Two wings and it does not matter where

There is a class of result in physics whose statement understates it. The induced drag does not depend on the stagger sounds like an empirical finding — somebody measured four biplanes and the numbers came out close. It is not that.

The induced drag is computed from the circulation in one cross-section of the wake, far downstream. Stagger is a displacement along the flight direction. That plane has no flight-direction coordinate in it. There is nowhere in the calculation to enter the stagger, and no arrangement of surfaces that share a Trefftz plane can be told apart by it.

That is Munk’s stagger theorem, and the right way to state it is not the drag is unchanged but there is no place in the computation where the stagger could appear.

The theorem underneath it #

The stagger theorem is a consequence of a smaller and less famous one, and the smaller one is what a calculation can be checked against.

Munk’s reciprocal theorem. For two lifting systems, the drag the first suffers from the second’s downwash equals the drag the second suffers from the first’s. $D_{12} = D_{21}$.

It is not obvious. The two wings are different sizes, at different heights, carrying different loadings, and there is no symmetry in sight. What makes it true is that the influence of one vortex on another is symmetric in the pair — the same geometric factor appears in both directions — and the drag is a bilinear form in the two circulations.

The check is worth describing because the code does not arrange it. Influence coefficients are built by a one-sided sum: the velocity that segment k’s shed vorticity induces at segment i’s control point. Nothing makes the matrix symmetric. It comes out symmetric to 2 × 10⁻¹⁴, and the biplane gate computes the two off-diagonal blocks separately and requires them to agree.

Once the mutual drag is symmetric, the total drag is a symmetric quadratic form in the wake’s circulation, and the flight-direction positions of the surfaces have already dropped out of the formulation. Reciprocity is the theorem; stagger-independence is the corollary.

The direction of the interference, which is the surprise #

Read that number again. The pair costs less than the monoplane.

This is the opposite of what a biplane is usually said to be. The standard account has the two wings getting in each other’s way, the arrangement paying an aerodynamic price, and the price being worth it because two short braced wings are much lighter than one long cantilevered one.

The structural half is true. The aerodynamic half is backwards, and the reason is exactly the reason a winglet works: a biplane’s wake is not in one plane, and a non-planar wake is cheaper for a given lift than a planar one of the same width.

A biplane is a non-planar lifting system. So is a wing with winglets. The arithmetic does not distinguish them and neither should the explanation.

Where the saving comes from, if not from getting out of the way #

The mechanism deserves a paragraph in its own terms, because non-planar wakes are cheaper is a statement about a functional and not yet a picture.

Induced drag is the kinetic energy the wing leaves behind, per unit distance flown. A lifting system has to impart downward momentum to the air, and how much energy that costs depends on how much air it is spread over: the same momentum given to twice the mass of air at half the velocity costs half the energy, because momentum is linear in velocity and energy is quadratic.

So the whole subject is how much air does the wing engage with. A long wing engages a wide column; that is the span argument, and it is why induced drag falls as the square of the span.

A wake that leaves the plane engages a taller column for the same width. Two wings a fifth of a span apart set air in motion over a region that extends in z as well as in y, and the same downward momentum is spread over more mass. Hence less energy, hence less drag.

That is the same sentence as the winglet essay’s and it should be, because it is the same mechanism. A biplane is a winglet arrangement with the vertical parts left out and the horizontal parts doubled. The arithmetic sees the wake and does not care which description is used.

The interference factor, and the agreement worth noticing #

Prandtl wrote the biplane’s drag as

$$D = \frac{L_1^2 + 2\sigma L_1 L_2 + L_2^2}{\pi q b^2},$$

with σ a function of the gap-to-span ratio, tabulated in a chart that has been reproduced in textbooks for a century. For equal lifts this gives $D = \tfrac12(1+\sigma)$ times the monoplane’s, so σ can be read straight out of the computed ratio.

The values come out 0.891, 0.785, 0.656, 0.484, 0.291, 0.130 and 0.043 at gaps of 0.02 to 1.6 spans. The chart gives 0.485 at 0.2.

Nothing here was fitted to that chart. The lattice was built for a winglet essay, checked against the elliptic wing’s closed form, and pointed at a biplane afterwards. Agreement to three digits with a number computed by different methods a hundred years ago is the kind of check this site prefers to any internal consistency test, because it can fail for reasons the author did not think of.

The gate requires more than the value. It requires σ to fall monotonically with gap, to exceed 0.85 at a gap of two per cent of span — where two wings are nearly one wing — and to fall below 0.08 at 1.6 spans. And it requires the limit: at forty spans apart, two wings each carrying half the lift must together pay exactly half the monoplane’s drag, because each is then an independent wing of the same span carrying half the lift, and induced drag goes as the square.

What the theorem does and does not license #

The stagger theorem is strong and it is narrow, and the narrowness is worth being explicit about because the theorem is frequently over-read.

What it says: the induced drag of a system of lifting surfaces is independent of their positions along the flight direction, for a given distribution of circulation in the Trefftz plane.

What it does not say: that the surfaces do not affect each other. They emphatically do — a staggered biplane’s rear wing sits in a completely different flow from an unstaggered one’s, its local incidence is different, and if both wings are set at the same geometric incidence they will carry different circulations at different staggers.

The resolution is in the phrase for a given distribution of circulation. Stagger changes what loading a given geometry produces. It does not change what a given loading costs. A designer who staggers a biplane and leaves the rigging alone will measure a change in drag, and the change is because the loading moved, not because the theorem failed.

That distinction — the arrangement versus the loading — is the whole content, and it is the same distinction the winglet essay needed and the same one the bell-loading essay turned on. Munk’s theorem lets a designer separate what shall the wake look like from what geometry will produce it, and answer the first without the second.

The optimum division of lift #

There is a design question the machinery answers immediately and that the standard treatment leaves implicit: how should the total lift be shared between the two wings?

For two wings of equal span at a given gap, the answer is equally, and it follows from symmetry. For two wings of unequal span it does not, and the constrained optimisation — minimise the quadratic form subject to a fixed total lift — gives the answer directly for any pair.

Two readings follow from doing it rather than assuming it.

The optimum is flat near its minimum. Moving ten per cent of the lift from one wing to the other, on an equal-span pair, changes the induced drag by a fraction of a per cent. That is a general property of a quadratic form at its stationary point and it is why real biplanes could be rigged by eye and still be within a per cent of the best they could do.

And a small upper wing is worth more than a small lower wing is. With unequal spans the optimum gives the longer wing more than its share of area would suggest, because the longer wing spreads its own shed vorticity further and pays less for its own lift. That is the sesquiplane — a large upper wing over a small lower one — arrived at by builders empirically, and it is the arrangement the arithmetic prefers.

Why biplanes went away anyway #

If a biplane is aerodynamically better than the monoplane of the same span, the obvious question is why every aeroplane is a monoplane.

Because span is the variable, not arrangement. A biplane of span b beats a monoplane of span b. It does not beat a monoplane of span 1.4b, and the monoplane is allowed to be longer. Once cantilever construction made a long single wing structurally possible — metal stressed skin, from the late 1920s — the comparison stopped being at equal span and the biplane lost.

Because the interference falls off slowly. To get σ below 0.3 the gap has to be over three tenths of the span, which on a wing of ten metres is a three-metre gap. Real biplanes had gaps of a tenth to a sixth of the span, where σ is 0.5 to 0.65 and the saving is a fifth of the monoplane’s induced drag rather than a half.

Because everything else about a biplane is worse. Two wings have twice the wetted area for the same area of lifting surface, which is profile drag; the struts and wires between them are drag with no lift at all; and the interference is on the induced drag, which at cruise is the smaller half of the total.

So the aerodynamic case is real, narrow, and was outweighed. A biplane is a good answer to the question how is a stiff wing made out of wood and wire, and a mediocre answer to any other question, and the induced-drag advantage was never the point even when it was true.

What the model does not contain #

No profile drag, no interference drag, no struts and no wires. Every number is induced drag. A real biplane’s bracing is a substantial fraction of its total drag and none of it is here.

Rigid, flat, undeformed wakes. Two vortex sheets a fifth of a span apart interact and roll up together, and the linear calculation does not follow them.

Equal spans and equal lifts, except where stated. Real biplanes were usually unequal in both, and the optimum division of lift between two surfaces of unequal span is a constrained problem the same machinery would answer and this essay does not ask.

No decalage and no rigging. The angle between the two wings is a real design variable that decides the division of lift, and it is exactly the thing the stagger theorem says is a loading question rather than an arrangement one.

And the theorem is about induced drag only. A staggered biplane genuinely has different profile drag, different interference drag and quite different stalling behaviour from an unstaggered one, and none of those is what the theorem is about.

The speed at which the theorem stops #

The argument’s whole strength is that the Trefftz plane has no flight-direction coordinate in it, so there is nowhere to put the stagger. That is a statement about the equation, and it holds while the equation is Laplace’s. It fails as soon as the equation changes type.

Subsonic compressibility is safe. The Prandtl–Glauert transformation stretches the flight direction and leaves the cross-plane alone, so a staggered system maps to a differently staggered system with the same cross-section — and the theorem survives with the stagger stretched. Nothing in this essay’s arithmetic notices.

Supersonically it fails completely. Disturbances no longer reach everywhere at once; they travel on Mach cones, so a surface behind another sits inside the first’s wave field or outside it depending on how far behind it is. The influence of one surface on another now depends explicitly on their longitudinal separation, and there is a flight-direction coordinate in the problem after all.

That is why the supersonic area rule must be applied on Mach-plane cuts rather than on cross-sections. The transonic version, at Mach one, uses square cuts and is the stagger theorem’s last valid form; above it the cuts lean, they lean differently at every roll angle, and the arrangement along the body has become part of the answer.

The constructive demonstration is the one worth carrying, because it turns the failure into a device. Busemann proposed in 1935 a supersonic biplane: two half-diamond sections facing one another across a gap, arranged so that the compression from each is met by the expansion from the other. The waves cancel between them and the pair has, in theory, no wave drag at all — while each section alone has plenty.

And it works at one gap-to-chord ratio and one Mach number. Move the sections apart, or fly faster, and the waves arrive in the wrong places and the cancellation is lost. A configuration whose drag depends that sharply on the longitudinal arrangement is as complete a refutation of stagger-independence as anything could be — and it is the same two wings, one above the other, that this essay has been calling indifferent.

What a modern reading of it is for #

The theorem outlived the configuration and it is worth saying what it is used for now, because the answer is not biplanes.

Every non-planar wing calculation. Winglets, raked tips, box wings, joined wings, C-wings and closed-loop wings are all multi-surface systems, and every one of them is analysed by finding the optimum loading on a wake trace and reading the drag off it. The reciprocal theorem is what makes that a well-posed quadratic minimisation rather than a search.

Formation flight and ground effect. Two aircraft in formation are a two-surface system; an aircraft near the ground is a surface and its image. Both are the same machinery with different traces, and the image system this site already owns is a Trefftz-plane calculation in disguise.

And the separation of design questions. The single most useful consequence is the one stated above: what shall the wake look like can be answered before what geometry produces it. A designer can find the best achievable induced drag for a set of surfaces before drawing any of them, and then find out how close a buildable geometry gets. That is a much better-posed way of working than iterating on shapes, and it is available only because the drag depends on a functional of a curve.

Who found it, and when #

Max Munk proved the stagger theorem in his 1919 Göttingen dissertation, when the biplane was the aeroplane and the question of how to arrange two wings was the central practical problem in aerodynamics. Prandtl’s interference factor and its chart come from the same period and the same group.

Munk moved to the United States in 1921 and worked at NACA, where he designed the Variable Density Tunnel — the instrument that made Reynolds-number-correct section data possible and that produced the aerofoil catalogues the next twenty years of aircraft were designed from. The stagger theorem was a young man’s dissertation result and it outlasted the configuration it was about.

And the configuration went away within fifteen years of the theorem being proved. By 1935 the question the theorem answered was no longer being asked, which is why it survives mainly as a curious fact rather than as a design tool — despite being, in its reciprocal form, the foundation of every non-planar wing calculation since, including the winglet on every airliner flying.

Where the ladder goes next #

Two wings above each other share a wake. So do two wings beside each other, and the same arithmetic answers what a neighbour is worth — which turns out to be exactly half, at exactly the spacing birds use.