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An aircraft is a vehicle capable of flight through the atmosphere. Its support can involve aerodynamic lift or buoyancy. A fixed-wing aircraft obtains aerodynamic lift from its aircraft wings as it moves relative to the surrounding air. An engine can supply thrust, but propulsion and the aerodynamic support of gravitational weight are distinct functions.^[ref]

For straight, level flight at constant velocity, neglect vertical thrust and align the remaining thrust with the flight direction. The force balance isHere $L$ is lift, $D$ is drag, $T$ is thrust and $W$ is gravitational weight. For a stipulated $1000\mathrm{kg}$ vehicle with $g=9.81m/{\mathrm{s}}^2$ and lift-to-drag ratio $L/D=12$, this requires $L=9810\mathrm{N}$ and $T=D=817.5\mathrm{N}$. At an air-relative speed of $50m/s$, $TV=40.875\mathrm{kW}$ is the useful propulsive mechanical power; an engine's input power can be larger because of losses.

An unpowered glider can instead maintain a steady descending velocity. If its glide angle below the horizontal is $\gamma$, resolving gravitational weight perpendicular and parallel to the flight path givesConsequently an assumed constant $L/D=12$ corresponds to $\gamma \simeq 4.76{}^{\circ }$ and 12 units of horizontal travel per unit of height lost in still air.^[ref] The loss of gravitational potential energy supplies the work against drag; the kinetic energy need not be falling during that steady descent.

These balances concern translation of the vehicle. Maintaining its orientation also requires a balance of torques about its centre of mass. An aircraft can satisfy an instantaneous vertical force balance while still having an unwanted pitching moment.

The area rule is an aerodynamic design principle relating transonic wave drag to the longitudinal distribution of the total cross-sectional area of an aircraft, including its fuselage and aircraft wings. A wing-body combination can create a rapid area increase even when each part appears streamlined in isolation. Redistributing the fuselage area near the wing can smooth the combined distribution. This does not say that any reduction in frontal area guarantees lower total drag.^[ref]

An original geometric model makes the redistribution explicit. Let $x$ increase along the fuselage, and let $A_b(x)$ and $A_w(x)$ denote body and added-wing areas, counting their overlap only once. Over a stipulated interval of length $\ell$, takeIf the desired combined distribution on that interval is $A_0$, the bookkeeping choice $A_b(x)=A_0-A_w(x)$ gives $A_b+A_w=A_0$. For $A_0=3.0{\mathrm{m}}^2$, $a=0.6{\mathrm{m}}^2$ and $\ell =4\mathrm{m}$, the body's area is $2.4{\mathrm{m}}^2$ at the midpoint. Relative to a constant-area body, the removed volume isThis invented example exposes a design tradeoff: smoothing an area distribution can consume internal volume. It is not a calculation of wave drag, and the rest of the vehicle still needs smooth transitions.

Richard T. Whitcomb's motivating test used a specially built wing-body model in a wind tunnel, before applying the shape change to a fighter configuration. Comparing measured drag through the transonic range tests the proposed area-distribution effect; appearance alone does not. Viscous stress, lift-induced drag, loading and off-design conditions still matter to the complete aircraft.^[ref]

A fixed-wing aircraft obtains most of its aerodynamic support from aircraft wings that do not continuously rotate around a shaft to generate lift. The word fixed distinguishes the lifting arrangement; it does not require every surface to be immovable or perfectly rigid. Both powered airplanes and an unpowered glider can use fixed wings.

Thrust and aerodynamic support are distinct. Relative motion through air produces lift and drag; an engine can supply the thrust needed to balance drag in steady level flight. The components must be resolved as vectors, since their directions change with the motion and attitude.^[ref]

For an original ideal turning model, suppose a point-mass aircraft maintains horizontal circular flight at constant airspeed $V$, with negligible vertical thrust and no net vertical acceleration. If its lift vector is inclined through a bank angle $\varphi$, thenThe horizontal component supplies centripetal acceleration. Eliminating $L$ gives $L/W=1/\cos \varphi$ and $R=V^2/(g\tan \varphi )$. With $V=40m/s$, $\varphi =30{}^{\circ }$ and $g=9.81m/{\mathrm{s}}^2$, the radius is about $282.5\mathrm{m}$ and the required lift is about $1.1547W$. The wing remains fixed to the body while its force direction changes in space.

This is a Newtonian force model, not a prediction that an arbitrary airplane can produce the required lift. The attainable lift coefficient, wing loading, propulsion and aerodynamic moments determine whether a particular steady condition can exist. In a steady unpowered descent, the balances instead lead to the glide angle relation.

A fuselage is the main body of a conventional aircraft, accommodating payload or equipment and connecting major assemblies such as the aircraft wings. It is part of the load-carrying structure as well as the aerodynamic shape. Not every aircraft has a separate conventional fuselage, and not every fuselage is pressurized.^[ref]

The structural role is more than enclosing a volume. Forces from attached assemblies and distributed gravitational weight must be transmitted through the body, producing bending, shear and axial loading. A pressurized body also carries loads from the pressure difference between its interior and the atmosphere. These are different loading mechanisms, even if they act on the same shell.

An original ideal model illustrates the pressure contribution. Approximate a closed cylindrical section as a thin-walled pressure vessel of radius $R$, wall thickness $t$ and internal excess pressure $\Delta p$. Force balance across a longitudinal cut of length $\ell$ gives $2R\ell \Delta p=2t\ell {\sigma }_{\theta }$. Balance of an end cap gives $\pi R^2\Delta p=2\pi Rt{\sigma }_z$. ThusThe first is hoop stress and the second axial stress. For $R=1.5\mathrm{m}$, $t=2\mathrm{mm}$ and $\Delta p=50\mathrm{kPa}$, they are $37.5$ and $18.75\mathrm{MPa}$. This stipulated membrane model excludes cutouts, joints, stiffeners and aerodynamic bending; it is not a complete fuselage stress analysis.

Shape also affects the combined flow around the body and wings. Richard T. Whitcomb's area rule work examined the longitudinal area distribution of the entire combination, rather than judging the wing and fuselage as isolated streamlined parts. That connection explains why structural packaging and aerodynamic design cannot always be optimized independently.

A glider is an aircraft designed for sustained flight without continuous engine thrust. Its aircraft wings obtain aerodynamic lift from motion relative to the air. It can descend steadily while its speed remains constant: loss of gravitational potential energy, rather than a continuous reduction of kinetic energy, can supply the work against drag.^[ref]

In still air and a steady unpowered glide, the mechanical power balance iswhere $V$ is airspeed and $w>0$ is downward speed. This follows from $D=W\sin \gamma$ and $w=V\sin \gamma$ in glide angle. Motion of the surrounding air changes the ground-relative energy account; rising air can offset a glider's descent relative to that air.

Two performance objectives should be distinguished. The minimum sink speed minimizes the downward rate, while the best glide speed maximizes horizontal distance per unit height lost in still air. They need not be the same speed.

For an original comparison, stipulate two steady operating points: $V_1=20m/s$ with $w_1=0.6m/s$, and $V_2=30m/s$ with $w_2=0.7m/s$. Their horizontal-distance ratios areAmong these two stipulated points, the slower one stays aloft longer for a given height loss, but the faster one covers more horizontal distance. From $1000\mathrm{m}$, the corresponding times are about $1666.7$ and $1428.6\mathrm{s}$. These invented points illustrate the competing objectives; they are not a measured performance curve.

A simple motivating experiment tracks a small unpowered model's position and height through a nearly steady portion of flight, then compares the vertical and horizontal motion. NASA identifies paper and lightweight model gliders as accessible examples.^[ref] Repeating with controlled geometry and launch conditions distinguishes the resulting motion from a claim based solely on a large wing aspect ratio.

Ira H. Abbott was an aerospace engineer whose Langley work connected airfoil geometry, wind tunnel measurements and corrections to aerodynamic data. NASA's institutional profile records his arrival at the Variable Density Tunnel in 1929, subsequent airfoil research and participation in tunnel design. The profile emphasizes that section data do not by themselves determine a complete three-dimensional aircraft wing's behavior.^[ref]

A specific primary contribution is the 1939 report with Eastman N. Jacobs on support interference in the variable-density wind tunnel. Its abstract states that the support effect made earlier drag coefficients and their apparent increase with airfoil thickness too large; the report supplies corrected data and methods. The apparatus supporting a model can alter the flow being measured, so its effect is not established merely by an empty-tunnel zero.^[ref]

For an original arithmetic illustration, stipulate measured drag coefficients of $0.012$ and $0.016$ for two sections, with separately established support corrections of $0.002$ and $0.005$. The corrected values are $0.010$ and $0.011$: the apparent difference falls from $0.004$ to $0.001$. At dynamic pressure $500\mathrm{Pa}$ and a shared reference area of $0.20{\mathrm{m}}^2$, these correspond to corrected drag values of $1.0$ and $1.1\mathrm{N}$. This invented example illustrates the experimental question; it does not replace the report's geometry-dependent correction method with a universal subtraction.

Lifting-line theory represents a finite aircraft wing by a spanwise bound vortex whose circulation varies with position. The trailing vortex filaments induce downwash, reducing the local effective angle of attack. In its classical form, the model assumes steady, incompressible potential flow, a comparatively large wing aspect ratio, small sweep and locally approximately two-dimensional section behavior.^[ref]

Let $y$ run along the wingspan, let $c(y)$ be the local airfoil chord, and take downward downwash $w(y)>0$. Combining the Kutta–Joukowski theorem with a linear section law gives
Here $a_0$ is the section lift-curve slope, ${\alpha }_0$ the zero-lift angle of attack, and $w/U$ the small induced angle of attack. Because the wake induced by the entire circulation distribution determines $w(y)$, these equations couple different sections. Solving independent two-dimensional airfoils and adding their lift misses that coupling.^[ref]

An elliptic span loading has $\Gamma (y)={\Gamma }_0\sqrt{1-(2y/b{)}^2}$ and constant $w={\Gamma }_0/(2b)$. Integration gives
This is the unit span efficiency factor case.^[ref] For stipulated $b=10\mathrm{m}$, $U=30m/s$, $\rho =1.2kg/{\mathrm{m}}^3$ and $L=6000\mathrm{N}$, it gives ${\Gamma }_0\simeq 21.22{\mathrm{m}}^2/s$, $w\simeq 1.061m/s$ and $D_i\simeq 212.2\mathrm{N}$.

A proposed test compares sectional surface pressure integration for airfoil lift with a traverse of the downstream velocity field. The two observations test the predicted loading and wake, rather than only one total force. Flow separation, strong sweep and compressibility require further modeling; the classical calculation does not predict all drag or aerodynamic stall.

The mean aerodynamic chord is a reference length that weights each local airfoil chord by the area of its spanwise strip. For a symmetric aircraft wing with wingspan $b$, local chord $c(y)>0$ and wing area $S$, its geometric definition isThe extra factor of $c(y)$ distinguishes this from the mean geometric chord $S/b$.^[ref]

The weighting has a simple interpretation: a strip has area $dS=c(y)dy$, so $\bar{c}=(1/S)\int cdS$ is an area-weighted mean. If a simplified section model has the same local pitching moment coefficient $c_m$ and dynamic pressure $q$ everywhere, its strip moment is $dM=q{c}_{m}c(y{)}^{2}dy$. Integration then gives $M=q{c}_{m}S\bar{c}$. This explains its use as a moment reference; an actual three-dimensional flow need not have a uniform section coefficient.

For the mean aerodynamic chord of a trapezoidal wing, take root chord $c_r$, tip chord $c_t=\lambda c_r$ and a linear distribution $c(y)=c_r[1-2(1-\lambda )y/b]$. Integrating givesThe stipulated wing planform $b=10\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=1\mathrm{m}$ has $S=15{\mathrm{m}}^2$, $S/b=1.5\mathrm{m}$ and $\bar{c}=14/9\mathrm{m}\simeq 1.5556\mathrm{m}$. The wider strips receive more weight.

The Cauchy-Schwarz inequality gives a useful general check:Equality holds for a constant chord almost everywhere. A reported mean aerodynamic chord shorter than $S/b$ therefore contradicts this definition for the same span and area convention. Its length alone does not specify a location: the reference position of that chord and the aerodynamic center are additional geometric or aerodynamic data.

The mean geometric chord, also called the standard mean chord, is the span-average airfoil chord of an aircraft wing:where $S$ is the wing area and $b$ the wingspan. The area is the projected wing planform, not the sum of the upper and lower skin areas.^[ref]

A useful exact identity explains its difference from the mean aerodynamic chord. Represent uniform spanwise averaging by a coordinate $Y$ with a continuous uniform distribution on $[-b/2,b/2]$, and put $C=c(Y)$. This is a mathematical averaging device, not a claim that the wing geometry is random. With $\mu =\mathbb{E}[C]=S/b>0$,Thus the relative excess is the squared coefficient of variation of the local chords. The mean aerodynamic chord can also be interpreted as the expected value under the corresponding size-biased distribution, since larger chords receive proportionally larger weights.

For the original linear taper from $2$ to $1\mathrm{m}$, uniform spanwise sampling makes $C$ uniform on $[1,2]\mathrm{m}$. Its expected value is $1.5\mathrm{m}$ and variance is $1/12{\mathrm{m}}^2$. Hence ${\overline{c}}_a=1.5+(1/12)/1.5=1.5556\mathrm{m}$, a relative excess of $1/27\simeq 3.70\%$. A constant-chord wing has zero variance, so its two averages coincide.

Equal $S/b$ does not determine how local chord varies or where the wing lies relative to a reference axis. Those additional data enter wing sweep and the mean aerodynamic chord reference position.

The National Advisory Committee for Aeronautics was the United States' aeronautical research organization established in 1915. Its laboratories and personnel formed the core of the National Aeronautics and Space Administration in 1958. Its technical legacy includes wind tunnels, systematic NACA airfoil families and published measurements that let designers compare aircraft wings.^[ref]

A particularly useful research strategy was to change geometry systematically while keeping test conditions comparable. Eastman N. Jacobs, Kenneth E. Ward and Robert M. Pinkerton varied airfoil thickness-to-chord ratio and airfoil camber in their related-section experiments, rather than testing only unrelated existing designs. The measurements connected an explicitly specified shape with lift coefficient, drag and pitching moment.^[ref]

An original similarity calculation illustrates why the apparatus mattered. A small model with airfoil chord $0.10\mathrm{m}$ in air of density $1.2kg/{\mathrm{m}}^3$, dynamic viscosity $1.8\times 10^{-5}\mathrm{Pa}\mathrm{s}$ and speed $30m/s$ has Reynolds numberA five-times-larger chord at the same conditions has $Re=10^6$. A variable-density wind tunnel can instead raise the model's air density fivefold to match that Reynolds number, approximately holding dynamic viscosity fixed. Matching it alone does not also establish matching Mach number, surface condition or wind-tunnel turbulence intensity.

The enduring contribution is a reusable experimental comparison: specify shape, operating conditions and how forces were measured, then publish results that can be checked against another experiment or computational fluid dynamics calculation.

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.

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.

The wing area $S$ is the area of an aircraft wing's projected wing planform. It is a reference area for such quantities as lift coefficient and wing aspect ratio. It is not the sum of the upper and lower curved surface areas.^[ref]

If the projected airfoil chord at spanwise position $y$ is $c(y)$, thenFor a symmetric trapezoidal wing planform this becomes twice the area under a linear half-wing chord distribution:With wingspan $b=10\mathrm{m}$, root chord $3\mathrm{m}$ and tip chord $1\mathrm{m}$, the result is $20{\mathrm{m}}^2$ for both halves together. Each half has area $10{\mathrm{m}}^2$. This factor of two matters when converting a half-wing drawing into a whole-wing reference area.^[ref]

The choice of reference area also affects reported coefficients. From $L=qS{C}_L$, where $q$ is dynamic pressure, a measured lift of $4000\mathrm{N}$ at $q=500\mathrm{Pa}$ and $S=20{\mathrm{m}}^2$ gives $C_L=0.4$. Using the half-wing area in that same whole-wing equation would incorrectly produce $C_L=0.8$. When comparing experiments or calculations, preserve the reference-area convention along with the measured force.

A wing sweep construction that shifts both projected edges aft by the same amount at each $y$ preserves $c(y)$ and therefore preserves wing area. The original comparison in wing planform illustrates this invariance. Changes to airfoil camber or thickness can likewise alter the three-dimensional surface and aerodynamic behavior without changing the projected outline.

The wing aspect ratio is the dimensionless measurewhere $b$ is wingspan and $S$ is projected wing area. For a rectangular wing planform, $S=bc$ and therefore $AR=b/c$, where $c$ is airfoil chord. For other wing planforms, dividing the span by a chosen root chord need not give the wing aspect ratio.^[ref]

Its aerodynamic importance comes from lift-induced drag. The wingtip vortices of a finite aircraft wing change the local flow direction; maintaining lift then entails an additional rearward force. A standard model writesHere $e$ is a span efficiency factor, $q$ is dynamic pressure and $C_L=L/(qS)$ is lift coefficient. The model compares a specified spanwise loading in a regime where this finite-wing approximation is appropriate; $e$ need not stay unchanged when the design changes.^[ref]

For an original example, take $S=16{\mathrm{m}}^2$, $q=500\mathrm{Pa}$, $C_L=0.6$ and $e=0.8$. Both wings then supply $L=4800\mathrm{N}$. Spans of 8 and 12 metres give $AR=4$ and $9$, respectively, and lift-induced drag approximately $286.48$ and $127.32\mathrm{N}$. Under those stipulated conditions, increasing span by 50% reduces that drag component by $1-(8/12{)}^2=5/9$, about 55.6%.

The boxed expression also exposes a common comparison error. At fixed $L$, $q$, $e$ and $b$, changing only the reference wing area changes both wing aspect ratio and lift coefficient; their effects cancel in this model's dimensional lift-induced drag. A larger numerical $AR$ is therefore not, by itself, a promise of smaller total drag. Structural loads, wing loading, surface losses and aerodynamic stall impose further design constraints.

A winglet is a small wing-tip lifting surface, often inclined upward out of the main wing planform. By changing the three-dimensional loading and flow near the tip, it can reduce lift-induced drag. Its winglet cant angle, shape and attachment must be considered with the entire aircraft wing; it is not a universal bolt-on reduction in total drag.^[ref]

Richard T. Whitcomb's 1976 report compared measured forces, moments and loads on a representative transport wing with and without wing-tip devices. A particularly useful comparison was a tip extension giving approximately the same increase of bending moment at the wing–fuselage junction as the winglets. That comparison asks how much aerodynamic benefit is obtained for a similar structural penalty, rather than comparing only nominal span.^[ref]

An original coefficient budget illustrates why component and total benefits differ. Keep the same reference wing area, dynamic pressure and lift coefficient. Suppose initiallywhere the second term is the lift-induced drag contribution. If a proposed modification reduces that term by $20\%$ but adds $0.001$ to other drag coefficient contributions, thenThe total reduction is about $3.33\%$, not $20\%$. This is an invented budget, not a prediction or a measured winglet efficiency. Increased loading, additional wetted area and an off-design operating point can change the result.

A convincing wind tunnel comparison measures total drag at matched lift, records moments and loading, and documents how the mounting correction and reference area are handled. A photograph of a weaker-looking wingtip vortex is not a measurement of total drag. Conversely, reducing lift-induced drag does not require all three-dimensional trailing vorticity to disappear.

A wing planform is the two-dimensional outline of an aircraft wing viewed from above. It is bounded by the projected leading edge, trailing edge and wing tips. A rectangular, tapered or swept wing planform can be combined with many different airfoil profiles; the outline alone does not specify the wing's thickness or airfoil camber.^[ref]

Let $x$ point aft and $y$ run along the wingspan. Describe the two edges by $x_{\mathrm{LE}}(y)$ and $x_{\mathrm{TE}}(y)$. Their separation is the projected local airfoil chord,The second equation computes wing area. The first retains information lost by that single area value: where the chord lies along the fore-and-aft direction.

For an original comparison, a rectangular wing with $b=10\mathrm{m}$ and $c=2\mathrm{m}$ has $S=20{\mathrm{m}}^2$. A symmetric trapezoidal wing of the same wingspan, root chord $3\mathrm{m}$ and tip chord $1\mathrm{m}$ also has $S=20{\mathrm{m}}^2$. Now shift both edges of each spanwise strip aft by $k|y|$: the local airfoil chord and wing area stay unchanged, while the wing sweep changes. This explains geometrically why even specifying wingspan, wing area and wing taper ratio does not uniquely determine a wing planform.

The construction compares geometry, not aerodynamic equivalence. The airfoil, angle of attack and three-dimensional fluid flow still matter when predicting lift and drag.

The wingspan $b$ of an aircraft wing is its lateral tip-to-tip extent. For a symmetric wing planform, a spanwise coordinate $y$ runs from $-b/2$ at one tip to $b/2$ at the other; the semispan is $b/2$. This geometrical distance differs from the local fore-and-aft airfoil chord.^[ref]

Together with wing area $S$, the wingspan defines the dimensionless wing aspect ratio,For a rectangular wing planform of constant airfoil chord $c$, $S=bc$, giving $AR=b/c$. For example, a wing with $b=12\mathrm{m}$ and $S=18{\mathrm{m}}^2$ has $AR=8$. A second wing with the same wing area but $b=9\mathrm{m}$ has $AR=4.5$. Equal wing area therefore does not imply equal lateral slenderness.

There is also a simple structural reason why the span matters. Suppose a symmetric wing supplies total upward lift $L$, distributed uniformly along its span. On one half-wing the resultant load is $L/2$ and acts $b/4$ from the center. The corresponding bending moment at the center isFor $L=4000\mathrm{N}$ and $b=12\mathrm{m}$ this gives $6000\mathrm{N}\mathrm{m}$. This is an idealized aerodynamic loading calculation: the actual load distribution, structural supports and component masses modify the wing's bending moment. It nevertheless shows why increasing wingspan affects the structure as well as the wing aspect ratio.

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]