Calculating Resonance Frequencies of Tubes

Tubes open at one end and closed at the other end.

This is effectively the case during the production of a vowel where the vocal tract is almost closed at the glottis and open at the mouth. Although the glottis during phonation is periodically opening and closing, the openings are very small, and when compared with the opening at the lips can for practical purposes be considered to be closed.

The lowest frequency resonance, the first formant can be calculated by:-

Higher frequency resonances can be calculated by determining all odd multiples of the first formant (ie. multiply by 1, 3, 5, 7, ...: F_z = F₁ x (2z -1)).

It can be readily seen that F₁ 1/L and so if we know the F₁ and L for the neutral vowel of one speaker and know that a second speaker has a vocal tract half the length of the first speaker then it can easily be seen that the second speaker will have an F₁ twice that of the first speaker.

Q. Assume that the medium through which the sound is propagated is air at 1 atm and 25°C and so the speed of sound is 346 ms^-1. Further, the speaker has an average male vocal tract 17.3 cm long and is producing a neutral vowel /3:/ for which the vocal tract is approximately of equal cross-sectional area along its entire length. What are the frequencies of the first 3 resonances for this sound produced in this vocal tract under these conditions?

Step 1: Convert length to the standard MKS unit of length:- 17.3 cm ÷ 100 = 0.173 m

Step 2: Determine F₁ F₁ = c/4L = 346 ms^-1 / (4 x 0.173 m) = 500 s^-1 (Hz)

Step 3: Determine F₂ and F₃ F₂ = F₁ x ((2x2) -1) = 500 x 3 = 1500 Hz F₃ = F₁ x ((2x3) -1) = 500 x 5 = 2500 Hz

Appendix: Calculating Resonance Frequencies

Tubes open at both ends and tubes closed at both ends

From the point of view of formulae for the calculation of tube resonances, there are two types of tubes. Tubes that are open at one end and closed at the other closely approximate the vocal tract during neutral vowel production and the formula that describes the calculation of the resonances of such tubes has been examined in the main body of the text, above. The other type of tube is one either open at both ends or closed at both ends (from the point of view of resonance calculations it make no difference whether both ends are open or both ends are closed). Tubes closed at both ends are approximated in the vocal tract when, for example, a voiced fricative is being produced. The vocal tract posterior to the constriction (ie. between the constriction and the glottis) is effectly closed at both the constriction point and the glottis. The nasal tract during production of heavily nasalised vowels, on the other hand, may be modelled by a tube open at both ends because the opening at the nostrils and at the velum are of similar cross-sectional areas when heavily nasalised vowels are being produced (nb. in this example the nasal tract tube is coupled to the oral tract tube, which is a tube open at one end and closed at the other).

For tubes open at both ends or closed at both ends:-

The lowest frequency resonance, the first formant can be calculated by:-

F₁ the lowest frequency resonance c the speed of sound L the length of the tube

Higher resonances consist of all multiples of this value. This formula assumes that the tube has a uniform cross-sectional area along its whole length.

Practice Question

A tube is 32 cm in length and has uniform cross sectional area along its full length. Assuming the speed of sound (c) is 330 ms^-2, what are the first five resonance frequencies of the tube if the tube is open at one end and closed at the other.

Answer:-

258, 773, 1289, 1805, 2320 Hz