Question

Explain how overtones are produced in a: 

(a) Closed organ pipe 

(b) Open organ pipe

Answer 1

(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. Antinode 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,

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,

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.

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 = (2n + 1)f₁ . Therefore, the frequencies of harmonics are in the ratio

f₁ : f₂ : f₃ : f₄ …… = 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 antinodes are formed at the open end, a node is 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

The frequency of the note emitted is

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 pipes. It has two nodes and three antinodes, and therefore,

is called first over tone. Since n = 2 here, it is called second harmonic.

The Figure (f) above shows the third mode of vibration having three nodes and four anti-nodes.

is called second over tone. Since n = 3 here, it is called the third harmonic. 

Hence, the open organ pipe has all the – harmonics and frequency of nth harmonic is f_n = nf₁. Therefore, the frequencies of harmonics are in the ratio 

f₁ : f₂ : f₃ : f₄ … = 1 : 2 : 3 : 4 : … …(6)