The equetion of a wave travelling on a string is
`y = 4 sin(pi)/(2)(8t-(x)/(8))`
if x and y are in centimetres, then velocity of waves 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"`

Answer

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 wave equation
The general form of a wave equation is:
\[ y = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Rewrite the given equation
The given equation can be rewritten as:
\[ y = 4 \sin\left(\frac{\pi}{2}(8t - \frac{x}{8})\right) \]
This can be expanded to:
\[ y = 4 \sin\left(4\pi t - \frac{\pi}{16} x\right) \]
Here, we can identify:
- \( \omega = 4\pi \)
- \( k = \frac{\pi}{16} \)

### Step 3: 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 of \( \omega \) and \( k \):
\[ v = \frac{4\pi}{\frac{\pi}{16}} \]

### Step 4: Simplify the expression
To simplify:
\[ v = 4\pi \times \frac{16}{\pi} \]
The \( \pi \) cancels out:
\[ v = 4 \times 16 \]
\[ v = 64 \text{ cm/s} \]

### Step 5: Determine the direction of wave propagation
Since the wave equation is in the form \( \sin(\omega t - kx) \), it indicates that the wave is traveling in the positive x-direction.

### Final Answer
The velocity of the wave is:
\[ \text{Velocity} = 64 \text{ cm/s} \text{ in the positive x-direction} \]

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