Chapter 6: The Source-Filter Theory of Vowels

Equation 6.1. Acoustic pressure, p, is the pressure created by sound waves; acoustic airflow, u, is the volume of air that moves into the vocal tract per unit of time. The ratio of these, z, is created. This is known as the acoustic impedance, which we can think of as the "effectiveness" of the tube at impeding the airflow.

Equation 6.2. This equation is derived from equation 6.1, in the special case where the sound wave travels in one direction only. In this case, the acoustic impedance is the ratio of air density, r, multiplied by sound velocity, c, divided by the cross-sectional area A.

Equation 6.3. The reflection coefficient, r, is a ratio of the difference between the acoustic impedances to the sum of the acoustic impedances. This equation reflects this ratio, with the impedances shown in terms of equation 6.2.

Equation 6.4. Usually, we can assume that the air density and sound velocity are constant throughout the airway. This reduces the reflection coefficient to the ratio of the difference between the areas to the sum of the areas between adjacent tubes.

Equation 6.5-8. For any given wave, there is a value t₀ which is the time it takes a wave to make a complete cycle. This is known as the transit time.

A standing wave is a waveform that has a finite, discrete number of nodes. Either endpoint of a standing wave is fixed at the zero crossing (halfway between the minimum and maximum amplitudes) or at the minimum or maximum amplitude.

Equation 6.9. Transit time can be expressed in terms of total distance (two tube lengths) divided by the speed of sound.

Equation 6.10. For resonance, we can take equation 6.8 and replace the term for transit time with its equivalent expression defined in equation 6.9.

Equation 6.11. Knowing that the frequency F is the inverse of the period T, the relation of frequency to length can be derived from equation 6.10, resulting in the equation above.

Equation 6.12. A series of frequencies can be calculated by using different odd integers for n. These frequencies, which are resonances of the vocal tract, are known as formants. It is convenient to distinguish the formants by placing a numerical subscript after the symbol F. For example, the second formant would be written as F₂. In order to have a formant n equate to its respective frequency, it is convenient to write equation 6.11 in terms of the formant index. The term n from equation 6.11 is replaced with the expression (2n-1), which results in equation 6.12, where n is now the formant index.

Equation 6.13. Recall earlier than a standing wave can occupy a tube with two closed ends, two open ends, or one of each. The last case has been shown in equation 6.12. However, the formant relation for a tube with two open or two closed ends is a bit simpler, as shown above.

Equation 6.14. To display the practicality of the formant relation, let us apply some typical values to the equation. First, assume that the average male vocal tract length is 17.5 cm. Sound generally travels through air at a velocity of 35,000 cm/s. Equation 6.14 hence shows a simple equation which can determine the frequency for an arbitrary formant. The first formant, F₁, is 500 Hz in this case. F₂ and F₃ follow at 1,500 Hz and 2,500 Hz, respecitvely.

Equation 6.15. As time and frequency are inversely related, it would follow suit that a duration in time is inversely proportional to the frequency bandwidth of the spectrum. Note that as duration of events increase, the representative bandwidth of frequency decreases.

Equation 6.16. This equation demonstrates a logarithmic property -- the logarithm of a product is equivalent to the sum of the logarithms of the two factors.