When a progressive wave is propagating in a medium, at a given instant, two particles which are separated by three wave lengths will have…..

Options

A. Different displacement in same direction
B. Different displacement in opposite direction
C. Same displacement in opposite direction
D. Same displacement in same direction

Answer

To solve the problem, we need to understand the behavior of particles in a progressive wave. Let's break down the solution step by step.

### Step 1: Understand the Wave Equation
The equation of a progressive wave can be expressed as:
\[ y = A \sin(\omega t + kx) \]
where:
- \( y \) is the displacement of the wave,
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is the time,
- \( k \) is the wave number, and
- \( x \) is the position along the wave.

### Step 2: Identify the Positions of the Two Particles
Let’s denote two particles in the medium:
- Particle 1 is located at position \( x \).
- Particle 2 is located at position \( x + 3\lambda \) (since they are separated by three wavelengths).

### Step 3: Write the Equations for Both Particles
For Particle 1, the displacement can be represented as:
\[ y_1 = A \sin(\omega t + kx) \]

For Particle 2, we replace \( x \) with \( x + 3\lambda \):
\[ y_2 = A \sin(\omega t + k(x + 3\lambda)) \]
This simplifies to:
\[ y_2 = A \sin(\omega t + kx + 3k\lambda) \]

### Step 4: Substitute the Value of \( k \)
The wave number \( k \) is defined as:
\[ k = \frac{2\pi}{\lambda} \]
Thus, \( 3k\lambda = 3 \cdot \frac{2\pi}{\lambda} \cdot \lambda = 6\pi \).

### Step 5: Simplify the Equation for Particle 2
Now, substituting \( 6\pi \) into the equation for Particle 2:
\[ y_2 = A \sin(\omega t + kx + 6\pi) \]

### Step 6: Use the Sine Function Property
Using the property of the sine function, we know:
\[ \sin(\theta + 2n\pi) = \sin(\theta) \]
Thus, we can write:
\[ y_2 = A \sin(\omega t + kx) \]
This shows that:
\[ y_2 = y_1 \]

### Conclusion
Since both particles have the same displacement at any given instant, they are in the same phase. Therefore, the answer to the question is that the two particles separated by three wavelengths will have the same displacement and will be in the same phase.

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