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

Explanation

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

**Given:**
\[ y = 3 \sin\left(\frac{\pi}{2} (50t - x)\right) \]

### Step 1: Identify the parameters from the wave equation
From the wave equation, we can identify:
- Amplitude \( a = 3 \) m
- Angular frequency \( \omega = \frac{\pi}{2} \times 50 = 25\pi \) rad/s
- Wave number \( k = \frac{\pi}{2} \) rad/m

### Step 2: Calculate the maximum particle velocity
The maximum particle velocity \( v_{\text{max}} \) in a simple harmonic wave is given by the formula:
\[ v_{\text{max}} = a \omega \]
Substituting the values we found:
\[ v_{\text{max}} = 3 \times 25\pi = 75\pi \, \text{m/s} \]

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

### Step 4: Calculate the ratio of maximum particle velocity to wave velocity
Now, we can find the ratio:
\[ \text{Ratio} = \frac{v_{\text{max}}}{v} = \frac{75\pi}{50\pi} = \frac{75}{50} = \frac{3}{2} \]

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

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