Question

If the equation of the stationary wave is $y=2a\,sin\left(\frac{2\pi ct}{\lambda}\right)cos\left(\frac{2\pi x}{\lambda}\right)$, then which of the following statements is incorrect?

• 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 $2\pi c/\lambda$ is same as that of $2\pi x/\lambda t$
• D. The unit of $c/\lambda$ is same as that of $x/\lambda$

Answer 1

Answer: (D)

Here, $\frac{2\pi ct}{\lambda}$ as well as $\frac{2\pi x}{\lambda}$ are dimensionless
($\because$ angle), i.e., $\left[\frac{2\pi ct}{\lambda }\right]=\left[\frac{2\pi x}{\lambda }\right]=\left[M^{0}L^{0}T^{0}\right]$
So, $(1)$ unit of $ct$ is same as that of $\lambda$
$(2)$ unit of $x$ is same as that of $\lambda$
$(3)$ $\left[\frac{2\pi c}{\lambda }\right]=\left[\frac{2\pi x}{\lambda t }\right]$
$(4)$ $\frac{x}{\lambda}$ is unit less. It is not the case with $\frac{c}{\lambda}$.