Which of the following statements are true for a stationary wave ?

Options

A. Every paticle has a fixed amplitude which is different from the amplitude of its nearest particle
B. All the particles cross their mean position at the same time
C. There are some particles which are alwayss at rest
D. There is not transfer of energy across any plane

Explanation

To determine which statements are true for a stationary wave, we can analyze each option based on the properties of stationary waves. Let's go through the solution step by step.

### Step-by-Step Solution

1. **Understanding the Equation of a Stationary Wave**:
The equation of a stationary wave is given by:
\[ y = a \sin(kx) \cos(\omega t) \]
where \(a\) is the amplitude, \(k\) is the wave number, and \(\omega\) is the angular frequency.

2. **Analyzing Option A**:
- **Statement**: Every particle has a fixed amplitude which is different from the amplitude of its nearest particle.
- **Analysis**: The amplitude of a stationary wave is given by \(a \sin(kx)\). This means the amplitude varies with position \(x\). Therefore, different particles at different positions have different amplitudes.
- **Conclusion**: Option A is **True**.

3. **Analyzing Option B**:
- **Statement**: All the particles cross their mean position at the same time.
- **Analysis**: The mean position corresponds to \(y = 0\), which occurs when \(\cos(\omega t) = 0\). This happens at \(t = \frac{(2n - 1) \pi}{2\omega}\) for integer \(n\). Since this time is the same for all particles, they all cross the mean position simultaneously.
- **Conclusion**: Option B is **True**.

4. **Analyzing Option C**:
- **Statement**: There are some particles which are always at rest.
- **Analysis**: The stationary wave has nodes where the displacement \(y\) is always zero. For example, at \(x = 0\) (where \(\sin(kx) = 0\)), the particles remain at rest. Thus, there are indeed particles that are always at rest.
- **Conclusion**: Option C is **True**.

5. **Analyzing Option D**:
- **Statement**: There is no transfer of energy across any plane.
- **Analysis**: In a stationary wave, energy is not transferred through the medium; it is confined between nodes. Thus, there is no net energy transfer across any plane in a stationary wave.
- **Conclusion**: Option D is **True**.

### Final Conclusion
All four options (A, B, C, and D) are true for a stationary wave.

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