Q.

How are the stationary waves formed in a closed pipe ? Explain the various modes of vibrations in a closed pipe and establish the relation between their frequencies.

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answer is 1.

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Detailed Solution

An organ pipe is a cylindrical tube having an air column. If one end is closed and the other is open, it is known as a closed organ pipe.
At closed end always a node and at open end an antinode will generate.
Harmonics in Closed Pipes :
i) The first harmonic in a closed pipe should have a node at the closed end and an antinode at the open end, then the length of the pipe

l=λ14;λ1=4l
The fundamental frequency n1=vλ1
n1=v4l .......(1)
ii) The next possible harmonic with two nodes and two antinodes it is the third harmonic, then the length of the pipe

l=3λ34;λ3=4l3
The frequency of the third harmonic (or) first overtone is given by n3=vλ3
n3=3v4l   [from equation (1)]
n3=3n1 ....(2)
iii) Similarly the next overtone in the closed pipe is only the fifth harmonic. It will have three antinodes and three nodes. Then the length of the pipe

λ5=4l5
Frequency of fifth harmonic (or) second overtone
n5=vλ5;n5=v45;n5=5v4n5=5n1..(3)
from equation(1),(2)(3) the ratio of fundamental and next harmonics are 1 : 3 : 5 : ....

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