The particles of a medium oscillate about their equilibrium position, whenever a wave travels through that medium. The phase difference between the oscillations of two such practices varies

Options

A. with time but not with distance separating them
B. with distance separation them but not with time
C. with distance separating them as well as with time
D. neither with distance separating them nor with time

Answer

To solve the question regarding the phase difference between the oscillations of two particles in a medium through which a wave is traveling, we can follow these steps:

### Step 1: Understand the Wave Equation
The wave can be represented by the equation:
\[ y = A \sin(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( t \) is time,
- \( x \) is the position.

### Step 2: Determine the Phase of Each Particle
For two particles located at positions \( x_1 \) and \( x_2 \) at times \( t_1 \) and \( t_2 \), the phases can be expressed as:
- For particle 1 at \( (x_1, t_1) \):
\[ \phi_1 = \omega t_1 - k x_1 \]
- For particle 2 at \( (x_2, t_2) \):
\[ \phi_2 = \omega t_2 - k x_2 \]

### Step 3: Calculate the Phase Difference
The phase difference \( \Delta \phi \) between the two particles is given by:
\[ \Delta \phi = \phi_2 - \phi_1 \]
Substituting the expressions for \( \phi_1 \) and \( \phi_2 \):
\[ \Delta \phi = (\omega t_2 - k x_2) - (\omega t_1 - k x_1) \]
This simplifies to:
\[ \Delta \phi = \omega (t_2 - t_1) - k (x_2 - x_1) \]

### Step 4: Define the Variables
Let:
- \( \Delta t = t_2 - t_1 \) (the time difference between the two particles)
- \( \Delta x = x_2 - x_1 \) (the distance separating the two particles)

Thus, we can rewrite the phase difference as:
\[ \Delta \phi = \omega \Delta t - k \Delta x \]

### Step 5: Analyze the Dependence of Phase Difference
From the equation \( \Delta \phi = \omega \Delta t - k \Delta x \), we can see that:
- The phase difference \( \Delta \phi \) depends on both the time difference \( \Delta t \) and the distance \( \Delta x \) separating the two particles.

### Conclusion
Based on the analysis, we conclude that the phase difference between the oscillations of two particles in a medium varies with both time and the distance separating them.

### Final Answer
The correct option is that the phase difference varies with both time and the distance separating the particles.

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