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WAVE MOTION

 

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

 

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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