Question

A closed organ pipe has length $L$. The air in it is vibrating in third overtone with maximum amplitude $a$. The amplitude at distance $\dfrac{L}{7}$ from closed end of the pipe is
A. $0$
B. $a$
C. $\dfrac{a}{2}$
D. data insufficient

Answer 1

Organ pipes are musical instruments which produce music when air is blown through it. There are two types of organ pipes, namely the closed organ pipes and opened organ pipes. Here, we need to find the amplitude at the third overtone in a closed pipe, using the given details , we can find the amplitude.

Formula used: $L=\dfrac{\lambda}{4}$
$K= \dfrac{2\pi}{\lambda}$

Complete step by step answer:
Organ pipes produce music due to the standing waves which are formed inside the length of the tubes. This happens when the vibrations of the air blown in the tube get reflected and interfere again with the main vibration. These interferences produce nodes, where the vibrations cross each other.
Here, we have a closed organ pipe, where one side of the pipe is closed. When the air is blown through the closed organ pipes then the air column vibrates, to give the fundamental node. If the length of the tube is $L$ , then
$L=\dfrac{\lambda}{4}$
Then at $\dfrac{L}{7}$, we have, $\lambda=\dfrac{4L}{7}$
Then the wave number $K$ is given as $K= \dfrac{2\pi}{\lambda}$
$\implies K=\dfrac{2\pi}{\left(\dfrac{4L}{7}\right)}$
$\implies K=\dfrac{7\pi}{2L}$
Look at the wave equation, $X=a\; sinkx$,
Then the wave equation at third overtone is given as $X= a\;sin\left(\dfrac{7\pi}{2L}\times\dfrac{L}{7}\right)$
$\implies X=a \;sin\left(\dfrac{\pi}{L}\right)$
Clearly, the amplitude $a$ remains the same as compared to the wave equation.

So, the correct answer is “Option B”.

Note: Since there are two types of organ pies, each has a different formula for the nodes. Hence one has to be careful when applying the formula of the node. However, the wave number and wave equation remain the same for both the pipes. Here, since the initial amplitude is seen again at the third overtone, we can conclude that the amplitude remains unchanged. However, this is not always the case.