Name: Open and Closed-End Air Column
Uploaded: 2019-07-19
Duration: 4 min 32 s
Description: Many musical instruments consist of an air column enclosed inside of a hollow metal tube. We will learn about the open- and closed-end air column in this video.

Question

Derive expressions for the fundamental and overtones of a closed organ pipe.

Accepted Answer

Clarinet is an example of a closed organ pipe. It is a pipe with one end closed and the other end open. If one end of a pipe is closed, the wave reflected at this closed end is 180° out of phase with the incoming wave. Thus, there is no displacement of the particles at the closed end. Therefore, nodes are formed at the closed end and anti-nodes are formed at open end.

Let us consider the simplest mode of vibration of the air column called the fundamental mode. Anti node is formed at the open end and node at closed end. From the figure, let L be the length of the tube and the wavelength of the wave produced.

For the fundamental mode of vibration, we have,

$L = \frac{λ_{1}}{4} o r λ_{1} = 4 L . . . . . . . . . (i)$

The frequency of the note emitted is

$f_{1} = \frac{ν}{λ_{1}} = \frac{ν}{4 L} . . . . . . . . . . . (i i)$

which is called the fundamental note.

The frequencies higher than fundamental frequency can be produced by blowing air strongly at open end. Such frequencies are called overtones.

The figure (b) shows the second mode of vibration having two nodes and two anti nodes.

$4 L = 3 λ_{2} \Rightarrow L = \frac{3 λ_{2}}{4} o r λ_{2} = \frac{4 L}{3}$

The frequency for this,

$f_{2} = \frac{ν}{λ_{2}} = \frac{3 ν}{4 L} = 3 f_{1}$

is called first over tone, since here, the frequency is three times the fundamental frequency it is called third harmonic.

The figure (c) shows third mode of vibration having three nodes and three anti-nodes.

We have, $4 L = 5 λ_{3}$

$\Rightarrow L = \frac{5 λ_{3}}{4} o r λ_{3} = \frac{4 L}{5}$

The frequency for this,

$f_{3} = \frac{ν}{λ_{3}} = \frac{5 ν}{4 L} = 5 f_{1}$

is called second over tone, and since $n = 5$ . This is called fifth harmonic.

Hence, the closed organ pipe has only odd harmonics and frequency of the nth harmonic is $f = (2 n + 1) f_{1}$

Therefore, the frequencies of harmonics are in the ratio $f_{1} : f_{2} : f_{3} : f_{4} \dots \dots = 1 : 3 : 5 : 7 : . . . . . . . . . . . .$

Derive expressions for the fundamental and overtones of a closed organ pipe.

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