The equation of a wave travelling on a string is `y=8 sin [pi/2 (4t-x/16)]`, where x, y are in cm and t in second. The velocity of the wave is

Options

A. `256 cms^(-1)`, in -x direction
B. `32 cms^(-1)`, in -x direction
C. `32 cms^(-1)`, in +x direction
D. `64 cms^(-1)`, in +x direction

Explanation

To find the velocity of the wave given by the equation \( y = 8 \sin \left( \frac{\pi}{2} (4t - \frac{x}{16}) \right) \), we can follow these steps:

### Step 1: Identify the wave equation format
The general form of a wave equation is:
\[ y = A \sin(\omega t - kx + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( \phi \) is the phase constant.

### Step 2: Rewrite the given wave equation
The given equation can be rewritten as:
\[ y = 8 \sin\left(2\pi t - \frac{\pi}{32} x\right) \]
This is done by recognizing that:
\[ \frac{\pi}{2}(4t - \frac{x}{16}) = 2\pi t - \frac{\pi}{32} x \]

### Step 3: Identify parameters from the equation
From the rewritten equation, we can identify:
- Amplitude \( A = 8 \)
- Angular frequency \( \omega = 2\pi \)
- Wave number \( k = \frac{\pi}{32} \)

### Step 4: Calculate the velocity of the wave
The velocity \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values we found:
\[ v = \frac{2\pi}{\frac{\pi}{32}} = 2\pi \times \frac{32}{\pi} = 64 \text{ cm/s} \]

### Step 5: Determine the direction of the wave
Since the equation has a negative sign in front of \( kx \), it indicates that the wave is moving in the positive x-direction.

### Final Answer
The velocity of the wave is \( 64 \text{ cm/s} \) in the positive x-direction.

---