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

Options

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

Explanation

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

### Step 1: Identify the angular frequency (\( \omega \)) and the wave number (\( k \))

The general form of a wave equation is given by:

\[ y = A \sin(\omega t - kx) \]

From the given equation, we can identify:

- \( \omega t = \frac{\pi}{2} \cdot 8t \) implies \( \omega = 4\pi \)
- \( kx = \frac{\pi}{2} \cdot \frac{x}{8} \) implies \( k = \frac{\pi}{16} \)

### Step 2: Use the relationship between velocity, angular frequency, and wave number

The velocity \( v \) of the wave can be calculated using the formula:

\[ v = \frac{\omega}{k} \]

### Step 3: Substitute the values of \( \omega \) and \( k \)

Now substituting the values we found:

\[ v = \frac{4\pi}{\frac{\pi}{16}} \]

### Step 4: Simplify the expression

To simplify this, we can multiply by the reciprocal of \( k \):

\[ v = 4\pi \cdot \frac{16}{\pi} = 64 \]

### Step 5: State the final answer with units

Since \( x \) and \( y \) are in centimeters and \( t \) is in seconds, the velocity will be in centimeters per second:

\[ v = 64 \, \text{cm/s} \]

### Final Answer

The velocity of the wave is \( 64 \, \text{cm/s} \).

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