# Geometry of the Vocal Tract and Properties of Phonation near Threshold: Calculations and Measurements

^{1}

^{2}

^{3}

^{*}

## Abstract

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## Featured Application

**The model enables a detailed investigation of phonation with regard to the conditions at phonation onset. Thereby, the goal is to identify advantageous ranges of material parameters that support an efficient vocal fold oscillation as the phonation source.**

## Abstract

_{th}) by its inertance. The latter depends on the geometrical boundary conditions prescribed by the vocal tract length (directly) and its cross-sectional area (inversely). Using Titze’s surface wave model (SWM) to account for the properties of the vocal folds, the influence of the vocal tract inertia is examined by two sets of calculations in combination with experiments that apply silicone-based vocal folds. In the first set, a vocal tract is constructed whose cross-sectional area is adjustable from 2.7 cm

^{2}to 11.7 cm

^{2}. In the second set, the length of the vocal tract is varied from 4.0 cm to 59.0 cm. For both sets, the pressure and frequency data are collected and compared with calculations based on the SWM. In most cases, the measurements support the calculations; hence, the model is suited to describe and predict basic mechanisms of phonation and the inertial effects caused by a vocal tract.

## 1. Introduction

_{VT}and length L

_{VT}, one can use Newton’s second law to derive the following formula for the pressure required to accelerate the air within the vocal tract:

_{g}/dt is the time derivative of the glottal flow rate U

_{g}, and I

_{VT}is the inertance of the air in the vocal tract. The inertance of the air is given by I

_{VT}= ρ L

_{VT}/ A

_{VT}, where ρ is the density of air.

_{th}. Titze also gave qualitative arguments that the pressures in the vocal tract should reinforce the effect of the pressures acting on the medial surfaces of the vocal folds, and thus enhance the action of these pressures. A consequence of this is that one would expect the inertial effects of the vocal tract to lower the pressure required for the phonation threshold. In order to examine the predictions of the SWM, Titze et al. [3,4,5] did a number of experiments. Of the most relevance to this work was the 2006 paper where Chan and Titze [5] examined the dependence of the threshold pressure on the prephonatory glottal half-width with and without a vocal tract. This comparison allows rigorous scrutiny of the effects of Equation (1), since it introduces no new free parameters. Fulcher and Scherer [6] showed that a calculation based on the SWM and using Equation (1) to account for the effects of the vocal tract gave a reasonable explanation of the threshold data that Chan and Titze collected. It is shown below in Section 2.1 that this fit can be improved by choosing a larger value for the surface wave velocity.

## 2. Models and Methods

#### 2.1. Surface Wave Model and the Vocal Tract

_{sub}is responsible for accelerating the air in the vocal tract as well as moving the air through the glottal duct created by the medial surfaces bounding the glottis, which is shown schematically in Figure 1. Thus, the subglottal pressure is related to the flow rate, its time derivative, and the glottal areas at the entrance A

_{1}(t) and at the exit A

_{2}(t) by the equation:

_{ent}is the entrance loss coefficient and k

_{ex}is the exit coefficient [23,24]. The dynamics of the vocal fold coordinate ξ in the SWM is described by an equation of motion [2,6] whose driving term includes a factor P

_{g}, which is the average of the pressure P(z) over the medial surface of the vocal fold, that is:

_{g}there denotes the glottal length (anterior–posterior direction), and T denotes the glottal thickness (inferior–superior direction). The dots in Equation (3) denote time derivatives.

_{Surf}(z,t) by the equation, A(z,t) = 2 L

_{g}[ξ

_{0}(z) + ξ

_{Surf}(z,t)], where ξ

_{0}(z) is the prephonatory shape of the glottal surface. Since the surface coordinate satisfies a wave equation, the combination of variables z and t takes a simpler form, that is, ξ

_{Surf}(z,t) = ξ

_{Surf}(t − z/c), where c is the velocity of the surface wave. Fundamental to the SWM is the expansion of ξ

_{Surf}(z,t) as a power series in z. This requires the assumption that the phase difference between the glottal entrance and the glottal exit is not too large.

_{g}

^{2}in Equation (4) by using Equation (2). Then Equation (4) takes the form:

_{ex}in Equation (6), specializing to the rectangular case where ξ

_{01}= ξ

_{02}= ξ

_{0}, and carrying out an expansion in inverse powers of ξ

_{0}there allows Equation (6) to be simplified to the following form:

_{0})

^{−1}are kept. Thus, the driving force on the right-hand side of Equation (3) simplifies to a term that has the effect of a negative damping term. Equating the coefficient of this term to that of the damping term on the left-hand side of Equation (3) yields:

_{t}= k

_{ent}, when k

_{ex}is neglected. Equation (8) requires careful interpretation. If the entrance loss coefficient k

_{ent}is assumed to be near one (k

_{ent}= 1.37 was a typical value for much of the earlier work), then Equation (8) predicts that the threshold pressure becomes very small in the limit of small glottal half-widths. However, none of the data collected in the experiments that Titze et al. did to test the predictions of the SWM showed this behavior [3,5]. Fulcher and Scherer [6] showed that the key to resolving this discrepancy was a careful analysis of the behavior of the entrance loss coefficient at small glottal widths, since this coefficient could become very large in this limit, where viscous effects become large. In fact, they showed that the behavior of the entrance loss coefficient in the limit of small glottal widths was consistent with an inverse relationship, that is:

_{g}= 2.22 cm and T = 1.1 cm, are determined from the dimensions of the vocal fold model used in the experiments.

_{VT}= 16.51 cm and A

_{VT}= 2.82 cm

^{2}. The results of the numerical solution are shown in Figure 2, where they are compared with Chan and Titze’s vocal tract measurements. The fits to the data collected with the vocal tract as well as those collected without the vocal tract are very good. The largest differences (about 13%) occur at a glottal width of 0.10 cm, and most of the differences are substantially smaller.

^{2}and m = 0.15 g. In order to make the calculation as simple as possible, the nonlinearity parameter was set equal to zero. It is worth noting that the use of Equations (2) and (4) for the vocal tract calculation eliminates the necessity for the inverse ξ

_{0}expansion, and thus removes the small amplitude requirement. However, the requirement of a reasonably small phase difference between the inferior and the superior edges of the medial surface remains.

#### 2.2. Larynx Model and Measuring Setup

^{2}to 11.7 cm

^{2}in the course of the experiments. The length of the vocal tract is 19.0 cm.

^{2}, which is the same as the smallest area used in the experiments with the variable cross-sectional dimensions. Each of the channel segments was fabricated by an Additive Manufacturing process controlled by an in-house, self-designed 3D printer. Due to its reliability and availability, fused layer modeling, which was invented by Scott Crump [34], was chosen to implement the additive manufacturing technology.

_{s}= 44.1 kHz.

#### 2.3. Onset Detection: Methodology

^{Δt}

_{sub}(t

_{m}) (equivalent to introducing a low-pass filter) defined in a time interval about t

_{m}as follows [17,36]:

_{s}= 10,000 counts the number of points in the sample. The time interval Δt is set to 0.23 s, so that the sum of Equation (10) includes pressure values from 32 oscillations for Model 113 and 25 oscillations for Model 114. The pressure P

^{Δt}

_{sub}(t

_{m}) is represented by the thick yellow curve of Figure 7. When the pressure is near 3.5 kPa, the character of the pressure oscillations changes because of a noticeable change of amplitude. This qualitative difference may be used to define the onset pressure. The change in the amplitude of the oscillations near the threshold is apparent, although the size of this amplitude is less than 100 Pa, which is about 3% of the size of the total pressure signal near the threshold. Pressure oscillations of this size are consistent with one’s expectation for small amplitude oscillations near the threshold. At the onset, the corresponding oscillation amplitude of the vocal folds is much smaller—less than 1 mm, as indicated in Lodermeyer et al. [36]. Furthermore, the typical convergent-to-divergent glottal duct shape evolution during an oscillation cycle is visible but less pronounced at the onset pressure.

^{Δ}

^{t}

_{sub}is between 10–15 Pa. Thus, the criterion used to define onset pressure is that σ

^{Δ}

^{t}

_{sub}> 50 Pa. This criterion gives P

_{sub}near 3.5 kPa for the measurements of Figure 7. Each flow rate increase and decrease cycle was repeated five times, so that the procedure used to determine the subglottal pressure lasted about 120 s. The values for P

_{sub}required to initiate each of the five oscillations were averaged. Thus, the measured value of the P

_{sub}was recorded as 3.52 kPa for Model 113 when the lateral channel wall displacement was D = 1.8 cm. Five more measurements of P

_{sub}were made with D = 2.8 cm, 3.8 cm, 4.8 cm, 5.8 cm, and 7.8 cm, respectively.

## 3. Results and Discussion

^{2}to 11.70 cm

^{2}are shown in Figure 8a. The value 1.0 is chosen for the entrance loss coefficient k

_{ent}, since the measurements of k

_{ent}for larger pressures [23] tend to approach 1.0. The geometrical parameters are determined from the dimensions of the experiments; that is, L

_{VT}= 19.0 cm, L

_{g}= 1.5 cm, and T = 0.35 cm. The parameters B and c are chosen so that the calculated result is the same at the measured pressure (3520 Pa) at A

_{VT}= 2.70 cm

^{2}, and the parameters k and m are chosen so that the calculated frequency fits the measured frequency (139 Hz) at A

_{VT}= 2.70 cm

^{2}. The prephonatory half-width is ξ

_{0}= 0.20 cm for each of the calculations shown in Figure 8. This value was not measured during the experiments, but it was considered to be reasonable from observations of the separation of the vocal folds at the pressure when sustained oscillations become possible [37]. The fit to the first four pressure measurements is very reasonable, but the measured pressures are substantially smaller than the calculations for the two larger areas.

_{VT}= 8.7 cm

^{2}is adequate to reduce the calculated pressure from 4460 Pa to 4040 Pa, which is enough to remove the discrepancy in Figure 8a. This value for the exit coefficient is similar to that calculated from the measurements at larger glottal widths in [23] (Figure 3). A similar reduction occurs at A

_{VT}= 11.7 cm

^{2}. Another possible source of the discrepancy is the nature of the supraglottal flow pattern, as shown in Figure 9. It was observed to change its character when the cross-section of the vocal tract was large in comparison with the smaller cross-sections [40]. The jet was found to be deflected to either lateral side for A

_{VT}= 2.7 cm

^{2}, due to the appearance of a large vortex that inhibited the jet from proceeding straight ahead. This large vortex increases the flow resistance in the vocal tract channel. As a consequence, the pressure increases in the supraglottal region immediately downstream of the vocal folds. Upon enlarging the vocal tract, the volume of the large vortex and thus its inertia rise, resulting in a further increase of the supraglottal pressure downstream to the vocal folds, and with it an increase of the subglottal threshold pressure.

_{VT}= 60.0 cm

^{2}, P

_{th}= 4960 Pa, and at A

_{VT}= 80.0 cm

^{2}, P

_{th}= 4980 Pa. The formula listed in Equation (8) yields P

_{th}= 4963 Pa for this limit, and thus, the calculations for these two larger areas are within 0.4% of the expected value.

^{−2}, which is also present in the calculation of Figure 8a. Adding a nonlinearity parameter is necessary because of the behavior of the amplitude of oscillation as the areas of Figure 8 increase. For example, at A

_{VT}= 2.70 cm

^{2}, the displacement of the vocal fold from equilibrium varies between −0.051 cm to 0.043 cm and at A

_{VT}= 11.70 cm

^{2}, the displacement varies from −0.0143 cm to 0.0135 cm. The amplitude decreases monotonically as the vocal tract area increases, and the effective stiffness of the vocal fold decreases, explaining the frequency decrease of Figure 8b. If the nonlinearity parameter is set to equal to zero, then the trend of the calculations does not match the trend of the data.

_{113}= 4.4 kPa and E

_{114}= 2.5 kPa). On the basis of these considerations, one would expect k = 65,900 g/s

^{2}for Model 114.

_{VT}= 2.70 cm

^{2}. The results based on Equations (12) and (13) are shown in Figure 10a, where they are compared with measurements. The fit of Figure 10a is of a similar quality to that of Figure 8a; that is, the calculated values for threshold pressure are in reasonable agreement with the measurements for the four smaller areas of Figure 10a. However, the same discrepancy appears at the two larger areas, and the cause of this discrepancy is likely to be the same as that for Figure 8a. Choosing the nonlinear parameter η = 190 cm

^{−2}gives a good fit to the frequency data presented in Figure 10b.

_{VT}= 19.0 cm and A

_{VT}= 2.7 cm

^{2}to be the same as that for the smallest area of Figure 8a. However, this is not the case. In order to account for the difference, the prephonatory half-width for Figure 11a is adjusted to 0.172 cm to fit the pressure (3061 Pa) at L

_{VT}= 19.0 cm. This difference in prephonatory half-width threshold pressure may have several causes, including slight differences in how the synthetic vocal folds were manually glued in the mounting device. The SWM calculations presented in Figure 11a produce a very reasonable fit to the trend of the threshold pressure data, but they are not especially accurate for either large or small values of L

_{VT}. Frequency measurements with the vocal tract of Figure 6 are shown in Figure 11b. Again, a small adjustment from the parameters of Figure 8 was required. In this case, the mass of the vocal folds was increased from 0.170 g to 0.184 g in order to decrease the frequency to 129 Hz at L

_{VT}= 19.0 cm and A

_{VT}= 2.70 cm. The calculations in Figure 11b reproduce the trends in the data at intermediate and large glottal lengths to a reasonable degree, but do not have the right trend at smaller lengths, where the difference becomes as large as 4% at the smallest length.

## 4. Limitations

## 5. Conclusions

_{VT}and inversely on the vocal tract area A

_{VT}. Thus, Equation (1) predicts how the vocal tract area and the vocal tract length should affect phonation properties near the threshold, such as the onset pressure and frequency near onset. In this work, Equation (1) is used in conjunction with the SWM developed by Titze [2], since the SWM introduces a smaller number of parameters than the two-mass model [1] or the three-mass model [46]. Qualitative arguments are presented in the Introduction chapter to show that increasing the vocal tract area should raise the onset pressure. These qualitative arguments are examined by the SWM calculations presented in Figure 8a and Figure 10a. Reasonable agreement between the calculations and measurements is presented in Figure 8a and Figure 10a for the four smaller areas, but the discrepancy is considerable for the two largest areas. An examination of the supraglottal flow reveals changes in the locations of the vortical structures downstream of the vocal folds, which may be the reason. However, a better understanding of the fluid dynamics would require a more fundamental calculation based on the Navier–Stokes equation to allow a more thorough examination of the discrepancy and further clarify the role that the inertial properties of the air in the vocal tract play in determining the phonation onset. The agreement between the SWM calculations and the measurements for the frequencies at threshold is much better, as shown in Figure 8b and Figure 10b, provided that the nonlinear stiffness effects are included.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**A schematic diagram showing the surface wave propagating along the medial surface of the vocal tract. The halfwidth at the glottal entrance is ξ

_{01}, and that at the glottal exit is ξ

_{02}.

**Figure 2.**(Color online) Measured and calculated phonation threshold pressures as functions of the glottal half-width with and without a vocal tract.

**Figure 3.**(Color online) (A) Schematic diagram of the synthetic larynx model including the mass flow generator, silencer, subglottal channel, and mounting device for the synthetic vocal folds (distances are in millimeters).

**Figure 4.**(Color online) Schematic diagram and photograph of the synthetic silicone model of one of the vocal folds (distances are in millimeters). R1: radius 1 mm; R2: radius 2 mm.

**Figure 5.**Schematic diagram of the supraglottal channel that allows for the variation of the lateral displacement D (distances are in millimeters).

**Figure 6.**Segmented vocal tract model used to vary the length of the vocal tract (distances are in millimeters).

**Figure 7.**(Color online) Subglottal pressure (black), its moving average (yellow), and its standard deviation (red) during one cycle of the variable flow rate.

**Figure 8.**(Color online) (

**a**) Measured and calculated threshold pressures for vocal tracts with different areas for vocal fold model 113. (

**b**) Measured and calculated frequencies for vocal tracts with different areas for vocal fold model 113.

**Figure 9.**Flow structures in the supraglottal channel for vocal tract areas (

**a**) 2.7 cm

^{2}, (

**b**) 11.7 cm

^{2}, and for (

**c**) no vocal tract. Additional information about the flow structures can be found in [40].

**Figure 10.**(Color online) (

**a**) Measured and calculated threshold pressures for vocal tracts with different areas for vocal fold model 114. (

**b**) Measured and calculated frequencies for vocal tracts with different areas for vocal fold model 114.

**Figure 11.**(Color online) (

**a**) Measured and calculated threshold pressures for model 113 as a function of vocal tract length. (

**b**) Measured and calculated frequencies as a function of vocal tract length.

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**MDPI and ACS Style**

Fulcher, L.; Lodermeyer, A.; Kähler, G.; Becker, S.; Kniesburges, S.
Geometry of the Vocal Tract and Properties of Phonation near Threshold: Calculations and Measurements. *Appl. Sci.* **2019**, *9*, 2755.
https://doi.org/10.3390/app9132755

**AMA Style**

Fulcher L, Lodermeyer A, Kähler G, Becker S, Kniesburges S.
Geometry of the Vocal Tract and Properties of Phonation near Threshold: Calculations and Measurements. *Applied Sciences*. 2019; 9(13):2755.
https://doi.org/10.3390/app9132755

**Chicago/Turabian Style**

Fulcher, Lewis, Alexander Lodermeyer, George Kähler, Stefan Becker, and Stefan Kniesburges.
2019. "Geometry of the Vocal Tract and Properties of Phonation near Threshold: Calculations and Measurements" *Applied Sciences* 9, no. 13: 2755.
https://doi.org/10.3390/app9132755