Explain how overtones are produced in a: (a) Closed organ pipe (b) Open organ pipe

Explanation

(a) Closed organ pipes: 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 = (lambda_(1))/(4)` or `lambda_(1) = 4 L " " ..... (1)`
The frequency of the note emitted is `f_(1) = (v)/(lambda_(1)) = (v)/(4L) " " ... (2)` 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 antinodes ,

`4L = 3lambda_2 `
` L = (3 lambda_2)/(4)` or `lambda_(2) = (4L)/(3)`
The frequency for this , `f_(2) = (v)/(lambda_(2)) = (3v)/(4L) = 3f_(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, `4L = 5 lambda_(3)`
`L = (5 lambda_(3))/(4)` or `lambda_(3) = (4L)/(5)`
The frequency , `f_(3) = (v)/(lambda_(3)) = (5v)/(4L) = 5 f_(1)`
is called second over tone, and since n = 5 here, this is called fifth harmonic. Hence the closed organ pipe has only odd harmonics and frequency of the `n^(th)` harmonic is `f_(n) = (2n +1) f_(1)` . Therefore, the frequencies of ham1onics are in the ratio `f_(1) : f_(2) : f_(3) : f_(4) : .... = 1 : 3 : 5 : 7: ... " " .... (3)`
(b) Open organ pipes: Flute is an example of open organ pipe. It is a pipe with both the ends open. At both open ends, anti-nodes are formed . Let us consider the simplest mode of vibration of the air column called fundamental mode . Since anti-node are formed at the mid point of the pipe .
From figure (d) , if L be the length of the tube , the wavelength of the wave produced is given by
`L = (lambda_(1))/(2)` or `lambda_(1) = 2L " " ... (4)`

The frequency of the note emitted is `f_(1) = (v)/(lambda_(2)) = (v)/(2L) " " ... (5)` which is called the fundamental note . The frequencies higher than fundamental frequency can be produced by blowing air strongly at one of the open ends . Such frequencies are called overtones .

The figure (e) shows the second mode of vibration in open pipe . It has two nodes and three anti-nodes , and therefore ,
`L= lambda_(2) or lambda_(2) = L`
The frequency = `(v)/(lambda_(1)) = (v)/(L) = 2 xx (v)/(2L) = 2f_(1)` is called first over tune . Since n = 2 here , it is called the second harmonic .
The figure (f) above shows the third mode of vibration having three nodes and four anti-nodes
`L = (3)/(2) lambda_(3) or lambda_(3) = (2L)/(3)`
The frequency , `f_(3) = (v)/(lambda_(3)) = (3v)/(2L) = 3 f_(1)` is called second over tone . Since n = 3 here , it is called the third harmonic .
Hence , the open organ pipe has all the harmonic and frequencies of `n^(th)` harmonic is `f_(n) = nf_(1)` Therefore the frequencies of harmonics are in the ratio
`f_(1) : f_(2) : f_(3) : f_(4) : ...... 1 : 2 : 3 : 4` ....