The equation of a simple harmonic progressive wave is given by y = A sin `(100 pi t - 3x)` . Find the distance between 2 particles having a phase difference of `(pi)/(3)`.

Options

A. `(pi)/(9)m`
B. `(pi)/(18)m`
C. `(pi)/(6)m`
D. `(pi)/(3)m`

Answer

To solve the problem, we need to find the distance between two particles in a wave that have a phase difference of \(\frac{\pi}{3}\). The wave equation given is:

\[ y = A \sin(100\pi t - 3x) \]

### Step 1: Identify the parameters from the wave equation
From the wave equation, we can identify:
- Angular frequency, \(\omega = 100\pi\)
- Wave number, \(k = 3\)

### Step 2: Relate wave number to wavelength
The wave number \(k\) is related to the wavelength \(\lambda\) by the formula:

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

Substituting the value of \(k\):

\[ 3 = \frac{2\pi}{\lambda} \]

### Step 3: Solve for the wavelength \(\lambda\)
Rearranging the equation to find \(\lambda\):

\[ \lambda = \frac{2\pi}{3} \]

### Step 4: Use the phase difference to find the distance
The phase difference \(\Delta \phi\) between two particles is given by:

\[ \Delta \phi = k \Delta x \]

Where \(\Delta x\) is the distance between the two particles. We know that the phase difference is given as \(\frac{\pi}{3}\):

\[ \frac{\pi}{3} = k \Delta x \]

Substituting the value of \(k\):

\[ \frac{\pi}{3} = 3 \Delta x \]

### Step 5: Solve for \(\Delta x\)
Rearranging the equation to find \(\Delta x\):

\[ \Delta x = \frac{\pi}{3 \cdot 3} = \frac{\pi}{9} \]

### Conclusion
The distance between the two particles having a phase difference of \(\frac{\pi}{3}\) is:

\[ \Delta x = \frac{\pi}{9} \, \text{meters} \]