What is the phase difference between the displacement wave and pressure wave in sound wave :-

Options

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

Answer

To determine the phase difference between the displacement wave and the pressure wave in a sound wave, we can follow these steps:

### Step-by-Step Solution:

1. **Write the Equation for the Displacement Wave**:
The displacement wave can be represented mathematically as:
\[ y(x, t) = S_m \sin(\omega t - kx) \]
where \( S_m \) is the amplitude, \( \omega \) is the angular frequency, and \( k \) is the wave number.

2. **Write the Equation for the Pressure Wave**:
The pressure wave can be expressed as:
\[ \Delta P(x, t) = \Delta P_m \cos(\omega t - kx) \]
where \( \Delta P_m \) is the maximum change in pressure.

3. **Convert the Pressure Wave Equation to Sine Form**:
We know that the cosine function can be expressed in terms of sine:
\[ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) \]
Therefore, we can rewrite the pressure wave equation as:
\[ \Delta P(x, t) = \Delta P_m \sin\left(\omega t - kx + \frac{\pi}{2}\right) \]

4. **Identify the Phase of Each Wave**:
- For the displacement wave, the phase \( \phi_1 \) is:
\[ \phi_1 = \omega t - kx \]
- For the pressure wave, the phase \( \phi_2 \) is:
\[ \phi_2 = \omega t - kx + \frac{\pi}{2} \]

5. **Calculate the Phase Difference**:
The phase difference \( \Delta \phi \) between the pressure wave and the displacement wave is given by:
\[ \Delta \phi = \phi_2 - \phi_1 = \left(\omega t - kx + \frac{\pi}{2}\right) - (\omega t - kx) \]
Simplifying this, we find:
\[ \Delta \phi = \frac{\pi}{2} \]

6. **Conclusion**:
Thus, the phase difference between the displacement wave and the pressure wave in a sound wave is:
\[ \Delta \phi = \frac{\pi}{2} \text{ radians} \]