The Physics of the Human Vocal Folds as a Biological Oscillator

Abstract

The human voice results from the vibration of air at the vocal folds (VF), which behave as a damped oscillator controlled by intraglottic pressure and tranglottic airflow. This chapter presents a complete synthesis of the physics of vocal dynamics (1) during a sustained oscillation, particularly with regard to the phase relationship between intraglottic pressure and glottal opening and closing; (2) during the onset of the oscillation, particularly with regard to the mechanism explaining the triggering of the initiation of the oscillation; and (3) during the decay of the damped oscillations during voice offset, particularly with regard to the effect of lung volume. The importance of air volume as an essential component of the vibratory system is highlighted. The experimental data are obtained in vivo by simultaneous measurement of the transglottic flow and the glottic surface, which allows the calculation of the intraglottic pressure and its interaction with the inertia of the vocal tract.

1. Introduction

The human voice results from the vibration of air at the vocal folds (VF), which behave as a damped oscillator driven by two forces: the transglottic airflow and the intraglottic pressure [1, 2].

Evidence for this oscillator concept is provided by the observation that in high-speed video recordings, at the end of a vocal utterance, when the airflow is not abruptly interrupted by laryngeal closure and the airway remains open. The vocal folds are abducting relative to the median phonatory position and a symmetrical, damped oscillatory motion can be observed on each VF after the last contact of both fold edges on the midline (Figures 1 and 2). The damping results from frictional forces that reduce the energy content of the oscillatory system, thereby reducing the amplitudes of the oscillations as soon as the driving force disappears [3, 4, 5, 6, 7, 8].

Figure 1.

Single-line scan (videokymogram: see “glottic morphometry”) at four levels of the glottis obtained from high-speed video record. Left is the more dorsal part of the vibrating glottis; right the more ventral part. Healthy male subject. End of a/a:/at comfortable pitch (124 Hz) and loudness. Time (about 200 ms for the whole picture) increases from top to bottom.

Figure 2.

Movements of the vocal fold edges, computed from the single-line scans of Figure 1. The damping phase extends over about eight cycles.

The damping dynamics reflect important mechanical properties of the VF, particularly those related to the efficiency of voice emission. Specifically, the decrease in amplitude from one cycle to the next reflects the energy input required to maintain a stable oscillation, since in this situation the work provided by the driving source (pulmonary pressure) exactly compensates for the energy lost in damping.

The concept of a damped oscillator has important clinical implications:

1. It has been shown that the mechanical properties of VF differ constitutionally between normal subjects [9]. Measures of damping could help clarify this concept and identify ‘robust’ (i.e. less fatiguing) voices (essential in professional voice users), or study the effects of training and ageing, for example.

2. It is expected that in several organic larynx pathologies, the properties of the vocal oscillator are somehow altered due to physical changes in the multilayered structure of the VF [10]. The damping characteristics could therefore reflect these changes, resulting in a decrease in vocal efficiency.

In the recent past, numerous mechanical, mathematical and computational models have been developed for analysing of the vocal fold oscillation, either separated or not from the vocal tract dynamics. The various aims pursued ranged from, e.g., deriving the VF oscillations directly from recorded voice signals to explaining mechanobiological processes at cellular and molecular levels [11, 12, 13, 14].

In contrast, the present chapter presents original experimental data obtained in vivo and comprehensive synthesis of the biophysics of the vocal dynamics (1) during the onset of oscillation, particularly as to the mechanism accounting for the start of the oscillation, (2) during sustained oscillation, particularly as to the phase relation between the intraglottal pressure and the glottal opening and closing, and (3) during the decay of the damped oscillations at voice offset, particularly as to the effect of lung volume. A crucial parameter, the intraglottal pressure, is obtained by simultaneous measurements of transglottal flow and glottal area.

2. Biophysics

2.1 Methodological aspects

2.1.1 Intraglottic pressure

Intraglottic pressure is the main driving force of vocal cord vibration [1]. When correlated with VF motion, it is the key parameter for understanding VF biomechanics.

Measuring it directly (via tracheal or transglottic puncture) is difficult and may interfere with spontaneous vocalisation. However, as pointed out by Titze [5, 15], if surface and airflow curves are accurately plotted, the cyclic velocity of air particles can be easily calculated, and in turn, using the principle of energy conservation, the intraglottic pressure during the open phase of the vibration cycle can be calculated. This methodology can be applied to in vivo measurements because it is non-invasive and allows real-time correlations with other parameters, either during regular phonation under controlled conditions or for the study of specific events, such as the voicing onset.

2.1.2 Morphometry of the glottis

High-speed video [16, 17, 18, 19, 20] requires laryngoscopy with an endoscope capable of providing sufficient illumination (300 W xenon lamp) for image frequencies of 2 to 4 kHz with a sufficient resolution (preferably at least 2000 x 2000 pixels). Single-line scanning (videokymography) is an imaging technique based on a special digital camera mounted on a 90° rigid (Wolf 4450.57; CE 0124) laryngeal telescope with a focusing handle [21, 22]. In high-speed mode, the video camera provides single-line images selected across the entire image at a rate of approximately 7875/7812.5 line images per second and with a resolution of 720 x 1/768 x 1 pixel. The resulting high-speed image, also called a ‘videokymogram’, shows the cycle-by-cycle oscillatory pattern of the selected small portion of the VF (Figure 1).

High-speed imaging combined with an analysis programme [19] greatly facilitates the measurement of glottal area parameters. However, even at 2000 frame/s, the definition remains limited to approximately 15 samples/cycle [23].

The glottal area can also be derived very accurately from a photometric record obtained by transilluminating the trachea. The luminous flux is detected by a photovoltaic transducer located in the pharynx. The transducer, a BP104 silicon photodiode (Vishay Precision Group, Malvern, PA), is glued to a small laryngoscopic mirror (No. 3) (Figure 3). The current generated by the photodiode is pre-amplified by a current-to-voltage converter with a linear and flat frequency response up to 2 kHz. As the VF vibrate, the photovoltaic transducer produces a current that is directly proportional to the light flux through the glottis, i.e. the glottic surface. Unlike airflow, light flux through the glottis is difficult to calibrate because the absolute value of the light intensity detected by the photodiode in the pharynx is very sensitive to minute changes in the position of the sensor relative to the glottis and cannot be reliably maintained in a perfectly stable position beyond the duration of a single utterance. Therefore, only individual utterances can be recorded. However, it can be assumed that the amplitudes within a short utterance, e.g. at the beginning or end of the utterance, can be validly compared. Depending on the type of search, light signals can be expressed as a percentage of the maximum signal value, between 0 and 100% [2].

Figure 3.

Diagram of combined flow, photometric and acoustic measurements.

2.1.3 Flow glottography

Rothenberg’s ‘flow-glottograph’ is a high-speed pneumotachograph (a fast differential pressure transducer) that consists of a specially designed mask (Figure 3) and an inverse filtration system. The ‘Rothenberg mask’ (MSIF2 Glottal Enterprises, Syracuse, NY) is widely used to analyse the waveform of glottic volume velocity. The effects of vocal tract resonances can be cancelled by filtering the oral airflow recorded at the lips. The mask is fitted with a compressible seal and must be pressed firmly against the subject’s face to prevent any air leakage [24, 25, 26, 27, 28].

2.1.4 Intraglottic pressure calculation

The intraglottic pressure P can be calculated from the transglottic flow rate and the air particle velocity (= flow/surface) based on Bernoulli’s law of energy [2, 11, 29]:

$$P+\mathrm{1/2}\rho v^2=constant$$E1

where ρ is the density of the fluid and v is the velocity of the particles (Figure 4) [2, 15].

Figure 4.

Graphic simulation of a single vibration cycle of the vocal folds in a typical normal phonation of a male subject (modal register). The two upper traces represent the glottic surface area and the airflow as a function of time. Both signals increase upwards. The middle trace represents the shape of the air particle velocity waveform, obtained by dividing the (skewed) airflow waveform by the displacement waveform. Only the open part of the vibration cycle is shown. The lower trace is the waveform of the intraglottic pressure, computed on the basis of Bernoulli energy law. (Adapted from Titze).

However, when the glottis is open, the intraglottic pressure is influenced by the supraglottic sound pressure, which changes the overall pressure distribution in the glottis [2]. In fact, the above equation applies to a convergent glottic duct, i.e., upstream of the glottic constriction, whereas for a divergent glottic duct, i.e., downstream of the constriction, where separation of the airflow from the wall and vortices may occur, the inertance equation is as follows:

$$P=IdU/dt$$E2

where I is the supraglottic acoustic inertance and U is the airflow (Figure 5). The inertance of an air column is defined as the density of the air multiplied by the length of the column (in the direction of acceleration or deceleration) divided by its cross-section (perpendicular to the acceleration or deceleration). Inertance is the effect of inertial forces opposing the transmission of vibrations by the supraglottic air column, i.e., the resistance to movement. It can be calculated as follows: [30].

Figure 5.

Match of the maximal glottic area during the vibration cycle with the calculated contour of an ellipse of similar major and minor axes. The glottal contour is traced out from a real videostroboscopic picture “frozen” shortly after a soft voice onset.

$$I=\rho L/S$$E3

where S is the cross-sectional area of the supraglottic air column, and L is its effective length. The units are g.cm⁻⁴ or kg.m⁻⁴ (1 g.cm⁻⁴ = 10⁵ kg.m⁻⁴). L and S can be considered constant during the emission of a sustained vowel, as was the case in our experiments. However, this divergent form of the glottic channel appears mainly at higher subglottic pressures, when the vibration cycle is characterised by a long-closed phase and a significant phase difference between the lower and upper edges of the VFs. When subglottic pressures are low, as in voice onset, the vertical glottic channel should be shorter [31, 32], and the shape differentiation (convergent/divergent) – including the phenomena of airflow separation from the glottic wall and vortex formation – should be less pronounced than during sustained modal phonation, or even absent. It is therefore reasonable to assume that the mean motor pressure (from bottom to top) is close to the Bernoulli pressure, estimated numerically over the glottic surface at the position where the glottis is narrowest. It can be shown that when the airflow curve is shifted to the right relative to the glottic area curve, the intraglottic pressure during the opening phase exceeds that of the closing phase (Figure 4) [2].

This asymmetry results from the compressibility of the air and the inertia of the vocal tract. In this context, the closed phase of the vibration cycle is irrelevant, and the calculation of the intraglottic pressure does not make sense; the limits of the open phase will be explained in the next section.

2.1.5 Vocal fold contact

Translaryngeal electrical impedance [33] is measured by using alternating current at a frequency larger than 100 kHz and monitors changes in the contact surface of the VFs (electroglottography; EGG). This method is non-invasive and does not interfere with vocalisation. It allows precise phonetic tasks to be performed under acoustic control. However, the sensitivity of detecting very small transglottic impedance changes (essential in this context) depends on the design of the electronic circuitry. The original design by Fourcin and Abberton [34] has been replaced by newer devices using a higher carrier frequency, more efficient oscillator control, multipolar filters with a sharper cut and flat bandwidth (e.g. F-J Electronics, Denmark; Laryngograph, UK; Synchrovoice Research, USA; etc.), resulting in a better signal-to-noise ratio and a greater sensitivity with wider bandwidth and linearity [35]. It has been shown [36] that the EGG signal can be as sensitive as the flow-glottogram in detecting very small vocal cord oscillations, but unlike the flow signal, it may not show the very first movements when there is no contact between the VFs. Improved devices can detect small sinusoidal EGG cycles before true contact occurs along the entire length of the VFs [20, 35]. These small sinusoidal EGG cycles probably correspond to small periodic impedance fluctuations at the acute angle of the VFs commissure.

2.1.6 Acoustic signal

A small condenser microphone (Ø 5.6 mm) can be attached to the side of the Rothenberg mask, fitting exactly into a mask opening in front of the pressure transducer (Figure 3). The sound pressure levels of the speech samples were evaluated using the PRAAT software (www.praat.org). The sound levels of the microphones were calibrated using a Wärtsilä 7178 sound level metre in a position corresponding to a direct measurement 10 cm from the lips.

2.1.7 Glottal area calculation

As explained above, the luminous flux of the transilluminated trachea is detected by a photovoltaic transducer located in the pharynx. The relative amplitudes of the oscillatory signal are sufficient for some types of studies, e.g., to deal with damping, but other experiments need absolute values of the glottic surface, which requires valid and accurate calibration [32].

The open part of the vibration cycle is the essential part, consisting of an opening phase and a closing phase. The beginning of the opening phase and the end of the closing phase must therefore be clearly defined because when the airflow and the glottal area are close to zero, their quotient, i.e., the airspeed, no longer makes sense and is therefore unusable. We considered, using the method of Gerratt et al. [37], that these limits occur when the ascending and descending traces intersect a horizontal line 90% below the positive peak. This line is parallel to a line drawn between the negative peaks before and after the maximum opening of the positive peak. It is not possible to obtain a fixed quantitative regression line between the current produced by the photodiode and the actual glottal zone, as the current also depends to a large extent on the precise position of the photodiode in the pharynx, which varies from one recording to another. However, as stated above, the relationship can be considered linear and stable for a single controlled voice utterance.

In order to measure the glottic surface and calibrate the photoglottographic signal, it is first necessary to know the ventrodorsal length of the vibrating glottis, which can be assumed – for a given vocalist – to be stable in the frequency range of 100–125 Hz. This ventrodorsal length of the glottis during a vibration cycle is constant for a modal vowel emission at controlled F₀ and can be measured (in mm) on a videostroboscopic image obtained from the same subject producing a similar vocal sound.

To obtain this reference, a rigid 90° Wolf laryngeal telescope (4450.57; CE 0124) and an ATMOS Strobo 21 LED strobe (Atmos Medizin Technik, Lenzkirch, Germany) were used. This telescope is equipped with a magnifying device and has a small depth of field and a critical sharpness adjustment; a piece of scale paper was filmed at the same focal length, with special care for maximum sharpness. This is based on Fex et al. [38], who used a microscope and calculated a maximum measurement error of 4.65 ± 3.10%.

With the 90° telescope used here and its magnifier option, the maximum range of sharpness was found to be 3–4 mm at a distance of 40–45 mm. The ventrodorsal length of the glottis was therefore estimated to be 13 mm, which corresponds to the values found by Larsson & Hertegard [39] using a laser triangulation method. The light signal can be expressed as a fraction of the maximum amplitude at full glottic aperture.

In computational fluid dynamics, an important parameter is the ‘equivalent diameter’ (compared to a cylindrical tube) [40]. The contour of the glottic image can be well fitted by an ellipse (see below, under the heading 2.3.1. Triggering the oscillation), whose major and minor axes are respectively the ventrodorsal length and the maximum width of the glottic image. This is illustrated in Figure 5.

In contrast to the length (which is stable in modal phonation), the maximum glottal width is strongly correlated (male subjects, modal register) with the intensity of the vocal emission [41, 42].

Maximum glottic area is calculated directly from the photometric signal after image-based calibration with an accuracy of 5–10%. The determination of the maximum closing velocity uses the first derivative of the glottal area, which requires a high-quality, noise-free signal and a high sampling frequency. On this point, our photometric method far outperforms imaging techniques. High-speed video imaging is limited by the number of pixels (resolution) but especially by the sampling frequency, as shown in the experiments of Horacek et al. [43], where, for example, at 2000 frames/s, 100 Hz F₀ and a closed quotient of 0.5, only 5 points can be measured during the closure phase.

2.2 Sustained oscillation dynamics

A positive energy transfer from the airflow to the tissue can only be achieved if the net aerodynamic driving force has a component that is in phase with the tissue velocity (i.e. the first derivative of the displacement, i.e. with a phase advance of 90° to the displacement) [32]. Using a model in which the intraglottic pressure P is calculated from the transglottic flow and the velocity of air particles based on Bernoulli’s energy law.

$$P+\mathrm{1/2}\rho v^2=constant$$E4

(ρ is the density of the fluid, and v is the velocity of the particles), it can be shown that when the curve of the airflow is tilted to the right with respect to the curve of the glottic surface (Figure 4), the intraglottic pressure during the opening phase is higher than that during the closing phase. This asymmetry is due to the compressibility of the air and the inertance of the vocal tract.

However, this is only a first approximation of the driving force on the tissue, as it assumes a laminar and incompressible flow that remains attached to the glottic wall and whose viscous loss is negligible. Bernoulli’s law is not valid for compressible flows (variable air density), but the glottic air flow can be considered incompressible for Mach numbers <0.3 [44]. In fact, the rate of air compression at the glottis is limited: for spoken voices, volume changes due to air compression are in the range of 1 to 2% (10–20 hPa subglottic pressures). More importantly, the separation of the wall flow and the formation of vortices in a divergent glottic duct also result in a deviation from Bernoulli’s law. Pressure does not fully recover in an expanding (divergent) duct (expansion angle >5°) [45].

In the adult larynx, during modal exhalation phonation, the glottis assumes three successive forms during each opening phase: convergent, uniform and divergent (Figure 6), the uniform form being only the brief transition between the convergent and divergent forms. The estimation of the intraglottic pressure according to Bernoulli’s law can be considered valid during the opening phase. If the flow is separating from the glottic wall and if vorticity occurs during the closing phase, the intraglottic pressure is influenced by the supraglottic sound pressure, which changes the overall distribution of pressure in the glottis. When flow separation and supraglottic sound pressure are taken into account, the intraglottic pressure can theoretically be divided into two parts, one upstream and one downstream of flow separation,

Figure 6.

Schematic frontal section at midpoint of the glottis showing the two parts of the open phase: The opening phase, during which the glottal duct is convergent, and the closing phase, during which it is divergent. The drawing also shows the lateral progression of the mucosal wave during the closing phase.

$$Upstream\left(convergent\right):P=P_s–k\left(1/2\right)\rho v^2$$E5

(derived from Eq. (1))

where P_s is the subglottic pressure (the pulmonary pressure generated by the contraction of the expiratory muscles and/or the recoil of the thoracic elastic elements), and k is a pressure loss coefficient for the glottic entrance and viscous drag. The value of k was set at 1.37 according to van den Berg et al. [46] and Fulcher et al. [47] for soft to moderate phonation.

$$Downstream\left(divergent\right):P=IdU/dt$$E6

where I is the supraglottic acoustic inertance and U is the airflow.

Downstream of the flow separation, dU/dt is mainly positive during the opening of the glottis (when the flow increases) and mainly negative during the closing of the glottis (when the flow decreases). As defined above, the inertance of an air column is defined as the density of the air multiplied by the length of the column and divided by its cross-section. The inertance I is given by:

$$I=\rho L/S$$E7

where S is the mean cross-sectional area of the epipharynx and L is the effective length of the air column over the vocal cords. Inertia can be thought of as the density of a column of air per unit length. L and S can be considered constant during the emission of a sustained vowel, as is the case in our experiments.

Expressed in terms of Newton’s second law of motion, which states that force = mass x acceleration, Eq. (6) means that:

$$Vocaltractinletpressure=\left(inertance\right)\times \left(aircolumnacceleration\right)$$E8

The force is the analogue of the inlet pressure, the mass is the analogue of the inertance, and the acceleration remains the same. The values of L and S were chosen according to Titze [30].

It is impossible to define exactly when and to what extent Eqs. (5) and (6) are applicable, but it can be expected that Eq. (6) will have a greater weight.

Since the work of van den Berg et al. [46, 48], much attention has been paid to the possibility of negative values of intraglottic pressure during the closing phase. However, from a mechanical point of view, the only condition for maintaining the vocal folds in oscillatory motion is that the driving force during closing (including tissue recoil) be less positive than during opening and that the net driving force over the entire cycle is sufficient to overcome the frictional forces. The important point is the asymmetry of the pressure curve between the opening and closing parts of the cycle. Intraglottic pressure values can be quantified during the opening phase under different intensity conditions using in vivo calibrated flow and surface measurements and applying Eq. (5) (upstream flow separation) and Eq. (6) (downstream flow separation).

A 4-channel Pico Scope 3403D (Pico Technology Ltd., St Neots, England, UK) was used for recording all signals, which were stored on a PC.

Figure 7 shows a typical example of such a polygraph recording (raw, uncorrected) during stable phonation (two cycles). In addition to the flow and glottal area signals, the sound oscillogram and electroglottogram are also shown, the latter confirming VF contact. The flow curve (flowglottogram) is slightly tilted to the right.

Figure 7.

Typical example of a (raw, uncorrected) polygraphic record during steady state phonation (two cycles). Beside the airflow and the glottic area traces, the sound oscillogram and the electroglottographic trace are also shown, the latter confirming vocal fold contact. The flow curve (FGG) is slightly skewed to the right.

Three typical emission conditions are selected for detailed analysis: 62.35, 68.60 and 74.70 dB (10 cm from the lips), at an average emission fundamental frequency of approximately 110 Hz.

The surface, flow and intraglottic pressure curves for the three typical conditions are shown in Figures 8–10. The surface curves define the separation between the opening and closing phases. The intraglottic pressure was calculated using Eq. (5) (upstream) and Eq. (6) (downstream). Regardless of the condition of vocal emission and the equation used, the average intraglottic pressure is systematically larger during the opening phase (convergent duct) than during the closing phase (divergent duct). The results confirm in vivo the data obtained by modelling: over a complete cycle, the driving force produces a net positive work that accounts for the sustained VF movement. When the flow curve is asymmetric and skewed to the right with respect to the glottic surface curve, the intraglottic pressure is systematically higher during the opening phase than during the closing phase, regardless of the glottic duct configuration, which is the essential condition for maintaining the oscillation. This asymmetry is due to the compressibility of the air and the inertance of the vocal tract. Quantitatively, the intraglottic pressure becomes negative during the closing phase [32]. It is important to note that the general tendency of the average intraglottic pressure to decrease during the open phase of the cycle corresponds to the tendency of the tissue velocity to decrease; in other words, the displacement of the tissue has a phase lag of π/2 radians (in ideal conditions, without friction) with respect to the driving force.

Figure 8.

Top to bottom: Glottic area, airflow and intraglottic pressure calculated by Eqs. (5) and (6). The closed phase is very short and limited skewing of the flow trace is visible. With Eq. (5) as well as with Eq. (6), the area under the pressure curve is obviously larger during the opening than during the closing phase, even becoming negative during the closing phase. Maximum intraglottic pressure is about 5 hPa.

Figure 9.

Top to bottom: Glottal area, airflow and intraglottic pressure computed by Eqs. (5) and (6). To be compared with Figure 8. The maximum intraglottic pressure is about 15 hPa.

Figure 10.

Top to bottom: Glottal area, airflow and intraglottic pressure computed by Eqs (5) and (6). To be compared with Figures 8 and 9. The maximum intraglottic pressure is about 21 hPa.

Regardless of the configuration of the glottic channel, calculations based on measured values of the glottic surface and air flow show that the integrated intraglottic pressure during the opening phase systematically exceeds that of the closing phase, which is the basic condition for maintaining vocal fold oscillation. The crucial point is that the airflow curve is right-skewed with respect to the glottic surface curve.

2.3 Dynamics of voice onset

It is generally accepted that there are three main categories of vocal onset (or ‘attack’): soft (or ‘coordinated’), hard and respiratory (or ‘aspirated’): The vibration of the VF can start from either a closed glottis (hard onset) or an open glottis (soft or breathy onset) [17, 49]. In a normal subject, the most commonly observed type of voice onset in spontaneous speech is the soft onset. A typical example of a soft onset (slightly breathy) at a comfortable pitch and volume, as observed in four videokymograms obtained from a high-speed video, is shown in Figure 11: the VF oscillation starts from a spindle-shaped glottis (Figure 12). It is not possible to determine whether the very first movement is medial (closing) or lateral (opening) because it is too small, but the VFs are never ‘sucked’ together. The amplitude of the glottic oscillations increases very gradually until there is a first contact between the edges of the VFs (Figure 11). After a very short initial contact on the midline, the duration of the closure phase gradually increases. This phenomenon is also visible on a single-line scan at the midpoint of the glottis length (Figure 13a). In Figure 13b (hard onset), the glottis is closed when the first oscillation occurs; again, the duration of the closing phase gradually increases. In such cases of hard onset, period irregularities or asymmetries are often observed in the first cycles. The number of cycles of a vocal onset can vary considerably, mainly depending on the degree of breathiness (aspiration), which results from both the importance of the expired airflow and the adduction speed of the VFs. In general, during a soft or breathy onset, the amplitude of the oscillations measured on the photometric signal gradually increases over 2 to more than 30 cycles before reaching the first clear closed plateau. As soon as contact occurs, the sinusoidal oscillation is interrupted and the amplitude of the signal decreases slightly. The differences between a soft and a hard onset are clearly visible in Figures 14 and 15. In the case of a hard onset (Figure 14), the amplitude of the oscillations also increases gradually, but the number of cycles with increasing amplitude is generally lower. In the case of a soft onset (Figure 15), the first oscillation is usually detected on the flow plot, immediately followed by the area plot, but it is not systematic: the air column and the vocal fold edge seem to form together a single oscillator. The airflow sensor may be slightly more sensitive. Changes in electrical impedance (contact with the vocal cords) occur later (Figure 15). In case of a breathy onset (Figure 16), the pattern is similar to that of a soft onset, but the progression is slower: the amplitude of the oscillations gradually increases over ten cycles and more.

Figure 11.

VKG at four equidistant levels of the vibrating glottis, obtained from highspeed video. Soft, somewhat breathy onset. Time increases from top to bottom. /a:/; healthy male subject (~125 Hz; 65 dB at 10 cm).

Figure 12.

Spindle-shaped glottis immediately before a soft onset (snapshot from high-speed video).

Figure 13.

VKG at midpoint of VF length, with corresponding images from highspeed video just before the onset of vibration. Left (a): Soft onset; right (b): Hard onset. A soft onset starts with an open glottis, and a hard onset with a closed glottis.

Figure 14.

Hard onset. From top to bottom: Flowglottogram, electroglottogram, photoglottogram and sound oscillogram. Comfortable pitch and loudness (125 Hz; 65 dB at 10 cm). Raw tracings. Oscillation starts from a closed glottis.

Figure 15.

Soft onset. From top to bottom: Photoglottogram, ultrasonic signal, electroglottogram and flowglottogram. Comfortable pitch and loudness (~125 Hz; 65 dB at 10 cm). Raw tracings. Oscillation starts from an open glottis.

Figure 16.

Breathy onset. From top to bottom: Flowglottogram, electroglottogram, photoglottogram. Comfortable pitch and loudness (~125 Hz; 65 dB at 10 cm). The pattern is similar to that observed in a soft onset, but the progression is slower: The amplitude of oscillations progressively increases over more than ten cycles. Raw tracings.

To a certain extent, the onset mirrors the offset, where an oscillatory movement of the damped vocal folds can be observed, unless the vocal emission is interrupted by a closure of the glottis.

The ratio between the intraglottic pressure during the opening phase and that during the closing phase must be >1, so that for a whole cycle, during the first free oscillations of the VFs, the pressure affects a positive work. In a soft onset, this ratio gradually increases before the first closed phase [32]. This is in line with the results obtained in steady-state phonation under different loudness conditions [2]: at low intensity (minimal closed plateau), the intraglottic pressure ratio is close to 1, while in loud voices it increases to about 6 when the closed quotient (duration of closed time/duration of period) exceeds 0.5.

Figure 17 gives an example of the progress of the intraglottic pressure ratio during the first six cycles in soft onset.

Figure 17.

Soft onset (125 Hz; 65 dB at 10 cm) From top to bottom: Glottal area, transglottic airflow and intraglottic pressure. An increasing phase lead (slightly less than 90°) of the intraglottic pressure with respect to the glottal opening is observed. Time is in ms.

In a state of stable oscillation, the movement of the tissue should show a phase shift of π/2 radians (under ideal conditions, without friction) with respect to the driving force. In fact, as soon as a closed phase appears, all the signals undergo a significant distortion compared to their original sinusoidal form, which masks the phase difference. However, during a soft onset, a gradual increase in the phase shift from about 0° to about 90° can be observed over a few cycles (Figure 17).

The underlying explanation lies in the gradual increase in asymmetry (skewing to the right) of the airflow curve relative to the glottic surface curve.

2.3.1 Triggering the oscillation

A critical moment in a soft voice onset is the transition from a motionless, spindle-shaped glottis (as in Figure 12), crossed by a continuous expired flow, to a damped oscillating system [29]. A key element is the occurrence of turbulence in the airflow, as indicated by the dimensionless Reynolds number. Low values of this number indicate that viscous forces are dominant and that the flow is laminar (sheet-like), characterised by steady and constant fluid motion. High values indicate that viscous forces are low and that the flow is essentially turbulent, dominated by inertial forces that tend to produce chaotic vortices, eddies and other instabilities. The Reynolds number is used to predict turbulence and the transition from laminar to turbulent flow.

The Reynolds number for flow in a duct or cylindrical pipe is calculated by (Eq. (9)):

$$\begin{array}{l}Re=velocity\left(m/s\right)\times pipeinternaldiameter\left(m\right)/\\ kinematicviscosityoffluid\left(m^2/s\right).\end{array}$$E9

In general, for values of Re < 2000, the flow is stationary and laminar. For values between 2000 and 4000, the system is in transition, characterised by increasing coherent two-dimensional vortices, merging of vortices, separation of vortices and a turbulent state in three dimensions. For values of Re > 4000, the fluid is non-stationary and three-dimensional [29, 50].

Since the shape of the glottis is not cylindrical, it is necessary to find an expression to calculate an equivalent diameter of the glottis to be introduced into Eq. (9). For this purpose, videostroboscopic recordings of the subject’s glottis were made under phonation conditions similar to those used for aerodynamic measurements. A fixed strobe image at the time of maximum aperture during phonation (i.e., as soon as the strobe is triggered by the sound emission) provides an adequate measurement of glottic dimensions at this critical time. After calibration, the ventrodorsal length of the glottis was 13 mm and the maximum width was 3 mm, corresponding to the values found by Larsson & Hertegard [39]. Actually, the contour of the glottis image can be well-fitted by an ellipse whose major and minor axes are the ventrodorsal length and maximum width of the glottic image, respectively. This is illustrated in Figure 6. In this case, the difference between the calculated area of the ellipse and the measured area of the glottis is less than 1%. The advantage of using an ellipse to describe the recorded curves is that from the glottic area at the time of onset, given by the photometric signal, and taking the constant ventrodorsal length of the glottis (13 mm) as the major axis of the ellipse, the minor axis can be easily calculated using the formula for the area of the ellipse.

The equivalent diameter (for an ellipse) is [51]:

$$ed.=1.55A^{0.625}/P^{0.25}$$E10

where A and P are the area and perimeter, respectively. Several formulas are commonly used to calculate the perimeter of the ellipse, but they give an approximate result, generally valid within a limited range of ratios of the two axes of the ellipse, narrower than the values obtained for the glottis. We therefore use an exact formula that uses an infinite number of terms. A handy online calculation tool based on such a formula is available at https://www.mathsisfun.com/geometry/ellipse-perimeter.html [52].

The result was used to calculate the equivalent diameter according to Eq. (10), which made it possible to calculate the Reynolds number according to Eq. (9), where the kinematic viscosity is 1.6 × 10⁻⁵ m²/s. This was done for each record.

2.3.2 Presence of turbulence in the intraglottic airflow

Figure 18 shows an example. From top to bottom: airflow (Rothenberg mask), electroglottogram and light signal (photoglottography), proportional to the glottic surface, during a soft onset. The total duration of the recording is 124 ms. On the area trace, the level at which the oscillation begins and the maximum amplitude of the area (100%) are indicated by vertical arrows.

Figure 18.

From top to bottom: Airflow, electroglottogram and glottic area (photoglottography) during a soft onset (area increases upwards). Total duration of the record: 124 ms. Vertical arrows on the area trace indicate the level at which oscillation starts and the maximal amplitude.

Figure 19 shows the same recording. In this case, the flow level at which the oscillation begins is indicated by the vertical arrow on the airflow plot, relative to the baseline reached when a complete closure of the glottis is observed.

Figure 19.

From top to bottom: Airflow, electroglottogram and glottic area (photoglottography) during a soft onset (same as Figure 18). Total duration of the record: 124 ms. The vertical arrow on the airflow trace indicates the flow level at which oscillation starts with respect to the baseline reached when complete glottal closure is observed.

The average Reynolds number calculated for 72 records is 3073 ± 479 (average ± SD).

Figure 20 shows a plot of the equivalent diameter (mm) of the glottis, calculated from the measured glottic area, as a function of the velocity of the air particles (m/s) at the start of the oscillation. A strong negative correlation (R = −0.80; p < .0001) is found, and the hyperbolic shape of the regression curve suggests that the velocity varies inversely with the equivalent diameter of the glottis. Their roughly constant product corresponds to a Reynolds number of approximately 3000.

Figure 20.

Plot of the equivalent diameter (mm) of the glottis, calculated from the measured glottic area, against the velocity of air particles (m/s) at the start of oscillation. A strong negative correlation (R = −0.80; p <.0001) is observed, and the hyperbolic shape of the regression curve suggests that the velocity varies as an inverse function of the equivalent glottal diameter. Their approximately constant product corresponds to a Reynolds number of around 3000.

2.3.3 The neutral (atmospheric) pressure at the intraglottic level at the onset of vibration

Figure 21 shows an example of the method used to estimate subglottic pressure. The upper trace is the sound recorded by a microphone, and the lower trace is the intraoral pressure measured by a Millar catheter. A single vocalisation – on a series of ten – is represented (repetitions of the syllable/pi/). As the lips open, the intraoral pressure drops suddenly to approximately atmospheric pressure and rises as the lips close. Some cycles of oscillation persist, even with greater amplitude, when the lips are already closed. This phenomenon is hidden when using a simple manometer but is made visible here thanks to the wide bandwidth of the Millar transducer. The phonation threshold pressure measured in this case is 2.46 hPa. The average phonation threshold pressure value for 15 soft onsets is 2.52 (standard deviation 1.78) hPa. This value is comparable to the mean value found by Jiang et al. [17] in normal subjects at low intensity: 2.38 ± 1.273 cm H₂O. It is therefore reasonable to assume that the estimated pulmonary pressure at the start of the oscillation is close to the phonation threshold pressure.

Figure 21.

Example of the method of estimation of subglottal lung pressure. The upper trace is the sound recorded by a microphone and the lower trace is the intraoral pressure measured by a Millar catheter. One single vocalization—out of a series of ten—is shown (repetitions of the syllable /pi/). At the moment of lip opening, the intraoral pressure suddenly drops to approximately the atmospheric pressure and increases as soon as the lips close again. A few oscillation cycles persist—even with a larger amplitude—when the lips are already closed. This phenomenon is hidden when a simple manometer is used, but it is made visible here due to the large bandwidth of the Millar transducer. The PTP measured in this case is 2.46 hPa. The average value of PTP for 15 soft onsets is 2.52 (SD 1.78) hPa.

The mean air velocity at the beginning of the oscillation is 16.74 (SD 1.81) m/s.

The resulting pressure drop is

$$\Delta P=-\left(16.74m/s^2.k.1.14kg/m^3\right)/2=-219Paor-2.19hPa\left(SD1.26\right)$$E11

which is close to the phonation threshold pressure value.

2.4 Vocal offset dynamics

At the end of a vocal utterance, a damped oscillatory movement can be observed on each vocal fold (VF) after the last phase of contact of VF edges on the midline [36]. The decrease in amplitude from one cycle to the next reflects the energy input required to maintain a stable oscillation. A rapid repetition (3 to 4 s⁻¹) of a vowel followed by an abrupt bilabial occlusion (e.g., /ɛpɛpɛpɛpɛpɛp/) at a comfortable fundamental frequency and intensity is a convenient protocol for analysis [53]. The oscillating system itself consists of two elements: the two VFs and the air masses of the lower and upper airways. The size of the vibrating mass of VF tissue can be roughly estimated based on magnetic resonance imaging. The thickness and width of each vibrating fold are approximately 4 and 5 mm respectively. The vibrating length, as seen on videostroboscopic images, is approximately 16 mm (male subject, modal register, comfortable pitch and volume). Thus, 0.5 g is a reasonable estimate of the upper limit of total vibrating tissue mass in vivo (2 VFs). In a female subject, 0.35 g can be expected. A rough hypothesis is that modal speech occurs with an average lung volume slightly above the upper limit of the current volume.

The volume of vibrating internal air thus represents about 50% of the vital capacity (half of 3000 to 4500 ml), to which must probably be added a large part of the residual volume (on average 1.1 to 1.2 l) and the supraglottic vocal tract (about 75 ml). The total weight of vibrating air can be estimated to be approximately 2.7 to 3.7 g (1.14 g/l), which is significantly higher than the high estimate of VF vibrating mass. Varying the volume of vibrating air would verify its importance for the damping characteristics. This can be done by comparing two conditions, high and low pulmonary volume voting, while applying the above protocol. The hypothesis is that increasing the volume of air vibrated by the VFs (by about 2.5 litres) should improve the mechanical quality of the whole vibrating system, resulting in lower damping when the driving force is abruptly removed.

Figure 22 shows an overview of a polygraphic recording of a single vocalisation/pɛp/ in the ‘high lung volume’ condition. The /pɛp/ is extracted from a sequence of /ɛpɛpɛpɛpɛpɛ…/ at a rate of three to four sound emissions per s. The vowel /ɛ/ is determined by the constraints of the oral and pharyngeal sensors. The F₀ is about 130 Hz and the intensity is about 64 dB (at 10 cm). The estimated subglottic pressure is 4.9 hPa.

Figure 22.

Global view of a polygraphic recording of a single vocalization /pɛp/ in the ‘high lung volume’ condition. The /pɛp/ is extracted from a /ɛpɛpɛpɛpɛp…/ sequence at a rhythm of three to four vocalizations per s. Fo is around 130 Hz and intensity around 64 dB (10 cm). Subglottal pressure (estimated) is 4.9 hPa.

The logarithmic decrement is defined as the natural logarithm of the ratio of the amplitudes of two consecutive positive peaks: (ln [x_n/x_n + 1]). The overall mean logarithmic decrement is 0.72 ± 0.31 in the ‘high lung volume’ condition (n = 212 logarithmic decrements) and 0.88 ± 0.26 in the ‘low lung volume’ condition (n = 133 logarithmic decrements). This difference is highly significant (p < .001).

It is thus possible, with the appropriate methodology, to control, normalise and quantify the damping characteristics of the vibrating system (VF tissue and air mass) during a physiological offset with sudden interruption of airflow. This allows the role of lung volume to be specifically investigated. The mechanical quality of the global oscillating system appears to be determined to a large extent by the lung volume: a reduction in air volume leads to a significant increase in the rate of decay of the vibrations, resulting in higher energy demand for voice emission [54].

2.5 What is the vocal oscillator actually made of, and what determines its properties?

The few experimental data available on the damping characteristics of the vocal folds outside the context of phonation, either in vivo [55] or on excised larynx [56, 57], indicate a high damping ratio after an external pulse (the oscillation stops after 2 cycles). This is in stark contrast to observations of phonation shifts recorded on high-speed film: Figure 1 shows a four-level (ventral to dorsal) videokymogram (single line scan) of the vibrating glottis obtained from high-speed video. The recording was made at the end of a supported /a:/ in a healthy male subject. Due to the persistence of some airflow, the total damping transient extends over at least 20 cycles, beginning with a gradual shortening of the closed phase, while the VF still make contact on the midline (Figures 23 and 24). The vocal context therefore seems to play an essential role.

Figure 23.

VKG at four levels of the vibrating glottis obtained from high-speed video. Left halves of pictures correspond to the more dorsal part of the vibrating glottis, and right halves to the more ventral part. Healthy male subject. End of a somewhat breathy /a:/at comfortable pitch and loudness. Due to persistence of some airflow and slow vocal fold abduction, the damping phase spans over at least 20 cycles, starting with a progressive shortening of the closed phase.

Figure 24.

Movements of the vocal fold edges, computed from the videokymograms of Figure 23. Upper traces correspond to the more dorsal part of the vibrating glottis, and lower traces to the more ventral part.

Another parameter that can intervene is morphological: ideally, the morphology of the oscillator should remain constant during the damping phase. In reality, from a certain degree of abduction, the morphology of the vibrating masses changes considerably, the lip shape of the VF disappears and the VF ‘flattens’ laterally. The magnitude of this non-linear change seems to depend mainly on the degree of abduction.

Recently, damping has also been observed during inspiratory phonation [58]: the characteristics seem similar to those of expiratory phonation.

It has been shown that the mechanical properties of VFs differ constitutionally between normal subjects [9]. Measurements of damping could help clarify this concept and to identify ‘robust’ voices (i.e. less prone to fatigue), which is essential in the field of professional voice use, or to study the effects of, for example, training and ageing.

It is also expected that in some organic VF pathologies, the mechanical properties of the vocal oscillator will change due to physical changes in the stratified structure of the VF [10]. Therefore, the damping characteristics could reflect these changes, with – as a consequence – a reduction in vocal efficiency.

Finally, the damping of the vocal oscillator is a non-conscious objective phenomenon that cannot be controlled voluntarily by the subject, and thus escapes simulation. This aspect makes it particularly interesting in a medical-legal context for individuals seeking compensation for loss of vocal function in the event of an injury or an occupational dysphonia [59].

3. Conclusion

Damping is probably an important issue in the pathophysiology of the voice and perhaps a valuable clinical parameter for various applications. Several methods have been used to record the damping phase of vocal cords oscillations at the end of vocal emission.

Currently, available non-invasive techniques (such as flow glottography and photoglottography) appear to be well-suited for this purpose. High-speed video and video-kymography (combined with image processing and analysis software) are clearly superior but are not widely used at present due to their high cost. In addition, recent developments (with a flexible transnasal scope) have made the examination much more comfortable for the subject/patient. However, the main problem for a reliable assessment of damping is not the technique but the variability of characteristics related to laryngeal and respiratory behaviour at the end of voice emission. The definition of a simple protocol to standardise voice emission seems illusory. The only solution therefore seems to be the integral recording by high-speed video of a standardised passage read by the subject, with automatic extraction a posteriori of all the damping phases and calculation of the average damping coefficient by ad hoc software. The question remains, of course, whether the information provided will make this sophisticated approach attractive to the clinician.