The equation of a simple harmonic wave is given by
`y = 3 sin"(pi)/(2) (50t - x)`
where `x` and `y` are in meters and `x` is in second .The ratio of maximum particle velocity to the wave velocity is

Options

A. `(3)/(2)pi`
B. `3pi`
C. `(2)/(3)pi`
D. `2pi`

Explanation

To solve the problem, we need to find the ratio of the maximum particle velocity to the wave velocity for the given wave equation:

### Step-by-Step Solution:

1. **Identify the wave equation**:
The given wave equation is:
\[ y = 3 \sin\left(\frac{\pi}{2}(50t - x)\right) \]
Here, \( A = 3 \) (amplitude), \( \omega = \frac{\pi}{2} \cdot 50 = 25\pi \) (angular frequency), and \( k = \frac{\pi}{2} \) (wave number).

2. **Calculate the maximum particle velocity**:
The particle velocity \( v_p \) is given by the partial derivative of \( y \) with respect to time \( t \):
\[ v_p = \frac{dy}{dt} = 3 \cdot \frac{d}{dt} \left(\sin(25\pi t - \frac{\pi}{2} x)\right) \]
Using the chain rule, we have:
\[ v_p = 3 \cdot 25\pi \cos(25\pi t - \frac{\pi}{2} x) \]
The maximum particle velocity occurs when \( \cos(25\pi t - \frac{\pi}{2} x) = 1 \):
\[ v_{p, \text{max}} = 3 \cdot 25\pi = 75\pi \, \text{m/s} \]

3. **Calculate the wave velocity**:
The wave velocity \( v_w \) is given by the formula:
\[ v_w = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v_w = \frac{25\pi}{\frac{\pi}{2}} = 25\pi \cdot \frac{2}{\pi} = 50 \, \text{m/s} \]

4. **Find the ratio of maximum particle velocity to wave velocity**:
Now, we can find the ratio:
\[ \text{Ratio} = \frac{v_{p, \text{max}}}{v_w} = \frac{75\pi}{50} = \frac{3\pi}{2} \]

### Final Answer:
The ratio of maximum particle velocity to wave velocity is:
\[ \frac{3\pi}{2} \]