,
Waves in Pipes
 

[closed pipes][open pipes][comparison][end correction]

 

 

Pipes produce standing waves similar to stretched strings. However it must be emphasized that in stretched strings the waves are transverse, while in pipes the waves are longitudinal.

Transverse - particles of the wave vibrate at right angles to the direction of travel of the wave.

Longitudinal - particles of the wave vibrate in the same line as the direction of travel.

Closed pipes

In the diagrams, P is the site of a node, while Q is at an antinode.

Nodes are always formed at the closed end of a pipe, where the air cannot move.
Antinodes are always formed at the open end of pipes.

As with stretched strings, the distance between node and antinode is 1/4 of a wavelength.

closed pipe fundamental frequency

The diagram above represents the Fundamental Frequency, where n=1. This is the 1st harmonic.

closed pipe - 1st overtone

The diagram above represents the 3rd harmonic, sometimes called the First Overtone.

closed pipe 2nd overtone

The diagram above represents the 5th harmonic, sometimes called the Second Overtone.

Looking at the different wavelengths in terms of the length of the pipe L ,

closed pipes - equation #1

we can then make wavelength the subject of each equation.

closed pipes equation #2

Using the wave equation and making the frequency f the subject:

wave velocity in terms of wavelength and frequency

Substituting the different values of wavelength to obtain different expressions for frequency:

closed pipes equation #3

Looking at the form of these equations it is observed that each is a multiple of fx (the Fundamental Frequency).

closed pipes - equation #4

closed pipes - equation #5

where n is 1, 3, 5, ... (odd)

 

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Open pipes

open pipes - fundamental frequency

The diagram above represents the Fundamental Frequency, where n=1. This is the 1st harmonic.

open pipes - 1st overtone

The diagram above represents the 2nd harmonic, sometimes called the First Overtone.

open pipes - the 2nd overtone

The diagram above represents the 3rd harmonic, sometimes called the Second Overtone.

Looking at the different wavelengths in terms of the length of the pipe L ,

open pipes - equation #1

We can then make wavelength the subject of each equation.

open pipes - equation #2

Using the wave equation and making the frequency f the subject:

wave velocity in terms of wavelength and frequency

We can now substitute the different values of wavelength to obtain different expressions for frequency:

open pipes - equation #3

Looking at the form of these equations it is observed that each is a multiple of fx (the Fundamental Frequency).

open pipes - equation #4

open pipes - equation #5

where n is 1, 2, 3, 4, 5, ... (odd + even)

 

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A comparison of 'closed' and 'open' pipes

1. Comparing expressions for the Fundamental Frequency (n=1) for closed and open pipes respectively,

closed and open pipe frequencies compared

For a pipe of the same length L, the open pipe frequency is twice that of the closed pipe frequency.

open pipe frequency is twice closed pipe frequency

2. For a given length of pipe, an open pipe gives more harmonics (odd & even) than a closed pipe (odd only). This results in a richer note from the open pipe.

 

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End correction

The 'end correction' (c) is a length that must be added on to the the length (Lo) of a pipe to take account of antinodes extending beyond the open end of the pipe.

End correction for a closed pipe

end correction for a closed pipe

The effective length (LE) is given by:

pipes - end correction equations

where r is the radius of the pipe

 

End correction for an open pipe

end correction for an open pipe

The effective length (LE) is given by:

 

open pipe end correction

where r is the radius of the pipe

 

 

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