The equation of the stationary wave is `y=2asin(2pict//lambda)cos(2pix//lambda)`, which of the following statements is wrong?

Options

A. The unit of ct is same as that of `lambda`
B. The unit of x is same as that of `lambda`
C. The unit of `2pic//lambda` is same as that of `2pix//lambdat`
D. The unit of `c//lambda` is same as that of `x//lambda`

Answer

To solve the problem, we need to analyze the given equation of the stationary wave and the statements provided to identify which one is incorrect.

### Step-by-Step Solution:

1. **Understand the Equation**: The equation of the stationary wave is given as:
\[ y = 2a \sin\left(\frac{2\pi ct}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right) \]
Here, \(a\) is the amplitude, \(c\) is the speed of the wave, \(t\) is time, \(x\) is the position, and \(\lambda\) is the wavelength.

2. **Identify the Units**:
- The argument of the sine and cosine functions must be dimensionless (unitless). Therefore, the units of \(ct\) and \(x\) must be the same as that of \(\lambda\).
- The unit of \(c\) (speed) is \(\text{m/s}\).
- The unit of \(t\) (time) is \(\text{s}\).
- The unit of \(x\) (position) is \(\text{m}\).
- The unit of \(\lambda\) (wavelength) is also \(\text{m}\).

3. **Check the Statements**:
- **Statement 1**: The unit of \(ct\) is the same as that of \(\lambda\).
- \(ct\) has units of \((\text{m/s})(\text{s}) = \text{m}\), which is the same as \(\lambda\). This statement is correct.

- **Statement 2**: The unit of \(x\) is the same as that of \(\lambda\).
- \(x\) has units of \(\text{m}\), which is the same as \(\lambda\). This statement is also correct.

- **Statement 3**: The unit of \(\frac{c}{\lambda}\) is the same as that of \(\frac{x}{\lambda}\).
- The unit of \(\frac{c}{\lambda}\) is \(\frac{\text{m/s}}{\text{m}} = \text{s}^{-1}\).
- The unit of \(\frac{x}{\lambda}\) is \(\frac{\text{m}}{\text{m}} = 1\) (dimensionless).
- These units are not the same, so this statement is incorrect.

- **Statement 4**: The unit of \(2\pi c/\lambda\) is the same as that of \(2\pi x/\lambda t\).
- The unit of \(2\pi c/\lambda\) is \(\frac{\text{m/s}}{\text{m}} = \text{s}^{-1}\).
- The unit of \(2\pi x/\lambda t\) is \(\frac{\text{m}}{\text{m} \cdot \text{s}} = \text{s}^{-1}\).
- These units are the same, so this statement is correct.

4. **Conclusion**: The wrong statement is the third one: "The unit of \(c/\lambda\) is the same as that of \(x/\lambda\)."

### Final Answer:
The wrong statement is: "The unit of \(c/\lambda\) is the same as that of \(x/\lambda\)."