A two-dimensional airfoil can carry one circulation value everywhere along its span because the span is idealized as infinite. A finite wing is different: its lift and bound circulation must fall to zero at the tips. That spanwise variation cannot terminate inside the fluid, so it continues downstream as trailing vorticity.

Choose coordinates with

• $x$ downstream;
• $y$ along the span, from $-b/2$ to $+b/2$;
• $z$ upward;
• free-stream velocity $\mathbf U_\infty=U_\infty\hat{\mathbf x}$.

Let the bound circulation distribution be $\Gamma(y)$, with positive circulation directed along $+y$. Then Kutta-Joukowski gives positive upward local lift per unit span:

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

Why varying bound circulation creates a wake sheet

Consider two neighboring spanwise stations $y$ and $y+dy$. Their bound circulations differ by

$$d\Gamma=\frac{d\Gamma}{dy}dy.$$

A vortex filament cannot simply change strength along one continuous line without additional vortex structure. The difference must therefore leave the lifting line as a downstream filament.

If $\gamma_w(y)$ denotes the circulation per unit span carried by the trailing vortex sheet, vortex-strength continuity gives

$$\gamma_w(y)dy=-d\Gamma.$$

Hence

$$\boxed{\gamma_w(y)=-\frac{d\Gamma}{dy}}.$$

A spanwise circulation gradient is therefore the source of the free vorticity shed into the wake.

For a symmetric lifting wing, $\Gamma$ is largest near the center and tends to zero at both tips. The resulting trailing sheet eventually rolls up into the familiar tip-vortex-dominated wake, but the distributed sheet is the more accurate starting model.

Downwash induced by one wake strip

A sheet strip of width $dy$ at span location $y$ carries circulation

$$d\Gamma_w=\gamma_w(y)dy.$$

Model it as a straight semi-infinite vortex filament extending downstream. At a point on the lifting line with span coordinate $y_0$, its perpendicular distance from the filament is

$$|y_0-y|.$$

Biot-Savart gives the induced vertical velocity

$$dw =\frac{\gamma_w(y)dy}{4\pi(y_0-y)},$$

where the signed denominator and circulation determine whether the induced velocity points upward or downward.

Using

$$\gamma_w=-\frac{d\Gamma}{dy},$$

we obtain

$$dw =-\frac1{4\pi} \frac{d\Gamma/dy}{y_0-y}dy.$$

Adding all wake strips gives the downwash distribution

$$\boxed{ w(y_0) =-\frac1{4\pi} \operatorname{PV}\int_{-b/2}^{b/2} \frac{d\Gamma/dy}{y_0-y},dy }.$$

The principal value is required because the idealized sheet has a singular kernel at $y=y_0$.

For an ordinarily loaded wing, $w<0$: the wake induces velocity downward through the wing plane.

Induced angle of attack

When

$$|w|\ll U_\infty,$$

the local incoming flow is tilted downward by the small induced angle

$$\boxed{\alpha_i(y)=-\frac{w(y)}{U_\infty}}.$$

Thus downwash corresponds to

$$\alpha_i>0.$$

If the geometric local angle of attack is $\alpha(y)$, the section actually sees

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

A finite wing therefore operates at a smaller effective angle of attack than an otherwise identical isolated two-dimensional section.

Worked qualitative sign check

On the right half of a symmetric wing,

$$y>0,$$

and circulation normally decreases toward the tip, so

$$\frac{d\Gamma}{dy}<0.$$

At the centerline $y_0=0$, the denominator for a right-half wake strip is

$$y_0-y=-y<0.$$

Therefore

$$\frac{d\Gamma/dy}{y_0-y}>0,$$

and the leading minus sign in the downwash integral makes that contribution negative:

$$dw<0.$$

The left half gives the same downward sign by symmetry. The wake therefore induces downwash at the wing center, exactly as expected physically.

The feedback loop behind finite-wing lift

The finite-wing problem is self-consistent rather than one-way:

1. $\Gamma(y)$ determines local lift through Kutta-Joukowski;
2. its spanwise gradient determines the trailing vortex sheet;
3. the wake sheet induces $w(y)$;
4. downwash creates $\alpha_i(y)$;
5. $\alpha_i$ changes the effective section angle of attack;
6. the changed section angle determines the circulation that can actually be sustained.

Prandtl's lifting-line theory closes this loop by combining the wake-induced angle with the local two-dimensional section lift law.