A travelling wave in a string is represented by `y=3 sin ((pi)/(2)t - (pi)/(4) x) `. The phase difference between two particles separated by a distance 4 cm is ( Take x and y in cm and t in seconds )

Options

A. `(pi)/(2) ` rad
B. `(pi)/(4)` rad
C. `pi` rad
D. 0

Answer

To find the phase difference between two particles separated by a distance of 4 cm in the given wave equation \( y = 3 \sin\left(\frac{\pi}{2} t - \frac{\pi}{4} x\right) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the wave equation**:
The wave is given by:
\[ y = 3 \sin\left(\frac{\pi}{2} t - \frac{\pi}{4} x\right) \]
This can be compared to the standard form of a wave equation:
\[ y = A \sin(\omega t - kx) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( k \) is the wave number.

2. **Extract values of \( \omega \) and \( k \)**:
From the equation, we have:
\[ \omega = \frac{\pi}{2} \quad \text{and} \quad k = \frac{\pi}{4} \]

3. **Determine the phase difference formula**:
The phase difference \( \Delta \phi \) between two points separated by a distance \( \Delta x \) is given by:
\[ \Delta \phi = k \Delta x \]

4. **Substitute the values**:
Here, \( \Delta x = 4 \) cm. Thus, substituting the values we have:
\[ \Delta \phi = k \Delta x = \left(\frac{\pi}{4}\right) \times 4 \]

5. **Calculate \( \Delta \phi \)**:
\[ \Delta \phi = \frac{\pi}{4} \times 4 = \pi \text{ radians} \]

### Final Answer:
The phase difference between the two particles separated by a distance of 4 cm is \( \pi \) radians.

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