The equation of the stationary wave is y= 2a sin (2πct/λ) cos (2πx/λ) , which of the following statements is wrong

To determine which statement is incorrect (wrong), we can analyze the given equation of the stationary wave and apply dimensional analysis to each option.

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, the arguments of trigonometric functions (sine and cosine) must be dimensionless because they represent angles. Therefore, both $\frac{2\pi ct}{\lambda }$ and $\frac{2\pi x}{\lambda }$ are dimensionless quantities. Since $2\pi$ is a dimensionless constant, we have:
1. $\frac{ct}{\lambda }$ is dimensionless, which means the unit of $ct$ must be the same as the unit of $\lambda$ (both represent length). Thus, the statement "The unit of ct is same as that of λ" is correct.
2. $\frac{x}{\lambda }$ is dimensionless, which means the unit of $xj$ must be the same as the unit of $\lambda$ (both represent length). Thus, the statement "The unit of x is same as that of λ" is correct.

Now let's examine the remaining options by checking their dimensions:

Since $x$ and $\lambda$ have the unit of length (L), and $t$ has the unit of time (T), the wave velocity $c$ must have the unit of velocity (LT^-1).

Let's check the third option: the unit of $\frac{2\pi c}{\lambda }$ and $\frac{2\pi x}{\lambda t}$.
- Dimension of $\frac{2\pi c}{\lambda }$ is $\frac{{\mathrm{LT}}^{-1}}{L}=T^{-1}$
- Dimension of $\frac{2\pi x}{\lambda t}$ is $\frac{L}{L·T}=T^{-1}$
Since both have the same dimension (T^-1), their units are the same. Thus, the statement "The unit of 2πc/λ is same as that of 2πx/λt" is correct.

Let's check the fourth option: the unit of $\frac{c}{\lambda }$ and $\frac{x}{\lambda }$.
- Dimension of $\frac{c}{\lambda }$ is $\frac{{\mathrm{LT}}^{-1}}{L}=T^{-1}$
- Dimension of $\frac{x}{\lambda }$ is $\frac{L}{L}=M^0{L}^0{T}^0$ (dimensionless)
Since $\frac{c}{\lambda }$ has the unit of frequency (s^-1) and $\frac{x}{\lambda }$ is dimensionless, their units are not the same. Therefore, this statement is wrong.

Thus, the incorrect statement is: "The unit of c/λ is same as that of x/λ".