in a plane progaressive harmonic wave particle speed is always less than the wave speed if.

Options

A. amplitude of wave is less than `lambda/(2 pi)`
B. amplitude of wave is greater than `lambda /(2 pi ) `
C. amplitude to wave is less than ` lambda`
D. amplitude of wave is greater than ` lambda / pi)`

Answer

To solve the problem, we need to analyze the relationship between particle speed and wave speed in a plane progressive harmonic wave.

### Step-by-Step Solution:

1. **Understanding Particle Speed and Wave Speed**:
- The particle speed (\(v_p\)) in a wave is the speed at which a particular point on the wave (a particle) moves up and down as the wave passes.
- The wave speed (\(v\)) is the speed at which the wave itself travels through space.

2. **Expression for Particle Speed**:
- The displacement of a particle in a wave can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase constant.
- To find the particle speed, we differentiate the displacement with respect to time:
\[ v_p = \frac{dx}{dt} = A \omega \cos(\omega t + \phi) \]
- The maximum particle speed (\(v_{p, \text{max}}\)) occurs when \(\cos(\omega t + \phi) = 1\):
\[ v_{p, \text{max}} = A \omega \]

3. **Expression for Wave Speed**:
- The wave speed is given by:
\[ v = \frac{\lambda}{T} \]
where \(\lambda\) is the wavelength and \(T\) is the period of the wave.
- We can express angular frequency \(\omega\) in terms of \(T\):
\[ \omega = \frac{2\pi}{T} \]
- Thus, the wave speed can also be expressed as:
\[ v = \lambda \cdot \frac{2\pi}{\lambda} = \frac{\lambda}{T} \]

4. **Condition for Particle Speed to be Less than Wave Speed**:
- We need to establish the condition under which the maximum particle speed is less than the wave speed:
\[ A \omega < v \]
- Substituting \(\omega\) and \(v\):
\[ A \cdot \frac{2\pi}{T} < \frac{\lambda}{T} \]
- Cancelling \(T\) from both sides (assuming \(T \neq 0\)):
\[ A \cdot 2\pi < \lambda \]
- Rearranging gives:
\[ A < \frac{\lambda}{2\pi} \]

5. **Conclusion**:
- Therefore, the condition for the particle speed to be always less than the wave speed is:
\[ A < \frac{\lambda}{2\pi} \]
- This corresponds to option (a) in the question.

### Final Answer:
The particle speed is always less than the wave speed if the amplitude \(A\) is less than \(\frac{\lambda}{2\pi}\).