Question 10: What are organ pipes? Show that an open organ pipe is richer in harmonics than a closed organ pipe.

An organ pipe is an instrument producing sound by means of vibrating air column.

There are two types of organ pipes, namely;

1. Closed Organ pipes
2. Open Organ pipes

Closed Organ Pipes

If one end of the pipe is closed and the other is open through which air is whistled into the pipe, it is called a closed organ pipe.
Consider the figure. Air column into the pipe is set into vibration by sending a narrow jet of air toward the lip of the pipe as shown. When the fast jet strikes the lip, compresses the air and as a result a compression is sent into the pipe. The compression proceeds forward and strikes the closed end. Here it is reflected back toward the open end where it pushes the air out and as a result a rarefaction is sent into the pipe toward the closed end. Now a rarefaction is reflected from here to the open end where it draws air into the pipe and hence a new compression is produced which goes toward the closed end like the previous case. In this way a vibration is set in the pipe and stationary wave is set up.
Since the closed end is fixed which stops further vibration of the air molecules, therefore, a ‘node’ is formed at the closed end. Similarly, air molecules easily move out in the open space at the open end, therefore, ‘anti-node’ is formed at the open end.

First Harmonic Since in a standing wave, the distance between an anti-node and the adjacent node is λ/4, therefore, if ‘L’ is the length of the pipe,then,

Here λ1 is the wavelength of vibration.

If ‘v’ is the speed of the wave into the pipe the frequency f1 of the vibration is

This frequency f1 is the fundamental frequency of first harmonic.

Second Harmonic

Suppose the air column is vibrating for the second harmonic with wavelength λ2 and frequency f2.We know that two full loops make a wavelength.

There are one and a half loops in the whole length of the pipe ‘L’. Therefore,

This means

And the frequency is

Put the value from equation (A)

Thus to get the second harmonic we have to increase the frequency 3 times.

Third Harmonic

Suppose the air column is vibrating for third harmonic. There would be two and a half loops in the pipe length, L(one full wavelength from two loops and one quarter wavelength from one loop). Thus;

Here λ3 is the wave length of oscillation. If f3 is the frequency then ;

Thus the frequency has to be increased 5 folds if the air column has to vibrate in third harmonic.

Generalization

We have the data from the above discussions:
For 1st, 2nd and 3rd harmonics, the frequencies are

Hence, to generalize for the nth harmonic, the frequency fn is given by,

Where n = 1, 2, 3, …

Open Organ Pipes

The pipe whose opposite end to the blowing end is open is called Open Organ Pipe.

Since both ends are open, there are anti nodes at the ends of the pipe and nodes in the middle.

Modes of vibration in open pipe

If the air column in an open organ pipe oscillates such that its frequency is in agreement with the first harmonic the stationary wave produced is of the shape as shown in the figure.
There are two half loops with a node at the middle. Now one half loop is one-fourth of wavelength. Let the length of the pipe is ‘L’. Then

The frequency f1 is calculated as;

Second Harmonic

If the air column in the open organ pipe oscillates for the second harmonic then there is one complete loop and two half in the pipe. Therefore,

Frequency f2 is given by,

(By using value of λ2from equation (3).)

Third harmonic

If the vibration set in the open organ pipe is of third harmonic then there are 2 complete and 2 half loops in the pipe of length L.

The frequency f3 is then;

Generalization

For first, second and third harmonics, the respective frequencies are;

Therefore, for nth harmonic, the frequency

Open organ pipes are richer in harmonics than closed organ pipes

Comparing equations (P) and (Q), we see that more harmonics can be produced with arrangement (Q). As equation (P)is derived for closed pipes and equation (Q) for open pipes, therefore, it can be said that open pipes are richer in harmonics than closed pipes.