A longitudinal wave is represented by
`x = x_(0) "sin"2pi("nt" - ("x")/(lambda))`
The maximum particle velocity will be four times the wave velocity if

Options

A. `lamda=(nx_(0))/(4)`
B. `lamda=2pix_(0)`
C. `lamda=(pix_(0))/(2)`
D. `lamda=4pix_(0)`

Explanation

To solve the problem step by step, we will analyze the given wave equation and derive the necessary relationships to find the condition under which the maximum particle velocity is four times the wave velocity.

### Step 1: Write down the wave equation
The longitudinal wave is represented by the equation:
\[ x = x_0 \sin(2\pi(nt - \frac{x}{\lambda})) \]

### Step 2: Identify parameters
From the wave equation, we can identify:
- \( x_0 \): Amplitude of the wave
- \( n \): Frequency of the wave
- \( \lambda \): Wavelength of the wave

### Step 3: Calculate the maximum particle velocity
The particle velocity \( v_p \) can be calculated by differentiating the displacement \( x \) with respect to time \( t \):
\[ v_p = \frac{dx}{dt} = \frac{d}{dt} \left( x_0 \sin(2\pi(nt - \frac{x}{\lambda})) \right) \]

Using the chain rule:
\[ v_p = x_0 \cdot 2\pi n \cos(2\pi(nt - \frac{x}{\lambda})) \]

The maximum particle velocity \( v_{p,\text{max}} \) occurs when \( \cos(2\pi(nt - \frac{x}{\lambda})) = 1 \):
\[ v_{p,\text{max}} = 2\pi n x_0 \]

### Step 4: Calculate the wave velocity
The wave velocity \( v \) is given by the relationship:
\[ v = n \lambda \]

### Step 5: Set up the condition
According to the problem, the maximum particle velocity is four times the wave velocity:
\[ v_{p,\text{max}} = 4v \]

Substituting the expressions we found:
\[ 2\pi n x_0 = 4(n \lambda) \]

### Step 6: Simplify the equation
Dividing both sides by \( n \) (assuming \( n \neq 0 \)):
\[ 2\pi x_0 = 4\lambda \]

### Step 7: Solve for \( \lambda \)
Rearranging gives:
\[ \lambda = \frac{\pi x_0}{2} \]

### Conclusion
Thus, the condition under which the maximum particle velocity is four times the wave velocity is:
\[ \lambda = \frac{\pi x_0}{2} \]

### Final Answer
The correct option is (C) \( \frac{\pi x_0}{2} \).

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