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. `64 cm//s "in"-ve x-"direction"`
B. `32 cm//s "in"-ve x-"direction"`
C. `32 cm//s "in"+ve x-"direction"`
D. `64 cm//s "in"+ve x-"direction"`

Explanation

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

### Step 1: Simplify the Wave Equation
The given wave equation can be rewritten for clarity:
\[ y = 4 \sin\left(4\pi t - \frac{\pi}{16} x\right) \]
This is done by distributing the \(\frac{\pi}{2}\) inside the sine function.

### Step 2: Identify Parameters
We can compare the simplified equation \( y = 4 \sin(4\pi t - \frac{\pi}{16} x) \) with the standard wave equation:
\[ y = a \sin(\omega t - kx) \]
From this comparison, we identify:
- Amplitude \( a = 4 \)
- Angular frequency \( \omega = 4\pi \)
- Wave number \( k = \frac{\pi}{16} \)

### Step 3: Calculate the Velocity of the Wave
The velocity \( v \) of a wave is given by the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{4\pi}{\frac{\pi}{16}} = 4\pi \times \frac{16}{\pi} \]
This simplifies to:
\[ v = 64 \text{ cm/s} \]

### Step 4: Determine the Direction of the Wave
In the standard wave equation \( y = a \sin(\omega t - kx) \), the sign of the term \( -kx \) indicates the direction of wave propagation. Since we have a minus sign, the wave travels in the positive x-direction.

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

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