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Elliptic span loading is a distribution of lift per unit span proportional to a half-ellipse, vanishing at both tips of a planar wing. For span $b$ and total lift $L$, writeThe normalization follows by setting $s=2y/b$ and using ${\int }_{-1}^1\sqrt{1-s^2}ds=\pi /2$. It specifies a load distribution, rather than the wing outline. Classical lifting-line theory identifies this loading as the minimum-induced-drag solution for fixed total lift and span in its ideal planar setting.^[ref]

The loading corresponds to circulation magnitude $\Gamma (y)={\Gamma }_0\sqrt{1-(2y/b{)}^2}$, with $L^{\prime }(y)=\rho U\Gamma (y)$ under the chosen positive-lift convention. Its ideal downwash at the lifting line is constant,Using dynamic pressure $q=\rho U^2/2$ recovers $D_i=L^2/(\pi q{b}^2)$ and the span efficiency factor $e=1$ in that normalization.

A Fourier series makes the minimum transparent. In the classical lifting-line expansion, put $y=(b/2)\cos \vartheta$ andAt fixed $C_L$ and $AR$, $A_1$ is fixed and every higher-order contribution to drag is nonnegative. The minimum has only $A_1$, yielding the half-ellipse. A higher-harmonic penalty in span loading quantifies departure from it: if $A_3=0.1A_1$ is the only additional term, $1/e=1+3(0.1{)}^2=1.03$, so $e\approx 0.971$ and induced drag rises three percent under the stated constraints.

For an original loading example, $L=6000\mathrm{N}$ and $b=10\mathrm{m}$ give a central load $763.94N/m$. At $y=b/4$ it is $0.8660$ of the central value. With $\rho =1.2kg/{\mathrm{m}}^3$ and $U=50m/s$, the model gives ${\Gamma }_0\approx 12.732{\mathrm{m}}^2/s$, $w\approx 0.6366m/s$ and $D_i\approx 76.39\mathrm{N}$.

A proposed test integrates measured surface pressure around several wing sections to estimate $L^{\prime }(y)$, then compares the normalized distribution and a wake survey with the model. An elliptic planform does not alone guarantee elliptic loading: section characteristics, incidence variation, deformation and operating conditions also enter. Nonplanar lifting systems, fixed bending-moment constraints and viscous or compressible effects change the optimization problem; the fixed-lift, fixed-span planar result should not be generalized beyond its assumptions.

A higher-harmonic penalty in span loading is the increase of ideal lift-induced drag caused by nonfundamental terms in a wing's circulation distribution at fixed total lift and span. In the classical planar lifting-line theory expansion, choose $y=(b/2)\cos \theta$ andHere $b$ is span, $U$ is freestream speed, and the coefficients $A_n$ are dimensionless. The primary force calculation uses this circulation expansion, the local lift law and sine orthogonality to obtain the lift and induced drag.^[ref]

Because $L^{\prime }(y)=\rho U\Gamma (y)$ and $|dy|=(b/2)\sin \theta d\theta$, the integral for total lift selects only $A_1$. The induced angle contains ${\sum }_{n}n{A}_n\sin (n\theta )/\sin \theta$. Integrating its product with the loading then givesThe orthogonality identity behind both reductions isA higher harmonic can thus have zero net lift contribution while still adding positive induced drag. Its effect on local loading has not disappeared merely because its span integral vanishes.

For nonzero $C_L$, division by the elliptic reference givesThe pure first harmonic gives elliptic span loading and $e=1$. At a fixed relative coefficient magnitude, a larger order carries a larger penalty. Changing a coefficient's sign can redistribute the loading without changing its own squared contribution in this ideal model.

For an original example, take $A_3=0.10A_1$, $A_5=0.05A_1$ and all other higher terms zero. The excess is $3(0.10{)}^2+5(0.05{)}^2=0.0425$, so $e=1/1.0425\approx 0.95923$. With $C_L=0.8$ and $AR=8$,At $q=500\mathrm{Pa}$ and $S=20{\mathrm{m}}^2$, the corresponding induced drags are about $254.65$ and $265.47\mathrm{N}$. The higher harmonics add $4.25\%$ to this induced contribution, rather than $4.25\%$ to every source of aircraft drag.

A truncation error in an inferred span-efficiency factor matters when fitting sparse load measurements. Omitting nonzero terms makes the ideal-model drag sum too small at a fixed known $A_1$, while fitting noisy high orders can inflate it because the terms are squared and weighted by order. If a justified bound is ${\sum }_{n\ge 2}n(A_n/A_1{)}^2\le \delta$, then $e\ge 1/(1+\delta )$. A fit residual alone does not supply that bound.

A proposed experiment reconstructs section lift or circulation from a spanwise pressure survey, fits a declared set of modes, and compares the predicted induced contribution with an independent wake-based estimate. Report sampling density, load uncertainty and truncation choice. Viscosity, compressibility, nonplanar geometry or constraints other than fixed lift and span change the physical optimization problem; the quadratic penalty belongs to the stated lifting-line model.