# Engineering Acoustics/Human Voice Production

## Physiology of Vocal Fold

Human vocal fold is a set of lip-like tissues located inside the
larynx, and is the source of
sound for a human and many animals. The Larynx is located at the top of
trachea. It is mainly composed of cartilages and muscles, and the
largest cartilage, thyroid, is well known as the \"Adam\'s Apple.\"

The organ has two main functions; to act as the last protector of the
airway, and to act as a sound source for voice. This page focuses on the
latter function. In the following image, the cross section of vocal cord
is shown. This three dimensional geometry is made using CT scan data.

!*Vocal fold cross
section*{width="400"}

Links on Physiology: Discover The
Larynx

## Voice Production

Although the science behind sound production for a vocal fold is
complex, it can be thought of as similar to a brass player\'s lips, or a
whistle made out of grass. Basically, vocal folds (or lips or a pair of
grass) make a constriction to the airflow, and as the air is forced
through the narrow opening, the vocal folds oscillate. This causes a
periodical change in the air pressure, which is perceived as sound.

Vocal Folds Video

When the airflow is introduced to the vocal folds, it forces open the
two vocal folds which are nearly closed initially. Due to the stiffness
of the folds, they will then try to close the opening again. And now the
airflow will try to force the folds open etc\... This creates an
oscillation of the vocal folds, which in turn, as I stated above,
creates sound. However, this is a damped oscillation, meaning it will
eventually achieve an equilibrium position and stop oscillating. So how
are we able to \"sustain\" sound?

As it will be shown later, the answer seems to be in the changing shape
of vocal folds. In the opening and the closing stages of the
oscillation, the vocal folds have different shapes. This affects the
pressure in the opening, and creates the extra pressure needed to push
the vocal folds open and sustain oscillation. This part is explained in
more detail in the \"Model\" section.

This flow-induced oscillation, as with many fluid mechanics problems, is
not an easy problem to model. Numerous attempts to model the oscillation
of vocal folds have been made, ranging from a single mass-spring-damper
system to finite element models. In this page I would like to use my
single-mass model to explain the basic physics behind the oscillation of
a vocal fold.

Information on vocal fold models: National Center for Voice and
Speech

## Models

### Single mass model

image: Singlemass - vocal folds motion model schematic (single
mass-spring-damper
system).png.png "wikilink")

`                                                       Figure 1: Schematics`

The most simple way of simulating the motion of vocal folds is to use a
single mass-spring-damper system as shown above. The mass represents one
vocal fold, and the second vocal fold is assumed to be symmetry about
the axis of symmetry. Position 3 respresents a location immediately past
the exit (end of the mass), and position 2 represents the glottis (the
region between the two vocal folds).

#### The Pressure Force

The major driving force behind the oscillation of vocal folds is the
pressure in the glottis. The Bernoulli\'s equation from fluid mechanics
states that:

$P_1 + \frac{1}{2}\rho U^2 + \rho gh = Constant$ \-\-\-\--EQN 1

Neglecting potential difference and applying EQN 1 to positions 2 and 3
of Figure 1,

$P_2 + \frac{1}{2}\rho U_2^2 = P_3 + \frac{1}{2}\rho U_3^2$ \-\-\-\--EQN
2

Note that the pressure and the velocity at position 3 cannot change.
This makes the right hand side of EQN 2 constant. Observation of EQN 2
reveals that in order to have oscillating pressure at 2, we must have
oscillation velocity at 2. The flow velocity inside the glottis can be
studied through the theories of the orifice flow.

The constriction of airflow at the vocal folds is much like an orifice
flow with one major difference: with vocal folds, the orifice profile is
continuously changing. The orifice profile for the vocal folds can open
or close, as well as change the shape of the opening. In Figure 1, the
profile is converging, but in another stage of oscillation it takes a
diverging shape.

The orifice flow is described by Blevins as:

$U = C\frac{2(P_1 - P_3)}{rho}$ \-\-\-\--EQN 3

Where the constant C is the orifice coefficient, governed by the shape
and the opening size of the orifice. This number is determined
experimentally, and it changes throughout the different stages of
oscillation.

Solving equations 2 and 3, the pressure force throughout the glottal
region can be determined.

#### The Collision Force

As the video of the vocal folds shows, vocal folds can completely close
during oscillation. When this happens, the Bernoulli equation fails.
Instead, the collision force becomes the dominating force. For this
analysis, Hertz collision model was applied.

$F_H = k_H \delta^{3/2} (1 + b_H \delta')$ \-\-\-\--EQN 4

where

$k_H = \frac{4}{3} \frac{E}{1 - \mu_H^2} \sqrt{r}$

Here $\delta$ is the penetration distance of the vocal fold past the
line of symmetry.

#### Simulation of the Model

The pressure and the collision forces were inserted into the equation of
motion, and the result was simulated.

image: matlab.png

`                                              Figure 2: Area Opening and Volumetric Flow Rate`

Figure 2 shows that an oscillating volumetric flow rate was achieved by
passing a constant airflow through the vocal folds. When simulating the
oscillation, it was found that the collision force limits the amplitude
of oscillation rather than drive the oscillation. Which tells us that
the pressure force is what allows the sustained oscillation to occur.

#### The Acoustic Output

This model showed that the changing profile of glottal opening causes an
oscillating volumetric flow rate through the vocal folds. This will in
turn cause an oscillating pressure past the vocal folds. This method of
producing sound is unusual, because in most other means of sound
production, air is compressed periodically by a solid such as a speaker
cone.

Past the vocal folds, the produced sound enters the vocal tract.
Basically this is the cavity in the mouth as well as the nasal cavity.
These cavities act as acoustic filters, modifying the character of the
sound. The acoustics of vocal tract have traditionally been described on
the basis of a source-filter theory.
Whereas the glottis produces a sound of many frequencies, the vocal
tract selects a subset of these frequencies for radiation from the
mouth. These are the characters that define the unique voice each person
produces.

### Two Mass Model

The basic two mass model is shown in Figure.3 and the two mass model of
vocal fold is shown in Figure.4.

!CGtypes\|500px\|Figure3. Two mass
model

!CGtypes\|350px\|Figure4. Smoothed two mass
model

### Three Mass Model

In Figure5, three mass model of vocal fold is shown. !Figure5. Three
mass model{width="400"}

### Rotating Plate Model[^1]

!Rotatingplate{width="400"} The
motion of vocal fold can be described with two degrees of freedom. First
rotation of mass **M2** and displacement **r**. The equation of motion
will be:

$$\ddot{I}+ B \dot{\theta} +k \theta= T$$

$$m\ddot{r}+b(\dot{r}-\dot{r_b})+k_2(r-r_b)=F$$

Where in these equations: *T* is the applied aerodynamic torque

$I_c$ is the moment of inertia for rotation cover

$B$ is the rotational damping

$k$ is the rotational stiffness

$k_2$is the translational stiffness

$b$ is the translational damping

$F$ is the force\
$M_2$is the cover mass\
$r_b$ is the displacement of the body

$r$ is the displacement of the cover

The equation of motion for the body mass can be written as:

$$M_1\ddot{r_b}+b(\dot{r_b}-\dot{r})+k_2(r_b-r)+K_1r_b+B\dot{r_b}=0$$
where:

$K_1$ is the body stiffness

$M_1$ is the body mass

$B$ is the body damping

### Finite Element Models

## Lumped-element flow circuit for the vocal tract

In the image below, the lumped element flow circuit for the vocal tract
airway is shown. The input impedance to the vocal tracts can be shown by
resistive and inertive lumped elements.[^2] According to the shown
circuit we have:

!lumped element circuit for vocal
tract{width="600"}

$$P_L-R_s u-I_s\dot{u} -\frac{1}{2} \rho u^2/a_g^2-R_eu-I_e \dot{u}=0$$
Where

$P_L$ is a steady lung pressure

$R_s$ is subglottal resistance

$I_s$ is subglottal inertance(epilaryngeal) input resistance

$R_e$ is supraglottal(epilaryngeal) input resistance\
$I_e$ is supraglottal(epilaryngeal) input inertance

## References

\[1\] Fundamentals of Acoustics; Kinsler *et al.*, John Wiley & Sons,
2000

\[2\] Acoustics: An introduction to its Physical Principles and
Applications ; Pierce, Allan D., Acoustical Society of America, 1989.

\[3\] Blevins, R.D. (1984). Applied Fluid Dynamics Handbook. Van
Nostrand Reinhold Co. 81--82.

\[4\] Horacek, J., Sidlof, P., Svec, J.G. Self-Oscillations of Human
Vocal Folds. Institute of Thermomechanics, Academy of Sciences of the
Czech Republic

\[5\] Lucero, J.C., Koenig, L.L. (2005). Simulations of temporal
patterns of oral airflow in men and women using a two-mass model of the
vocal folds under dynamic control, Journal of the Acoustical Society of
America 117, 1362--1372.

\[6\] Titze, I.R. (1988). The physics of small-amplitude oscillation of
the vocal folds. Journal of the Acoustical Society of America 83,
1536--1552

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[^1]: Ingo R. Titze, \"The myoelastic aerodynami theory of phonation\"
    2006

[^2]:
