Standing Waves in Air Columns


Standing waves also form in air columns. The compressions and rarefactions of the sound wave are reflected by both open and closed ends of a tube. This can allow resonance to occur in the tube and the length of the tube determines the frequency of the resonating sounds. 

Two situations are considered:

  • An air column with openings at both ends
  • An air column with an opening at one end and the other end closed

Nodes form at positions where there is a closed end and antinodes form at positions where there is an opening.


Many of the same definitions from standing waves in strings apply here:

Standing or stationary waves in air columns vibrate at the resonant frequencies of the air column. The resonant frequencies produced by these vibrations which produce standing waves are known as harmonics. The simplest form of vibration is called the fundamental frequency or first harmonic. The other modes of vibration are known as the second harmonic, third harmonic and so on. Other than the first harmonic, all other harmonics are known as overtones


The resonant frequencies or harmonics in an air column can be determined by the relationship between the length of the air column l, and the wavelength \lambda, of the corresponding standing wave.

For open air columns:

\lambda =\cfrac { 2l }{ n }

where:

\lambda = the wavelength in metres (m)

l = the length of the air column in metres (m)

n = the number of harmonics, which is also the number of antinodes (1, 2, 3, 4 etc)

The relationship between \lambda l and n is illustrated below:

Another equation derives from the relationship: v=f\lambda . This equation allows the frequency of the standing wave to be determined:

f=\cfrac { nv }{ 2l }

where:

f = the frequency of the wave in Hertz (Hz)

v = the velocity of the wave in { m }/{ s }

l = the length of the string in metres (m)

n = the number of harmonics, which is also the number of antinodes (1, 2, 3, 4 etc)

For air columns closed at one end:

\lambda =\cfrac { 4l }{ n }

f=\cfrac { nv }{ 4l }

Note: only odd harmonics exist for closed air columns.


Example 1:

An open-ended pipe 54cm in length has sound travelling in it at 320{ m }/{ s }.

a) What is the wavelength of the fourth harmonic?

b) What is the frequency of the fourth harmonic?

Answers:

a) using: \lambda =\cfrac { 2l }{ n }

\lambda =\cfrac { 2\times 0.54 }{ 4 }

\lambda =0.27m

b) using: f=\cfrac { nv }{ 2l }

f=\cfrac { 4\times 320 }{ 2\times 0.54 }

f=1185.2\;Hz


Example 2:

An closed-ended pipe has sound travelling in it at 320{ m }/{ s }. It produces a fundamental frequency of 150Hz.

a) What is the length of the air column?

b) What is the length of the next possible harmonic above the fundamental frequency produced by this air column?

Answers:

a) using: f=\cfrac { nv }{ 4l }

150=\cfrac { 1\times 320 }{ 4l }

l=\cfrac { 1\times 320 }{ 4\times 150 }

l=53.3\;cm

b) using: \lambda =\cfrac { 4l }{ n } and n=3 because only odd number harmonics can exist for a closed air column:

\lambda =\cfrac { 4l }{ n }

\lambda =\cfrac { 4\times 53.3 }{ 3 }

\lambda =71.1\;cm

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