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

Options

A. The unit of ` c t` is same as that of `lambda`.
B. The unit of ` x` is same as that of `lambda`.
C. The unit of `2 pi c // lambda` is same as that of ` 2 pi x // lambda t`.
D. The unit of ` c// lambda` is same as that of ` x // lambda`.

Explanation

To solve the problem, we need to analyze the given equation of a stationary wave and determine which of the provided statements is incorrect. The equation given is:

\[ y = 2A \sin\left(\frac{2\pi ct}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right) \]

### Step 1: Understand the terms in the equation
The equation consists of:
- \( A \): Amplitude of the wave (a constant)
- \( c \): Speed of the wave
- \( t \): Time
- \( x \): Position
- \( \lambda \): Wavelength

### Step 2: Analyze the dimensionless nature of the arguments of sine and cosine
The arguments of the sine and cosine functions must be dimensionless. Therefore, both \( \frac{2\pi ct}{\lambda} \) and \( \frac{2\pi x}{\lambda} \) must be dimensionless.

### Step 3: Check the dimensions of \( ct \) and \( \lambda \)
Since \( \frac{2\pi ct}{\lambda} \) is dimensionless, we can say:
- The dimensions of \( ct \) must be the same as the dimensions of \( \lambda \).

### Step 4: Establish the relationship between \( ct \) and \( x \)
From the previous step, we can conclude:
- \( ct \) has the same dimensions as \( \lambda \).
- \( x \) also has the same dimensions as \( \lambda \).

Thus, we can say:
\[ ct \text{ has the same unit as } x \]

### Step 5: Compare the ratios \( \frac{ct}{\lambda} \) and \( \frac{x}{\lambda} \)
Since both \( ct \) and \( x \) have the same dimensions as \( \lambda \), we can write:
\[ \frac{ct}{\lambda} \text{ has the same unit as } \frac{x}{\lambda} \]

### Step 6: Identify the incorrect statement
The question asks which of the following statements is wrong. Based on our analysis:
- \( ct \) has the same unit as \( x \).
- Therefore, \( \frac{ct}{\lambda} \) has the same unit as \( \frac{x}{\lambda} \).

If any statement suggests otherwise, it is incorrect.

### Conclusion
The incorrect statement among the options provided is the one that claims \( ct \) does not have the same unit as \( x \).