Chapter 4: Vocal Fold Oscillation
Equation 4.1. The frequency of oscillation, F0,
is the number of back-and-forth movements made per second. Since
1/2p0 is a constant, and since the square roots
of large numbers are larger than the square roots of smaller
numbers, we can see that increasing the stiffness k will
increase frequency, while increasing the mass m will decrease
Equation 4.2. Here we have a mathematical definition
of a periodic (i.e., repeating) waveform. If the waveform lasts
for a pre-determined time T, then the waveform must be
the same at the time t + T as it was at the time t.
The quantity T is known as the period; it is the
duration of one cycle of the waveform.
Simple Harmonic Motion: Equations 4-3 through 4-12
The next ten equations, which all lead up to 4-12, cover simple
harmonic motion, also known as sinusoidal motion.
The first four equations (4.3 through 4.6) are simple definitions
of the trigonometric functions sine and cosine:
Equation 4.7. This equation makes clear the various
ways in which circular motion can be expressed. Once around a
circle is equal to all the quantities shown.
Equation 4.8. Frequency is the inverse of period.
High-frequency oscillations have short periods, and vice versa.
Equation 4.9. Oscillation can also be measure in terms
of angular velocity, measured in radians per second. This
is just a simple conversion; multiply F0 by 2p to
get the angular velocity in radians.
Equation 4.10. Here's the same expression, but now we're
solving for the angle covered during a given period of time by
the oscillation, in radians. We simply multiply the radian frequency
by the elapsed time.
Equation 4.12. This is the general formula for simple
harmonic, or sinusoidal, motion. As we can now see, there are
three key quantities involved; the amplitude A, the phase q0,
and the frequency F0.
Equation 4.13. Here we are solving
for the mean intraglottal pressure, which is the pressure
inside the glottis over the medial surfaces of the vocal folds.
You may refer to the Tutorial
for Chapter 4 to see the quantities in this equation illustrated
in an animated model.
Equation 4.14. The inertance of an air column
is defined as the air density times the length of the column,
divided by its cross-sectional area.
Vocal tract input pressure = (inertance) x (acceleration
of the air column)
Equation 4.15. This is Equation 4-14 written out in
terms of Newton's second law of motion, which states that force
= mass times acceleration. Force is analagous to the vocal tract
input pressure, mass is analagous to inertance, and acceleration
remains the same. If we restate this again with mathematics,
Equation 4.16. I is once again the inertance, dU/dt is
the rate of change of the flow of air in the air column, and Pi,
which we are solving for here, is the input pressure to the vocal
tract. From this, we can see that the input pressure will be
positive if the glottis is opening and the rate of change of
the flow (acceleration) is positive.
Equation 4.17: Bernoulli's energy law. Refer
to Equation 3-6.
3 Equations Equations