# Flow Dynamics and Acoustics from Glottal Vibrations at Different Frequencies

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{0}), while the normal pressure force increased nonlinearly with the frequency (~f

_{0}

^{1.85}). Instantaneous pressure signals were irregular at low vibration frequencies (10 and 20 Hz) but became more regular with increasing frequencies in the pressure profile, periodicity, and wavelet-transformed parameters. The acoustic characteristics specific to the glottal vibration frequency were explored in temporal and frequency domains, which may be used individually or as a combination in diagnosing vocal fold dysfunction, snoring, sleep apnea, or other breathing-related diseases.

## 1. Introduction

^{2}. As high as 1568 Hz has been reported by Echternach et al. [11] using a high-speed trans-nasal endoscope. Even for the same person, glottal vibrations can differ significantly in magnitude and waveform [12]. Some of these differences are related primarily to differences in vocal fold anatomy, while others may arise from how the vocal folds are stressed. In addition, both complete and incomplete glottal closure was observed during the phonation [10].

## 2. Materials and Methods

#### 2.1. Study Design

_{0}, a wide range of f

_{0}was considered, including 10, 20, 50, 100, 200, and 500 Hz. The transient flows induced by the vibrating glottis were characterized in terms of velocity, stream traces, and vortex structures. The pressure forces acting on the glottal wall were examined on their roles in either promoting or suppressing the glottal vibration. To understand the glottal vibration frequency on flow-induced acoustics, the instantaneous pressure values at 0.3 m from the mouth opening were analyzed using different transform algorithms, as presented in Section 2.4.

#### 2.2. Dynamic Glottal Model

^{2}and the minimal area being 96.5 mm

^{2}(Figure 1a). This ratio fell within the range of 1.16–1.8 in normal adults [32,33,34]. To make sure that the dynamic glottal aperture transited smoothly to the static airway walls both upstream and downstream, sinusoidal functions were implemented to connect with the larynx 4 mm above and 6 mm below the glottis (Figure 1a), which conformed with the glottis model described in Wu and Zhang [35].

#### 2.3. Numerical Methods

_{m}is the average velocity (i.e., 0.66 m/s), r is the radius from the mouth opening center, and R is the mouth opening radius (i.e., 11 mm). The outflow distributions for the twenty-three bronchiolar outlets followed the measurement by Cohen et al. [36]. Note that the lung model in this study was reconstructed from a cadaver-based lung cast used in [36]. The respiratory flows were solved using large eddy simulations (LES), which have been demonstrated to accurately capture flow evolutions and turbulent-laminar transitions [37,38]. The subgrid-scale variables were resolved using the wall-adapting local eddy-viscosity (WALE) model [37].

_{ij}was the Lighthill stress tensor. The variables n

_{i}, δ(f), and H(f) were the normal vector of the wall, Dirac delta function, and Heaviside function, respectively. The three groups on the right-hand side represented the monopole, dipole, and quadrupole sources, respectively. Fast Fourier transformation (FFT) was applied to transform the time series pressure signals into the frequency domain. The sound pressure level (SPL, unit: dB) was calculated as SPL = 20 log(p/p

_{ref}), with the reference pressure p

_{ref}= 2 × 10

^{−5}Pa [40]. The A-weighted sound pressure level (unit: dBA) was calculated using an A-scale function to better mimic the perception of human ears, which are more sensitive to signals between 500 and 6000 Hz [41]. The power spectral density (PSD, unit: Pa

^{2}) denotes the signal power distribution in the frequency domain and is defined as the Fourier transform of the autocorrelation function [42].

^{−6}s for all test cases and the pressure was recorded every 1.0 × 10

^{−6}s at 0.3 m from the mouth opening [46]. A UDF-controlled dynamic mesh model was implemented in the glottal region, with the glottis shape and computational mesh updated at each time step. For each case, 20 vibration cycles were simulated. The SPL and PSD were computed from the recorded time series of pressure signals as a post-processing procedure in ANSYS Fluent using the fast Fourier transform module.

#### 2.4. Hilbert and Wavelet Transform Analyses

_{P}into varying scales to seek correlations, which are not that obvious in the original C

_{P}profiles. A series of wavelets were implemented to scale and shift the signal f(t):

## 3. Results

#### 3.1. Airflow Dynamics

_{g}using the inhalation flow rate Q

_{inh}(15 L/min) and vibration frequency f

_{0}: [(Q

_{g}− Q

_{inh})/(Q

_{inh}f

_{0})]. The normalized (scaled) flows were asymmetric relative to the baseline, which skewed toward the main flow direction and most likely resulted from the flow inertia. In addition, the glottal flow fluctuations (Q

_{g}− Q

_{inh}) were approximately linear with the glottal vibration frequency.

#### 3.2. Forces on the Glottal Walls

_{0}

^{1.85}, which successfully collapsed the scatters of all frequencies (Figure 5c). The exponent of 1.85 could be attributed to the dominant flow inertia effect (i.e., V

^{2}) vs. less dominant viscous effects.

_{0}

^{0.1}) and a stronger dependence on f when f

_{0}≥ 100 Hz (f

_{0}

^{0.14}). As discussed in Figure 2, this difference might result from the more complex vortex patterns in the near-wall region at high frequencies.

#### 3.3. Sound Pressure and Acoustic Analyses

#### 3.3.1. Pressure Signals at the Receiver (Mouth Opening)

#### 3.3.2. Hilbert and Wavelet Transform Analyses

_{h}and holder exponent h for higher frequencies. Similar observations were made in Figure 9b, with a wider range of the scaling exponent λ at a higher vibration frequency. The increase rate of the λ-range was nonlinear, which increased faster at low frequencies and slower at high frequencies (Figure 9b).

_{0}), the sound pressure dropped quickly along the frequency f axis, and there was apparent periodicity along the time axis. Meanwhile, high levels of disturbances existed, presumably due to turbulence-generated fluctuations.

#### 3.3.3. Fast Fourier Transform (FFT) Analyses

_{0}= 10–500 Hz. The reference pressure was 2 × 10

^{–5}Pa. Both normal and semi-logarithmic plots were presented. When f

_{0}≥ 20 Hz, harmonics occurred at the low end of acoustics frequency (f) with descending amplitudes, while no obvious harmonic was detected when f

_{0}= 10 Hz. At the high end of acoustics frequency (f > 2000), the SPL magnitudes became random regardless of the glottal vibration frequency f

_{0}(Figure 11a–f).

_{0}= 20, 100, and 500 Hz). For a given f

_{0}, the first SPL peak occurred at the vibration frequency (f

_{0}, forcing frequency). It was followed by three or more harmonics with descending sound energy. Similar observations were made in the plots of the power spectral density (PSD, lower panels of Figure 12a–c). Note the large differences in the PSD magnitude among the three cases, which was three orders of magnitude higher when increasing f

_{0}from 10 to 100 Hz and two orders of magnitude higher when increasing f

_{0}from 100 to 500 Hz.

_{0}) increased the perceived loudness both at individual acoustic frequencies (f) and total sound loudness (i.e., area under the curve between 20 and 20,000 Hz).

_{0}= 20 Hz were decomposed into f-specific components. Figure 14a,b show the f-localized signals that are 2, 3, and 4 times that of the forcing frequency f

_{0}(i.e., f/f

_{0}= 2, 3, 4) at normal and semi-logarithmical scales, while Figure 14c, d show those at f/f

_{0}= 50, 100, 200. At lower acoustic frequencies (f/f

_{0}= 2, 3, 4), the sound signals had larger amplitudes, smaller variations in magnitude, and were periodic; while these were not true at high acoustic frequencies (f/f

_{0}= 50, 100, 200), exhibited high levels of variations. In addition, the amplitude of the first harmonic (f/f

_{0}= 2) is higher than all other f-localized signals.

## 4. Discussion

#### 4.1. Flow Responses to Glottal Vibrations

_{G}− Q

_{0})/Q

_{0}f) nicely collapsed the Q

_{G}data into approximately one curve, indicating that the inertance term ωI was dominant in the glottal region. Theoretically, the inertance I is a fluid-geometrical property and can be expressed as ρL/A, with ρ being the air density, and L and A being the glottal length and cross-sectional area, respectively [54].

#### 4.2. Is the Glottal Vibration Self-Sustained?

_{g}is proportional to the flow change rate, i.e., p

_{g}= I·dQ/dt, where I is the glottal inertance. At the initial stage of glottal contraction (first quartile, Figure 15), the flow rate increased (i.e., dQ/dt > 0), leading to a positive intra-glottal pressure p

_{g}, and therefore opposing the glottal contraction. In the second quartile, the glottal contraction continued, but the flow rate decreased (i.e., dQ/dt < 0) to the inertance-induced phase shifted; this resulted in a negatively valued p

_{g}, which assisted the glottal contraction to the minimum aperture. In the third quartile, the glottis started to expand, and the glottal flow rate decreased (i.e., dQ/dt < 0), leading to a negative p

_{g}and resisting the expansion process. The glottis continued to expand in the fourth quartile, but the flow rate started to increase (dQ/dt > 0), and the positive p

_{g}promoted the expansion to the maximal aperture (Figure 15). These theoretical results were verified by the numerically predicted pressure forces acting on the glottal wall in Figure 5c, which opposed the glottal motion in the first and the third quartiles and assisted the motion in the second and fourth quartiles of the vibration cycle. Considering that the negative work in the first/third quartiles and the positive work in the second/fourth quartiles may cancel each other, reaching a net zero work, a self-sustained vibration is possible. In this case, an existing vibration, regardless of the forces that started it, will continue and remain self-sustained till an anti-work slows it down or a positive work speeds it up.

#### 4.3. Effect of Glottal Vibrations on Acoustics

_{0}, signature features were obtained for certain ranges of f

_{0}. The 10 Hz multifractal spectrum had a much narrower range than other vibration frequencies (Figure 9). Likewise, the power spectral density, and to a lesser degree the sound power level, was closely related to the glottal vibration frequency, especially for f

_{0}= 100 Hz and above (Figure 12). A variation in vibration frequency is often a direct result of alterations in the structure and/or material properties, as demonstrated by Lodermeyer et al. [27] and Kimura et al. [58]; as such, the vibration-frequency-specific responses are promising to be used as a diagnostic tool, individually or as a combination, in vocal fold dysfunction, snoring, sleep apnea, or other breathing-related diseases [59].

_{0}(source of the perturbation, 10–500 Hz herein), and the other was the acoustic frequency (f

_{a}or simply f) of the flow-generated sound (0–20,000 Hz). In Figure 14, acoustic-frequency-specific (f-localized) signals were also decomposed from the acoustic spectrum at f

_{0}= 20 Hz. Because of the two-way structure–vibration association, such acoustic f-localized signals have also been widely applied in diagnostics, prognosis, and monitoring [60].

#### 4.4. Limitations

## 5. Conclusions

_{0}), while the normal pressure force increased nonlinearly with the frequency (~f

_{0}

^{1.85}).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Computational airway model and validation against in vitro experiments: (

**a**) Upper respiratory airway model with a rhythmically contracting/expanding glottis, (

**b**) model validation against the experimental measurement of pressure in the airway cast with three glottal apertures.

**Figure 2.**Respiratory flow dynamics in the upper airway with different glottal vibration frequencies (f

_{0}): (

**a**) 10 Hz, (

**b**) 20 Hz, (

**c**) 50 Hz, and (

**d**) 200 Hz.

**Figure 3.**Flow evolution during a glottal vibration cycle at f

_{0}= 200 Hz: (

**a**) contraction, (

**b**) expansion, and (

**c**) time evolution of the flow rate at different slices in comparison to the sinusoidal waveform. There was a phase advance of the flow during expansion and a phase lag during contraction relative to the glottal sinusoidal waveform.

**Figure 4.**Dynamics of glottal flow at different vibration frequencies: (

**a**) glottal flow rate increases with the vibration frequency, and (

**b**) normalized glottal flow rate relative to the frequency.

**Figure 5.**Pressure forces acting on the glottal walls: (

**a**) diagram of the normal forces acting on the right and left lateral walls, (

**b**) the pressure force in the first and third quartiles of a vibration cycle facilitates the glottis motion (self-sustaining), while that in the second and third quartiles work against the motion (anti-self-sustaining). (

**c**) The pressure force has a nonlinear relationship with the vibration frequency (f

_{0}

^{1.85}).

**Figure 6.**Shear forces acting on the glottal walls: (

**a**) the variation in the shear force acting on the glottal walls in one vibration cycle, with the shear force visualized at three instants (arrows) of the 200 Hz case. The shear force, τ does not obey a continuous relationship with the glottal vibration frequency, (

**b**) with a very weak τ dependence on f when f ≤ 50 Hz (f

_{0}

^{0.1}) and (

**c**) a much higher dependence on f

_{0}when f ≥ 100 Hz (f

_{0}

^{0.14}).

**Figure 7.**Pressure signals (blue line) at 0.3 m from the mouth opening computed using the FW-H model: (

**a**) 10 Hz, (

**b**) 20 Hz, (

**c**) 50 Hz, (

**d**) 100 Hz, (

**e**) 200 Hz, and (

**f**) 500 Hz. The red line is the sinusoidal motion of the glottis.

**Figure 8.**Phase space based on Hilbert transform of the sound signals at 0.3 m from the mouth opening with different glottal vibration frequencies: (

**a**) 10 Hz, (

**b**) 20 Hz, (

**c**) 50 Hz, (

**d**) 100 Hz, (

**e**) 200 Hz, and (

**f**) 500 Hz. The red ellipse in (

**a**) is the sinusoidal waveform.

**Figure 9.**Multifractal analyses of the pressure signals for varying glottal vibration frequencies: (

**a**) multifractal spectrum and (

**b**) distribution of scaling exponents.

**Figure 10.**Waterfall spectrogram of the sound pressure signals at different glottal vibration frequencies: (

**a**) 10 Hz, (

**b**) 20 Hz, (

**c**) 50Hz, and (

**d**) 100 Hz.

**Figure 11.**Sound pressure level (SPL) of the pressure signals at different glottal vibration frequencies: (

**a**) 10 Hz, (

**b**) 20 Hz, (

**c**) 50 Hz, (

**d**) 100 Hz, (

**e**) 200 Hz, and (

**f**) 500 Hz.

**Figure 12.**Sound pressure level (

**upper panel**) and power spectral density (

**lower panel**) in the low-frequency range at different glottal vibration frequencies: (

**a**) 20 Hz, (

**b**) 100 Hz, and (

**c**) 500 Hz.

**Figure 13.**A-weighted sound power level (unit: dBA) of the sound signals generated at different glottal vibration frequencies at (

**a**) normal scale and (

**b**) logarithmical scale.

**Figure 14.**Frequency localization of the acoustic signals generated at the glottal vibration frequency of 20 Hz: f-localized signals that are 2, 3, and 4 times that of the forcing frequency at (

**a**) normal scale and (

**b**) logarithmical scale, and those that are 50, 100, and 200 times that of the forcing frequency at (

**c**) normal and (

**d**) logarithmical scale. The amplitude of the first harmonic (f/f

_{0}= 2) is higher than all other f-localized signals.

**Figure 15.**Responses of the glottal flow rate and intra-glottal pressure (P

_{g}) to one cycle of the glottal vibration at f

_{0}= 200 Hz.

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## Share and Cite

**MDPI and ACS Style**

Xi, J.; Talaat, M.; Si, X.; Dong, H.
Flow Dynamics and Acoustics from Glottal Vibrations at Different Frequencies. *Acoustics* **2022**, *4*, 915-933.
https://doi.org/10.3390/acoustics4040056

**AMA Style**

Xi J, Talaat M, Si X, Dong H.
Flow Dynamics and Acoustics from Glottal Vibrations at Different Frequencies. *Acoustics*. 2022; 4(4):915-933.
https://doi.org/10.3390/acoustics4040056

**Chicago/Turabian Style**

Xi, Jinxiang, Mohamed Talaat, Xiuhua Si, and Haibo Dong.
2022. "Flow Dynamics and Acoustics from Glottal Vibrations at Different Frequencies" *Acoustics* 4, no. 4: 915-933.
https://doi.org/10.3390/acoustics4040056