Let's solve the problem **step by step**: ### **Step 1: Express \( u \) and \( v \) in Terms of the Panel Parameters** We are to express the velocity components \((u, v)\) at a point \((x, y)\) due to a **uniform source panel** of strength \( \lambda \), length \( l \), oriented at angle \( \Phi \), with center at \((x_0, y_0)\). For a horizontal panel (from lectures or standard sources), the induced velocity components at a field point \((x', y')\) in the panel-local coordinates are: \[ u' = \frac{\lambda}{2\pi} \ln\left(\frac{R_2}{R_1}\right) \] \[ v' = \frac{\lambda}{2\pi} (\theta_2 - \theta_1) \] where: - \( R_1 \), \( R_2 \) are distances from the point to the ends of the panel, - \( \theta_1 \), \( \theta_2 \) are angles from the panel to the point. ### **Transformation to Global Coordinates** The panel is not necessarily horizontal. Its local axes are rotated by angle \( \Phi \). The transformation from global coordinates \((x, y)\) to panel-local coordinates \((x', y')\) is: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\Phi & \sin\Phi \\ -\sin\Phi & \cos\Phi \end{pmatrix} \begin{pmatrix} x - x_0 \\ y - y_0 \end{pmatrix} \] So, \[ x' = (x-x_0)\cos\Phi + (y-y_0)\sin\Phi \] \[ y' = -(x-x_0)\sin\Phi + (y-y_0)\cos\Phi \] The panel spans from \( x' = -l/2 \) to \( x' = +l/2 \). #### **Distances to Panel Ends** - Endpoint 1: \( (x_1', y_1') = (-l/2, 0) \) - Endpoint 2: \( (x_2', y_2') = (l/2, 0) \) Thus, \[ R_1 = \sqrt{(x' + l/2)^2 + (y')^2} \] \[ R_2 = \sqrt{(x' - l/2)^2 + (y')^2} \] \[ \theta_1 = \arctan\left(\frac{y'}{x' + l/2}\right) \] \[ \theta_2 = \arctan\left(\frac{y'}{x' - l/2}\right) \] #### **Transforming \( (u', v') \) to \( (u, v) \):** The induced velocities in global coordinates are: \[ \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} \cos\Phi & -\sin\Phi \\ \sin\Phi & \cos\Phi \end{pmatrix} \begin{pmatrix} u' \\ v' \end{pmatrix} \] --- ### **Step 2: Plug in the Given Values** Given: - \( \lambda = 2\pi \) m/s - \( l = 2 \) - \( \Phi = \pi/3 \) - \( x_0 = 1, y_0 = 1 \) - \( x = 3, y = 4 \) #### **Compute local coordinates:** \[ x' = (3-1)\cos\left(\frac{\pi}{3}\right) + (4-1)\sin\left(\frac{\pi}{3}\right) \\ x' = (2)\left(\frac{1}{2}\right) + (3)\left(\frac{\sqrt{3}}{2}\right) \\ x' = 1 + \frac{3\sqrt{3}}{2} \approx 1 + 2.598 = 3.598 \] \[ y' = -(3-1)\sin\left(\frac{\pi}{3}\right) + (4-1)\cos\left(\frac{\pi}{3}\right) \\ y' = -2\left(\frac{\sqrt{3}}{2}\right) + 3\left(\frac{1}{2}\right) \\ y' = -\sqrt{3} + 1.5 \approx -1.732 + 1.5 = -0.232 \] #### **Compute \( R_1, R_2 \):** \[ R_1 = \sqrt{(x' + l/2)^2 + (y')^2} = \sqrt{(3.598 + 1)^2 + (-0.232)^2} = \sqrt{(4.598)^2 + 0.0538} \] \[ (4.598)^2 \approx 21.134 \] \[ R_1 \approx \sqrt{21.134 + 0.0538} = \sqrt{21.1878} \approx 4.602 \] \[ R_2 = \sqrt{(x' - l/2)^2 + (y')^2} = \sqrt{(3.598 - 1)^2 + (-0.232)^2} = \sqrt{(2.598)^2 + 0.0538} \] \[ (2.598)^2 \approx 6.748 \] \[ R_2 \approx \sqrt{6.748 + 0.0538} = \sqrt{6.8018} \approx 2.609 \] #### **Compute \( \theta_1, \theta_2 \):** \[ \theta_1 = \arctan\left(\frac{-0.232}{3.598 + 1}\right) = \arctan\left(\frac{-0.232}{4.598}\right) \approx \arctan(-0.0505) \approx -0.0505\ \text{rad} \] \[ \theta_2 = \arctan\left(\frac{-0.232}{3.598 - 1}\right) = \arctan\left(\frac{-0.232}{2.598}\right) \approx \arctan(-0.0893) \approx -0.0891\ \text{rad} \] #### **Plug into \( u', v' \):** \[ u' = \frac{2\pi}{2\pi}\ln\left(\frac{4.602}{2.609}\right) = \ln(1.763) \approx 0.567 \] \[ v' = \frac{2\pi}{2\pi}(-0.0505 + 0.0891) = 0.0386 \] #### **Transform back to global coordinates:** \[ u = u'\cos\Phi - v'\sin\Phi \] \[ v = u'\sin\Phi + v'\cos\Phi \] \[ \cos\Phi = \cos\left(\frac{\pi}{3}\right) = 0.5,\ \sin\Phi = \frac{\sqrt{3}}{2} \approx 0.866 \] \[ u = 0.567 \cdot 0.5 - 0.0386 \cdot 0.866 \approx 0.284 - 0.0334 = 0.2506 \] \[ v = 0.567 \cdot 0.866 + 0.0386 \cdot 0.5 \approx 0.491 + 0.0193 = 0.5103 \] --- ### **Final Answer** \[ \boxed{ u \approx 0.25, \quad v \approx 0.51 } \] --- #### **Summary of Steps:** 1. **Transform** the field point \((x, y)\) to panel coordinates \((x', y')\). 2. **Calculate** \(R_1, R_2, \theta_1, \theta_2\). 3. **Compute** \(u', v'\) in panel coordinates. 4. **Transform** \(u', v'\) back to global coordinates \(u, v\). 5. **Plug in** the given values and evaluate numerically.