Name: Closed Organ Pipe
Uploaded: 2019-07-19
Duration: 3 min 44 s
Description: An organ pipe is closed at one end and open at the other end. Bottle, whistle, etc., are examples of a closed organ pipe. Let's learn about its different mod...

Question

What is meant by harmonics. Giving necessary diagram, show that only odd harmonics are produced in a closed end organ pipe.

Accepted Answer

If the frequencies of the fundamental modes and overtone produced by a sound source are in harmonic progression, then these tones are called harmonic. Also, fundamental tone is called first harmonic, other overtone which are produced are harmonics. If the overtone frequency is a multiple of fundamental frequency then it will be that harmonic. e.g. if fundamental frequency $= 200 Hz$ then $120 Hz$ is $6 t h$ harmonic.

Consider a closed organ pipe of length $l$ , along $x$ -axis. Closed end of pipe at $x = 0$ , and open end at $x = l$ .

When air is flown into the pipe then longitudinal waves are reflected back from rigid support. Equations for incident wave and reflected waves are as follows:

Incident wave $y_{1} = a \sin \frac{2 π}{λ} (v t + x)$

Reflected wave $y_{2} = a \sin \frac{2 π}{λ} [(vt - x) + π]$

On superposition:

$y = y_{1} + y_{2}$ $= 2 a \cos (\frac{2 πvt}{λ}) \sin (\frac{2 πx}{λ}) . . . . . . . . . . . . . . . . . . (i)$

At the closed end $x = 0,$ so $y = 0$

$A$ node is formed always at closed end.

At the open end $x = l$

$y = 2 a \cos (\frac{2 πvt}{λ}) \sin (\frac{2 π l}{λ})$

The displacement $y$ will be always be maximum, if $\sin (\frac{2 πl}{λ}) = \pm 1$

so $\Rightarrow \frac{2 πl}{λ} = (2 m - 1) \frac{π}{2} \Rightarrow λ = \frac{4 l}{2 m - 1} . . . . . . . . . . . . . . . . . . . . . . . . (i i)$

Nodes are formed at closed ends and antiniodes are formed at open ends.

This mode is shown in fig. $(a)$ . It is also called fundamental mode of first harmonic.

Frequency of fudamental mode $= \frac{v}{4 l} = n_{1}$

Second mode of vibration- It is shown in fig. $(b)$

From eq. $2$ $λ_{2} = \frac{4 l}{3} \Rightarrow l = \frac{3 λ_{2}}{4}$

So, $n_{2} λ_{2} = v \Rightarrow n_{2} = \frac{v}{λ_{2}} o r n_{2} = \frac{3 v}{4 l} = 3 n_{1}$

$n_{2}$ frequency of second overtone is three times the frequency of fundamental. It is known as second overtone or third harmonic.

Third mode of vibration- It is shown in fig. $(c)$ . There are three nodes and three antinodes.

From equation, $i i$ ,

$λ_{3} = \frac{4 l}{5}$ for $m = 3$

In this third overtone is formed.

$l = \frac{5 λ_{3}}{4}$

Frequencies of third overtone $= n_{3}$

So, $n_{3} = \frac{v}{λ_{3}} = \frac{5 v}{4 l} = 5 n_{1}$

Hence fifth harmonic is formed.A pipe is closed at one end forms only odd harmonics.

What is meant by harmonics. Giving necessary diagram, show that only odd harmonics are produced in a closed end organ pipe.

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