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Why the second overtone (third harmonic) of closed organ pipe is given by 5v/4L and also why the general formula for the same is (2n-1)v/4L ?

Why the second overtone (third harmonic) of closed organ pipe is given by 5v/4L and also why the general formula for the same is (2n-1)v/4L ?

Grade:11

3 Answers

Arun
25763 Points
2 years ago
Closed organ pipe:If the air is blown lightly at the open end of the closed organ pipe, then the air column vibrates 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, thenn1 = 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
 
 
Rajdeep
231 Points
2 years ago
HELLO THE
Rajdeep
231 Points
2 years ago
HELLO THERE!
 
Though your question cannot be answered exactly to the point, here I can give you some details about a closed organ pipe:
 
  • The air at the closed end of the pipe must be a node (not moving), since the air is not free to move there and must be able to be reflected back.
  • There must also be an antinode where the opening is, since that is where there is maximum movement of the air.
  • Notice how even though it has been flipped left-to-right and it looks squished and stretched a bit to fit, this is still ¼ of a wavelength.
  • Since this is the smallest stable piece of a wave I can fit in this pipe, this is the Fundamental, or First Harmonic.

Thus, we can get our first formula from here, 

 

L = \frac{1}{4}\lambda

 

And,

 

f = \frac{v}{4L}

which is the first harmonic.

 

Closed organ pipe always produces odd harmonics, so the general formula is:

f = (2n-1)\frac{v}{4L}

 

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