The amplitude of a wave produced in a string is 5 cm. The wave is moving along positive direction of X-axis with a speed of 150 m/s and 5 vibrations are completed in 3 m length of the string. The equation representing the wave will be

Options

A. `0.05 sin (5t - 100 t)`
B. `0.05 sin (10.4x - 1571 t)`
C. `0.09 sin (10.9 x - 2000 t)`
D. `0.05 sin (9x - 1000 t)`

Explanation

To find the equation representing the wave, we will follow these steps:

### Step 1: Identify the given values
- Amplitude (A) = 5 cm = 0.05 m (convert to meters)
- Speed of the wave (v) = 150 m/s
- Number of vibrations = 5
- Length of the string (L) = 3 m

### Step 2: Calculate the wavelength (λ)
The number of vibrations (cycles) in a given length helps us find the wavelength. If 5 vibrations are completed in 3 m, we can find the wavelength as follows:

\[ \text{Wavelength} (\lambda) = \frac{\text{Length of the string}}{\text{Number of vibrations}} = \frac{3 \, \text{m}}{5} = 0.6 \, \text{m} \]

### Step 3: Calculate the frequency (f)
Using the wave speed and wavelength, we can find the frequency using the formula:

\[ f = \frac{v}{\lambda} \]

Substituting the values:

\[ f = \frac{150 \, \text{m/s}}{0.6 \, \text{m}} = 250 \, \text{Hz} \]

### Step 4: Calculate the wave number (k)
The wave number (k) is given by:

\[ k = \frac{2\pi}{\lambda} \]

Substituting the value of λ:

\[ k = \frac{2\pi}{0.6} \approx 10.47 \, \text{m}^{-1} \]

### Step 5: Calculate the angular frequency (ω)
The angular frequency (ω) is given by:

\[ \omega = 2\pi f \]

Substituting the value of f:

\[ \omega = 2\pi \times 250 \approx 1570.8 \, \text{rad/s} \]

### Step 6: Write the wave equation
The general form of the wave equation moving in the positive x-direction is:

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

Substituting the values we have calculated:

\[ y(x, t) = 0.05 \sin(10.47x - 1570.8t) \]

### Final Wave Equation
Thus, the equation representing the wave is:

\[ y(x, t) = 0.05 \sin(10.47x - 1570.8t) \]