According to Prandtl's classical lifting-line theory, a planar finite wing with an elliptic planform lift distribution produces an induced downwash velocity $w$ along its span that is:

(A) Strictly constant across the entire wingspan ($w = w_0 = \text{constant}$)

(B) Maximum at the wing root and zero at the wingtips

(C) Zero at the wing root and maximum at the wingtips

(D) Inversely proportional to local chord length

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Integral:** Downwash $w(y_0) = \frac{1}{4\pi} \int_{-b/2}^{b/2} \frac{(d\Gamma/dy)}{y_0 - y} dy$. 2. **Stage 2: Elliptic Circulation:** For $\Gamma(y) = \Gamma_0 \sqrt{1 - (2y/b)^2}$, transforming to angular coordinate $\theta$ where $y = -(b/2)\cos\theta$ gives Fourier coefficient $A_1 = \frac{\Gamma_0}{2 b V_\infty}$ and all $A_n = 0$ for $n > 1$. 3. **Stage 3: Constant Downwash:** The general downwash formula $w(\theta) = V_\infty \sum_{n=1}^\infty n A_n \frac{\sin(n\theta)}{\sin\theta} = V_\infty A_1 = \frac{\Gamma_0}{2b} = \text{constant}$. 4. **Stage 4: Aerodynamic Significance:** Because $w$ is constant, induced angle of attack $\alpha_i = w/V_\infty$ is uniform across the span, yielding Oswald efficiency $e = 1.0$ (minimum induced drag). 5. **Stage 5: Official Reference:** John D. Anderson Jr., *Fundamentals of Aerodynamics*, Section 5.3.

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