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 $y=(b/2)\cos\theta$ and
$$
\Gamma(\theta)=2bU\sum_{n\ge1}A_n\sin(n\theta),\qquad 0\le\theta\le\pi.
$$
Here $b$ 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>.<Force Calculations for Lifting Line (16.100) by MIT OpenCourseWare>[printed pp. 1–3, circulation expansion and force integration]{ref}

Because $L'(y)=\rho U\Gamma(y)$ and $|dy|=(b/2)\sin\theta\,d\theta$, the integral for total <lift> selects only $A_1$. The induced angle contains $\sum_n nA_n\sin(n\theta)/\sin\theta$. Integrating its product with the loading then gives
$$
C_L=\pi AR A_1,\qquad C_{D_i}=\pi AR\sum_{n\ge1}nA_n^2,\qquad AR=b^2/S.
$$
The <orthogonality> identity behind both reductions is
$$
\int_0^\pi\sin(m\theta)\sin(n\theta)\,d\theta=\frac\pi2\delta_{mn}.
$$
A 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 gives
$$
\boxed{\frac{C_{D_i}}{C_L^2/(\pi AR)}=\frac1e
=1+\sum_{n\ge2}n\left(\frac{A_n}{A_1}\right)^2.}
$$
The 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 $e=1/1.0425\approx0.95923$. With $C_L=0.8$ and $AR=8$,
$$
C_{D_i,\mathrm{elliptic}}\approx0.025465,\qquad
C_{D_i}\approx0.026547.
$$
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 $\sum_{n\ge2}n(A_n/A_1)^2\le\delta$, then $e\ge1/(1+\delta)$. 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.


 Higher-harmonic penalty in span loading (source code)