Describe the various modes of vibration of closed and open ended pipes
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Answer:
Vibrations of Air Column in Pipes
Stationary Waves in a Closed Pipe
Musical wind instruments like flute, clarinet etc. are based on the principle of vibrations of air columns. Due to the superposition of the incident wave and the reflected wave, longitudinal stationary waves are formed in the pipe.
Organ pipes
Organ pipes are musical instruments which are used to produce musical sound by blowing air into the pipe. Organ pipes are two types (a) closed organ pipes, closed at one end (b) open organ pipe, open at both ends.
(a) 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.
Overtones in a Closed Pipe
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
Stationary Waves in an Open Pipe
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,
Overtones in an Open Pipe
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 ....
Resonance air column apparatus
The resonance air column apparatus consists of a glass tube G about one metre in length (as shown in figure) whose lower end is connected to a reservoir R by a rubber tube.
The glass tube is mounted on a vertical stand with a scale attached to it. The glass tube is partly filled with water. The level of water in the tube can be adjusted by raising or lowering the reservoir.
A vibrating tuning fork of frequency n is held near the open end of the tube. The length of the air column is adjusted by changing the water level. The air column of the tube acts like a closed organ pipe. When this air column resonates with the frequency of the fork the intensity of sound is maximum.
Here longitudinal stationary wave is formed with node at the water surface and an antinode near the open end. If l1 is the length of the resonating air column
Resonance Air Column Apparatus
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