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

Explanation

Closed organ pipes : If one end of a pipe is closed, the wave reflected at this closed end is `180^(@)` out of phase with the incoming wave. So, there is no displacement of the particles at the closed end. Hence nodes are formed at the closed end anti-closed are formed at open end.

Consider the simplest mode of vibration of the air column callede the fudamental mode. Anit-node is formed at the open end and node at closed end. From the above 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)`
Wave length `lambda_(1)=4L`
The frequency of the note emitted is
`f_(1)=(v)/(lambda_(1))=(v)/(4L)`
which is called the fundamentla note.
The frequencies higher than fundamental frequency can be produced by blowing air strongly at open end. Such frequencies are called overtones.
The second mode of vibration having two nodes and two antinodes is showing in the figure.

`4L=3 lambda_(2)`
`L=(3lambda_(2))/(4) (or) lambda_(2)=(4L)/(3)`
The frequency for this
`f_(2)=(v)/(lambda_(2))=(3y)/(4L)=3f_(1)`
is called first over tone, since here , the frequency is three times the fundamental frequency it is called thir harmonic.
The figure shows third mode of vibration having three nodes and three anti-nodes.

We have, `4L=5 lambda_(3)`
`L=(5lambda_(3))/(4)=5f_(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)`. Hence the frequency of harmonics are in the ratio,
`f_(1):f_(2):f_(3):f_(4):.....=1:3:5:7:...`
Flute is an example of open organ pipe. It is a pipe with both the ends are opened. At both open ends, anit-nodes are formed. Consider the simplest mode of vibration of the air column called fundamental mode. Since anit-nodes are formed at the open end, a node is formed at the mid-point of the pipe.

From above Figure. if L be length of the tube, the wavelenght of the ave produced is given by
`L=(lambda_(1))/(2) (or) (lambda_(1)=2L`
The frequency of the note emitted is
`f_(1)=(v)/(lambda_(1))=(v)/(2L)`
That 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 second mode of vibration in open pipes is shown in figure. It has two nodes and three anti-nodes,
`:.L lambda_(2)(or) lambda_(2)=L`
The freuency,
`f_(2)=(v)/(lambda_(2))=(v)/(L)=2xx(v)/(2L)=2f_(1)`
is called first over tone. Since n=2 here, it is called the second harmonic.

The figure above shows the third mode of vibration having three nodes four anti-nodes
`L:(3)/(2) lambda_(3) (or) lambda_(3)=(2L)/(3)`
The frequency,.
`f_(3)=(v)/(lambda_(3))=(3v)/(2L)=3f_(1)`
is called second over tone. since n=3, here, it is called the third harmonic.
Hence, the open organpipe has all the harmonics and frequency of `n^(th)` harmonic is `f_(n)=nf_(1)`. Hence the frequency of harmonics are in the ratio
`f_(1):f_(2):f_(3):f_(4):... =1:2:3:4:...`