Mechanical properties of tissues - human voice and human voice production

First, let's talk about what voice actually is. As it is from the moment it arises in our body until the moment we perceive it again, it is a sound propagating in a gaseous, thus ``liquid'' environment (When we are, for example, under water in a swimming pool. However, our description would not change, by the way, if we were to generalize to liquid environments and then only consider the case of zero viscosity' for the gas.). Which is so, because it is created by the ``bumping'' of the oscillating membranes of vocals into the molecules of the elastic environment (which exert a force on each other, therefore we can speak of an elastic environment where a wave can propagate), which they reflect and are condensed in some places and diluted in others (compared to the equilibrium distribution). This creates a pressure change which, thanks to the flexible environment, can spread through space over time. We classically speak of such a phenomenon as sound. Such a fleeting consideration therefore gives us some insight into how the description of this phenomenon, such as the propagation of a sound wave, should look like - the function we are looking for should probably be somehow periodic and 'dependent on time and position in space. And now, with the acquired motivation, let's explore it in a little more detail.

Propagation of sound
If we want to somehow mathematize this process, we have to decide what variables we need. For now, we will consider a very simple case where we focus on a single dimension, and we will also assume that we are far from the source, or that at least the

$$\frac{\sqrt{S}}{r}$$

S is the ``receiver surface'' (for example, an eardrum),

r is the distance from the source (a very small number).

Then the wave front differs a little from the plane and we can approximate it like this. We will certainly be interested in how the air was displaced, so the displacement $$\chi $$ of the air in the wave will certainly be important. So we express the motion of the wave with the function $$ \chi (x,t)$$, which will tell us how the displacement changes with time and position. Additionally, we would need to know how the density changes with displacement. Even the pressure changes, so that will be another variable as well. The average pressure of the environment $$p_{0} [Pa] $$ changes only very slowly depending on the atmospheric conditions. Let's call the difference between it and the actual gas pressure in our case the acoustic pressure $$ p [Pa] $$. Changes in acoustic pressure, on the other hand, take place very quickly (we can probably expect it to be of the same order as the movement of molecules in the environment) so that no significant heat exchange between individual gas particles can occur in this short time. Therefore, the mentioned changes can be considered adiabatic and we can use the equation of state for the adiabatic process, namely:


 * $$ p_{ref}V^{\kappa}_{0} = (p_{ref}+p)(V_{0}+\Delta V)^{\kappa}$$

Where $$ p_{ref} $$ is some reference pressure at the beginning of the process, $$ V_{0} $$ is the corresponding element volume and $$ \kappa $$ is the Poisson constant. We modify this equation by "cross multiplication" to:


 * $$ 1+ \frac{p}{p_{ref}}=(1+\frac{\Delta V}{V_{0}})^{-\kappa} $$

Since $$ \frac{\Delta V}{V_{0}} $$ is very small, we can approximate it to the equation:


 * $$ \frac{p}{p_{ref}}=-\kappa \frac{\Delta V}{V_{0}} $$

And to express the pressure: $$ p=-A \frac{\Delta V}{V_{0}} $$

Where $$ A=\kappa p_{ref} $$ is simply the characteristic constant. For us, the module of flexibility of the environment can be important.



For our one-dimensional problem we can write: $$ \Delta V= \frac{\part \chi}{\part x} \Delta x $$ And so with respect to $$ V_{0}=\Delta x $$, we substitute the ratio $$ \frac{\Delta V}{V_{0}}=\frac{\part \chi}{\ part x} $$ into our pressure equation and we get: $$ p=-A \frac{\part \chi}{\part x} $$

The pressure changes from $$ p $$ to $$ p+\frac{\part p}{\part x}\Delta x$$, the difference $$ \frac{ \part p}{\part x}\Delta x$$ of the pressures will then cause the wave to move. Expressing ``mass as $$ \rho \Delta x$$ and ``acceleration as $$ \frac{\part^{2} \chi}{\part t^2} $$, using Newton's law we have: \rho \Delta x \frac{\part^{2} \chi}{\part t^2}=-\frac{\part p}{\part x}\Delta x

We divide this equation by $$ \rho \Delta x $$ and partially differentiate with respect to x, so we get: $$ \frac{\part^{2} \chi \part \chi}{\part x \part t^2}=-\frac{1}{\rho}\frac{\part^2 p}{\part x^2}$$ When we partially 'derivate our good old equation for pressure twice with respect to time, we get the form: $$\frac{\part^{2} p}{\part t^2}=-A\frac{\part \chi \part^2 \chi}{\part x \part t^2 }$$ Well, comparing the two previous equations, we have: $$ \frac{\part^{2} p}{\part t^2}=c^2 \frac{\part^2 p}{\part x^2}$$ Where $$ c=\sqrt\frac{A}{\rho}$$ is the 'velocity of wave propagation in the given medium - air. By substituting for pressure, we can adjust it to speak of deflection: $$ \frac{\part^{2} \chi}{\part t^2}=c^2 \frac{\part^2 \chi}{\part x^2}$$

And so we get the desired 'wave equation', which every function describing the wave propagation in space must satisfy. Such a function that satisfies it is then called a wave function'. So far, we have limited ourselves to only one dimension, but we can generalize this equation to all three. For simplicity, we introduce the Laplace operator as $$ \Delta=\frac{\part^2}{\part x^2}+\frac{\part^2}{\part y^2} +\frac{\part^2}{\part z^2} $$.

Then, since the situation is equivalent in all dimensions, we can simply write the wave equation as:

 $$\frac{1}{c^2}\frac{\part^2 \chi}{\part t^2}=\Delta \chi $$ 

.

The speed of sound
The derivation of the ``wave equation led us to a relationship that, at normal pressure, ``relates the ``wave speed and the ``rate of change of pressure with density. $$ c^2 = \frac{d p}{d t}$$

In this calculation it is important to know how the temperature changes'. Newton, who was the first to perform it, assumed that the changes are so fast that they take place ``isothermally''. But such a calculation is wrong, i.e. very imprecise. The correct result was given only by Laplace, who assumed that pressure and temperature change ``adiabatically''. The heat flow from the condensed to the diluted part is negligible if the wavelength is large compared to the mean free path. Under these conditions, the tiny flow does not affect the speed of sound, even though a small energy flow is actually there and causes the sound energy to be absorbed. Such absorption will increase as the wavelength approaches the mean free path. But such lengths are a million times smaller than the length of audible sound. The actual change of pressure with density is one in which no heat flows, this corresponds to adiabatic, in which $$ pV^{\kappa}=const. $$. If density varies inversely with volume, then the relationship between pressure and it is:

$$ p=konst. \rho ^{\kappa} $$.

Chilli $$ \frac{dp}{d \rho}=\kappa \frac{p}{\rho} $$.

So for the speed of sound we have:

$$c=\sqrt\frac{\kappa p}{\rho}$$

Using that

$$pV=NkT$$ and $$c=\sqrt\frac{\kappa pV}{\rho V}=\sqrt\frac{\kappa pV}{Nm}$$

we can write:

$$c=\sqrt\frac{\kappa k T}{m}$$ ,

$$ m $$ is the mass of a molecule'

$$ k $$ is the Boltzmann constant.

So we got a relationship from which it is clear that the speed of sound depends only on temperature' and not on density or pressure. We also know that

$$ kT=\frac{1}{3}m$$

$$$$ is the root mean square velocity, with which we can express the speed of sound as:

$$c=\sqrt\frac{\kappa}{3}$$

And so we find that the speed of sound is of the same order as the average speed of air molecules, which is in good agreement with common sense and our ordinary reality.

Emergence of Voice
We have two elastic ligaments that are stretched like a membrane from front to back between the thyroid cartilage and the vocal cords. These vocal cords are covered by a mucous membrane that forms the vocal folds - vocal cords. Between the two vocal cords is a passage (vocal slit).

'During breathing, the muscle ligaments are lax' and the glottis is open', air can flow freely during breathing.

When speaking or singing, the tiny muscles of the larynx tense' the vocal cords, thereby 'narrowing'' the glottis. Exhaled air flows through the glottis and vibrates the vocal cords in the same way as the reeds in a whistle vibrate. The ripples further resonate in the airways. This is how a voice is created.



The formation of sounds composed into syllables and words is given by the interplay of the vocal organ, upper respiratory tract, tongue, oral cavity, teeth and lips. From the acoustic point of view, speech is a sequence of sounds of different "composition" and different ""intensities"" that arise in the human vocal organ.



The sound of the voice is produced by the vibrating vocal cords rhythmically interrupting the flow of air coming from the trachea. The laryngeal voice has no human timbre. It acquires this only after passing through the space above the glottis - the laryngeal cavity above the vocal cords, pharynx, oral cavity and nasal cavity and the nasopharynx, collectively called the extension vocal tube. Resonance, which depends on the size and shape of the relevant space, the weight of the air contained in it, and the dimensions and arrangement of the inlet and outlet openings, plays a significant role in the timbre of the voice. The resonator amplifies some frequencies and suppresses others. With the help of articulation, we change the shape, size and mutual ratio of the spaces in which resonance occurs. For the interpretation of acoustic phenomena during the production of vowels, two theories were created, which agree well with the experiments performed:



1. Helmholtz' - the sound emanating from the larynx is a compound tone that contains harmonic notes. Through the resonators, only the harmonic tone of the same (or similar) frequency as the natural frequency of the resonator is amplified, other tones are suppressed. By changing the speakers, the shape and volume of the resonators change, and this results in the amplification of other tones, which manifests itself as a different color of the vowel.



2. Herrmann' - considers the laryngeal voice to be a series of short impulses that are emitted when the vocal fold opens into the extension tube. The pressure shock vibrates the air contained in the resonators, and each resonator responds with a short tone, the so-called formant (shapes and creates a sound), when the vocal slit is briefly opened. The formant frequency is given by the natural frequency of the resonator.

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 * Speech communication disorders and swallowing disorders/PGS