Thin-airfoil theory replaces a slender airfoil by an idealized vortex sheet distributed along its chord. The sheet strength is chosen so that the outer inviscid flow is tangent to the airfoil camber line and satisfies the Kutta condition at the trailing edge.

Let

• $c$ be the chord length;
• $x$ measure distance from the leading edge along the chord;
• $z_c(x)$ be the camber-line height;
• $\alpha$ be a small angle of attack, measured in radians;
• $U_\infty$ be the free-stream speed;
• $\gamma(x)$ be the vortex-sheet strength, defined as upper-surface tangential velocity minus lower-surface tangential velocity.

The theory assumes a thin profile, small camber slope, and small angle of attack, so products of small perturbations can be neglected.

Collapse the airfoil to a line

At leading order, the actual upper and lower surfaces are replaced by the camber line. The vortex sheet is placed along

$$0<x<c.$$

Thickness no longer appears directly in the idealized geometry. The camber-line slope

$$\frac{dz_c}{dx}$$

retains the leading effect of shape on the tangency condition.

Velocity induced by a sheet element

A small sheet element at position $\xi$ has counterclockwise circulation

$$d\Gamma=-\gamma(\xi),d\xi.$$

A point vortex of circulation $d\Gamma$ induces, on the chord line at $x$, a normal velocity

$$dw =\frac{d\Gamma}{2\pi(x-\xi)}.$$

Therefore

$$dw =-\frac{\gamma(\xi)}{2\pi(x-\xi)},d\xi.$$

Adding all sheet elements gives the induced normal velocity

$$\boxed{ w_i(x) =-\frac1{2\pi},\operatorname{PV} \int_0^c\frac{\gamma(\xi)}{x-\xi},d\xi }.$$

The symbol PV denotes the Cauchy principal value: the singular contribution immediately to either side of $\xi=x$ is taken symmetrically so that the physically meaningful finite induced normal velocity is retained.

Linearized no-penetration condition

For small angles, the free-stream normal component relative to the chord is approximately

$$U_\infty\alpha.$$

A flow tangent to the camber line must have normal velocity approximately

$$U_\infty\frac{dz_c}{dx}.$$

Thus

$$U_\infty\alpha+w_i =U_\infty\frac{dz_c}{dx}.$$

Substituting the sheet-induced velocity gives the thin-airfoil integral equation

$$\boxed{ U_\infty \left(\alpha-\frac{dz_c}{dx}\right) -rac1{2\pi}\operatorname{PV} \int_0^c\frac{\gamma(\xi)}{x-\xi},d\xi =0 }.$$

This equation determines the sheet distribution required to make the ideal outer flow tangent to the prescribed camber line.

Kutta condition

The tangency equation alone still allows an inappropriate trailing-edge singularity. For the ideal thin airfoil with a cusped trailing edge, the Kutta condition requires finite smooth departure of the upper and lower flows.

In the vortex-sheet representation this gives

$$\boxed{\gamma(c)=0}.$$

The sheet strength may become large near an ideal infinitely thin leading edge, but it must vanish at the trailing edge.

Pressure jump from sheet strength

Write the upper and lower tangential velocities as small perturbations about $U_\infty$:

$$u_{upper}=U_\infty+\tilde u_{upper},$$

$$u_{lower}=U_\infty+\tilde u_{lower}.$$

For small perturbations, Bernoulli gives

$$C_p\approx-2\frac{\tilde u}{U_\infty}.$$

Since

$$\gamma=\tilde u_{upper}-\tilde u_{lower},$$

the pressure-coefficient difference is

$$\boxed{ \Delta C_p \equiv C_{p,lower}-C_{p,upper} =\frac{2\gamma(x)}{U_\infty} }.$$

Positive $\gamma$ therefore corresponds to lower pressure on the upper side and positive upward lift under this convention.

Total lift and circulation

The lift per unit span from the pressure difference is approximately

$$L' =\int_0^c(p_{lower}-p_{upper}),dx.$$

Using

$$p_{lower}-p_{upper} =\frac12\rho U_\infty^2\Delta C_p,$$

we get

$$L' =\rho U_\infty\int_0^c\gamma(x),dx.$$

Define

$$G\equiv\int_0^c\gamma(x),dx.$$

Then

$$\boxed{L'=\rho U_\infty G}.$$

Because the counterclockwise circulation convention gives

$$\Gamma=-\int_0^c\gamma(x),dx=-G,$$

this is exactly the Kutta-Joukowski result

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

Thus the vortex-sheet formulation connects local and global descriptions:

• $\gamma(x)$ determines the local pressure difference;
• its integral determines the total circulation;
• the circulation determines the total lift.

What remains to solve

The central mathematical problem of thin-airfoil theory is now clear: solve the singular tangency equation for $\gamma(x)$ subject to the Kutta condition.

For a symmetric thin airfoil, the solution has a closed form. For a cambered airfoil, the geometry can be encoded through a trigonometric expansion after a convenient change of variable along the chord.