The equation of a stationary wave is `y=2A sin((2pict)/lambda) cos ((2pix)/lambda)`
Which of the following is correct?

Options

A. The unit of ct is same as that of `lamda`
B. The unit of x is same as that of `lamda`
C. The unit of `(2pic)/lamda` is same as that of `(2pix)/(lamdat)`
D. The unit of `c/lamda` is same as that of `x/lamda`

Answer

To solve the problem regarding the stationary wave equation given by \( y = 2A \sin\left(\frac{2\pi ct}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right) \), we will analyze the dimensions of the terms involved and determine which of the provided options is correct.

### Step-by-Step Solution:

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

2. **Analyzing the Argument of the Sine and Cosine Functions**:
- The argument of the sine function is \( \frac{2\pi ct}{\lambda} \).
- The argument of the cosine function is \( \frac{2\pi x}{\lambda} \).
- Both sine and cosine functions require their arguments to be dimensionless.

3. **Dimensionless Condition**:
- For \( \frac{2\pi ct}{\lambda} \) to be dimensionless:
\[ [ct] = [\lambda] \]
This implies that the units of \( ct \) must be the same as the units of \( \lambda \).
- For \( \frac{2\pi x}{\lambda} \) to be dimensionless:
\[ [x] = [\lambda] \]
This implies that the units of \( x \) must also be the same as the units of \( \lambda \).

4. **Dimensions of the Terms**:
- Since \( c \) is the speed, its dimensions are \( [L][T^{-1}] \).
- Therefore, \( ct \) has dimensions:
\[ [ct] = [L][T^{-1}][T] = [L] \]
- Thus, \( ct \) has the same dimensions as \( \lambda \) (which is also \( [L] \)).
- For \( x \), since it is a position, its dimensions are also \( [L] \).

5. **Evaluating the Options**:
- **Option 1**: The unit of \( ct \) is the same as that of \( \lambda \).
- **Correct**: From our analysis, \( [ct] = [\lambda] \).
- **Option 2**: The unit of \( x \) is the same as that of \( \lambda \).
- **Correct**: Since both \( x \) and \( \lambda \) have dimensions of length.
- **Option 3**: The unit of \( \frac{2\pi c}{\lambda} \) is the same as that of \( \frac{2\pi x}{\lambda t} \).
- **Incorrect**: \( \frac{2\pi c}{\lambda} \) has dimensions of \( [T^{-1}] \) while \( \frac{2\pi x}{\lambda t} \) is dimensionless.
- **Option 4**: The unit of \( \frac{c}{\lambda} \) is the same as that of \( \frac{x}{\lambda} \).
- **Incorrect**: \( \frac{c}{\lambda} \) has dimensions of \( [T^{-1}] \) while \( \frac{x}{\lambda} \) is dimensionless.

6. **Conclusion**:
The correct options are **Option 1** and **Option 2**.