Question

Explain how overtone is produced in a Open organ pipe.

Answer 1

The flute is an example of an open organ pipe. It is a pipe with both ends are opened. At both open ends, anti-nodes are formed. Consider the simplest mode of vibration of the air column called fundamental mode. Since anti-nodes are formed at the open end, a node is formed at the mid-point of the pipe.

Antinodes are formed at the open end and a node is formed at the middle of the pipe.

From above Figure, if L be the length of the tube, the wavelength of the wave produced is given by

L = `lambda_1/2` (or) λ₁ = 2L

The frequency of the note emitted is

`"f"_1 = "v"/lambda_1 = "v"/(2"L")`

That is called the fundamental note.

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

Second mode of vibration in open pipes having two nodes and three anti-nodes

The second mode of vibration in open pipes is shown in the figure. It has two nodes and three anti-nodes,

∴ L = λ₂ (or) λ₂ = L

The frequency,

`"f"_2 = "v"/lambda_2 = "v"/"L" = 2 xx "v"/(2"L") = 2"f"_1`

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

Third mode of vibration having three nodes and four anti-nodes

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

∴ L = `3/2 lambda_3` (or) `lambda_3 = (2"L")/3`

The frequency,

`"f"_3 = "v"/lambda_3 = (3"v")/("2L") = 3_1`

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 the frequency of the n^th harmonic is f_n = nf₁.

Hence, the frequencies of harmonics are in the ratio f₁ : f₂ : f₃ : f₄ = 1 : 2 : 3 : 4 : …