Equation of a progressive wave is given by
`y=0.2cospi(0.04t+0.02x-(pi)/(6))`
The distance is expressed in cm and time in second. The minimum distance between two particles having the phase difference of `pi//2` is................

Answer

To solve the problem, we need to find the minimum distance between two particles that have a phase difference of \( \frac{\pi}{2} \) in the given wave equation:

\[ y = 0.2 \cos \left( \pi (0.04t + 0.02x) - \frac{\pi}{6} \right) \]

### Step 1: Identify the wave parameters

The wave equation can be compared with the standard form:

\[ y = A \cos(\omega t + kx + \phi) \]

From the given equation, we can identify:

- Amplitude \( A = 0.2 \) cm
- Angular frequency \( \omega = \pi \times 0.04 = 0.04\pi \) rad/s
- Wave number \( k = \pi \times 0.02 = 0.02\pi \) rad/cm

### Step 2: Calculate the wavelength

The wavelength \( \lambda \) can be calculated using the relationship between wave number \( k \) and wavelength:

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

Rearranging gives:

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

Substituting \( k = 0.02\pi \):

\[ \lambda = \frac{2\pi}{0.02\pi} = \frac{2}{0.02} = 100 \text{ cm} \]

### Step 3: Relate phase difference to distance

The relationship between phase difference \( \Delta \phi \) and distance \( \Delta x \) is given by:

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

For a phase difference of \( \frac{\pi}{2} \):

\[ \frac{\pi}{2} = k \Delta x \]

### Step 4: Solve for \( \Delta x \)

Substituting \( k = 0.02\pi \):

\[ \frac{\pi}{2} = (0.02\pi) \Delta x \]

Dividing both sides by \( \pi \):

\[ \frac{1}{2} = 0.02 \Delta x \]

Now, solving for \( \Delta x \):

\[ \Delta x = \frac{1/2}{0.02} = \frac{1}{0.04} = 25 \text{ cm} \]

### Final Answer

The minimum distance between two particles having a phase difference of \( \frac{\pi}{2} \) is:

\[ \Delta x = 25 \text{ cm} \]

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