Wave Motion Theory Part4

Physics Helpline
L K Satapathy
Wave Motion Theory-4
Vibration of Air Columns
Open & Closed Pipes
Fundamental frequency & Overtones
End Correction
Resonance Tube

Physics Helpline
L K Satapathy
Vibration of air columns in closed pipes :
Consider an air column trapped in a pipe of length L , closed at one end. The open
end is always an antinode and the closed end is always a node.
Fundamental mode : In this mode , there is no node between the
node and antinode at the ends.
Let the wavelength frequency1 1f
P
v



L
From the fig. we have 1
1 4
4
L L

  
 Fundamental frequency is 1
1
1
4 4
v v P
f
L L

 
  
Speed of sound 1 loop =  / 2

Physics Helpline
L K Satapathy
First overtone : In this mode , there is one node between the
2
3 4
4 3
L
L

  
 Frequency of 1st overtone is 2 1
2
3 1
3 3
4 4
v v P
f f
L L

 
 
     
 
 1st overtone = 3rd Harmonic
P
v


Speed of sound

Physics Helpline
L K Satapathy
Second overtone : In this mode , there are two nodes between the
3
5 4
4 5
L
L

  
 Frequency of 2nd overtone is 3 1
3
5 1
5 5
4 4
v v P
f f
L L

 
 
     
 
 2nd overtone = 5th Harmonic
P
v



 In general (n)th overtone = (2n+1)th Harmonic
Frequencies obtained in a closed pipe are : [ Odd harmonics ]1 1 1, 3 , 5 . . .f f f
Speed of sound

Physics Helpline
L K Satapathy
Vibration of air columns in open pipes :
Consider an air column trapped in a pipe of length L , open at both ends. Hence
antinodes are formed at the two open ends.
Fundamental mode : In this mode , there is one node between
the two antinodes at the ends.
P
v



1 2
2
L L

  
 Fundamental frequency is 1
1
1
2 2
v v P
f
L L

 
  
L
Speed of sound

Physics Helpline
L K Satapathy
First overtone : In this mode , there are two nodes between the
two antinodes at the ends.
P
v



From the fig. we have 2L 
 Frequency of 1st overtone is 2 1
2
1
2 2
2
v v P
f f
L L

 
 
     
 
 1st overtone = 2nd Harmonic
Speed of sound

Physics Helpline
L K Satapathy
Second overtone : In this mode , there are three nodes between
the two antinodes at the ends.
P
v



From the fig. we have 3 3
3 2
2 3
L
L    
 Frequency of 2nd overtone is 3 1
3
3 1
3 3
2 2
v v P
f f
L L

 
 
     
 
 2nd overtone = 3rd Harmonic
 In general (n)th overtone = (n+1)th Harmonic
Frequencies obtained in an open pipe are : [ All harmonics ]
Speed of sound
1 1 1, 2 , 3 . . .f f f

Physics Helpline
L K Satapathy
End correction : It is observed that the antinode formed at an open end is not
formed exactly at the open end but at a small distance (e) outside the open end.
The end correction (e) depends on the diameter (d) of the pipe as e = 0.3d
This correction is to be applied at each open end of the pipe
Effective length = L + e
For a closed pipe of length L For an open pipe of length L
Effective length = L + 2e
L
e
L
e
e

Physics Helpline
L K Satapathy
Speed of sound in air [ Resonance tube ]
The length of air column in a pipe can be adjusted by
controlling the water level in it. The air column is made to
vibrate in resonance with an excited tuning fork kept over
the mouth of the pipe. Let us keep an excited tuning fork
of frequency f at the mouth of the tube and gradually
decrease the level of water in the tube from top.
1L
2L
1st and 2nd resonances occur at lengths : 1 2L and L
Let wavelength =  and speed of sound = v
1 2
2 1 2 1
3
&
4 4
2( )
2
L e L e
L L L L
 


    
     
2 12 ( )v f f L L   

Physics Helpline
L K Satapathy
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