The lifting-line Fourier representation makes one spanwise loading especially important. If only the first sine mode is present,

$$A_1\ne0,$$

$$A_2=A_3=\cdots=0,$$

then the circulation distribution is

$$\Gamma(\theta)=2bU_\infty A_1\sin\theta.$$

Using

$$y=-\frac b2\cos\theta,$$

we have

$$\sin\theta =\sqrt{1-\cos^2\theta} =\sqrt{1-\left(\frac{2y}{b}\right)^2}.$$

Therefore

$$\boxed{ \Gamma(y) =2bU_\infty A_1 \sqrt{1-\left(\frac{2y}{b}\right)^2} }.$$

Since local lift is proportional to circulation, the lift distribution has the same elliptic spanwise shape.

Uniform downwash

The lifting-line induced-angle formula is

$$\alpha_i(\theta) =\sum_{n=1}^{\infty} nA_n\frac{\sin(n\theta)}{\sin\theta}.$$

With only $A_1$ present,

$$\boxed{\alpha_i=A_1}.$$

Thus elliptic loading produces the same induced angle at every span station.

Since

$$C_L=\pi AR A_1,$$

we can also write

$$\boxed{ \alpha_i=\frac{C_L}{\pi AR} }.$$

The effective angle of attack is reduced uniformly across the wing.

Minimum induced drag at fixed lift and aspect ratio

The general lifting-line result is

$$C_{D_i} =\pi AR\sum_{n=1}^{\infty}nA_n^2.$$

At fixed total lift,

$$C_L=\pi AR A_1,$$

so $A_1$ is fixed. Every higher mode contributes an additional positive quantity

$$nA_n^2>0.$$

Therefore the smallest induced drag occurs when

$$\boxed{A_n=0\qquad(n\ge2)}.$$

Hence elliptic loading is the minimum-induced-drag lifting-line distribution for fixed lift and span.

For this ideal distribution,

$$\boxed{e=1},$$

and

$$\boxed{ C_{D_i}=\frac{C_L^2}{\pi AR} }.$$

This result does not say that every practical wing should have an elliptical planform. Twist, taper, section choice, structure, stall behavior, and operating conditions can be adjusted to produce near-elliptic loading with many different planform shapes.

Elliptic planform with uniform section properties

A particularly simple exact lifting-line case uses an elliptic chord distribution

$$\boxed{c(\theta)=c_0\sin\theta},$$

where $c_0$ is the centerline chord.

For an untwisted wing with constant section lift slope $a_0$ and constant zero-lift angle $\alpha_{L=0}$, suppose the circulation contains only $A_1$.

The lifting-line equation becomes

$$\alpha-\alpha_{L=0} =A_1\sin\theta \left[ \frac{4b}{a_0c_0\sin\theta} +\frac1{\sin\theta} \right].$$

The $\sin\theta$ factors cancel, giving the same equation at every span station:

$$\boxed{ \alpha-\alpha_{L=0} =A_1\left(\frac{4b}{a_0c_0}+1\right) }.$$

Thus the elliptic loading is self-consistent across the entire wing.

Relating chord geometry to aspect ratio

The area of an elliptical planform with span $b$ and maximum chord $c_0$ is

$$\boxed{S=\frac{\pi b c_0}{4}}.$$

Therefore

$$AR=\frac{b^2}{S} =\frac{4b}{\pi c_0},$$

so

$$\boxed{\frac{4b}{c_0}=\pi AR}.$$

The lifting-line equation becomes

$$\alpha-\alpha_{L=0} =A_1\left(1+\frac{\pi AR}{a_0}\right).$$

Hence

$$A_1 =\frac{\alpha-\alpha_{L=0}} {1+\pi AR/a_0}.$$

Using

$$C_L=\pi AR A_1,$$

we obtain

$$\boxed{ C_L =\frac{a_0}{1+a_0/(\pi AR)} \left(\alpha-\alpha_{L=0}\right) }.$$

The finite-wing lift-curve slope is therefore

$$\boxed{ a=\frac{a_0}{1+a_0/(\pi AR)} }.$$

Because the denominator exceeds one,

$$\boxed{a<a_0}.$$

Downwash makes a finite wing less sensitive to geometric angle of attack than an isolated two-dimensional section.

As

$$AR\to\infty,$$

$$a\to a_0,$$

recovering the two-dimensional limit.

Worked example

Take a symmetric wing whose airfoil sections have the thin-airfoil slope

$$a_0=2\pi,\mathrm{rad^{-1}}$$

and wing aspect ratio

$$AR=8.$$

The finite-wing lift slope is

$$a =\frac{2\pi}{1+2\pi/(8\pi)} =\frac{2\pi}{1.25} \approx\boxed{5.03,\mathrm{rad^{-1}}}.$$

At

$$\alpha=5^\circ=0.0873,\mathrm{rad},$$

with

$$\alpha_{L=0}=0,$$

the lift coefficient is

$$C_L=(5.03)(0.0873) \approx\boxed{0.439}.$$

The induced angle is

$$\alpha_i =\frac{0.439}{\pi(8)} \approx0.01745,\mathrm{rad} =\boxed{1.00^\circ}.$$

Thus the section effective angle is about

$$5^\circ-1^\circ=4^\circ.$$

Indeed,

$$a_0(4^\circ\text{ in radians}) =2\pi(0.0698) \approx0.439,$$

consistent with the finite-wing solution.

The ideal induced-drag coefficient is

$$C_{D_i} =\frac{(0.439)^2}{\pi(8)} \approx\boxed{7.66\times10^{-3}}.$$

Design interpretation

The elliptic result separates two related ideas:

• elliptic loading is the circulation/lift distribution that minimizes induced drag for fixed lift and span within lifting-line theory;
• an elliptic planform with uniform section properties is one geometry that produces that loading naturally.

Real wing design often seeks the beneficial loading rather than copying one exact outline. Taper, twist, airfoil variation, structural constraints, and multiple design conditions can all make a nonelliptic planform preferable while still achieving high span efficiency.