A single horseshoe vortex of circulation $\Gamma$ models a planar finite wing of span $b$. The bound vortex extends along the $y$-axis from $y = -b/2$ to $y = +b/2$, and two semi-infinite trailing vortices extend downstream to $x = +\infty$. According to the Biot-Savart law, the downwash velocity $w$ induced by the two trailing vortices at the midpoint of the bound vortex (origin $x = 0, y = 0$) is:

(A) $w = \frac{\Gamma}{\pi b}$

(B) $w = \frac{\Gamma}{2\pi b}$

(C) $w = \frac{2\Gamma}{\pi b}$

(D) $w = \frac{\Gamma}{4\pi b}$

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### 1. Strategy & Theoretical Principles By the Biot-Savart law, each semi-infinite straight trailing vortex line induces a velocity at a perpendicular distance $h$ given by: $$w_i = \frac{\Gamma}{4\pi h}$$ At the midpoint of the wing ($y = 0$): - The left trailing vortex is located at $y = -b/2$, so its perpendicular distance is $h = b/2$. - The right trailing vortex is located at $y = +b/2$, so its perpendicular distance is also $h = b/2$. - Both trailing vortices have opposite circulation signs that induce downwash in the downward ($-z$) direction at the center. ### 2. Analytical Formulation & Calculation Each trailing vortex induces: $$w_1 = \frac{\Gamma}{4\pi (b/2)} = \frac{\Gamma}{2\pi b}$$ $$w_2 = \frac{\Gamma}{4\pi (b/2)} = \frac{\Gamma}{2\pi b}$$ Total downwash induced by both trailing vortices at the origin: $$w = w_1 + w_2 = \frac{\Gamma}{2\pi b} + \frac{\Gamma}{2\pi b} = \frac{\Gamma}{\pi b}$$ (Note: The bound vortex itself induces zero velocity along its own axis by symmetry). ### 3. Distractor Forensics - **Option A ($w = \frac{\Gamma}{\pi b}$):** Correct sum of both semi-infinite trailing vortex filaments. - **Option B ($w = \frac{\Gamma}{2\pi b}$):** Considers only a single trailing vortex. - **Option C:** Uses infinite vortex formula for each line. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Distinguish downwash at the wing bound vortex ($w = \Gamma / \pi b$) from far downstream in the Trefftz plane where trailing vortices are effectively doubly infinite ($w_\infty = 2\Gamma / \pi b$). - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 5.

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