A sinusoidal wave is propagating in negative x-direction in a string stretched along x-axis. A particle of string at `x=2` cm is found at its mean position and it is moving in positive y-direction at `t=1` s. the amplitude of the wave, the wavelength and the angular frequency of the wave are `0.1m,pi//4m` and `4pi rad//s`, respectively.
The equation of the wave is

Options

A. `y=0.1 sin(4pi(t-1)+8(x-2))`
B. `y=0.1 sin((t-1)-(x-2))`
C. `y=0.1 sin(4pi(t-1)-8(x-2))`
D. none of these

Explanation

To find the equation of the sinusoidal wave propagating in the negative x-direction, we will follow these steps:

### Step 1: Identify the given parameters
- Amplitude \( A = 0.1 \, \text{m} \)
- Wavelength \( \lambda = \frac{\pi}{4} \, \text{m} \)
- Angular frequency \( \omega = 4\pi \, \text{rad/s} \)
- Position of the particle \( x = 2 \, \text{cm} = 0.02 \, \text{m} \) (convert cm to m)
- Time \( t = 1 \, \text{s} \)

### Step 2: Calculate the wave number \( k \)
The wave number \( k \) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{\frac{\pi}{4}} = 8 \, \text{m}^{-1} \]

### Step 3: Write the general form of the wave equation
For a wave propagating in the negative x-direction, the wave equation is given by:
\[ y(x, t) = A \sin(\omega t + kx + \phi) \]

### Step 4: Determine the phase constant \( \phi \)
We know that at \( x = 0.02 \, \text{m} \) and \( t = 1 \, \text{s} \), the particle is at its mean position (which corresponds to \( y = 0 \)) and moving in the positive y-direction. This means the sine function must be at its maximum slope, which occurs at \( \sin(\omega t + kx + \phi) = 0 \).

Since the particle is at the mean position, we can set:
\[ \omega t + kx + \phi = 0 \]

Substituting \( t = 1 \, \text{s} \) and \( x = 0.02 \, \text{m} \):
\[ 4\pi(1) + 8(0.02) + \phi = 0 \]
\[ 4\pi + 0.16 + \phi = 0 \]
\[ \phi = -4\pi - 0.16 \]

### Step 5: Substitute the values into the wave equation
Now we can write the wave equation:
\[ y(x, t) = 0.1 \sin(4\pi t + 8x - (4\pi + 0.16)) \]
Simplifying this gives:
\[ y(x, t) = 0.1 \sin(4\pi t + 8x - 4\pi - 0.16) \]
\[ y(x, t) = 0.1 \sin(8x + 4\pi t - 4\pi - 0.16) \]

### Final Equation
Thus, the equation of the wave is:
\[ y(x, t) = 0.1 \sin(4\pi t + 8x - 4\pi - 0.16) \]