# Squeezed waveguides as a framework to study vowel-like acoustic resonances

Al Oualid Eliraki, Florian Vixège, Xavier Pelorson, Annemie van Hirtum

► To cite this version:

Al Oualid Eliraki, Florian Vixège, Xavier Pelorson, Annemie van Hirtum. Squeezed waveguides as a framework to study vowel-like acoustic resonances. Journal of the Acoustical Society of America, 2026, 159 (1), pp.272 - 284. 10.1121/10.0042185. hal-05458610

HAL Id: hal-05458610

https://hal.science/hal-05458610v1

Submitted on 16 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Squeezed waveguides as a framework to study vowel-like acoustic resonances

A. Eliraki,¹ F. Vixege,¹ X. Pelorson,¹ and A. Van Hirtum¹

Univ. Grenoble Alpes, CNRS, Grenoble INP, LEGI, Grenoble,
France

(Dated: 27 November 2025)

The relationship between vocal tract geometry and acoustic resonances is often examined using speaker-specific data, limiting physical studies to a narrow set of static, rigid configurations and frequently producing discontinuous area functions that are poorly suited for studying dynamic vocal-tract behavior. To address these constraints, this study introduces a theoretical Squeezed Waveguide Vocal Tract (SWVT) framework for analyzing the relationship between acoustic resonances (up to 4-5 kHz) and smooth area functions that characterize vowel-like vocal tract configurations with one or two constrictions. The relevance of seven SWVT parameters – constriction positions, degrees, extents, and waveguide length – is established through theoretical analysis and experimental validation. Experiments are performed using both rigid, static waveguides and a deformable, molded waveguide. The molded waveguide facilitates experimental investigation of constriction degrees up to full closure and provides a basis for future studies on the (aero-)acoustic-geometry relationship in dynamic SWVT configurations.
## I. INTRODUCTION

The study of acoustic resonances in the vocal tract (VT) during vowel production, referred to as formants, has been a central focus in spoken language research¹⁻⁷. Driven by continuing advances in medical imaging and machine learning, a growing body of literature documents benchmark datasets of VT area functions and corresponding formants in order to describe how articulatory movements dynamically shape VT geometry and thereby influence vowel acoustics⁸⁻¹². However, VT configurations are known to vary widely due to inter- and intra-speaker factors (e.g., age, gender, posture, body size¹,¹³⁻¹⁶) and contextual influences (e.g., prosody, duration, environment⁵,¹⁶⁻¹⁹). To generalize beyond limited speaker-specific data³⁹, efforts have been made to formalize the relationship between VT geometry and vowel production through the development of parameterized VT models, grounded in biomechanical, articulatory and data reduction principles¹⁵,²⁰⁻²⁹. Nevertheless the necessary geometric detail of the VT remains debated³⁶. As a result, physical studies in vowel sound production mostly rely on a narrow set of static VT geometries, which are frequently characterized by discontinuous area functions⁶,¹²,³⁴⁻³⁶. This reliance not only restricts the generalizability of such studies, but also hinders their extension to dynamic VT configurations and aero-acoustic conditions.

To address these limitations, we introduce a novel parameterized framework for exploring vowel-like acoustics using deformable squeezed waveguides, i.e., cylindrical tubes compressed by up to two sets of parallel bars (pinchers). This framework builds on preliminary findings⁴² showing that continuous area functions can be obtained analytically as a function of a lim-
36 ited set of physically meaningful parameters. Such a formulation would enable systematic
37 parametric investigations, thereby in the long run facilitating physical aeroacoustic studies
38 of VT dynamics, provided that acoustic vowel features can be accurately reproduced.

39 Therefore, in this work, the resonance space of squeezed waveguide VT-like (SWVT) ge-
40 ometries is compared to formant spaces¹,²,⁴,¹²,¹⁴,⁴⁰ from American English and French vowels
41 (Fig. 1 of supplementary materials (SM)). Resonances are computed using a simplified one-
42 dimensional acoustic model, which has proven effective for predicting vowel formants up to
43 approximately 4-5 kHz⁶,²⁰,²¹,³³–³⁵,³⁷,³⁸. This frequency range covers the first few formants,
44 which define the formant spaces critical for vowel perception and, consequently, for vowel
45 classification³³,³⁴.

46 In the following, the SWVT framework is outlined in section II. The theoretical resonance
47 space of the seven-parameter SWVT framework is compared to vowel formant data in sec-
48 tion III. Selected SWVT area functions are considered in section IV. Theoretical results of
49 the SWVT framework are experimentally validated in section VI for the vowels /i/, /a/
50 and /u/, using both rigid and deformable waveguides, in accordance with the experimental
51 approach detailed in section V. The influence of constriction features is further explored in
52 section VII. Conclusions are formulated in Section VIII.

53
## II. INTRODUCING THE SWVT FRAMEWORK

The squeezed waveguide seven-parameter VT-like framework, or SWVT framework, is proposed to investigate the relationship between SWVT geometries (Section II A) and its acoustic resonances (Section II B).

### A. SWVT framework: the waveguide

The squeezed VT-like waveguide, the SWVT geometry, is illustrated schematically in Fig. 1. A circular, uniform waveguide with variable length $L_0$ and wall thickness $d_0 = 3$ mm is aligned along the x-axis, with its cross-section lying in the $yz$-plane. The internal diameter is $D_0 = 25$ mm, yielding a cross-sectional area $A_0 = 490 \text{ mm}^2$, consistent with values reported for unconstricted or open VT regions during adult vowel production$^{6,12,34}$. The cross-section is initially circular, with both minor ($b_0$ along the z-axis) and major ($a_0$ along the y-axis) axes equal to $D_0/2$. Deformation is introduced via one or two sets of opposing pincher bars (indexed by $i \in 1, 2$). Each pincher consists of two symmetrical, rounded bars (radius $r_p = 3.2$ mm) featuring a central flat segment of length $l_c \ge 0$ mm. The total pincher length along the x-axis is $2r_p + l_c \ge 6.4$ mm, and the bars are aligned parallel to the y-axis. Pinching induces a local constriction along the x-axis, reducing the cross-sectional area $\hat{A}(x) \le A_0$ over a length of up to $4 \times D_0$, symmetrically centered at $x_c^{41,42}$.

72

Following Van Hirtum et al.$^{41}$, the effect of a single pincher with $l_c = 0$ mm for a waveguide of length $L_0$ is described by two parameters: the constriction position $x_c$ and
<figure><img src="image_2.png" /><figcaption>FIG. 1. The squeezed waveguide VT-like (SWVT) framework. A cylindrical tube of length $L_0$ and diameter $D_0$ is squeezed between one or two sets (indexed by $i \in 1, 2$) of rounded parallel pincher bars ($r_p = 3.2$ mm), imposing constrictions defined by the parameter set ($l_c, x_c, \mathcal{P})_i$ (in green), where $l_{c,i}$ is the pincher extent, $x_{c,i}$ the constriction position, and $\mathcal{P}_i$ the constriction degree. The deformation results in stadium-shaped cross-sections with minor ($\hat{b} \le b_0$) and major ($\hat{a} \ge a_0$) axes, indicated for both undeformed ($\mathcal{P} = 0\%$) and deformed ($0 < \mathcal{P} \le 100\%$) cases.</figcaption></figure>

the constriction degree $0 \le \mathcal{P} < 100\%$, where $\mathcal{P} = 1 - \beta$ and $\beta = b(x_c)/b_0$. The de-
formed cross-sections are approximated as rounded stadium rings, with a reduced minor
axis $\hat{b} < b_0$ and increased major axis $\hat{a} > a_0$. The waveguide geometry, including the area
function $\hat{A}(x)$ and axes $\hat{a}(x)$ and $\hat{b}(x)$, can be expressed analytically as functions of the two-
parameter set $(x_c, \mathcal{P})$. Eliraki et al.⁴² extended and validated this model using mechanical
principles, demonstrating that it can be applied to configurations with two pinchers (each
with $l_{c,1} = l_{c,2} = 0$ mm). In this case, modeled quantities such as $\hat{A}(x)$ are defined by a
four-parameter set $(x_{c,1}, \mathcal{P}_1, x_{c,2}, \mathcal{P}_2)$, with pincher indices $i \in 1, 2$, as shown in Fig. 1. Two
 pinchers can produce up to two constrictions, enabling for two- to five-tube configurations
depending on constriction positions $(x_{c,1}, x_{c,2})$ and degrees $(\mathcal{P}_1, \mathcal{P}_2)$. Unlike traditional con-
catenated tube representations$^{6,20,21,33,34}$, the resulting area functions more closely resemble those of human speakers and exhibit smoothness, which facilitates subsequent studies with airflow, e.g. those involving fluid-structure interaction at the glottal source.

88

Considering the constriction length as a parameter is motivated, as during vowel production$^{12}$ (Fig. 1 of SM), the constricted region may extend from a few millimeters (e.g., /u/) up to several dozen millimeters (e.g., /o/). This variability led to the explicit inclusion of constriction length as a parameter in traditional tube models$^{6,20,21,33,34}$. For each pincher, the minimum area is confined to a waveguide segment of fixed length, determined by its width $2r_p + l_c \ge 6.4$ mm, or more precisely, by the flat segment length $l_c$. When $l_c = 0$ mm, the minimum area is restricted to a fixed short length of $2r_p = 6.4$ mm. The constricted segment length can be increased either explicitly, by setting $l_c > 0$ mm to create a uniform constriction of length $l_c$ with a single pincher, or indirectly, by positioning two pinchers close enough that the region between $x_{c,1}$ and $x_{c,2}$ remains constricted, albeit non-uniformly. It is assumed that the analytic geometric waveguide model, validated for one and two pinchers with $l_c = 0$ mm$^{41,42}$, also holds for pinchers with $l_c > 0$ mm, when the minimum area is extended over the length $l_c$. Thus, the modeled SWVT waveguide geometry – and derived functions such as $\hat{A}(x)$, $\hat{b}(x)$ and $\hat{a}(x)$ – is defined by a six-parameter set ($l_{c,1}$, $x_{c,1}$, $P_1$, $l_{c,2}$, $x_{c,2}$, $P_2$), where $i \in 1, 2$ indexes the pinchers (see Fig. 1). Including the waveguide length $L_0$ yields a general seven-parameter framework: ($L_0$, $l_{c,1}$, $x_{c,1}$, $P_1$, $l_{c,2}$, $x_{c,2}$, $P_2$), which represents up to two area reduction portions in a waveguide with a fixed maximal area $A_0 = 490$ mm$^2$, as
b₀ = D₀/2 is held constant.

107

108

B. SWVT framework: theoretical acoustic resonances

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

The acoustic pressure within the waveguide is computed using a theoretical one dimensional acoustic model based on spatial transmission line theory³⁷,³⁸,⁴¹. The waveguide geometry is represented by its one-dimensional area function $\hat{A}(x)$, discretized with a spatial step of 0.3 mm ($\Delta x = 0.3$ mm). This step size was selected to ensure the theoretical model's output is independent of the discretization and to assure an approximately smooth geometry variation (see Section II A). A flanged infinite baffle exit condition is implemented in order to approximate the presence of the face at the VT outlet ($x = 0$). The one-dimensional theory neglects higher order acoustic mode propagation within the waveguide. At room temperature (21 °C), the cut-on frequency of these modes for a uniform ($P = 0\%$) waveguide portion is approximately 8.0 kHz and decreases with increasing $P$ due to the increase of the major axis $a(x)$ along the constricted section (see Fig. 1)⁴¹,⁴³,⁴⁴. For waveguides pertinent to the VT (length $L_0 \approx 180$ cm, area $A_0 \approx 490$ mm²), the first three acoustic resonance frequencies typically occur below 4-5 kHz⁴¹,⁴². Therefore, the framework's resonance space defined by the first two theoretical resonances $\hat{F}_i$ with $i \in \{1, 2\}$ can be investigated. Resonance frequencies $\hat{F}_i$ are estimated as peak frequencies in the theoretical transfer function $H(f)$ (in dB) between the amplitude of the acoustic volume flow rate $U$ at the inlet ($U_{-L_0}$ at
125 $x = -L_0$) and outlet ($U_0$ at $x = 0$) as:

$$ \forall f, \quad H(f) = 20 \log_{10} \frac{|U_0(f)|}{|U_{-L_0}(f)|} \qquad (1) $$

126 Frequencies from 90 Hz to 6 kHz are considered in 10 Hz increments to cover the frequency range of interest.

127

128

For a waveguide with fixed length $L_0$ and a single pincher of length $l_c$, theoretical resonances can be expressed as functions of two SWVT parameters, $\hat{F}_i(x_c, \mathcal{P})$, assuming a single constriction (i.e., $\mathcal{P}_2 = 0\%$ so that pincher index $i = 1$ can be omitted for clarity). As an example to illustrate the influence of constriction parameters, theoretical resonance frequencies $\hat{F}_{i \in \{1,2\}}(x_c, \mathcal{P})$ for $L_0 = 180$ mm (reference VT length) and $l_c = 0$ mm are visualised in Fig. 2. Note that theoretical resonances cannot be visualized in this manner when more than two SWVT parameters are varied. In the following, constriction position(s) $x_{c,i}$ are varied from the inlet ($x = -L_0$) to the outlet ($x = 0$) in 5 mm steps, while constriction degree(s) $\mathcal{P}_i$ range from 0% (fully open) to 99% (near closure), using 5% increments for $0\% \le \mathcal{P} \le 80\%$ and 1% increments for $80\% < \mathcal{P} \le 99\%$. The following sections compare the SWVT framework with literature data, focusing on vowel formants (Section III) or area functions $A(x)$ (Section IV).

142
<figure><img src="image_3.png" /><figcaption>FIG. 2. Theoretical resonance frequencies $\hat{F}_{i \in \{1,2\}}$ for a single pincher as function of position $x_c$ ($-L_0$ to 0) and constriction degree $\mathcal{P}$ (0-99%) for $L_0 = 180$ mm and $l_c = 0$ mm.</figcaption></figure>

### III. SWVT VS LITERATURE: RESONANCE SPACES

The vowel space defined by the first two formants ($F_1, F_2$) of American English (gray ellipsoids) and French (red ellipsoids) vowels, spanning $200 \le F_1 \le 1000$ Hz and $500 \le F_2 \le 2500$ Hz (Fig. 1 of SM), is compared to the theoretical resonance space ($\hat{F}_1, \hat{F}_2$) generated from various SWVT parameter subsets symbols.

#### A. One constriction: SWVT subset ($L_0, l_c, x_c, \mathcal{P}$)

The theoretical resonance space ($\hat{F}_1, \hat{F}_2$) for the SWVT subset ($L_0, l_c, x_c, \mathcal{P}$), considering one constriction (omitting pincher index $i = 1$ for clarity), is compared to the vowel formant space ($F_1, F_2$) in Fig. 3. The effects of constriction parameters ($x_c, \mathcal{P}$) are analyzed using reference values of $L_0 = 180$ mm and $l_c = 0$ mm (see Fig. 2), as well as for varying $L_0$ with $l_c = 0$ mm (Fig. 3(a)) and varying $l_c$ with $L_0 = 180$ mm (Fig. 3(b)).
<figure><img src="image_4.png" /><figcaption>FIG. 3. Theoretical SWVT resonances ($\hat{F}_1$, $\hat{F}_2$) vs formants ($F_1$, $F_2$) for $-L_0 \le x_c \le 0$ mm and $0\% \le P \le 99\%$: a) SWVT subset ($L_0, l_c = 0$ mm, $x_c, P$) for $L_0 \in \{131, 171, 180\}$ mm. Relationships characterizing ($\hat{F}_1, \hat{F}_2$) statistics for $L_0 \in [100, 340]$ mm, $\hat{F}_{2\text{stat}}(\hat{F}_{1\text{stat}})$ with $\text{stat} \in \{\text{min, max, mean}\}$ are indicated (black lines). Reference curve for a uniform waveguide (dotted yellow line); b) SWVT subset ($L_0 = 180$ mm, $l_c, x_c, P$) for $l_c \in \{0, 15, 30, 60\}$ mm.</figcaption></figure>

1. Reference: ($L_0 = 180$ mm, $l_c = 0$ mm, $x_c, P$)

Theoretical SWVT resonances $\hat{F}_1(x_c, P)$ and $\hat{F}_2(x_c, P)$ are plotted in Fig. 2. Theoretical $\hat{F}_1$ varies between 250 Hz and 500 Hz (overall increase of 100%) and $\hat{F}_2$ between 800 Hz and 1500 Hz (overall increase of 77%). Consequently, vowels in Fig. 3 with formants outside these ranges can not be reproduced with the reference SWVT subset. Smallest and greatest resonance frequencies $\hat{F}_{i \in 1,2}$ in Fig. 2 are all observed in the parameter region associated with large constriction degrees $P$.

In this high $P$-range, $\hat{F}_1$ (Fig. 2(a)) decreases with constriction position $x_c$ which is a tendency not accounted for in the thumb rule³⁴ (i.e., the empirical rule associated with the vowel trapezoid, stating that $F_1$ primarily increases with greater constriction degree, while $F_2$ increases with a forward shift of the constriction from the pharyngeal region toward the
168 lips), where $x_c$ is solely associated with $F_2$. Variation of $\hat{F}_1$ occurs for rather large $\mathcal{P} > 90\%$.

169 Consequently, extreme $x_c$ near the outlet and inlet as well as extreme constriction degrees $\mathcal{P} > 90\%$ are needed to exploit the full range of $\hat{F}_1$. The increase of $\hat{F}_1$ for decreasing constriction degree $\mathcal{P}$ (close to open following the thumb rule³⁴) is retrieved for $x_c$ away from the inlet up to the waveguide's outlet. However, for the $x_c$-range near the inlet, the tendency is reversed as $\hat{F}_1$ decreases for decreasing $\mathcal{P}$, which thus contradicts the thumb rule for formant $F_1$.

175 In the high $\mathcal{P}$-range, $\hat{F}_2$ (Fig. 2(b)) exhibits two successive maxima and minima for increasing $x_c$. Nevertheless, when not considering the extrema near the inlet and outlet, $\hat{F}_2$ increases with $x_c$ (back to front). This trend is consistent with the rule of thumb³⁴ for formant $F_2$.

179 This increase or decrease of $\hat{F}_{1,2}$ with $x_c$ for high enough $\mathcal{P}$ is not necessarily linear, but can be concentrated in a compact parameter region. For constriction degrees $\mathcal{P}$ lower than the high $\mathcal{P}$-range, the variation of resonance frequencies with either $x_c$ or $\mathcal{P}$ is limited, suggesting that a multitude of VT geometries are associated with the same ($\hat{F}_1, \hat{F}_2$).

## 2. Influence of $L_0$, subset ($L_0, l_c = 0 \text{ mm}, x_c, \mathcal{P}$)

185 Theoretical SWVT resonances ($\hat{F}_1, \hat{F}_2$) for waveguide lengths $L_0 \in \{131, 171, 180\}$ mm and $l_c = 0$ mm are plotted in Fig. 3(a). These values correspond to typical VT lengths ranging from those of a child to an adult producing different vowels³⁴, e.g., for a single male adult speaker¹² from 161 mm for /æ/ to 196 mm for /u/. Assuming the reference VT length
<figure><img src="image_5.png" /><figcaption>FIG. 4. Descriptive statistics of theoretical SWVT resonance space ($\hat{F}_1$, $\hat{F}_2$) for subset ($L_0$, $l_c$ = 0 mm, $x_c$, $\mathcal{P}$) as a function of waveguide length $L_0$ (in mm) with $100 \le L_0 \le 340$ mm. Fitted quadratic approximations of $\hat{F}_{1stat}$ and $\hat{F}_{2stat}$ (in Hz) with $stat \in \{\text{min, max, mean}\}$.</figcaption></figure>

of $L_0$ = 180 mm, the theoretical resonance space is confined to the range $250 \le F_1 \le 500$ Hz and $0.8 \le F_2 \le 1.5$ kHz (see also Fig. 2). This resonance space, defined by ($\hat{F}_1$, $\hat{F}_2$), forms a tilted, butterfly-shaped isosceles trapezium. As $L_0$ increases, the resonance space shifts toward lower frequencies and narrows, leading to a decrease in both $\hat{F}_1$ and $\hat{F}_2$, as well as in their difference $\hat{F}_2 - \hat{F}_1$.

These tendencies are also observed in the behavior of descriptive statistics (in Hz) as a function of waveguide length $L_0$ (in mm); specifically, the minima $\min(\hat{F}_{1,2})(L_0)$, maxima $\max(\hat{F}_{1,2})(L_0)$ and mean $\overline{\hat{F}}_{1,2}(L_0)$. Statistics are computed across 12 different waveguide lengths $L_0$ (in mm) within the range $100 \le L_0 \le 340$ mm, covering typical human VT lengths. The resulting trends are accurately approximated (coefficient of determination $R^2 \ge 0.99$) by continuous quadratic functions, as shown in Fig. 4. Notably, these curves exhibit the greatest variation for $L_0$ values within the range typical of human articulation (161-196 mm). This indicates that SWVT resonance characteristics are most sensitive to small changes in $L_0$ within this range, thereby promoting greater resonance variation in the
204 SWVT space ($\hat{F}_1, \hat{F}_2$).

205

206 The relationships between the statistical quantities $\hat{F}_{2 stat}$ and $\hat{F}_{1 stat}$ (Fig. 3(a)), where
207 $stat \in \{\min, \max, \text{mean}\}$, for $L_0 \in [100\ 340]$ mm, is accurately approximated as:

$$ 
\left\{
\begin{aligned}
\hat{F}_{2\min}(\hat{F}_{1\min}) &= -0.46 \cdot \hat{F}_{1\min}^2 + 1.98 \cdot \hat{F}_{1\min} + 48, \\
\hat{F}_{2\max}(\hat{F}_{1\max}) &= 0.016 \cdot \hat{F}_{1\max}^2 + 0.28 \cdot \hat{F}_{1\max} + 260, \\
\hat{F}_{2\text{mean}}(\hat{F}_{1\text{mean}}) &= 0.096 \cdot \hat{F}_{1\text{mean}}^2 + 2.99 \cdot \hat{F}_{1\text{mean}} + 0.3.
\end{aligned}
\right.
\qquad (2) $$

208 The theoretical resonance space ($\hat{F}_1, \hat{F}_2$) is bounded by the curves $\hat{F}_{2 stat}(\hat{F}_{1 stat})$ representing the extrema ($stat \in \{\min, \max\}$), while $\hat{F}_{2 mean}(\hat{F}_{1 mean})$ illustrates the average trend.
209 Overall, this confinement indicates that the theoretical resonance space associated with the
210 SWVT subset ($L_0, l_c = 0$ mm, $x_c, \mathcal{P}$) overlaps only with the central region of the vowel
211 formant space ($\hat{F}_1, \hat{F}_2$), and supports the observed trend that $\hat{F}_1, \hat{F}_2$ and their difference
212 $\hat{F}_2 - \hat{F}_1$ all decrease as $L_0$ increases. Consequently, varying $L_0$ allows to shift along the
213 direction defined by the mean statistical trend, consistent with Eq. (2), which shows that
214 $\hat{F}_{2 mean} \approx 3\hat{F}_{1 mean}$ and that $\hat{F}_{1 mean}$ decreases with increasing $L_0$. The alignment between
215 the mean trend (black full line) and the reference curve (yellow dotted line in Fig. 3(a)),
216 which represents the first two resonances of a uniform tube$^{43,44}$ of length $L_0$ and diameter
217 $D_0$, supports the conclusion that the mean is governed solely by $L_0$.
218

219
### 3. Influence of $l_c$, subset ($L_0 = 180$ mm, $l_c, x_c, \mathcal{P}$)

Theoretical SWVT resonances ($\hat{F}_1, \hat{F}_2$) for pincher length $l_c \in \{0, 15, 30, 60\}$ mm and $L_0 = 180$ mm are shown in Fig. 3(b). Increasing $l_c$ expands the theoretical SWVT resonance space ($\hat{F}_1, \hat{F}_2$), increasing the overlap with the vowel formant space ($F_1, F_2$) observed for $l_c = 0$ mm. The overall position of ($\hat{F}_1, \hat{F}_2$) remains largely unchanged, confirming the previous section's statement (section III A 2) that the position is primarily determined by the fixed length $L_0 = 180$ mm. The expansion is most pronounced in the direction perpendicular to the statistical mean trend $\hat{F}_{2\text{mean}}(\hat{F}_{1\text{mean}})$ (see Fig. 3(a)), such that the theoretical resonance space remains its tilted, butterfly-shaped isosceles trapezium form, with its width increasing as $l_c$ increases. Thus, overall, $l_c$ influences the extrema statistics $\hat{F}_{2\text{stat}}(\hat{F}_{1\text{stat}})$, where $\text{stat} \in \{\text{min, max}\}$. Compared to $l_c = 0$ mm, this can result in higher values of either $\hat{F}_1$ or $\hat{F}_2$, but not both simultaneously. As a result, the overlap with the vowel formant space ($F_1, F_2$) remains confined to a subregion of that space.

### B. Two constrictions: SWVT subset ($L_0 = 180$ mm, $l_{c,1}, x_{c,1}, \mathcal{P}_1, l_{c,2}, x_{c,2}, \mathcal{P}_2$)

This section examines how the theoretical resonance space observed for a single constriction (Section III A) expands in the case of two constrictions (pincher index $i = \{1, 2\}$ in Fig. 1). The theoretical space ($\hat{F}_1, \hat{F}_2$) for the SWVT subset ($L_0 = 180$ mm, $l_{c,1}, x_{c,1}, \mathcal{P}_1, l_{c,2}, x_{c,2}, \mathcal{P}_2$) is compared to the vowel formant spaces ($F_1, F_2$).

Fig. 5(a) shows the theoretical SWVT resonance space ($\hat{F}_1, \hat{F}_2$) for $l_{c,1} = l_{c,2} = 0$ mm
<figure><img src="image_6.png" /><figcaption>FIG. 5. Theoretical SWVT resonances ($\hat{F}_1$, $\hat{F}_2$) vs formants ($F_1$, $F_2$) for $-L_0 \le x_{c,1}, x_{c,2} \le 0$ mm and $0\% \le P_1, P_2 \le 99\%$ ($L_0 = 180$ mm): a) SWVT subsets for $l_{c,1} = l_{c,2} = 0$ mm, one constriction ($P_2 = 0\%$) vs two constrictions ($P_{1,2} \neq 0\%$); b) SWVT subset for $l_{c,1}, l_{c,2} \in \{0, 30\}$ mm for two constrictions.</figcaption></figure>

($P_{1,2} \neq 0\%$), alongside the resonance space for a single constriction ($P_2 = 0\%$). The overall position of the resonance space, whether for one or two constrictions, remains unchanged. This confirms the previous section's (Section III A) finding that the position is primarily determined by the fixed length $L_0 = 180$ mm. Nevertheless, the size of the resonance space is expanded in all directions, resulting in a significant improvement in its overlap with the vowel formant space ($F_1, F_2$), although it remains localized based on $L_0$.

Fig. 5(b) illustrates the effect of increasing the pincher extent $l_{c,1}, l_{c,2}$, or both from 0 mm to 30 mm. This leads to a further expansion of the resonance space in all directions, improving its overlap with the vowel formant space ($F_1, F_2$), even though the waveguide length $L_0$ remains fixed at 180 mm. Notably, the theoretical SWVT resonance space ($\hat{F}_1, \hat{F}_2$) either reaches the boundary of the French vowel resonance space or overlaps at least partially with American English vowels in the ($F_1, F_2$) space. This is clearly observed for vowels such as
253 /i/, /a/ and /ɛ/ in Fig. 5(b).

254

255

## IV. SWVT VS LITERATURE: AREA FUNCTIONS

256

As shown in Section III, the SWVT framework can approximate the formants ($F_1, F_2$) of American English and French vowels using $L_0 = 180$ mm. The analysis below identifies the subsets of SWVT parameters ($L_0 = 180$ mm, $l_{c,1}, x_{c,1}, P_1, l_{c,2}, x_{c,2}, P_2$) that achieve this approximation, along with the corresponding SWVT area functions $\hat{A}(x)$.

257

258

259

260

The most direct method for selecting SWVT parameter subsets and corresponding area functions $\hat{A}(x)$ for a given vowel is to ensure that the resulting theoretical resonances ($\hat{F}_1, \hat{F}_2$) lie within the target vowel's space ($F_1, F_2$). For French vowels², this method is referred to as the **Calliope criterion**, and for American English vowels¹, the **Peterson criterion**. The ellipsoids representing vowel spaces exhibit substantial variation due to intra- and inter-speaker differences, as noted in the introduction (Section I). As a result, SWVT parameter selection is not unique. To address this, for American English vowels, an alternative criterion is proposed based on area functions $A(x)$ and formant frequencies derived from a single adult male speaker, as reported by Story¹². This approach is referred to as **MRI₀** (for area functions) and **Story** (for formants), and is detailed in Section IV A. For French vowels, however, only the Calliope criterion is used.

261

262

263

264

265

266

267

268

269

270

271
## A. SWVT-tailored area functions and resonance invariants

Compared to the Peterson criterion, which uses a range of vowel formant frequencies, the **Story criterion**, provides fixed, well-defined formant values¹². However, these values were obtained under specific, and possible variable, conditions (MRI environment, temperature). On the other hand, the corresponding VT area functions $A(x)$ ($\mathbf{MRI}_0$) vary in length $L$ from 161 mm to 196 mm. In this section, these MRI-based VT area functions $A(x)$ ($\mathbf{MRI}_0$) are adapted to fit within the SWVT domain. This is done by reshaping them to satisfy the condition $\hat{A}(x) \le A_0$ over the interval $-L_0 \le x \le 0$, where the reference length is $L_0 = 180$ mm and the maximum cross-section area is $A_0 = 490$ mm² (see Fig. 1). This approach allows for the derivation of a reference set of theoretical resonance invariants, offering an additional criterion for selecting SWVT parameters.

### 1. SWVT tailored area functions: $MRI_I$, $MRI_{II}$

The first type of SWVT-tailored area functions, denoted $\mathbf{MRI_I}$ (SM Fig. 2 for /i/, /a/, /u/ and /ε/), is obtained by rescaling the longitudinal dimension with a factor $\alpha_x = L_0/L$, ensuring that the adjusted length equals the reference length $L_0$; either stretching ($\alpha_x > 1$) or compressing ($\alpha_x > 1$) the x-dimension. Additionally, any cross-sectional area exceeding $A_0$ is capped at $A_0$. This thresholding preserves the original cross-section where $A(x) \le A_0$, while reducing values that exceed $A_0$.

The second type, denoted $\mathbf{MRI_{II}}$ (SM Fig. 2 for /i/, /a/, /u/ and /ε/), also rescales
the x-dimension by $\alpha_x$, but further adjusts the cross-sectional areas by a factor $\alpha_A =$
$A_0/\max(A(x))$. As a result, the MRI<sub>II</sub> functions conform to the full SWVT domain, with
both the length equal to $L_0$ and the maximum cross-sectional area equal to $A_0$. Unlike
MRI<sub>I</sub>, this method modifies the cross-section along the entire x-range; either enlarging it
($\alpha_A > 1$) or reducing it ($\alpha_A < 1$), depending on the original area function MRI<sub>0</sub>.

297

## 2. Theoretical resonance invariants of MRI<sub>I</sub>, MRI<sub>II</sub>

298

299

300

301

302

303

304

305

306

307

308

309

310

311

Section III outlined that theoretical resonances ($\hat{F}_1, \hat{F}_2$) depend on the degree of area reduction (impact $P_{1,2}$), the length of the constricted portion (impacts $l_{c,1}$ and $l_{c,2}$), and the waveguide length (impact $L_0$). The SWVT-tailoring of $\text{MRI}_0$, as described in Section IV A 1, involves both area scaling – which influences the degree of constriction – and longitudinal scaling – which affects both the waveguide length and the length of the constricted region(s). As a result, the theoretical resonances for $\text{MRI}_0$, $\text{MRI}_I$ and $\text{MRI}_{II}$ may differ.

The theoretical resonance space ($\hat{F}_1, \hat{F}_2$) for area functions $\text{MRI}_0$, $\text{MRI}_I$ and $\text{MRI}_{II}$, along with reference formant values ($F_1, F_2$) (Story$^{12}$) for 10 vowels, is shown in Fig. 6(a). While all four data points for each vowel lie within the same general region, noticeable differences are present. For MRI-based geometries, these discrepancies may be linked to the magnitude of the scaling factors ($\alpha_x, \alpha_A$), as seen for vowels like /a/ and /u/. As mentioned previously, discrepancies between theoretical resonances obtained for $\text{MRI}_0$ and formants can be partly
<figure><img src="image_7.png" /><figcaption>FIG. 6. For MRI₀, MRII and MRIII: a) Theoretical resonance frequencies (F̂₁, F̂₂) and formants (F₁, F₂) (×, Story¹²); b) Relative accuracy Er₁ and Er₂ w.r.t. formants (F₁, F₂).</figcaption></figure>

attributed to different measurement conditions.

Observed differences (Fig. 6(a)) between theoretical resonances F̂ᵢ for MRI₀, MRII and MRIII and formants Fᵢ for each vowel are quantified considering the mean relative absolute difference (Er, relative difference in short) with respect to N formants Fᵢ with index i in the set {i₁, ..., iₙ} obtained as,

$$Er_{i_1 \dots i_n} = \frac{1}{N} \sum_{i \in \{i_1, \dots, i_n\}} \frac{|F_i - F_{i-1}|}{F_i} \quad (3)$$

For each vowel, relative differences Er₁ and Er₂ are shown in Fig. 6(b). The difference Er₂ remains below 15%, while Er₁ can reach up to 25%. Overall, discrepancies between vowel formants and theoretical resonance frequencies are all of the same order of magnitude, including those from the SWVT-tailored area functions MRII and MRIII. Therefore, these theoretical resonances F̂ᵢ, defined under well-specified conditions, serve as **resonance invariants** and can be effectively used for SWVT parameter selection for a given American
<figure><img src="image_8.png" /></figure>

FIG. 7. For /a/: a) Relative differences $Er_{12}$ for theoretical resonances of SWVT parameter subset ($L_0$ = 180 mm, $l_c$ = 0 mm, $x_c$, $\mathcal{P}$) w.r.t. formants ($F_1, F_2$) (Story$^{12}$) with $\min(Er_{12})$ = 17% for $x_c$ = -142 mm and $\mathcal{P}$ = 99% (arrow in zoom); b) minima of relative differences $Er_{12}$ as a function of $l_c$ w.r.t. three criteria.

English vowel. In the next section, **MRI<sub>II</sub> resonance invariants** are used to select SWVT parameter for a single pincher imposing one constriction.

### 3. SWVT parameter selection w.r.t resonance invariants from MRI<sub>II</sub>

Relative differences $Er_{12}$ between the theoretical resonances of the SWVT reference subset ($L_0$ = 180 mm, $l_c$ = 0 mm, $x_c$, $\mathcal{P}$) and the vowel formants ($F_1, F_2$) (i.e., the Story criterion$^{12}$) for /a/ are shown in Fig. 7 (and in SM Fig. 3 for /i/, /a/ and /u/). The SWVT parameter set minimizing $Er_{12}$ is highlighted in the zoomed-in range $15\% \le Er_{12} \le 30\%$ and the selected SWVT parameters ($x_c$, $\mathcal{P}$) are indicated. In general, selected configurations exhibit severe constrictions degrees $\mathcal{P} \ge 98\%$ and constriction positions shift from front to
back in agreement with vowel area functions $A(x)$ associated with $\text{MRI}_0$.

Using vowel formants ($F_1, F_2$) (i.e., the **Story** criterion) to compute the relative differences is considered a poorly defined selection criterion due to inter- and intra-speaker variability. Therefore, relative differences $Er_{12}$ with respect to theoretical resonance frequencies $\hat{F}_i$ from $\text{MRI}_0$ (the **MRI**$_0$ criterion) and resonance invariants from $\text{MRI}_{II}$ (the **MRI**$_{II}$ criterion) (see Section IVA 2) are assessed as well. The comparison of minimum relative differences obtained for these three selection criteria on the SWVT subset ($L_0 = 180$ mm, $l_c, x_c, \mathcal{P}$) for constriction lengths $l_c \le 60$ mm is plotted for /a/ in Fig. 7(b) (and in SM Fig. 4 for /i/, /a/ and /u/). All three selection criteria exhibit consistent trends with respect to the choice of $l_c$, confirming that the resonance invariants from $\text{MRI}_{II}$ are suitable for SWVT parameter selection. The overall minimum of the relative differences $Er_{12}(l_c)$ occurs at a shorter $l_c \approx 20$ mm for /u/, compared to $l_c \ge 50$ mm for /i/ and /a/.
This is consistent with reported area functions $\text{MRI}_0^{12}$.

## B. SWVT area function selection for vowels /i/, /a/ and /u/

SWVT area functions $\hat{A}(x)$ are selected by minimizing the relative differences $Er_{12}$ between theoretical SWVT resonance frequencies ($\hat{F}_1, \hat{F}_2$) from the SWVT parameter subset and reference frequencies for each vowel. The **MRI**$_{II}$ criterion is applied for American English vowels, so that reference vowel frequencies are the resonance invariants from $\text{MRI}_{II}$ (Section IVA 2). Results obtained from selected SWVT parameters for **American English**.
<figure><img src="image_9.png" /><figcaption>FIG. 8. Theoretical resonances ($\hat{F}_1$, $\hat{F}_2$) of selected SWVT area functions $\hat{A}(x)$ ($L_0$ = 180 mm) for American English (black) and French (red) vowels /i/, /a/, and /u/ from SWVT subsets: a) ($l_c, x_c, \mathcal{P}$) with $l_c$ = 0 mm (●), $l_c$ = 30 mm (+) and $l_c$ = 60 mm (*); b) ($l_{c,1}, x_{c,1}, \mathcal{P}_1, l_{c,2}, x_{c,2}, \mathcal{P}_2$) with $l_{c,1} = l_{c,2} = 0$ mm (●), $l_{c,1} = l_{c,2} = 30$ mm (*) and $l_{c,1}, l_{c,2} \in \{0, 30\}$ mm with $l_{c,1} \neq l_{c,2}$ (+).</figcaption></figure>

**glish** vowels are plotted in **black**, in accordance with the gray-shaded ellipsoids in the vowel space. For French vowels, the **Calliope criterion** is applied. Results obtained from selected SWVT parameters for **French vowels** are plotted in **red**, in accordance with the red-shaded ellipsoids in the vowel space ($F_1, F_2$). To lighten notations, the used criterion and formant space coloring is omitted from figure captions and legends in this section. Selected configurations for subsets of the SWVT parameter space corresponding to configurations with one (Section IV B 1) or two constrictions (Section IV B 2) are considered. Results are presented for the American English and French vowels /i/ (front constriction in $\text{MRI}_0$), /a/ (back constriction in $\text{MRI}_0$) and /u/ (center and front constriction in $\text{MRI}_0$). Following Section IV, $L_0$ = 180 mm is held constant and the length is not further considered.
367

# 1. One constriction: SWVT selections in ($L_0 = 180$ mm, $l_c, x_c, \mathcal{P}$)

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

Theoretical resonances ($\hat{F}_1, \hat{F}_2$) associated with selected SWVT area functions $\hat{A}(x)$ (Fig. 5 of SM) for the parameter subset ($L_0 = 180$ mm, $l_c = 0$ mm, $x_c, \mathcal{P}$) are shown (•) in Fig. 8(a). Overall, the tendencies for American English and French vowels are similar. For the vowel /a/, the constriction is located near the back (the glottis) at $-165 < x_c < -144$ mm, with $\mathcal{P} = 96\%$. For /i/, the constriction is near the front (the lips) at $-57 < x_c < -51$ mm, with $96 < \mathcal{P} < 99\%$. For /u/, the constriction lies near the center of the waveguide at $-124 < x_c < -113$ mm, with $\mathcal{P} = 99\%$. These observations are consistent with literature. However, selected SWVT area functions $\hat{A}(x)$ exhibit constrictions slightly upstream (i.e. further back) compared to those observed in VT area functions $A(x)$ (MRI$_0$). As in Section III A, the theoretical resonances ($\hat{F}_1, \hat{F}_2$) do not reach the target vowel spaces for /i/ and /a/. However, for /u/, the resonances fall within the expected vowel space for both American English and French, supporting the relevance of the proposed selection criteria.

Theoretical resonances ($\hat{F}_1, \hat{F}_2$) associated with selected SWVT area functions $\hat{A}(x)$ (Fig. 6 of SM) for the parameter subset ($L_0 = 180$ mm, $l_c \in \{30, 60\}$ mm, $x_c, \mathcal{P}$) are shown (+, *) in Fig. 8(a). Thus, the influence of $l_c > 0$ mm is assessed. Similar to the case of $l_c = 0$ mm, the same general tendencies are observed for both American English and French vowels. For the vowel /a/, the constriction is located near the back (the glottis) between $155 < x_c < 130$ mm with $96 < \mathcal{P} < 99\%$. For /i/, the constriction is positioned
near the front (the lips) at $52 < x_c < 58$ mm with $93 < P < 96\%$. For /u/, the con-
striction lies toward the center of the waveguide, at $140 < x_c < 125$ mm with $P = 99\%$.

As with $l_c = 0$ mm, all SWVT area functions $\hat{A}$ show constrictions located slightly upstream (i.e., toward the back) relative to those observed in the VT area functions $A(x)$ (MRI₀). As found in Section III A, increasing the constriction extent ($l_c \in \{30, 60\}$ mm), enables theoretical resonances ($\hat{F}_1, \hat{F}_2$) to reach the target vowel spaces for /u/ as well as for /i/, and /a/. The resonances obtained for $l_c = 30$ mm or $l_c = 60$ mm are closely adjacent.

395

396

## 2. Two constrictions: SWVT selections in ($L_0 = 180$ mm, $l_{c,1}$, $x_{c,1}$, $\mathcal{P}_1$, $l_{c,2}$, $x_{c,2}$, $\mathcal{P}_2$)

Theoretical resonances ($\hat{F}_1, \hat{F}_2$) associated with selected SWVT area functions $\hat{A}(x)$ (Fig. 7 of SM) for the parameter subset ($L_0 = 180$ mm, $l_{c,1} = 0$ mm, $x_{c,1}, \mathcal{P}_1, l_{c,2} = 0$ mm, $x_{c,2}, \mathcal{P}_2$) are shown (•) in Fig. 8(b). As in Section III B, the theoretical resonances ($\hat{F}_1, \hat{F}_2$) do not reach the target vowel spaces for /i/, /a/, though they do fall within the correct region for /u/ in both American English and French. The influence of adding a second constriction varies by vowel:

397

398

399

400

401

402

403

404

405

- For /a/, where the VT area function $A(x)$ (MRI₀) typically shows a single constricted portion, the SWVT area function $\hat{A}(x)$ forms two closely spaced constrictions, effectively increasing the extent of the constricted region (though not uniformly).

406

407

- For /u/ and /i/, which show up to two constrictions in the VT area function $A(x)$ (front and back for /i/; center and front for /u/), SWVT area functions $\hat{A}(x)$ either
display two closely spaced constrictions – broadening the constricted region – or two
more widely spaced constrictions, which replicate the presence of a second constriction.

As with the single-constriction SWVT configurations (see Section IV B 1), all SWVT area
functions $\hat{A}(x)$ show constrictions located slightly upstream (i.e., further back) compared
to those observed in the VT area functions $A(x)$ (MRI₀). Importantly, adding a second
constriction does not significantly improve the accuracy of the results compared to single-
constriction models, particularly when $l_c > 0$ mm.

Theoretical resonances ($\hat{F}_1, \hat{F}_2$) associated with selected SWVT area functions $\hat{A}(x)$
(Fig. 8 of SM) for the parameter subset ($L_0 = 180$ mm, $l_{c,1}, x_{c,1}, P_1, l_{c,2}, x_{c,2}, P_2$) for either
$l_{c,1} = l_{c,2} = 30$ mm or $l_{c,1}, l_{c,2} \in \{0, 30\}$ mm with $l_{c,1} \neq l_{c,2}$ are shown ($\ast, +$) in Fig. 8(b).
Increasing the constriction length allows the theoretical resonances ($\hat{F}_1, \hat{F}_2$) to reach the
target vowel spaces, which confirms findings in Section III B. As for $l_{c,1} = l_{c,2} = 0$ mm,
the second constriction is either adjacent to the first one in order to extent the constricted
region or widely spaced in order to reproduce a second constriction location.

# V. EXPERIMENTAL METHODS

## A. Experimental setup and data analysis

The setup in Fig. 9 is designed to measure acoustic pressure at various positions $x_p$
($x_p \in \{-5, -37, -60\}$ mm) along the centerline of the waveguide under test (rigid or de-
<figure><img src="image_10.png" /></figure>

FIG. 9. Setup (xz-plane) for measuring the acoustic pressure ($V_p(t, x_p)$, Volt) along the centerline of a waveguide ($\mathcal{P}, x_c, l_c)_i$, pincher index $i \in \{1, 2\}$) at probe position $x_p$, following a source excitation signal $V_s(t)$ (Volt) transmitted via an adapter. The wall displacement along the major axis $\Delta a$ (mm) is measured using a laser transceiver at position $x_L$.

formable)$^{42}$. The waveguide is positioned within an insulation chamber (Décibel France, working dimensions 118 × 65 × 95 cm, volume 0.73 m$^3$). A compression chamber (Monacor KU-916T) is connected to its inlet at $x = -L_0$ via an adapter featuring a centrally positioned communication hole (2 mm in diameter and 6 mm in length) through which a 30-second linear frequency sweep (frequency range from 0.1 up to 6 kHz) is repeatedly emitted into the waveguide. A rigid Plexiglas rectangular baffle (365 × 352 mm in the yz-plane, with a thickness of 5 mm) is attached at the waveguide exit located at $x = 0$. This baffle approximates the infinite baffle assumption used in the theoretical acoustic model (see Section II B) and, unless stated differently, imposes a circular exit condition to the waveguide.

Data generation (sweep signal $V_s$ in Volt) and acquisition of acoustic pressure ($V_p$ in Volt), and wall displacement ($\Delta a \ge 0$ in mm) occurs as outlined in Eliraki et al.$^{42}$, so that the acoustic pressure amplitude ($|P_{x_p}(f)|$) as a function of frequency $f$ at each probe position $x_p$ can be obtained$^{41,42}$. Measured acoustic waveguide resonances, denoted $\bar{f}_i$, are identified
fied at frequencies corresponding to peaks in the acoustic pressure amplitude (in dB). The temperature during experiments (21 °C to 28 °C) can differ from 21 °C accounted for in the theoretical acoustic model (Section II B). Subsequent differences between measured and theoretical resonance frequencies can reach up to 2%.

446

## B. Experimental waveguides

### 1. Rigid (R) and molded (M) waveguides

Rigid (R) waveguides are 3D fabricated using 3D printing (RAISE3D Pro3, accuracy ±0.01 mm), ensuring precise reproduction of the intended waveguide geometry. This includes rigid waveguides based on VT area functions $A(x)$ (MRI₀), SWVT-tailored VT area function geometries (MRI_I, MRI_II, see Section IV A 1) and selected SWVT area functions $\hat{A}(x)$ for the American English vowels /i/, /a/, /u/ (see Section IV B).

454

Deformable molded (M) uniform cylindrical waveguides ($L_0$ = 180 mm, $D_0$ = 25 mm, $d_0$ = 3 mm), matching the SWVT dimensions shown in Fig. 1, are molded using a custom-designed mold and elastomer silicone mixture (Dragonskin (Smooth-On Inc.))⁴². SWVT parameters can be applied to these deformable waveguides using a custom-built, controllable squeezing apparatus⁴²,⁴⁵, which is installed within the insulation chamber (Fig. 9). The pinchers, designed to impose $l_c \in \{0, 30, 60\}$ mm, are 3D printed (RAISE3D Pro3, accuracy ±0.01 mm) to match the required SWVT parameters. Selected SWVT area functions $\hat{A}(x)$
462 for the American English vowels /i/, /a/, /u/ (see Section IV B) can be replicated using the
deformable molded waveguide.
463

464

In contrast to the theoretical exploration of the SWVT parameter space presented in
Section III and Section IV, the experimental setup imposes physical constraints near the
inlet (at $x = -L_0$ mm) and outlet (at $x = 0$ mm) due to the connection of the waveguide
with the source (at $x = -L_0$ mm) and with the baffle (at $x = 0$ mm). For a waveguide
with $L_0 = 180$ mm, this reduced the theoretical $x_c$-range from $-180 \le x_c \le 0$ mm to the
experimental $x_c$-range:
465
466
467
468
469
470

471

• $-155 + l_c/2 \le x_c \le -15 - l_c/2$ mm for molded (M) deformable waveguides,

472

• $-180 \le x_c \le -15 - l_c/2$ mm for rigid (R) waveguides.

473

474

475

476

477

478

479

480

Constriction degrees up to full closure can be applied for deformable (M) waveguides. How-
ever, for rigid (R) waveguides, fabrication is not feasible for constriction degrees exceeding
$P > 96\%$. This constrains the range of experimentally achievable SWVT configurations and
thus restricts the experimentally assessable resonance space ($\hat{F}_1, \hat{F}_2$) (Fig. 9 of SM). In the
case that selected SWVT parameter subsets for a given American English vowel, based on
the $\text{MRI}_{II}$-criterion (see Section IV B), are experimentally not feasible, SWVT parameters
are re-selected using the formant space-based Peterson-criterion (Table 1 of SM).
<figure><img src="image_11.png" /><figcaption>FIG. 10. Mean relative error (in %) as a function of $\mathcal{P}$ (in %) between the measured major axis $a \approx \Delta a + a_0$ and theoretical major axis $\hat{a}$ for $l_c \in \{0, 30, 60\}$ mm.</figcaption></figure>

## 2. Major axis validation for molded waveguides

The analytical model used to represent the SWVT waveguide geometry (see Section II A) was previously validated for deformable molded (M) waveguides with one or two pinchers, using $l_{c,1} = l_{c,2} = 0$ mm$^{11,42}$. In this section, the validation is extended to cases where $l_c \ge 0$ mm, by comparing SWVT theoretical values and experimental measurements of wall displacement $\Delta a$ as a function of constriction degree $\mathcal{P}$ and normalized position $(x_c - x_L - l_c/2)/a_0$ (Fig. 10 of SM). The mean relative error between the theoretical major axis $\hat{a}$ and the measured values $a \approx a_0 + \Delta a$ (see Fig. 1) as a function of $\mathcal{P}$ for $l_c \in \{0, 30, 60\}$ mm is shown in Fig. 10. For $l_c = \{0, 30\}$ mm, the experimental and theoretical results show good agreement across all values of $\mathcal{P}$, with relative difference remaining below 5% and following similar trends. For $l_c = 60$ mm, good agreement is observed for $\mathcal{P} \le 70\%$, with relative differences around 5%. However, for $\mathcal{P} > 70\%$, the discrepancy increases, likely due to physical constraints discussed in Section VB 1. Nevertheless, since all mean differences remain below 10%, the theoretical geometrical waveguide model is considered to accurately represent the deformation of molded (M) waveguides using the
experimental setup described in Section V for $l_c \ge 0$ mm.

497

498 VI. SWVT VALIDATION: /I/, /A/ AND /U/

499

500

501

502

Experimental resonances $\bar{f}_i$ of rigid (R) and molded (M) waveguides detailed in Section VB1 are compared to theoretical resonance frequencies $\hat{F}_i$ in order to validate the SWVT framework. As VT area functions $A(x)$ (MRI$_0$) are available for American English only, the experimental validation focuses on the American English vowels /i/, /a/ and /u/.

503

504 A. Rigid MRI₀ waveguides

<figure><img src="image_12.png" /><figcaption>FIG. 11. Experimental resonance frequencies ($\bar{f}_1, \bar{f}_2$) of rigid 3D printed MRI₀ waveguides (MRI₀<sup>exp</sup>); theoretical resonances ($\hat{F}_1, \hat{F}_2$) for MRI₀, MRI_I, and MRI_II; formant spaces ($F_1, F_2$) for the American English vowels /i/, /a/, /u/, along with reference formants (×, Story).</figcaption></figure>
Experimental resonance frequencies ($\overline{f}_1, \overline{f}_2$) of rigid 3D printed MRI₀ waveguides (MRI₀<sup>exp</sup>) are plotted in Fig. 11. For all three vowels, experimental values validate theoretical values $\hat{F}_i$ for MRI₀ as discrepancies are small. This provides an additional validation of the theoretical acoustic model used in the SWVT framework for the shown frequency range shown (up to 2.5 kHz). In the next section, rigid (R) and molded (M) waveguides obtained with selected SWVT parameters are considered.

## B. Rigid and deformable SWVT waveguides

As outlined in Section VB 1, rigid (R) and deformable molded (M) waveguides shaped with selected SWVT parameters are experimentally evaluated for each of the American English vowels /i/, /a/ and /u/. Multiple waveguides are assessed per vowel (Table 1 of SM), depending on the specific SWVT parameter subset used for selection (Section IV B) and the experimentally accessible parameter range (see Section VB 1). Waveguides are labeled as $/a^*_{(s,)l_c}/$, $/i^*_{(s,)l_c}/$, $/u^*_{(s,)l_c}/$ with superscript * ∈ {M, R} indicating the wall material and, where appropriate, s indexing selections from similar SWVT parameter subsets.

For deformable molded waveguides (M), the first resonance frequency $\overline{f}_{1,M}$ is plotted. This frequency is defined to reflect the effects of vibro-acoustic coupling with a wall vibration frequency ($\approx$ 200 Hz) specific to the experimental setup, which only occurs in molded waveguides<sup>42</sup>. This coupling leads to amplitude modulation that shifts the first resonance to a lower frequency $\overline{f}_{1,l}$ (around 210 Hz) and generates an additional resonance $\overline{f}_{1,h}$ (around
<figure><img src="image_13.png" /><figcaption>FIG. 12. Theoretical ($\hat{F}_1, \hat{F}_2$) and measured resonance frequencies ($\bar{f}_{1,M}, \bar{f}_2$) for deformable (M) SWVT waveguides and ($\bar{f}_1, \bar{f}_2$) for rigid (R) waveguides (SWVT, $\text{MRI}_0^{\text{exp}}$) with length $L_0 = 180$ mm; formant spaces ($F_1, F_2$): a) one constriction ($\mathcal{P} = 0\%$) cases selected from SWVT subset ($L_0, l_c \in \{0, 30, 60\}$ mm, $x_c, \mathcal{P}$); b) one ($\mathcal{P}_2 = 0\%$) and two ($\mathcal{P}_{1,2} \neq 0\%$) constriction cases, selected from SWVT subsets ($L_0, l_c \in \{0, 30\}$ mm, $x_c, \mathcal{P}$) and ($L_0, l_{c,1} \in \{0, 30\}$ mm, $x_{c,1}, \mathcal{P}_1, l_{c,2} \in \{0, 30\}$ mm, $x_{c,2}, \mathcal{P}_2$) with $l_{c,1} = l_{c,2}$.</figcaption></figure>

640 Hz). For molded waveguides, this modulation is accounted for by defining the first resonance as $\bar{f}_{1,M} = |\bar{f}_{1,h} - \bar{f}_{1,l}|$. Thus, ($\bar{f}_{1,M}, \bar{f}_2$) are reported for molded deformable waveguides, while ($\bar{f}_1, \bar{f}_2$) are reported for rigid waveguides.

1. One constriction SWVT configurations

Experimentally measured resonance frequencies ($\bar{f}_{1,(M)}, \bar{f}_2$) for one constriction configurations are presented in Fig. 12(a) (full and zoomed view in Fig. 11 of SM), based on
the SWVT parameter subset ($L_0 = 180$ mm, $l_c \in \{0, 30, 60\}$ mm, $x_c, P$). Overall, the measured resonances for rigid waveguides (+) closely match or even overlap with the theoretical SWVT resonances (□). Since the SWVT parameters were selected to approximate the target vowel formants, this agreement validates the SWVT framework for one constriction configurations with $L_0 = 180$ mm and $l_c \le 60$ mm.

Discrepancies are observed between the measured resonances for deformable waveguides (●) and both the theoretical SWVT resonances (□) and the ones measured for rigid waveguides (+). While these discrepancies remain limited, or even negligible, for the American English vowels /a/ (e.g., $a_0^M$ in proximity to $a_0^R$) and /u/ (e.g., $u_{60,0}^M$ near $u_{60,0}^R$), they become pronounced for /i/. For instance, comparing resonances for $i_{0,30,69}^M$ and for $i_{0,30,60}^R$, this discrepancy is entirely due to a shift in the first resonance frequency, i.e. between $\bar{f}_1$ and $\bar{f}_{1,M}$. Since the first resonance frequency associated with selected SWVT parameters for /i/ is relatively low compared to /a/ or /u/, it is hypothesized that vibro-acoustic coupling prevents the first resonance frequency $\bar{f}_{1,M}$ to reach low $\hat{F}_1 \lesssim 350$ Hz. Nevertheless, the measured results support the validity of the SWVT framework for configurations characterized by $\hat{F}_1 \gtrsim 350$ Hz.

## 2. One and two constriction SWVT configurations

Experimentally measured resonance frequencies ($\bar{f}_{1,M}, \bar{f}_2$) for both one ($P_2 = 0\%$) and two ($P_{1,2} \neq 0\%$) constriction configurations are presented in Fig. 12(b). These are based on
554 selections from the SWVT parameter subset ($L_0 = 180$ mm, $l_c \in \{0, 30\}$ mm, $x_c, P$) for one constriction, and ($L_0 = 180$ mm, $l_{c,1} \in \{0, 30\}$ mm, $x_{c,1}, P_1, l_{c,2} \in \{0, 30\}$ mm, $x_{c,2}, P_2$) with $l_{c,1} = l_{c,2}$ for two constrictions.

557

All measurements are obtained using deformable molded waveguides, with $\bar{f}_{1,M} \gtrsim 350$ Hz, consistent with the findings for one constriction configurations described in Section VIB 1. Overall, $350 \lesssim \bar{f}_{1,M} \lesssim 480$ Hz is observed. Notably, when comparing, for example, $i_0^M (\circ, P_{1,2} \neq 0)$ and $i_{1(2),0}^M (\bullet, P_2 = 0)$ or $u_{30}^M (\circ, P_{1,2} \neq 0)$ and $i_{1,30}^M (\bullet, P_2 = 0)$, it is evident that introducing two constrictions extends the experimentally accessible $\bar{f}_2$-range to both higher (as observed for $i^M$) and lower (as observed for $u^M$) frequencies. This confirms the theoretical trend described in Section III B.

565

## VII. SWVT VALIDATION: CONSTRiction FEATuRES

### A. Influence of pincher extent, $l_c$: one constriction

Both the theoretical SWVT resonances (Section III) and experimental results (Section VIB) support the conclusion that the constriction length $l_c$ contributes to achieving the target vowel formant space. Therefore, the influence of the constriction extent $l_c \le 60$ mm is experimentally investigated using a molded waveguide with a fixed length $L_0 = 180$ mm, for one constriction configurations. The constriction position is fixed at $x_c = -90$ mm, centered within the waveguide, to minimize the potential impact of
<figure><img src="image_14.png" /><figcaption>FIG. 13. Resonance frequencies of a molded waveguide ($L_0 = 180$ mm) with one constriction at $x_c = -90$ mm for $l_c \in \{0, 30, 60\}$ mm and $0 \le P \le 99\%$: a) $\bar{f}_{1,M}(P)$, b) $\bar{f}_2(P)$.</figcaption></figure>

physical constraints at the inlet or outlet, as outlined in Section VB1. For each assessed $l_c$, the constriction degree $P$ is varied from 0% (no constriction) to 99% (near full closure). The measured values of $\bar{f}_{1,M}(P)$ and $\bar{f}_2(P)$ for this SWVT parameter subset ($L_0 = 180$ mm, $l_c \in \{0, 30, 60\}$ mm, $x_c = -90$ mm, $P$) are plotted in Fig. 13. The general trend observed for both $\bar{f}_{1,M}(P)$ and $\bar{f}_2(P)$ is not altered by variations in $l_c$.
The first resonance frequency $\bar{f}_{1,M}(\mathcal{P})$ gradually increases (despite that fluctuations are observed), reaching a maximum around $\mathcal{P} \approx 80\%$, after which it declines rapidly with further increases in $\mathcal{P}$. This peak is more pronounced for $l_c = 60$ mm, while the subsequent decline is steeper for $l_c = 30$ mm. The frequency range $435 \lesssim \bar{f}_{1,M}(\mathcal{P}) \lesssim 465$ Hz observed for $l_c = 0$ mm extends by $\approx 70$ Hz to $400 \lesssim \bar{f}_{1,M}(\mathcal{P}) \lesssim 470$ Hz for $l_c > 0$ mm. For the second resonance frequency $\bar{f}_2(\mathcal{P})$, a general decrease is observed with increasing $\mathcal{P}$, with the most pronounced decline occurring at $l_c = 30$ mm. The range $1250 \lesssim \bar{f}_2(\mathcal{P}) \lesssim 1400$ Hz for $l_c = 0$ mm also expands by $\approx 70$ Hz to $1180 \lesssim \bar{f}_2(\mathcal{P}) \lesssim 1400$ Hz for $l_c > 0$ mm. These results confirm that increasing the constriction length broadens the frequency ranges associated with both the first and second resonance frequencies.

590

591

## B. Influence of constriction spacing, $S_c$: two constrictions

592

593

594

595

596

597

598

599

600

Both the theoretical SWVT resonances (Section III) and experimental results (Section VIB) support the conclusion that, in two constriction configurations, the spacing between constrictions contributes to achieving the target vowel formant space. Therefore, the influence of constriction spacing, defined as $S_c = |x_{c,2}| - |x_{c,1}|$ (in mm), is experimentally assessed. Concretely, resonance frequencies are measured using a molded waveguide ($L_0 = 180$ mm, $l_{c,1} = l_{c,2} = 0$ mm) with $\mathcal{P}_2$ fixed either at 77% or 90%, while $\mathcal{P}_1 \in \{77, 85, 90, 93, 96\}\%$ is varied for each spacing value in $S_c \in \{30, 60, 120\}$ mm. Large constriction degrees are assessed, as they characterize the selected SWVT parameters used to represent vowels (and typical vowels area reduction degrees $\min(A(x))/A_0 \approx 93 - 97\%$
<figure><img src="image_15.png" /><figcaption>FIG. 14. Resonance frequencies of a molded waveguide ($L_0 = 180$ mm, $l_{c,1} = l_{c,2} = 0$ mm) for two constrictions spaced symmetrically around $x = -56.23$ mm. $P_2$ is fixed at 77% or 90%, while $P_1 \in \{77, 85, 90, 93, 96\}$ (in %) is varied for each constriction spacing value $S_c \in \{30, 60, 120\}$ mm: a) $\bar{f}_{1,M}(P_1; S_c, P_2)$, b) $\bar{f}_2(P_1; S_c, P_2)$.</figcaption></figure>

601

(Fig. 1 in SM)). The constriction spacing is symmetrical centered around $x = -56.23$ mm.

602

The measured values of $\bar{f}_{1,M}(\mathcal{P})$ and $\bar{f}_2(\mathcal{P})$ are presented in Fig. 14.

603
The impact of $S_c$ on the first resonance frequency $\bar{f}_{1,M}(\mathcal{P}_1)$ becomes more noticeable as $S_c$ increases. However, for $S_c \le 120$ mm, the overall effect on the frequency range of $\bar{f}_{1,M}$ remains limited, with an extension of no more than $\lesssim 50$ Hz, regardless of $\mathcal{P}_2$. It is also observed that varying $\mathcal{P}_1$ can either decrease $\bar{f}_{1,M}$ (e.g., when increasing $\mathcal{P}_1$ from 85% to 90%) or increase it (e.g., when increasing $\mathcal{P}_1$ from 90% to 95%). Thus, compared to the influence of $l_c$ on $\bar{f}_{1,M}$ – which was found (Section VII A) to extend the frequency range by up to $\lesssim 70$ Hz – the influence of $S_c$ is similar or slightly smaller magnitude. Therefore, findings for $\bar{f}_{1,M}$ support the statement in Section IV B 2, that adding a second constriction does not significantly improve the accuracy of the results compared to single-constriction models, particularly when $l_c > 0$ mm.

For closely spaced constrictions ($S_c \in \{30, 60\}$ mm), the second resonance frequency $\bar{f}_2$ remains fairly constant, varying by no more than approximately 70 Hz, regardless of $\mathcal{P}_2$ and $\mathcal{P}_1$. In contrast, for widely spaced constrictions ($S_c = 120$ mm) $\bar{f}_2$ decreases by approximately 200 Hz with increasing $\mathcal{P}_1$, independent of $\mathcal{P}_2$. Furthermore, increasing the spacing from 30 mm to 120 mm reduces $\bar{f}_2$ by around 300 Hz for $\mathcal{P}_1 = 77\%$, and by up to 600 Hz for $\mathcal{P}_1 = 96\%$, regardless of $\mathcal{P}_2$. These results demonstrate that, at high constriction degrees, the spacing between constrictions can have a substantial impact on the observed second resonance frequency $\bar{f}_2$, thereby supporting the conclusion in Section V B 2 and illustrated in Fig. 12 (comparing $i_0^M$ where $\mathcal{P}_{1,2} \ne 0\%$, with $i_{1(2),0}^M$, where $\mathcal{P}_2 = 0\%$).
625

C. Measured resonance frequencies: influence of $l_c$ and $S_c$

<figure><img src="image_16.png" /><figcaption>FIG. 15. Measured resonance frequencies ($\bar{f}_{1,M}, \bar{f}_{2}$): influence of pincher extent length $l_c$ for one constriction configurations (black symbol and $\mathcal{P}$) and constriction spacing $S_c$ for two constriction configurations with $l_{c,1} = l_{c,2} = 0$ mm (non-black symbol and $S_{c\mathcal{P}_2^1}$).</figcaption></figure>

626

Fig. 15 (zoomed views in Fig. 12 of SM) presents an overview of the measured resonance frequencies ($\bar{f}_{1,M}, \bar{f}_{2}$), illustrating the influence of constriction length $l_c$ (one constriction configurations, black) and constriction spacing $S_c$ (two constriction configurations, non-black), as discussed in Sections VII B and VII C, respectively. Overall, the color clusters indicate that these parameters allow exploration of distinct regions within the resonance space ($\bar{f}_{1,M}, \bar{f}_{2}$). This demonstrates the relevance and utility of these parameters within the

627

628

629

630

631
632 SWVT framework.

633

634 The measured first resonance $\bar{f}_{1,M}$ varies from 390 Hz up to 470 Hz, while the second
635 resonance $\bar{f}_2$ spans from 1100 Hz up to 1800 Hz. This frequency range allows to reach
636 up to the ellipsoids associated with American English vowels. However, the region where
637 $\bar{f}_{1,M} \gtrsim 420$ Hz and $\bar{f}_2 \gtrsim 1500$ Hz coincide is not explored by the experimentally assessed
638 parameters. This omission is partly expected, given that the experiments were conducted
639 using a waveguide of fixed length $L_0 = 180$ mm. As discussed in relation to Fig. 3(a) (Sec-
640 tion III A 2), varying $L_0$ enables exploration of the vowel formant space along the direction
641 corresponding to the mean trend $\hat{F}_{2mean}(\hat{F}_{1mean})$, which aligns with the unexplored region
642 observed in Fig. 15.

643

644 VIII. CONCLUSION

645 Overall, the theoretical SWVT resonance space ($\hat{F}_1, \hat{F}_2$) for one constriction, defined by
646 the SWVT subset ($L_0, l_c, x_c, \mathcal{P}$), forms a tilted, butterfly-shaped isosceles trapezium. Its
647 position and overlap with the formant space ($F_1, F_2$) are influenced by the waveguide length
648 $L_0$, which shifts the mean position of ($\hat{F}_1, \hat{F}_2$), and by the minimum constriction length $l_c$,
649 which affects the spacing of the extrema and thus the width of the SWVT resonance space.
650 The same tendencies were observed considering two constrictions. It was demonstrated that
651 when increasing the extend length of at least one of the two pinchers, the SWVT framework
approximates the formants ($F_1, F_2$) of American English vowels using $L_0$ = 180 mm.

This motivates the investigation of selected SWVT parameter subsets for these vowels by introducing a well-defined objective selection criterion (MRI<sub>II</sub>) based on the theoretical resonance frequencies, or resonance invariants, derived from the VT area functions reported by Story, adapted to the geometrical domain of a cylindrical uniform waveguide with a fixed length of $L_0$ = 180 mm. The general features of the SWVT area functions corresponding to the selected parameters under the MRI<sub>II</sub>-criterion, such as severe constriction degree and constriction location, are consistent with those observed in the VT area functions.

However, to fully cover the vowel formants space, the SWVT parameter subset must either include constriction lengths $l_c \ge 0$ mm within one constriction configurations, be extended to include two constriction configurations, or account for both (i.e., two constriction configurations with $l_{c,1(2)} \ge 0$ mm). This underscores the relevance of the parameters in the SWVT framework for examining the relationship between acoustic resonance frequencies and area functions. For future studies it is of interest to compare selected SWVT configurations with other geometric VT models of vowels.

Experimental validation was assessed using rigid and deformable waveguides for selected SWVT parameter sets representing American English vowels /i/, /a/, and /u/. Overall, the experimental results align with theoretical resonance frequencies, thereby validating the selected SWVT parameter sets for both waveguide types. Notably, validation on deformable waveguides is particularly valuable, as it enables the assessment of constriction degrees up to
99%, a range not easily achievable with 3D printed rigid waveguides, and offers the potential for dynamic variation of SWVT parameters for (aero-)acoustic studies.

Beyond the immediate perspectives discussed above, this work opens several promising directions for future research. The SWVT framework provides a systematic tool for exploring the relationship between vocal tract geometry and acoustic resonances, and it can be readily extended to additional vowels, consonants, or dynamic articulatory gestures. Its simple yet flexible parametric representation offers a robust foundation for researchers developing computational or physical models of the vocal tract. Furthermore, integration of the SWVT approach with inverse-modeling or machine-learning techniques may enable prediction of articulatory configurations from acoustic data, with potential impact on speech synthesis, articulatory phonetics, and clinical speech-therapy applications. The methodological concept demonstrated here—controlling resonance frequencies through one or two localized deformations of an otherwise cylindrical waveguide—also suggests applications beyond speech science. Similar principles could inform the design of ducts, pipes, and acoustic enclosures in automotive or aeronautical engineering, where targeted resonance control is essential for noise reduction, sound-quality enhancement, or rapid prototyping. Overall, the SWVT framework not only advances theoretical and experimental studies of speech production but also provides a versatile tool with broader relevance to engineering acoustics.
IX. SUPPLEMENTARY MATERIAL

See supplementary material (SM) at [URL will be inserted by AIP] for a comprehensive
overview of used literature on vowel formants and area functions, additional theoretical
results for the SWVT framework and details of its experimental validation.

697 **X. AUTHOR DECLARATIONS**

698 The authors have no conflicts to disclose.

699 XI. DATA AVAILABILITY

700 Data available on request from the authors.

701 ACKNOWLEDGMENTS

702 Work partly supported by the Full3DTalkingHead project (No. ANR-20-CE23-0008-03).

703 REFERENCES

704 ¹G. Peterson and H. Barney, “Control methods used in a study of the vowels,” J Acoust Soc Am **24**, 175–184 (1952).

706 ²L. Calliope and G. Fant, *La parole et son traitement automatique* (Masson Paris, 1989).

707 ³D. Kewley-Port and C. Watson, “Formant-frequency discrimination for isolated english vowels,” J Acoust Soc Am **95**, 485–496 (1994).
709

710

⁴J. Hillenbrand, L. Getty, M. Clark, and K. Wheeler, “Acoustic characteristics of american english vowels,” J Acoust Soc Am **97**, 3099–3111 (1995).

711

712

713

⁵C. Gendrot and M. Adda-Decker, “Impact of duration on F1/F2 formant values of oral vowels: an automatic analysis of large broadcast news corpora in French and German.,” in *Interspeech 2005* (2005), pp. 2453–2456.

714

715

⁶G. Fant, *Acoustic theory of speech production: with calculations based on X-ray studies of Russian articulations* (Mouton & Co, 1971).

716

717

⁷G. Fant, “The relations between area functions and the acoustic signal,” *Phonetica* **37**, 55–86 (1980).

718

719

720

⁸M. Labrunie, P. Badin, D. Voit, A. Joseph, J. Frahm, L. Lamalle, C. Vilain, and L.-J. Boë, “Automatic segmentation of speech articulators from real-time midsagittal MRI based on supervised learning,” *Speech Communication* **99**, 27–46 (2018).

721

722

723

⁹I. Douros, Y. Xie, C. Dourou, K. Isaieva, P.-A. Vuissoz, J. Felblinger, and Y. Laprie, “3D dynamic spatiotemporal atlas of the vocal tract during consonant–vowel production from 2D real time MRI,” *J Imaging* **8**, 227 (2022).

724

725

726

¹⁰K. Isaieva, F. Odille, Y. Laprie, G. Drouot, J. Felblinger, and P.-A. Vuissoz, “Super-resolved dynamic 3D reconstruction of the vocal tract during natural speech,” *J Imaging* **9**, 233 (2023).

727

728

729

¹¹V. Ribeiro, K. Isaieva, J. Leclere, J. Felblinger, P.-A. Vuissoz, and Y. Laprie, “Automatic segmentation of vocal tract articulators in real-time magnetic resonance imaging,” *Comput Methods Programs Biomed* **243**, 107907 (2024).
730

731

¹²B. Story, "Comparison of magnetic resonance imaging-based vocal tract area functions obtained from the same speaker in 1994 and 2002," J Acoust Soc Am **123**, 327-335 (2008).

732

733

¹³P. Lieberman, "The evolution of human speech: Its anatomical and neural bases," Curr Anthropol **48**, 39-66 (2007).

734

735

¹⁴J. Eichhorn, R. Kent, D. Austin, and H. Vorperian, "Effects of aging on vocal fundamental frequency and vowel formants in men and women," J Voice **32**, 644-e1 (2018).

736

737

738

¹⁵B. Story, H. Vorperian, K. Bunton, and R. Durtschi, "An age-dependent vocal tract model for males and females based on anatomic measurements," J Acoust Soc Am **143**, 3079-3102 (2018).

739

740

¹⁶M. DiNino, "Age and masking effects on acoustic cues for vowel categorization," JASA Express Lett **4**, 060001 (2024).

741

742

¹⁷B. Story, I. Titze, and E. Hoffman, "The relationship of vocal tract shape to three voice qualities," J Acoust Soc Am **109**, 1651-1667 (2001).

743

744

745

¹⁸K. Tom, I. Titze, E. Hoffman, and B. Story, "Three-dimensional vocal tract imaging and formant structure: Varying vocal register, pitch, and loudness," J Acoust Soc Am **109**, 742-747 (2001).

746

747

¹⁹J. Tanner, "Prosodic and durational influences on the formant dynamics of japanese vowels," JASA Express Lett **3**, 085202 (2023).

748

749

²⁰T. Chiba and M. Kajiyama, *The vowel, its nature and structure* (Phonetic society of Japan, 1958).
750

751

²¹H. Dunn, "The calculation of vowel resonances, and an electrical vocal tract," J Acoust Soc Am **22**, 740–753 (1950).

752

753

²²E. L. Saltzman and K. G. Munhall, "A dynamical approach to gestural patterning in speech production," Ecol psychol **1**, 333–382 (1989).

754

755

756

²³S. Maeda, "Compensatory articulation during speech: Evidence from the analysis and synthesis of vocal-tract shapes using an articulatory model," in *Speech production and speech modelling*, pp. 131–149.

757

758

²⁴C. Browman and L. Goldstein, "Articulatory phonology: An overview," *Phonetica* **49**, 155–180 (1992).

759

760

²⁵R. Wilhelms-Tricarico, "A biomechanical and physiologically-based vocal tract model and its control," J Phonetics **24**, 23–38 (1996).

761

762

²⁶B. Story and I. Titze, "Parameterization of vocal tract area functions by empirical orthogonal modes," J Phonetics **26**, 223–260 (1998).

763

764

765

²⁷P. Mokhtari, T. Kitamura, H. Takemoto, and K. Honda, "Principal components of vocal-tract area functions and inversion of vowels by linear regression of cepstrum coefficients," J Phon **35**, 20–39 (2007).

766

767

768

²⁸J. Gaines, K. Kim, B. Parrell, V. Ramanarayanan, S. Nagarajan, and J. Houde, "Discrete constriction locations describe a comprehensive range of vocal tract shapes in the maeda model," JASA express lett **1**, 124402 (2021).

769

770

²⁹J. Lucero, "Parametrization of the vocal tract area function using a subset selection approach," J Acoust Soc Am **149**, 4036–4038 (2021).
771 $^{30}$B. S. Atal, J. Chang, M. Mathews, and J. Tukey, “Inversion of articulatory-to-acoustic
transformation in the vocal tract by a computer-sorting technique,” J Acoust Soc Am **63**,
1535–1555 (1978).

774 $^{31}$S. Ouni and Y. Laprie, “Modeling the articulatory space using a hypercube codebook for
acoustic-to-articulatory inversion,” J Acoust Soc Am **118**, 444–460 (2005).

776 $^{32}$S. Panchapagesan and A. Alwan, “A study of acoustic-to-articulatory inversion of speech
by analysis-by-synthesis using chain matrices and the maeda articulatory model,” J Acoust
Soc Am **129**, 2144–2162 (2011).

779 $^{33}$J. Flanagan, *Speech analysis synthesis and perception* (Springer, 1965).

780 $^{34}$K. Stevens, *Acoustic phonetics* (MIT press, 2000).

781 $^{35}$R. Blandin, M. Arnela, R. Laboissière, X. Pelorson, O. Guasch, A. Van Hirtum, and
X. Laval, “Effects of higher order propagation modes in vocal tract like geometries,” J
Acoust Soc Am **137**, 832–843 (2015).

784 $^{36}$M. Arnela, S. Dabbaghchian, R. Blandin, O. Guasch, O. Engwall, A. Van Hirtum, and
X. Pelorson, “Influence of vocal tract geometry simplifications on the numerical simulation
of vowel sounds,” J Acoust Soc Am **140**, 1707–1718 (2016).

787 $^{37}$J. Kelly and C. Lochbaum, “Speech synthesis,” in *Proc 4th Int congr Acoust* (1962), pp.
1–4.

789 $^{38}$M. Sondhi and J. Schroeter, “A hybrid time-frequency domain articulatory speech synthesizer,” IEEE Trans Acoust **35**, 955–967 (1987).
791

792

³⁹B. De Boer and W. Tecumseh Fitch, “Computer models of vocal tract evolution: An overview and critique,” *Adapt Behav* **18**, 36–47 (2010).

793

794

⁴⁰G. Yeung, S. Lulich, J. Guo, M. Sommers, and A. Alwan, “Subglottal resonances of american english speaking children,” *J Acoust Soc Am* **144**, 3437–3449 (2018).

795

796

⁴¹A. Van Hirtum, R. Blandin, and X. Pelorson, “Validation of an analytical compressed elastic tube model for acoustic wave propagation,” *J Appl Phys* **118**, 224905 (2015).

797

798

⁴²A. Eliraki, F. Vixege, X. Pelorson, and A. Van Hirtum, “Resonance frequencies in rigid and non-rigid squeezed waveguides,” *Revised* (2025).

799

⁴³P. Morse and K. Ingard, *Theoretical acoustics* (Princeton University Press, 1986).

800

801

⁴⁴A. Pierce, *Acoustics: an introduction to its physical principles and applications* (Springer, 2019).

802

803

⁴⁵A. Van Hirtum, R. Blandin, and X. Pelorson, “A setup to study aero-acoustics for finite length ducts with time-varying shape,” *Appl Acoust* **105**, 83–92 (2016).