Prandtl's lifting-line theory models a finite wing by a spanwise distribution of bound circulation $\Gamma(y)$ whose trailing vorticity induces downwash. The unknown circulation must satisfy two descriptions of the same local lift:

1. Kutta-Joukowski relates lift to circulation;
2. the local airfoil section relates lift coefficient to effective angle of attack.

Matching those descriptions closes the finite-wing problem.

Local section law

At span location $y$, let

• $c(y)$ be local chord;
• $a_0(y)$ be the two-dimensional section lift-curve slope in radians$^{-1}$;
• $\alpha(y)$ be the local geometric angle of attack, including any geometric twist;
• $\alpha_{L=0}(y)$ be the local zero-lift angle;
• $\alpha_i(y)$ be the induced angle from the trailing wake.

The effective angle is

$$\alpha_{eff}(y)=\alpha(y)-\alpha_i(y).$$

In the linear attached-flow regime,

$$\boxed{ c_l(y)=a_0(y) \left[\alpha(y)-\alpha_i(y)-\alpha_{L=0}(y)\right] }.$$

The same local lift from circulation

Kutta-Joukowski gives

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

Using

$$q_\infty=\frac12\rho U_\infty^2,$$

the local section lift coefficient is

$$c_l(y) =\frac{L'(y)}{q_\infty c(y)} =\boxed{\frac{2\Gamma(y)}{U_\infty c(y)}}.$$

Equating the two expressions for $c_l$ gives

$$a_0(y) \left[\alpha(y)-\alpha_i(y)-\alpha_{L=0}(y)\right] =\frac{2\Gamma(y)}{U_\infty c(y)}.$$

Therefore

$$\boxed{ \alpha(y)-\alpha_{L=0}(y) =\frac{2\Gamma(y)}{a_0(y)U_\infty c(y)} +\alpha_i(y) }.$$

The downwash itself is determined by the unknown circulation distribution:

$$\boxed{ \alpha_i(y) =\frac{1}{4\pi U_\infty} \operatorname{PV}\int_{-b/2}^{b/2} \frac{d\Gamma/dy_0}{y-y_0},dy_0 }.$$

Together, these equations form Prandtl's lifting-line problem.

Spanwise angular coordinate

It is convenient to map the span to

$$\boxed{y=-\frac b2\cos\theta},$$

with

$$0\le\theta\le\pi.$$

The left tip is $\theta=0$, the centerline is $\theta=\pi/2$, and the right tip is $\theta=\pi$.

Because circulation must vanish at both tips, expand it as a sine series:

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

Every sine mode vanishes automatically at

$$\theta=0,\pi.$$

Induced angle from the sine coefficients

Substituting the circulation series into the downwash integral gives

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

The same coefficients therefore determine both the bound circulation and the downwash it creates.

Substitution into the local section relation gives the standard lifting-line equation:

$$\boxed{ \alpha(\theta)-\alpha_{L=0}(\theta)

\sum_{n=1}^{N}A_n\sin(n\theta) \left[ \frac{4b}{a_0(\theta)c(\theta)} + \frac{n}{\sin\theta} \right] }.$$

For specified wing geometry, twist, and section properties, the unknowns are the coefficients

$$A_1,A_2,\ldots,A_N.$$

Solving by collocation

A practical low-order solution is obtained by enforcing the lifting-line equation at $N$ selected span stations. This produces $N$ linear algebraic equations for the $N$ Fourier coefficients.

For a symmetric wing with symmetric geometry and angle of attack, only odd sine modes are needed because the circulation distribution is symmetric about the centerline.

Worked one-mode approximation

Consider an untwisted rectangular wing with

$$b=10,\mathrm m,$$

$$c=1.0,\mathrm m,$$

and section lift slope

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

Let

$$\alpha-\alpha_{L=0}=5^\circ=0.0873,\mathrm{rad}.$$

Use only the first mode,

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

and enforce the equation at the centerline,

$$\theta=\frac\pi2.$$

There,

$$\sin\theta=1,$$

so

$$0.0873 =A_1\left(\frac{4b}{a_0c}+1\right).$$

Now

$$\frac{4b}{a_0c} =\frac{40}{2\pi} \approx6.37.$$

Hence

$$A_1 =\frac{0.0873}{7.37} \approx\boxed{0.0118}.$$

If

$$U_\infty=30,\mathrm{m/s},$$

the predicted centerline circulation is

$$\Gamma\left(\frac\pi2\right) =2(10)(30)(0.0118) \approx\boxed{7.1,\mathrm{m^2/s}}.$$

A one-mode approximation forces an elliptical-shaped circulation distribution, so it is only a rough model for a rectangular wing. Adding higher modes allows the spanwise loading to adapt to the actual chord, twist, and section properties.

What lifting-line theory adds beyond 2D airfoil theory

A two-dimensional section law tells how a section responds to its local effective angle. Lifting-line theory determines that effective angle self-consistently from the entire wing's trailing vortex system.

The theory therefore captures the essential three-dimensional feedback:

$$\boxed{ \text{spanwise loading} \rightarrow\text{wake vorticity} \rightarrow\text{downwash} \rightarrow\text{reduced effective angle} \rightarrow\text{spanwise loading} }.$$

Its Fourier coefficients also provide direct formulas for total lift and induced drag.