Stationary Wave in Open Organ Pipe

As we have already discussed, in open organ pipe, both ends of the pipe are opened. When air is blown through the flute (open organ pipe), wave travels through the tube from one end to another end which is then reflected back on striking the free air. When the waves collide, superposition of wave forms the stationary wave. When the wave vibrates with necessary amplitude, we hear the sound of different frequency and harmonics.

The various modes of vibration in open organ pipe are as follows:

Different mode of vibrations in an open pipe

Fundamental mode:

In this mode of vibration, there is one node at the middle of two antinodes as shown in first figure above.   Let λ be the wavelength of the stationary wave. Now,

$$L = \frac{\lambda }{2}$$

$$\text{or,} \lambda = 2L$$

Now, the frequency of fundamental mode or first harmonic is given by:

$$f_1 = \frac{v}{\lambda } = \frac{v}{2L}$$

It is the lowest frequency of the open organ pipe. This frequency is also known as fundamental frequency.

Overtones in open organ pipe

When a strong air is blown in the open organ pipe, notes of higher frequency are obtained which are called overtones.  Some of the overtones are given below:

First overtone (Second mode):

In this mode of vibration, there are two antinodes at the open end, but inside the pipe, there is one antinode and two nodes as shown in second figure above.

Let λ be the wavelength of the stationary wave. Now,

L = λ

So, the frequency of the first overtone or second harmonics is:

$$f_2 = \frac{v}{\lambda } = \frac{v}{L} = 2\frac{v}{2L} = 2f_1$$

$$\therefore f_2 = 2f_1$$

Second overtone (Third mode)

In this mode of vibration, two antinodes are produced at both the open ends and there are three nodes and two antinodes inside the pipe as shown in third figure above.

Let λ be the wavelength of the stationary wave. Now,

$$L = \frac{3 \lambda }{2}$$

$$\text{or,} \lambda = \frac{2 L}{3}$$

Now, the frequency of the second overtone or third harmonics is:

$$\text{or,} f_3 = \frac{v}{\lambda} = \frac{v}{\frac{2L}{3}} = \frac{3v}{2L} = 3\frac{v}{2L} = 3f_1$$

$$\therefore f_3 = 3f_1$$

nth overtone or (n-1)th mode:

In general, the frequency of the nth mode or nth harmonic or (n-1)th overtone is given by:

fn = nfo

where, n = 1,2,3……etc

Conclusions:  From the above expressions, we can say that:

• The frequencies of various modes of vibration are integral multiple of fundamental vibration.
• The frequency of first overtone is two times the fundamental frequency.
• The frequency of second overtone is three times the fundamental frequency.
• All harmonics are present.
• The fundamental frequency of an open organ pipe is twice that of the closed organ pipe of the same length.

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