it appears past the critical Mach number, and it is the one aerodynamic quantity in this essay whose Mach dependence is not a factor, not a stretch, and not smooth.

One consequence of the induced-drag result deserves a sentence on its own, because it settles an argument pilots have. If induced drag at a given lift coefficient does not depend on Mach number, then the whole of the drag rise at high subsonic speed is profile and wave drag, and none of it is the wing working harder to hold the aeroplane up. Flying faster reduces the induced drag only through the lift coefficient, which is the same reduction a lighter aeroplane gets, and neither has anything to do with compressibility.

Who found it, and the honest history #

Prandtl had the two-dimensional rule by 1922 in lectures, Glauert published it in 1928, and both worked in the two-dimensional case where the transformation looks like a factor. Göthert’s 1940 report is the one that did it properly in three dimensions and pointed out that the transformed planform is a different planform — a paper written in the middle of a war and translated a decade later, by which time transonic aircraft had made the point unavoidable.

The connection worth keeping is with the essay on asking for a pressure distribution. Both are cases of the same manoeuvre: a boundary-value problem is turned into another boundary-value problem by moving the boundary rather than by changing the equation. In the inverse problem the boundary is unknown and the pressure is given; here the pressure is unknown and the boundary is stretched. In both cases the equation is Laplace’s, in both cases the difficulty is entirely in the geometry, and in both cases the statement that looks like a formula turns out to be a statement about shape.

The delightful part is that this was not obvious to the people who found it. Glauert’s derivation treats the transformation as a substitution in the equation and never draws the stretched body; the picture — the same section, fatter — appears only when somebody asks what the incompressible problem being solved is about.

One more consequence deserves stating, because it is a rule of thumb with a derivation. Since the compressible slope is $2\pi/(\beta + 2/AR)$, the fraction of the correction a wing receives is $\beta AR/(\beta AR + 2)$ divided by the same thing at $\beta = 1$ — which means a wing receives essentially the full section correction once $AR$ is large compared with $2/\beta$, and essentially none once it is small compared with it. The crossover is at an aspect ratio of about three at Mach 0.7. Below that, a designer may treat compressibility as a second-order effect on lift-curve slope; above it, the section rule is nearly right. Between them lies most of general aviation.

Where the ladder goes next #

The rung above is the transonic small-disturbance equation, where the nonlinear term is kept because it is what makes the pocket, and where the similarity parameter is $K = (1-M^2)/\tau^{2/3}$ rather than $\beta$ alone — a rule that says which pairs of section and Mach number behave alike rather than how one behaves.

The one beside it is supersonic, where the equation is hyperbolic and the same affine map turns it into the wave equation instead. That is where the area rule lives, and where a slender body’s loading turns into a statement about its cross-sectional area distribution and nothing else.