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 find the phase difference between the displacement wave and the pressure wave in sound waves, we can follow these steps:

### Step 1: Understand the displacement wave equation
The displacement of particles in a sound wave can be described by the equation:
\[ s = s_0 \sin(\omega t - kx) \]
where:
- \( s \) is the displacement,
- \( s_0 \) is the amplitude of displacement,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( t \) is time,
- \( x \) is the position.

### Step 2: Understand the pressure wave equation
In a sound wave, pressure varies with time and position. The pressure wave can be expressed as:
\[ \Delta p = \Delta p_0 \cos(\omega t - kx) \]
where:
- \( \Delta p \) is the pressure variation,
- \( \Delta p_0 \) is the amplitude of pressure variation.

### Step 3: Relate the pressure wave to the displacement wave
The pressure wave can be rewritten in terms of sine function:
\[ \Delta p = \Delta p_0 \sin(\omega t - kx + \frac{\pi}{2}) \]
This transformation shows that the pressure wave is a sine wave that is phase-shifted by \( \frac{\pi}{2} \) radians (or 90 degrees) compared to the displacement wave.

### Step 4: Determine the phase difference
The phase difference \( \Delta \phi \) between the displacement wave and the pressure wave can be calculated as:
\[ \Delta \phi = \frac{\pi}{2} \]
This indicates that the pressure wave reaches its maximum value a quarter cycle (90 degrees) after the displacement wave.

### Conclusion
Thus, the phase difference between the displacement wave and the pressure wave in sound waves is:
\[ \Delta \phi = \frac{\pi}{2} \]

### Final Answer
The phase difference is \( \frac{\pi}{2} \) radians.

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