
An air
column in the tubes or pipes closed at one end or open at both ends can
be set into vibrations with tuning forks of suitable frequencies.
The air column in the tubes have a natural
frequency of vibration. The vibrating tuning fork held at its mouth, sets
the air column into forced vibrations.
If the frequency of the tuning fork equals the natural frequency of the
tube, resonance occurs and a loud sound is emitted.
Pipe Closed at One End
In these pipes, there is a node at the closed end and an antinode at
the open end.
 For
the simplest mode of vibration, there is a single node and antinode.
\ Length of air column
equals the distance between the node and its successive antinode.
l =

l _{1}


4


fundamental frequency n_{1} =

v


l_{1}


=

v


4l



For the second
mode of vibration, l =

3l_{2}


4




\ l_{2} =

4l


3




\ frequency of first overtone n_{2} =

v


l_{2}


=

3v


4l


=3n_{1}


Similarly,
the frequencies of the subsequent modes are 5n_{1}, 7n_{1},
… etc.
 These
frequencies consist of the fundamental frequency and its odd
multiples.
 Only
odd harmonics are present in the vibrations of an air column in a
pipe closed at one end.
Pipe
Open at Both Ends
In
these pipes, there is an antinode formed at both the ends.
 At
points of antinodes, there is no change in pressure.
 The
air column in these tubes vibrate in different ways or modes,
subject to the condition that there must be an antinode at each open
end.
 In
the simplest node of vibration,
n_{1}=

v


l


=

v


2l


= fundamental frequency or first harmonic

 For
the subsequent nodes,
n_{2}=

v


l_{1}


= second harmonic (first overtone)



= 2n_{1}

n_{3}=

v


l_{3}


=

3v


2l


= 3n_{1} = third harmonic or second
overtone.


 The
air column in a pipe, open at both ends, can vibrate with
frequencies n_{1}, 2n_{1}, 3n_{1},
… etc., i.e. both the odd and even multiples of the
fundamental frequency are present i.e. odd and even harmonics are
present.
 End Correction: Since the air particles are not free to vibrate
with maximum amplitude at the open end, the antinode is formed at a
distance 'e' slightly above the
open end. The distance 'e' is termed as the end correction.
e = 0.3 d
where d: internal diameter of
the pipe.
\ Corrected length l_{1} = l + e = l + 0.3 d
Relation between the fundamental
frequencies in an open tube and closed tube:
Fundamental frequencies
For open tube f_{0} =

V


4l


Closed tube f_{c} =

V


2l


\ f_{c} = 2 f_{0}

