Once Prandtl's lifting-line equation has been solved, the Fourier coefficients of the circulation distribution determine the wing's total lift and induced drag directly.

Write

$$\boxed{ \Gamma(\theta) =2bU_\infty \sum_{n=1}^{N}A_n\sin(n\theta) },$$

with

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

Let

• $b$ be wing span;
• $S$ be wing planform area;
• $q_\infty=\tfrac12\rho U_\infty^2$;
• $AR=b^2/S$ be the aspect ratio.

Total lift

The local lift per unit span is

$$L'(y)=\rho U_\infty\Gamma(y).$$

Total lift is

$$L=\int_{-b/2}^{b/2}L'(y),dy.$$

Using

$$dy=\frac b2\sin\theta,d\theta,$$

we obtain

$$L =\rho U_\infty \int_0^\pi \left[ 2bU_\infty\sum_{n=1}^NA_n\sin(n\theta) \right] \left(\frac b2\sin\theta\right)d\theta.$$

Orthogonality gives

$$\int_0^\pi\sin(n\theta)\sin\theta,d\theta=0$$

for

$$n\ne1,$$

while

$$\int_0^\pi\sin^2\theta,d\theta=\frac\pi2.$$

Therefore only the first coefficient contributes to total lift:

$$\boxed{L=\frac\pi2\rho U_\infty^2b^2A_1}.$$

The wing lift coefficient is

$$C_L=\frac{L}{q_\infty S},$$

so

$$\boxed{C_L=\pi AR,A_1}.$$

Higher modes redistribute lift spanwise but do not change total lift when $A_1$ is held fixed.

Why downwash produces induced drag

The local aerodynamic force associated with bound circulation is approximately perpendicular to the local effective flow, not exactly perpendicular to the undisturbed free stream.

Downwash tilts that effective flow downward by the small induced angle $\alpha_i$. The corresponding force vector tilts rearward, producing an induced drag component.

For small induced angle,

$$\boxed{D_i'(y)\approx L'(y)\alpha_i(y)}.$$

Using

$$L'=\rho U_\infty\Gamma$$

and

$$\alpha_i(\theta) =\sum_{n=1}^NnA_n\frac{\sin(n\theta)}{\sin\theta},$$

the total induced drag becomes

$$D_i =\int_{-b/2}^{b/2}\rho U_\infty\Gamma(y)\alpha_i(y),dy.$$

Substituting the Fourier series and using sine orthogonality gives

$$\boxed{ D_i =\frac\pi2\rho U_\infty^2b^2 \sum_{n=1}^{N}nA_n^2 }.$$

Therefore

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

Unlike total lift, induced drag receives a positive contribution from every nonzero Fourier mode.

Span efficiency

Since

$$C_L=\pi AR A_1,$$

separate the first mode from the induced-drag sum:

$$C_{D_i} =\pi AR A_1^2 \left[ 1+\sum_{n=2}^{N} n\left(\frac{A_n}{A_1}\right)^2 \right].$$

Define

$$\boxed{ \delta =\sum_{n=2}^{N} n\left(\frac{A_n}{A_1}\right)^2 \ge0 }.$$

Then

$$\boxed{ C_{D_i} =\frac{C_L^2}{\pi AR}(1+\delta) }.$$

It is common to define the span efficiency factor

$$\boxed{e=\frac{1}{1+\delta}}.$$

Thus

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

Within classical lifting-line theory,

$$0<e\le1.$$

A loading with higher Fourier modes has

$$e<1$$

and more induced drag than the first-mode-only distribution at the same lift and aspect ratio.

Induced drag is not viscous profile drag

Induced drag exists in the ideal lifting-line model even though the outer flow is inviscid. It is associated with the kinetic energy and momentum carried by the three-dimensional trailing-vortex system and with the rearward tilt of the local lift vector.

This is physically distinct from

• skin-friction drag caused directly by viscous shear;
• pressure drag associated with viscous separation;
• wave drag in compressible flow.

A real wing can experience all of these simultaneously.

Worked example

Suppose a symmetric lifting-line solution for a wing of aspect ratio

$$AR=10$$

has only two appreciable odd coefficients:

$$A_1=0.0134,$$

$$A_3=0.00122.$$

The lift coefficient is

$$C_L=\pi(10)(0.0134) \approx\boxed{0.421}.$$

The induced-drag coefficient is

$$C_{D_i} =\pi(10) \left[A_1^2+3A_3^2\right].$$

Numerically,

$$C_{D_i} \approx\boxed{5.78\times10^{-3}}.$$

The nonellipticity parameter is

$$\delta =3\left(\frac{0.00122}{0.0134}\right)^2 \approx0.0249,$$

so

$$e=\frac1{1+0.0249} \approx\boxed{0.976}.$$

The wing is close to the ideal first-mode loading, but the small third mode still increases induced drag by about $2.5%$ relative to the minimum possible value at the same $C_L$ and $AR$.

The coefficient formulas expose an important design principle: $A_1$ sets total lift, while higher spanwise modes cost induced drag without increasing that total lift.