The physics teacher, while teaching the topic ‘organ pipes’, explained to her students the
reason, for the change in the fundamental frequency (and the frequencies of harmonics),
when one end of an open organ pipe is closed. She went on say that we can expect a
similar ‘change pattern’ when we tend to give up an ‘open approach’ to our learning.
She advised her students to ‘keep open’ all avenues of learning so that their learning can
be built upon all aspects of their fundamental training. She also advised them to work hard in a dedicated way so that their extra effort pushes up their ‘learning Wavelengths’
in much the same way. As an increase in the length of an organ pipe does to 1st
fundamental wavelength. State what in your opinion, are the two values conveyed by
the teacher, to her students through her lecture. Also state the value of the ‘fundamental
mode wavelength’ associated with (i) an open organ pipe of length L, (ii) a closed
organ pipe of length L.
State how the wavelengths of the permitted normal modes, of the above two types of
organ pipes, are related to their fundamental wavelengths
 

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Dear student,

Closed organ pipe

If the air is blown lightly at the open end of the closed organ pipe, then the air column vibrates (as shown in figure) in the fundamental mode. There is a node at the closed end and an antinode at the open end. If l is the length of the tube,

l = λ1/4 or λ1 = 4l               …... (1)

If n1 is the fundamental frequency of the vibrations and v is the velocity of sound in air, then

n1 = v/λ1 = v/4l                  …... (2)

If air is blown strongly at the open end, frequencies higher than fundamental frequency can be produced. They are called overtones. Fig.b & Fig.c shows the mode of vibration with two or more nodes and antinodes.

l = 3λ3/4     or λ3 = 4l/3              …... (3)

Thus, n3 = v/λ3 = 3v/4l = 3n1      …... (4)

This is the first overtone or third harmonic.

Similarly, n5 = 5v/4l = 5n1          …... (5)

This is called as second overtone or fifth harmonic.

Therefore the frequency of pth overtone is (2p + 1) n1 where n1 is the fundamental frequency. In a closed pipe only odd harmonics are produced. The frequencies of harmonics are in the ratio of 1 : 3 : 5.....

(b) Open organ pipe

When air is blown into the open organ pipe, the air column vibrates in the fundamental mode as shown in figure. Antinodes are formed at the ends and a node is formed in the middle of the pipe. If l is the length of the pipe, then

l = λ1/2    Or   λ1 = 2l                 …... (1)

v = n1λ1 = n12l

The fundamental frequency,

n1 = v/2l                      …... (2)

In the next mode of vibration additional nodes and antinodes are formed as shown in Fig.b and Fig.c.

l = λ2 or v = n2λ2 = n2 (l)

So,  n2 = v/l = 2n1         …... (3)

This is the first overtone or second harmonic.

Similarly, 

 n3 = v/λ3 = 3v/2l = 3n1     …... (4)

This is the second overtone or third harmonic

Therefore the frequency of Pth overtone is (P + 1) n1 where n1 is the fundamental frequency.

The frequencies of harmonics are in the ratio of 1 : 2 : 3 ....