Question

The equation of the stationary wave is
$y=2A\sin \left( \dfrac{2\pi ct}{\lambda } \right)\cos \left( \dfrac{2\pi x}{\lambda } \right)$
Which of the following statements is wrong?
a. The unit of ct is the same as that of $\lambda $.
b. The unit of x is the same as that of $\lambda $.

Answer 1

Hint : An equation of the stationary wave is given. A stationary wave is also known as a standing wave. It can be defined as a wave which can oscillate but whose peak amplitude doesn’t move in space. And the arguments of sine and cosine are unit less. Hence the terms in the bracket must have the same dimensions.

Complete step-by-step solution:
The equation of the stationary wave is
$y=2A\sin \left( \dfrac{2\pi ct}{\lambda } \right)\cos \left( \dfrac{2\pi x}{\lambda } \right)$
Arguments of sine and cosine are unit less.
Hence,
$\dfrac{ct}{\lambda }=\dfrac{x}{\lambda }$
Both have the dimension $\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]$
Therefore we can say that $Ct,x\And \lambda $have the same dimensions.
Hence both the statements (a) and (b) are correct.

Note: The standing wave was first noticed by Michael Faraday. He observed a standing wave on the surface of liquid in a vibrating container. This phenomenon occurs when the medium is moving in the opposite direction to the wave. The common cause for standing waves is due to resonance.