A sound wave of frequency 160 Hz has a velocity of 320 m/s. When it travels in air, the particles having a phase difference of `90^(@)`, are separated by a distance of

Options

A. 25 cm
B. 50 cm
C. 75 cm
D. 1 m

Explanation

To solve the problem of finding the distance between particles with a phase difference of 90 degrees in a sound wave, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Frequency (f) = 160 Hz
- Velocity (v) = 320 m/s
- Phase difference (Δφ) = 90 degrees

2. **Convert Phase Difference to Radians:**
- Since calculations are often easier in radians, convert 90 degrees to radians:
\[ \Delta \phi = 90^\circ = \frac{\pi}{2} \text{ radians} \]

3. **Calculate the Wavelength (λ):**
- The wavelength can be calculated using the formula:
\[ \lambda = \frac{v}{f} \]
- Substitute the known values:
\[ \lambda = \frac{320 \text{ m/s}}{160 \text{ Hz}} = 2 \text{ m} \]

4. **Relate Phase Difference to Path Difference:**
- The relationship between phase difference (Δφ) and path difference (Δx) is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
- Rearranging the formula to find Δx:
\[ \Delta x = \frac{\Delta \phi \cdot \lambda}{2\pi} \]

5. **Substitute the Values:**
- Substitute Δφ and λ into the equation:
\[ \Delta x = \frac{\left(\frac{\pi}{2}\right) \cdot 2 \text{ m}}{2\pi} \]
- Simplifying this gives:
\[ \Delta x = \frac{\pi}{2} \cdot \frac{2}{2\pi} = \frac{1}{2} \text{ m} \]

6. **Convert to Centimeters:**
- Since the question asks for the distance in centimeters:
\[ \Delta x = \frac{1}{2} \text{ m} = 50 \text{ cm} \]

7. **Conclusion:**
- The distance between particles having a phase difference of 90 degrees is **50 cm**.

### Final Answer:
The particles having a phase difference of 90 degrees are separated by a distance of **50 cm**.

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