During propagation of longitudinal plane wave in a medium the two particles separated by a distance equivalent to one wavelength at an instant will be/have

Options

A. in phase, same displacement
B. in phase, different displacement
C. different phase, same displacement
D. different phase, different displacement

Answer

To solve the question regarding the behavior of two particles separated by a distance equivalent to one wavelength during the propagation of a longitudinal plane wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Wave Equation**:
The equation of a sinusoidal wave can be expressed as:
\[ y(x, t) = A \sin(\omega t + kx) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, \( k \) is the wave number, and \( x \) is the position.

2. **Identifying Two Particles**:
Let’s consider two particles in the medium. The first particle is located at position \( x \) and the second particle is at position \( x + \lambda \), where \( \lambda \) is the wavelength.

3. **Writing the Wave Equations for Both Particles**:
- For the first particle at position \( x \):
\[ y_1 = A \sin(\omega t + kx) \]
- For the second particle at position \( x + \lambda \):
\[ y_2 = A \sin(\omega t + k(x + \lambda)) \]

4. **Simplifying the Second Particle's Equation**:
Substitute \( x + \lambda \) into the wave equation:
\[ y_2 = A \sin(\omega t + kx + k\lambda) \]
Since \( k = \frac{2\pi}{\lambda} \), we can substitute \( k\lambda \):
\[ k\lambda = 2\pi \]
Thus, the equation for the second particle becomes:
\[ y_2 = A \sin(\omega t + kx + 2\pi) \]

5. **Using the Trigonometric Identity**:
We know from trigonometric identities that:
\[ \sin(\theta + 2\pi) = \sin(\theta) \]
Therefore:
\[ y_2 = A \sin(\omega t + kx) \]

6. **Conclusion**:
Since both \( y_1 \) and \( y_2 \) are equal:
\[ y_1 = y_2 \]
This means that both particles are at the same displacement and are in the same phase. Thus, during the propagation of a longitudinal wave, two particles separated by a distance of one wavelength will have the same displacement and will be in the same phase.

### Final Answer:
The two particles separated by a distance equivalent to one wavelength will have the same displacement and will be in the same phase.

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