Wing sweep ($\Lambda$)

Wing sweep describes the inclination of a chosen spanwise line in a wing planform relative to the direction perpendicular to the aircraft's longitudinal axis. It must name the reference line: a leading-edge sweep angle and a quarter-chord sweep angle can differ on the same aircraft wing. A swept wing therefore needs a reference convention, not just an unqualified angle.^[ref]

Let $x$ increase aft, let $y$ increase from root toward one tip, and define the straight leading edge and linearly varying airfoil chord byThe line at chord fraction $f$ has $x_f(y)=x_{\mathrm{LE}}(y)+fc(y)$. Differentiating gives the exact geometric relationFor $b=10\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=1\mathrm{m}$ and ${\Lambda }_{\mathrm{LE}}=20{}^{\circ }$, the quarter-chord sweep angle is $\arctan (\tan 20{}^{\circ }-0.05)\simeq 17.43{}^{\circ }$. This reference-line dependence of wing sweep is purely geometric.

There is also a useful high-speed approximation. Resolving the incoming velocity perpendicular to a swept straight leading edge gives the normal Mach number for a swept leading edge, $M_n=M\cos {\Lambda }_{\mathrm{LE}}$. Thus $M=0.9$ and ${\Lambda }_{\mathrm{LE}}=30{}^{\circ }$ give $M_n\simeq 0.779$. This component is a guide to compressibility effects, not an exact prediction of shock wave formation on a finite wing with thickness and a fuselage.

In a 1947 experiment, Richard T. Whitcomb compared unswept wings with forward and backward sweep in the Langley 8-foot high-speed wind tunnel. Pressure measurements and wake surveys over Mach numbers from 0.60 to 0.96 showed shock-associated changes occurring later for the swept configurations. The reported shifts were not exactly the simple cosine prediction.^[ref]

The normal Mach number for a swept leading edge is the incoming velocity component perpendicular to that edge divided by the local speed of sound. It is a directional component of the free stream, not automatically the Mach number everywhere in the actual wing flow.

Let $\mathbf{e}$ be a unit tangent along a straight leading edge, let $\mathbf{U}$ be the incoming velocity, and let $a$ be the speed of sound. Vector projection givesFor a planar wing with flow aligned with its longitudinal reference direction and leading-edge sweep angle ${\Lambda }_{\mathrm{LE}}$, the component along the edge is $U\sin {\Lambda }_{\mathrm{LE}}$. ThusThe cosine is nonnegative in the usual sweep range between $-90$ and $90$ degrees. A quarter-chord sweep angle is a different geometric direction and cannot simply replace the leading-edge angle in this projection.

For an original example, $M=1.2$ and ${\Lambda }_{\mathrm{LE}}=45{}^{\circ }$ give $M_n\approx 0.848528$. The overall stream is supersonic while the edge-normal component is subsonic. For $M>1$, the boundary $M_n=1$ occurs at $|{\Lambda }_{\mathrm{LE}}|=\arccos (1/M)$, about $33.5573{}^{\circ }$ for $M=1.2$. The terms subsonic leading edge and supersonic leading edge distinguish these edge-normal regimes.

Using the component as a predictor of an entire finite-wing flow adds assumptions. Thickness, finite span, angle of attack and interference with the fuselage can all matter. Whitcomb's pressure and wake measurements found that shock-associated changes with sweep did not follow the simple cosine shift exactly.^[ref] The projection itself is exact for the stated vectors; the reduction of a three-dimensional aerodynamic problem to that single component is the approximation.

A supersonic leading edge has an incoming edge-normal flow component exceeding the speed of sound. For a straight planar edge at leading-edge sweep angle ${\Lambda }_{\mathrm{LE}}$ in the convention of normal Mach number for a swept leading edge, the criterion isIt classifies a direction of the incoming velocity, rather than asserting that the same local Mach number holds over the whole aircraft wing.^[ref]

For an original example with $M=2$ and ${\Lambda }_{\mathrm{LE}}=30{}^{\circ }$, $M_n=\sqrt{3}\simeq 1.732$. The condition can persist at larger sweep: at $M=2.2$, even a $60{}^{\circ }$ leading-edge sweep gives $M_n=1.1$: large sweep by itself does not establish a subsonic leading edge.

For $M>1$, the Mach angle is $\mu =\arcsin (1/M)$. On an aft-swept half-wing the edge's angle from the downstream streamwise direction is $\theta =90{}^{\circ }-{\Lambda }_{\mathrm{LE}}$. Thusso the leading edge lies outside the downstream Mach cone from its apex in this ideal geometry. At $M=2$, $\mu =30{}^{\circ }$ whereas the example edge has $\theta =60{}^{\circ }$. Equality is the sonic boundary and belongs to neither strict inequality.

The classification helps select a high-speed flow model. It does not determine shock wave strength, drag or flow separation without the section shape and operating conditions. A measured pressure coefficient distribution or flow visualization can test those additional predictions; naming the regime alone cannot.

A subsonic leading edge has an incoming edge-normal flow component below the speed of sound. It can occur even when the free stream is supersonic: the relevant component is the normal Mach number for a swept leading edge,Using a quarter-chord sweep angle instead would project onto a different direction and can change the classification incorrectly.^[ref]

For an original example, $M=1.6$ and ${\Lambda }_{\mathrm{LE}}=60{}^{\circ }$ give $M_n=0.8$. If the same aircraft wing is exposed to $M=2.2$ with unchanged sweep, $M_n=1.1$ and the edge is instead a supersonic leading edge. This is a flow-condition classification, not an immutable geometrical name.

For $M>1$ and aft sweep between zero and $90{}^{\circ }$, the threshold isAt $M=1.6$, the boundary is about $51.32{}^{\circ }$. Equivalently, the edge's angle from the downstream streamwise direction is less than the Mach angle, placing it inside the apex Mach cone. The normal Mach number for a swept leading edge article derives the underlying vector projection; no assumption about a measured surface pressure is needed for that geometry.

Subsonic edge-normal inflow does not imply that every part of the three-dimensional flow is subsonic or free of shock waves. Thickness, angle of attack and interference with a fuselage affect the surface flow. The regime is useful precisely when kept separate from those further aerodynamic predictions.

Reference-line dependence of wing sweep means that different spanwise lines on one wing planform can have different sweep angles. Naming the line is therefore necessary to specify the geometry. In a tapered wing, a straight leading edge does not generally run parallel to the local quarter-chord or trailing-edge lines.

Let $x$ increase aft and $y$ run outward on one half-wing. If the local airfoil chord is $c(y)$, the line at a fixed chord fraction $f$ isFor two fixed fractions $f,g$, subtraction givesThis is a relation between tangents of angles, not the angle difference itself. Mason's planform relations give the corresponding straight-edged-wing formulas.^[ref]

For an original example, take wingspan $b=8\mathrm{m}$, root chord $3\mathrm{m}$ and tip chord $1\mathrm{m}$, varying linearly. Then $c^{\prime }=(1-3)/(8/2)=-0.5$. Choose a leading-edge slope $x_{\mathrm{LE}}^{\prime }=0.125$. Its leading-edge sweep angle is $\arctan (0.125)\approx 7.1250{}^{\circ }$, but its quarter-chord sweep angle obeys $\tan {\Lambda }_{1/4}=0.125-0.125=0$. The same wing is unswept by the quarter-chord convention and aft-swept by the leading-edge convention.

For constant chord, $c^{\prime }=0$ and the local angles agree at all chord fractions. For curved edges or nonlinear taper, the angles can also vary with $y$, making a local tangent distinct from a root-to-tip secant. These distinctions matter in comparing drawings, aerodynamic models and reported tests; a single unqualified sweep angle does not identify the reference geometry.

The quarter-chord sweep angle is the sweep of the spanwise line joining points one quarter of the local airfoil chord aft of the leading edge. It is a wing-planform reference direction, not the angle of attack and not the mean aerodynamic chord's quarter-point location.

For chordwise coordinate $x$ increasing aft,The reference-line dependence of wing sweep gives the general relation. On a straight-edged trapezoidal wing with full wingspan $b$, root chord $c_r$ and tip chord $c_t$,Mason explicitly relates sweep at different chord fractions in the planform geometry.^[ref]

For an original inverse calculation, let $b=8\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=0.8\mathrm{m}$ and ${\Lambda }_{1/4}=15{}^{\circ }$. The leading-edge tangent is $\tan 15{}^{\circ }+0.075$, giving ${\Lambda }_{\mathrm{LE}}\approx 18.9294{}^{\circ }$. With the root leading edge at $x=0$, the root quarter-point is at $0.5\mathrm{m}$. The tip quarter-point is $0.5+4\tan 15{}^{\circ }\approx 1.57180\mathrm{m}$ aft of that origin, while the tip leading edge is $0.2\mathrm{m}$ farther forward.

This calculation uses the full span in the chord-gradient formula and the semispan in the root-to-tip displacement. It also shows why the normal Mach number for a swept leading edge needs its own leading-edge direction: a reported fifteen-degree quarter-chord sweep does not make the leading edge fifteen degrees.

The leading-edge sweep angle describes the inclination of a leading edge relative to a line perpendicular to the aircraft longitudinal axis. On one half-wing, take $x$ positive aft and $y$ positive outward. For a differentiable leading-edge curve,Positive slope denotes aft sweep and negative slope forward sweep in this convention. A coordinate reference must be supplied before the sign has meaning.^[ref]

For a straight edge on a symmetric wing, let its tip be displaced aft from the root by $\Delta x$ and let $b$ be the full wingspan. ThenFor an original example, $b=12\mathrm{m}$ and $\Delta x=2.4\mathrm{m}$ give ${\Lambda }_{\mathrm{LE}}=\arctan 0.4\approx 21.8014{}^{\circ }$. Incorrectly using the full span as the root-to-tip run would give $\arctan 0.2\approx 11.3099{}^{\circ }$. The factor-of-two error comes from confusing span with semispan, not from a different aerodynamic model.

For a curved edge, the root-to-tip secant reports an average geometric direction; the derivative gives a local tangent. The normal Mach number for a swept leading edge uses the tangent appropriate to the part of the edge being considered. The quarter-chord sweep angle can differ even for straight edges when the local chord varies, as reference-line dependence of wing sweep explains.

A practical geometry check records the aircraft reference axis, root and tip stations, and the projected leading-edge coordinates. Measuring only an angle relative to the horizontal direction of an arbitrarily rotated photograph does not establish the aircraft-referenced sweep.