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Grade: 12th Pass

                        

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 ?

one year ago

Answers : (3)

Arun
25384 Points
							
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
 
 
one year ago
Rajdeep
231 Points
							HELLO THE
						
one year ago
Rajdeep
231 Points
							
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}

 
one year ago
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