When sound wave of wavelength `lambda` is propagating in a medium, the maximum velocity of the particle is equal to the wave velocity. The amplitude of wave is

Options

A. `lambda`
B. `(lambda)/(2)`
C. `(lambda)/(2pi)`
D. `(lambda)/(4pi)`

Answer

To solve the problem, we need to find the amplitude of a sound wave given that the maximum velocity of the particle is equal to the wave velocity. Let's break down the solution step by step.

### Step 1: Understand the relationship between particle velocity and wave velocity
The maximum velocity of a particle in a wave can be expressed as:
\[ v_{p_{\text{max}}} = A \omega \]
where:
- \( A \) is the amplitude of the wave,
- \( \omega \) is the angular frequency of the wave.

### Step 2: Express wave velocity in terms of wavelength and angular frequency
The wave velocity \( v \) can be expressed in terms of wavelength \( \lambda \) and angular frequency \( \omega \) as:
\[ v = \frac{\omega}{k} \]
where \( k \) (the wave number) is given by:
\[ k = \frac{2\pi}{\lambda} \]
Thus, we can rewrite the wave velocity as:
\[ v = \omega \cdot \frac{\lambda}{2\pi} \]

### Step 3: Set the maximum particle velocity equal to wave velocity
According to the problem, we have:
\[ v_{p_{\text{max}}} = v \]
Substituting the expressions we derived:
\[ A \omega = \frac{\omega \lambda}{2\pi} \]

### Step 4: Cancel out \( \omega \) from both sides
Assuming \( \omega \neq 0 \), we can divide both sides of the equation by \( \omega \):
\[ A = \frac{\lambda}{2\pi} \]

### Step 5: Conclusion
Thus, the amplitude \( A \) of the wave is:
\[ A = \frac{\lambda}{2\pi} \]

Therefore, the correct answer is option **C: \( \frac{\lambda}{2\pi} \)**.

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