The equation of a simple harmonic wave is given by $y=3\sin\frac{\pi}{2}(50t-x)$ where $x$ and $y$ are in mete

Question

The equation of a simple harmonic wave is given by $y=3\sin\frac{\pi}{2}(50t-x)$ where $x$ and $y$ are in meters and $t$ is in seconds. The ratio of maximum particle velocity to the wave velocity is:

Accepted Answer

1. **Identify the Given Wave Equation:** The equation is $y = 3\sin\frac{\pi}{2}(50t - x)$, which can be rewritten as $y = 3\sin(25\pi t - \frac{\pi}{2}x)$.
2. **Compare with the Standard Wave Equation:** The standard progressive wave equation is $y = A\sin(\omega t - kx)$ . 
   By comparing, we get:
   - Amplitude, $A = 3 	ext{ m}$
   - Angular frequency, $\omega = 25\pi 	ext{ rad/s}$
   - Wave number, $k = \frac{\pi}{2} 	ext{ m}^{-1}$
3. **Calculate Maximum Particle Velocity ($v_{\max}$):** The maximum particle velocity is given by $v_{\max} = A\omega = 3 	imes 25\pi = 75\pi 	ext{ m/s}$.
4. **Calculate Wave Velocity ($v$):** The wave velocity is given by $v = \frac{\omega}{k} = \frac{25\pi}{\pi/2} = 50 	ext{ m/s}$ .
5. **Find the Ratio:** The ratio of maximum particle velocity to the wave velocity is:
   $$\frac{v_{\max}}{v} = \frac{75\pi}{50} = \frac{3}{2}\pi$$
   *(Alternatively, this ratio can be directly found as $Ak = 3 	imes \frac{\pi}{2} = \frac{3}{2}\pi$)*.

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