A transverse wave is represented by the equation
`y = y_(0) sin (2pi)/(lambda) (vt-x)`
For what value of `lambda` is the particle velocity equal to two time the wave velocity ?

Options

A. `lambda = 2pi y_(0)`
B. `lambda = 2pi y_(0)//3`
C. `lambda = piy_(0)//2`
D. `lambda = piy_(0)`

Explanation

To solve the problem, we need to find the value of \(\lambda\) for which the particle velocity is equal to two times the wave velocity.

### Step-by-Step Solution:

1. **Understand the Wave Equation**:
The transverse wave is represented by the equation:
\[ y = y_0 \sin\left(\frac{2\pi}{\lambda}(vt - x)\right) \]
Here, \(y_0\) is the amplitude, \(\lambda\) is the wavelength, \(v\) is the wave velocity, and \(x\) and \(t\) are the position and time variables, respectively.

2. **Find the Particle Velocity**:
The particle velocity \(v_p\) can be found by differentiating the displacement \(y\) with respect to time \(t\):
\[ v_p = \frac{dy}{dt} \]
Using the chain rule, we differentiate:
\[ v_p = y_0 \cdot \cos\left(\frac{2\pi}{\lambda}(vt - x)\right) \cdot \frac{d}{dt}\left(\frac{2\pi}{\lambda}(vt - x)\right) \]
The derivative of \((vt - x)\) with respect to \(t\) is \(v\), so:
\[ v_p = y_0 \cdot \cos\left(\frac{2\pi}{\lambda}(vt - x)\right) \cdot \frac{2\pi v}{\lambda} \]

3. **Maximum Particle Velocity**:
The maximum value of the cosine function is 1, so the maximum particle velocity \(v_{p_{max}}\) is:
\[ v_{p_{max}} = y_0 \cdot \frac{2\pi v}{\lambda} \]

4. **Set Particle Velocity Equal to Two Times Wave Velocity**:
We are given that the particle velocity is equal to two times the wave velocity:
\[ v_{p_{max}} = 2v \]
Substituting the expression for \(v_{p_{max}}\):
\[ y_0 \cdot \frac{2\pi v}{\lambda} = 2v \]

5. **Simplify the Equation**:
We can cancel \(v\) from both sides (assuming \(v \neq 0\)):
\[ y_0 \cdot \frac{2\pi}{\lambda} = 2 \]
Rearranging gives:
\[ \frac{2\pi y_0}{\lambda} = 2 \]

6. **Solve for \(\lambda\)**:
Multiplying both sides by \(\lambda\) and dividing by 2 gives:
\[ \lambda = \pi y_0 \]

### Final Answer:
The value of \(\lambda\) for which the particle velocity is equal to two times the wave velocity is:
\[ \lambda = \pi y_0 \]