Let's solve the problem step-by-step as requested. --- ## **Step 1: Write the General Wave Equation** A traveling wave moving to the right can be written as: \[ y(x, t) = A \sin(kx - \omega t + \phi) \] Where: - \( A \) = amplitude = \( y_{max} = 2. \) m - \( k \) = wave number = \( \frac{2\pi}{\lambda} \) - \( \omega \) = angular frequency = \( \frac{2\pi}{T} \) - \( \phi \) = phase constant Given: - Period \( T = 6. \) s - At \( t = \), \( x_ = 2. \) m, \( y_ = 1.2 \) m First, calculate \( \omega \): \[ \omega = \frac{2\pi}{T} = \frac{2\pi}{6.} = \frac{\pi}{3} \ \text{rad/s} \] **Explanation:** We set up the general form and insert values for amplitude and angular frequency based on the problem statement. --- ## **Step 2: Find Wave Number \( k \) and Phase \( \phi \)** ### **Find the wavelength \(\lambda\) and \(k\):** From the graph, notice the distance between two peaks (full cycle) is the wavelength. The graph shows two peaks from \( x = \) to \( x = 8 \) m, so \( \lambda = 8. \) m. \[ k = \frac{2\pi}{\lambda} = \frac{2\pi}{8.} = \frac{\pi}{4} \ \text{rad/m} \] ### **Find phase constant \( \phi \):** At \( t = \), \( x = x_ = 2. \) m, \( y_ = 1.2 \) m: \[ y_ = A \sin(kx_ + \phi) \] \[ 1.2 = 2. \sin\left(\frac{\pi}{4} \cdot 2. + \phi\right) \] \[ 1.2 = 2. \sin\left(\frac{\pi}{2} + \phi\right) \] \[ .6 = \sin\left(\frac{\pi}{2} + \phi\right) \] But \(\sin(\frac{\pi}{2} + \phi) = \cos(\phi)\), so: \[ .6 = \cos(\phi) \implies \phi = \arccos(.6) \approx .93 \] **Explanation:** We use the position at a known \( x_ \) and \( y_ \) to solve for the phase shift \( \phi \). --- ## **Final Answer** \[ \boxed{ y(x,t) = 2. \sin\left(\frac{\pi}{4} x - \frac{\pi}{3} t + .93 \right) } \] Where: - \( \frac{\pi}{4} \) is the wave number, - \( \frac{\pi}{3} \) is the angular frequency, - \( .93 \) is the phase angle (rounded to two significant figures). --- **Summary of Steps:** 1. **Wrote the general wave equation and substituted amplitude and frequency.** 2. **Calculated wavelength, wave number, and solved for phase using initial conditions.** 3. **Assembled the final function.**