Question

In a stationary wave, the distance between a node and the next antinode is $10cm$. What is the wavelength?

Answer 1

Hint: Node is a point of minimum displacement of the standing wave and antinode is the point of maximum displacement of the standing wave. To solve this question, we must know the distance between a node and an antinode, and then, how to relate the distance between them to the wavelength of the wave.

Complete step by step solution:
While observing a standing wave, we can see that the distance between a node and an antinode is one-half the distance between a crest and a trough.
So, next comes the relation between wavelength and the distance between a node and an antinode.
We know the distance between a crest and a trough is one-half the wavelength. Therefore, the distance between a node and an antinode is one-fourth the wavelength.
Hence, mathematically, we can write, assuming $l$ to be the distance between a node and an antinode:
$l=\dfrac{\lambda }{4}$
According to the question:
$l=10cm$
$\Rightarrow \dfrac{\lambda }{4}=10cm$
$\Rightarrow \lambda =40cm$

Therefore, the wavelength of the standing wave is $40cm$.

Note: We must be very careful in writing the mathematical relation of the distance between a node and an antinode and the wavelength, as while writing this relation is a very common silly mistake. We must not confuse between a standing wave and a stationary wave; they are the same wave.
Sometimes, there will be a problem like two waves traveling in opposite directions coincide, the distance of the node and antinode of the resultant wave is, say $xcm$. And we are supposed to the wavelength of the resultant wave. We must know that a standing wave is formed by the interference of two waves. Therefore, we have to find the same thing as we did in this question.