Regulation of laryngeal resistance and maximum power transfer with semi-occluded airway vocalization

Abstract

Steady airflow resistances in semi-occluded airways as well as acoustic impedances in vocalization are quantified from the lungs to the lips. For clinical and voice training applications, the primary focus is on two airway conditions, an oral semi-occlusion and a semi-occlusion above the vocal folds. Laryngeal airflow resistance is divided into glottal airflow resistance and epilaryngeal airway resistance. Maximum aerodynamic power is transferred to the vocal tract if the glottal airflow resistance is reduced while the epilaryngeal airway resistance is increased. A semi-occlusion at the lips helps to set up this condition. For the acoustic power transfer, the epilaryngeal airway also serves to match the impedance of the source to the impedance of the vocal tract.

I. INTRODUCTION

Human vocalization begins with a newborn cry. The cry is often characterized by high intensity on the order of 99–120 dB(A) at 46 cm (18 in.) from the mouth (Carney, 2014; Calderon et al., 2016). The loud cry is also associated with roughness, an added factor for demanding attention. The emitted sound shows evidence of a chaotic vibration of tissues (Mende et al., 1990). If high intensity is a primary survival requirement for an infant cry, it would be expected that motor organization for obtaining the maximum acoustic output need not be learned (Hirschberg et al., 2009). It would be innate. Nevertheless, it is appropriate to question if this motor organization remains intact into adulthood. Speech, the primary context for vocalization in adulthood, is likely organized for maximum information transfer rather than maximum power transfer (Shannon, 1948; Nyquist, 1924).

Maximum power output in vocalization depends on at least two factors: (1) the power produced at the source and (2) the effectiveness of transmitting this power to the receiver. In voice production, both aerodynamic power and acoustic power are transmitted from the larynx through a portion of the airway prior to emission from the mouth. Unvoiced labial, dental, and labiodental consonants require aerodynamic power to be delivered to the terminal end of the vocal tract (Krane, 2005), whereas vowels require acoustic power to be delivered to the input of the vocal tract. It is not yet clear whether aerodynamic (non-oscillatory) power transfer and acoustic (oscillatory) power transfer are based on similar airway organization. In simple language, we ask if the ability to produce a powerful stream of air out of the mouth (as in blowing out a candle) is related to the ability to produce a loud sound?

The pharynx in infants is significantly shorter than that in adults. In reference to adult anatomy, the infant has no oropharynx but mainly a hypo-pharynx, which, here, is called the epilaryngeal airway because it is bordered anteriorly along its entire length by the epiglottis, which is considered part of the larynx. The epiglottis in infants and young children is relatively long, floppy, and narrow. It reaches all the way from the thyroid cartilage to nearly the velum (Truby and Lind, 1965). With the mouth wide open in a cry, it appears that the primary supraglottal airflow resistance is in the epilaryngeal airway. When not crying, the infant explores with other airway semi-occlusions, such as /m/, /b/, and lip trills, all of which provide airflow resistance during voicing.

The typical duration of an infant cry is about 2 s (Wilder and Baken, 1978). In this cry, approximately 20% is inspiration time and 80% is expiration time. The typical subglottal pressures in a cry are in the range 2–10 kPa and the typical crying vital capacity is 0.13 L for an infant 50 cm in length (Sutherland and Ratcliff, 1961; Doershuk and Mathhews, 1969). If this crying vital capacity is expelled in 1.6 s (80% of 2 s), the mean airflow is 0.081 L/s. The total airway resistance can, therefore, range between 2.0 kPa/(0.081 L/s) = 25 kPa per L/s and, possibly, up to 100 kPa per L/s. This is an extremely high value in comparison to the airway resistances in adult humans. Glottal resistances range on the order of 1–10 kPa per L/s in adults (Konnai et al, 2017), an order of magnitude lower. A rough estimate of the infant glottal resistance can be obtained by assuming a total kinetic pressure drop across the glottis,

$$R_g=\hspace{0.17em}½\rho \left|U\right|/A_g^2,$$

where ρ is the air density (0.001 g/cm³), U is the glottal airflow, and A_g is the mean glottal area in the vibration. Assuming the mean glottal area of the infant to be the product of a 0.05 cm mean glottal width and 0.4 cm glottal length, the glottal area A_g is 0.02 cm². With these estimates, the glottal order-of-magnitude resistance is estimated as (0.5)(0.001)(80)/(0.02)² = 100 (dyn/cm²)/(cm³/s) = 10 kPa per L/s. This is at the low end of the range of the total estimated airway resistance in a cry, suggesting that a vocal tract resistance (and, specifically, an epilaryngeal airway resistance) may contribute substantially to the total airway resistance.

The basic postulate in this paper is that matching the supraglottal airway resistance to the glottal resistance for a maximum power transfer is phylogenetically established in humans and other mammals. Whereas adults do not need a maximum power transfer in conversational-level speech, they are likely to be able to access the motor patterns quickly in a call, a shout, or while singing. It does require a strong interaction between the sound source and resonator, however, this is not a necessity for conversational-level speech and is, therefore, not practiced much.

The linear source-filter theory, validated primarily for adult male speech production, postulates that the source of sound and the vocal tract filter are independent, and there is no strong interaction between them. Such independence can be an advantage in speech because it allows large variations in the filter for vowel and consonant articulation with minimal disturbance of the sound source. The source-filter independence has a disadvantage, however, for vocalizations in which a maximum acoustic output power is desirable, such as in calling, shouting, and many forms of singing. It is well known from the design of signal and energy transmission systems that maximum energy is transferred when the system blocks are interactive (Kong, 1995). In the language of a transmission-line theory, maximum power is transferred from the source to the load (the termination where the power is needed) when the source impedance is a match to the load impedance (Titze, 2002). The current paper is a reexamination and extension of this principle of maximum power transfer introduced by this author nearly 20 years ago. Basic aspects of the nonlinear source-filter theory have since been quantified (Titze, 2008), which invites a new look at power transfer. The frequency spectra have been calculated for various degrees of interaction, but aerodynamic and acoustic power transfer calculations have only recently been reexamined (Krane, 2005; Zhang, 2016; Titze, 2018).

Classical aerodynamic investigations in voice and speech science have focused primarily on pressure-flow relations across portions of the airway, such as the glottis, velum, lips, and other constricted sections along the airway. It is known that, in adults, lung pressures on the order of 1–3 kPa produce volumetric airflows on the order of 0.1–1 L/s during vocalization or whisper (Rothenberg, 1977; Hixon, 1987; Holmberg et al., 1988; Konnai et al., 2017). These airflows are constant from lungs to lips for a stationary airway configuration because the airway walls are nonporous. The pressures inside the airway can vary greatly, however, with different vocal tract configurations. Airflow resistances have been characterized for the glottis (van den Berg et al., 1957; Scherer et al., 1983; Bielamowicz et al., 1993; Alipour and Jaiswal, 2009; Herbst et al., 2015; Zhang, 2016), the velum (Netsell et al., 1991), and the larynx, including the false folds (Agarwal et al., 2003; Agarwal et al., 2004). In addition, aerodynamic power produced by the lungs has been quantified (Schutte, 1980). Some of this aerodynamic power is transferred to the vocal folds to sustain vibration (Thomson et al., 2005), but much of it is absorbed by resistances in the vocal tract. Little is known to date about the distribution of aerodynamic power from the lungs to lips and how that distribution might affect the acoustic power transfer. The aerodynamic power transfer may be a clue toward better understanding the highly successful semi-occluded vocal tract (SOVTs) exercises for vocalization (Stemple et al., 1994; Titze, 2006b; Laukkanen et al., 2008; Laukkanen et al., 2012, Titze and Abbott, 2012).

For relatively steady airflow, the aerodynamic power transfer from the lungs to lips is estimated to be some fraction of the lung pressure multiplied by the airflow. This fraction can range from a few milliwatts to more than a watt (Schutte, 1980), depending on the lung pressure and pressure distribution along the airway. It is currently not clear what aerodynamic pressure and power distributions are most supportive of the sound source. In speech, every vowel and consonant affects these distributions to some degree, but we ask if there are downstream resistances that support self-sustained oscillation of the vocal folds. Is there an optimal relation between the aerodynamic power transfer and acoustic power transfer? Some speech and voice trainers promote so-called “flow phonation” (Patel et al., 2020; Sundberg, 2020), an attempt to increase the acoustic power and efficiency by increasing the peak-to-peak airflow through the glottis and, therewith, the maximum flow declination. Others promote semi-occlusion of the airway as an exercise to make the source more efficient (Kapsner-Smith et al., 2015; Guzman et al., 2015; Croake et al., 2017; Meerschman et al., 2017). Both may be an attempt to get a better impedance match between the source and airway.

The questions addressed in this paper are (1) can a controlled airway resistance at the lips facilitate a balance between the resistances of the glottis and epilaryngeal airway for a maximum aerodynamic (non-oscillatory) power transfer from the source to vocal tract?, and (2) does the resulting aerodynamic impedance match provide a better acoustic match between the source and airway? The paper is an extension of the paper by Titze (2002), wherein the maximum power transfer concept was introduced for vocalization. The current paper is divided into two parts because the methodologies are rather different for the two questions. Of particular interest in the impedance matching strategy are the mechanisms of control available to the vocalist for altering airway resistances and the power transfer.

II. PART 1. AERODYNAMIC POWER TRANSFER

The aerodynamic power transfer through the vocal tract is relevant for any expiratory activity that involves one or more airway resistances downstream of the lungs. It involves steady airflow, but vorticity, jet formation, and turbulence are included in the empirical pressure-flow relations. Not included is the oscillatory airflow, resulting from the vocal fold vibration. Thus, in the language of speech science, everything is unvoiced, such as whispered speech and all unvoiced consonants in normal speech. Here, the unvoiced condition is guaranteed by using a one-mass vocal fold model with a uniform (rectangular) glottis in conjunction with a purely resistive (nonreactive) vocal tract. For this condition, the oscillation threshold pressure is infinite (Titze, 1988), resulting in no self-sustained oscillation.

A. Methods

The methods of this investigation are based on the aerodynamic resistances cascaded in series. In this approach, airflow in a duct or pipe is analogous to the electric current in a circuit, pressure is analogous to the voltage, and airflow resistance is analogous to the electrical resistance in the current flow. This method is, at best, an approximation to solutions that involve the laws of momentum conservation and mass transfer (Krane, 2005; Bailly et al., 2010), but it simplifies the mathematics considerably. Specific velocity fields are not calculated; rather, empirical relations are used for pipe airflow. Figure 1 shows a simple aerodynamic circuit for the airways, where P_L is the lung pressure, R_t is the tracheal resistance, R_g is the variable glottal resistance, R_e is the epilaryngeal resistance, R_s is the supra-layngeal resistance (pharynx to lips), and R_L is the additional lip resistance that can represent a semi-occlusion (Maxfield et al., 2015; Smith and Titze, 2017; da Silva et al., 2019). There is no radiation resistance for steady airflow.

FIG. 1.

The flow circuit diagram for the airway, lungs to lips.

The sum R_t + R_g is considered to be the source resistance, and the sum R_e + R_s + R_L is considered to be the vocal tract resistance. If all resistances were constant (not time or airflow dependent), the maximum power transfer theorem (Kong, 1995), also known as Jacobi's law, would apply. This theorem states that for constant resistances, maximum power is transferred from the source to the vocal tract if

$$R_{\mathrm{t}}+R_g=\hspace{0.17em}{R}_e+\hspace{0.17em}{R}_s+\hspace{0.17em}{R}_L.$$ (1)

The resistances are not constant, however. There is generally a viscous component and a kinetic component, the second of which is flow dependent. Furthermore, the glottal resistance is time dependent because it changes with the vocal fold displacement. In this first analysis (part 1), we are looking for the steady airflow conditions such that self-sustained oscillation of the vocal folds is suppressed. As stated above, this condition is guaranteed for a one-mass vocal fold model with uniform tissue displacement and a vocal tract that is resistive only (no air inertance or compliance).

Smith and Titze (2017) have measured the airflow resistance of the cylindrical tubes for dimensions and pressures in the range of those of human airways, 1.8–9.7 mm inner diameters, 3.0–24.0 cm lengths, 0–7.0 kPa applied pressure, and 0–1.5 L/s resulting airflow. The full collection of measurements was represented by the empirical formula

$$R=\left(3.7631\times {10}^{-7}\frac{Lt}{D^{4.4997}}+1.0268\times {10}^{-6}\frac{1}{D^{4.0416}}\right)U+\left(3.9913\times {10}^{-9}\frac{Lt}{D^{5.0089}}+8.0169\times {10}^{-7}\frac{1}{D^{3.7696}}\right),$$ (2)

where Lt is the tube length (m), D is the tube diameter (m), and U is the airflow (L/s). The first term is the kinetic component and the second term is the viscous component. The density and viscosity of the air are numerically included in the coefficients. The resistance R is expressed in Pa per L/s with a mean accuracy of ±6%.

The glottal resistance is dominated by the kinetic component because the glottal length is relatively short. It has been quantified by Scherer et al. (1983) as

$$R_g=\hspace{0.17em}{k}_{t}1/2\rho \left|U_g\right|/A_g^2,$$ (3)

where k_t is a transglottal pressure coefficient, ρ is the air density (0.00114 g/cm³), U_g is the glottal airflow, and A_g is the glottal area. For a uniform (parallel surface) glottis, the glottal area is further quantified in terms of the vocal fold length L, postural glottal half-width x₀, and vocal fold surface displacement x(t),

$$A_g=2L\left[x_0+\hspace{0.17em}x\left(t\right)\right].$$ (4)

According to Scherer and Guo (1990), the transglottal pressure coefficient k_t for a uniform glottis of diameter D and transglottal pressure P_tg can be quantified as

$$k_t=\hspace{0.17em}\left(0.049D^{-1.487}\right)P_{tg}^{(0.2334\hspace{0.17em}\text{lnD}+0.346)},$$ (5)

where

$$P_{tg}=\hspace{0.17em}{P}_L-\left(R_t+R_e+R_s+R_L\right)U.$$ (6)

In Eq. (4), the glottal area is a function of the postural glottal width x₀ and the time-varying glottal width x(t), which expresses the expansion and contraction and becomes a constant value when the steady state has been reached after the application of a constant lung pressure.

With these resistances, the steady airflow equation becomes

$$P_L=\left(R_t+\hspace{0.17em}{R}_g+R_e+\hspace{0.17em}{R}_s+\hspace{0.17em}{R}_L\right)U,$$ (7)

and the vocal fold displacement equation is

$$M\hspace{0.17em}{d}^{2}x/d{t}^2+B\hspace{0.17em}dx/dt+Kx=P_{g}LT,$$ (8)

where M is the mass, B is the damping coefficient, K is the stiffness, and T is the thickness of the vocal folds. P_g is the driving pressure applied to the medial surface LT. The driving pressure is taken as the average between the glottal entry pressure P_L – (R_t + R_en)U and glottal exit pressure P_L – (R_t + R_g)U, where R_en is the entry resistance from the trachea to the entrance of the glottis. According to Scherer et al. (1983), this entry resistance is approximated as

$$R_{\text{en}}=1.371/2\rho \left|U_g\right|/A_g^2.$$ (9)

The driving pressure on the vocal folds now reduces to

$$P_g=P_L-\left(R_t+\hspace{0.17em}0.5\left(R_{\text{en}}+\hspace{0.17em}{R}_g\right)\right)U.$$ (10)

The damping coefficient B is further defined in terms of a damping ratio ζ, the mass M, and stiffness K of the vocal folds,

$$B=2\zeta {\left(MK\right)}^{0.5}.$$ (11)

The numerical values for the parameters in Eq. (11) are specified in Table II, based on measurements summarized in Titze (2006a, Chap. 2). Finally, the power W₁ that is delivered to the source is

$$W_1=U^2(R_t+R_g),$$ (12)

and the power W₂ that is delivered to the vocal tract is

$$W_2=U^2\left(R_e+R_s+R_L\right).$$ (13)

The tracheal resistance R_t and supra-laryngeal resistance R_s are small in comparison to the laryngeal resistances R_g and R_e, but they are included in the calculations for completeness. The lip resistance R_L can be very small (mouth open) or very large (small diameter tube between the lips or a bilabial fricative).

Equations (2)–(13) above equations were programmed in matlab. The differential equation [Eq. (8)] was split into two first-order equations and solved with a conventional fourth-order Runge-Kutta solver at a sampling rate of 44 100 Hz. The lung pressure P_L was turned on gradually with a 10 ms cosine window. All calculations were performed in the centimeter-gram-second (CGS) system of units because they match the dimensions of the system, but lung pressures, airflows, and airflow resistances are reported in units typically used in respiratory physiology and voice science (kPa, L/s, and kPa per L/s).

Only a few cycles of damped vocal fold oscillation were expected because the one-mass vocal fold model has no vertical phase difference in movement, the vocal tract has no inertive reactance component, and the glottal width is relatively large. All three of these conditions raise the phonation threshold pressure (Titze, 1988). Hence, no self-sustained oscillation was expected. Steady-state conditions were reached in 50 ms as shown in Fig. 2. Here, the steady-state lung pressure was 1.5 kPa and a flow-resistant lip tube of 10 cm length and 6.2 mm diameter was used to get a rough balance between the vocal tract load resistance with the source resistance [Fig. 2(c)]. The initial transient oscillations resulted from the second-order equation [Eq. (8)], which describes the movement of the vocal folds. This condition resembles a whisper or an unvoiced fricative with a narrow mouth opening.

FIG. 2.

(Color online) The time response of the aerodynamic flow circuit for a lip tube diameter of 6.2 mm and lung pressure of 1.5 kPa.

The natural frequency of the damped oscillation was 160 Hz, which is verifiable by calculating f_o = (K/M)^0.5/(2π), according to the stiffness and mass values given in Table I. Note that with this vocal tract resistance (essentially, the lip tube resistance), the glottal width increased and airflow reached 0.65 L/s. The source and tract resistances were in the range reported by Konnai et al. (2017) for a whisper. The power delivered to the tract was slightly greater than the power consumed by the source. In subsequent graphs in Sec. II B, the final steady-state values of the glottal width, airflow, resistances, and aerodynamic power at 0.05 s are reported. The nominal values of the circuit parameters are given in Table I, based on previous low-dimensional modeling (Story and Titze, 1995). All cross-sectional areas were assumed to be circular except for the glottal area, which was rectangular.

TABLE I.

The nominal values for the variable parameters and constant values. L, vocal fold length; T, vocal fold thickness; M, vocal fold mass; K, vocal fold stiffness; ζ, damping ratio; x₀, glottal half-width; P_L, lung pressure; D_t, tracheal diameter; D_e, epilaryngeal diameter; D_s, supralaryngeal (pharyngeal entry) diameter; D_L, lip or tube diameter; L_t, tracheal length; L_e, epilaryngeal length; L_s, supralaryngeal vocal tract length; L_L, lip tube length.

L	T	M	K	ζ	x0	PL	Dt	De	Ds	DL (tube)	Lt	Le	Ls	LL
1.0 cm	0.8 cm	0.3 g	3.0 e5 dyn/cm	0.08	0.065 cm	1.5 kPa	2.0 cm	0.8 cm	1.0 cm	0.6 cm	14 cm	2.5 cm	17 cm	10 cm

B. Results

There are several parameters in the model that can be under direct control by the vocalist. These were explored one by one over a range of values. The parameters that were under less control were kept constant as listed in Table I. The first parameter to be explored was the lip tube diameter D_L, the primary control variable in SOVT training and therapy protocols.

1. Lip tube diameter variation

The airflow resistance of a lip tube of 10 cm length was calculated with Eq. (2) for a range of diameters between 2.6 and 8.0 mm. The lung pressure in the circuit of Fig. 1 was 1.5 kPa. This pressure is higher than for conversational speech but is typical for lip tube vocalizations (Maxfield et al., 2015). The epilaryngeal airway diameter was kept at the nominal 8 mm value (0.5 cm² cross-sectional area). All other parameters were also kept nominal as listed in Table I. Figure 3 shows the results of this parameter variation. Figure 3(a) shows the glottal width, which increased with the applied pressure from the pre-pressure value of 1.3 mm (2x₀) but decreased with less vocal tract resistance (larger lip tube diameter). Thus, the vocal folds were spread apart with all of the values of the tube diameter but more with smaller diameters. This is in agreement with previous computer simulation results with a five-mass vocal fold model (Titze et al., 2021). Figure 3(b) shows the glottal airflow, which increased with the lip tube opening as expected, but the increase saturated because the glottal width decreased, restricting the airflow. Figure 3(c) shows three resistances, two of which are labeled (tract, solid curve, and source, dashed line). The third (unlabeled) is the epilaryngeal resistance R_e in isolation (solid, near the baseline). It is very low but increases with an increased lip tube diameter. The decreasing tract resistance is the sum R_e + R_s + R_L, which is dominated by the lip tube resistance R_L for tube diameters less than about 6 mm. In the larger diameter region, the source resistance (dashed line) begins to approach the tract resistance, changing in the opposite direction. The source resistance R_t + R_g is dominated by R_g. It decreases with the glottal width but increases with the glottal airflow [Eq. (3)]. Hence, the overall change is not dramatic, remaining in the 0–1 kPa per L/s range.

FIG. 3.

The steady-state responses of the flow circuit with a varying lip tube diameter. The curves for (a) glottal width, (b) glottal flow, (c) flow resistances, and (d) power distribution are shown. The lung pressure was 1.5 kPa.

It will be shown that for open-mouth conditions, the epilaryngeal airway resistance R_e can be raised to match the source resistance, assisted by the additional supra-laryngeal tract resistance R_s for an open-mouth vowel. R_s is on the same order of magnitude as R_e (greater average diameter but also greater length). Together, they can facilitate the maximum power transfer.

Figure 3(d) shows the change in the power delivered to the vocal tract (solid line) and power consumed by the source (dashed line). Note that a lip tube diameter of 7 mm divides the power equally, but the maximum power (nearly 0.7 W) is transferred to the vocal tract with a 5.5 mm tube diameter. This minor departure from the maximum power transfer theorem is the result of the fact that the resistances are flow dependent, not constant.

2. Lung pressure variation

In a linear circuit, the airflow would increase in proportion to the lung pressure, the resistances would remain constant, and the power delivered would be proportionate to the square of the lung pressure. These relations are not exactly met in a nonlinear circuit. Figure 4 shows the results for a variation of the lung pressure from 0.7 to 2.5 kPa. The lip tube diameter was kept constant at 6 mm, which is the nominal value in Table I.

FIG. 4.

The steady-state responses of the flow circuit with varying lung pressure. The curves for (a) glottal width, (b) glottal flow, (c) flow resistances, and (d) power distribution are shown.

Note that there is a nearly linear increase in both the glottal width and glottal airflow with increased lung pressure. These two increases also produce a linear growth in the tract resistance. Again, the top solid line in Fig. 4(c) is the total vocal tract resistance (still dominated by lip tube resistance) and the bottom solid line is the epilaryngeal airway resistance in isolation. In Fig. 4(d), we see that the power to the vocal tract grows in a quadratic fashion, but the power consumed by the source varies more linearly with the lung pressure. These results show that the lung pressure variation does not produce new maxima in the power distribution, but it shifts the laryngeal resistance somewhat.

3. Epilaryngeal airway diameter variation

With the lip tube resistance removed and the mouth opened to a modest 10 mm diameter (1.7 cm² cross-sectional lip area as for a slightly lip-rounded vowel), there is no extra lip resistance R_L added to the supraglottal resistance R_s, and control of the vocal tract resistance shifts to the epilaryngeal airway diameter. The results are shown in Fig. 5. For comparison to the lip tube case in Fig. 3, the lung pressure was kept at 1.5 kPa. Note that the glottal width and airflow are not remarkably different from those produced with the lip tube resistance, but the epilaryngeal resistance R_e rises dramatically with a smaller diameter [lower solid curve in Fig. 5(c)]. It now becomes the dominant part of the total vocal tract resistance (upper solid curve). Furthermore, the total vocal tract resistance with the increased R_e becomes a match to the source resistance R_g at a 6 mm epilaryngeal diameter. Figure 5(d) shows that the power is divided equally for this epilaryngeal diameter of 6 mm, corresponding to a cross-sectional area of 0.28 cm². The false fold glottis alone could provide this resistance (Agarwal et al., 2003), but distribution of resistance over a longer constriction that includes the laryngeal vestibule would require less false fold adduction and thereby less chance of vibrating the false folds. The maximum power delivered to the vocal tract, about 0.7 W, occurs for a slightly smaller diameter, D_e = 5 mm. The 0.7 W is about the same amount of power delivered to the lip tube with a similar diameter (recall Fig. 3).

FIG. 5.

The steady-state responses of the flow circuit with a varying epilaryngeal airway diameter. The curves for (a) glottal width, (b) glottal flow, (c) flow resistances, and (d) power distribution are shown. The lung pressure was 1.5 kPa.

4. Glottal width variation

Vocalists have control over the glottal width with the use of adductor muscles. Hence, the computations were repeated with the glottal width (2x₀) as the parameter. No lip tube was attached, so the comparison is with Fig. 5 at an epilaryngeal diameter of 6 mm. The glottal width was varied from 1.5 to 2.7 mm with the nominal lung pressure of 1.5 kPa (Fig. 6). The post-pressure glottal width was not much different from the pre-pressure glottal width. This is shown in Fig. 6(a).

FIG. 6.

The steady-state responses of the flow circuit with a varying glottal width. The curves for (a) glottal width, (b) glottal flow, (c) flow resistances, and (d) power distribution are shown. The lung pressure was 1.5 kPa.

The glottal airflow in Fig. 6(b) shows a linear increase with this glottal expansion as would be expected. The glottal resistance in Fig. 6(c) decreased more than fourfold (dashed line). In addition, the epilaryngeal resistance and total tract resistance increased (solid lines), which allowed the power to the vocal tract to rise dramatically [Fig. 6(d)]. This result shows clearly that a low glottal source resistance is beneficial to match the epilaryngeal resistance for the maximum aerodynamic power transfer. Specifically, the source resistance matches R_e at a glottal width of about 2.4 mm [Fig. 6(c)]. The sourec power matches the tract power at a glottal width of 2.2 mm [Fig. 6(d)].

III. PART 2. ACOUSTIC POWER TRANSFER

A. Basic theory

When pressures and airflows are oscillating in the airway, the pressure/flow ratios become complex impedances, exhibiting both a resistance R and a reactance X,

$$Z=R+iX.$$ (14)

The maximum power transfer theorem is still applicable (Kong, 1995) but is now written as

$$Z_s^{*}=Z_e,$$ (15)

where Z_s is the impedance of the source and Z_e is the impedance of the vocal tract at the epilaryngeal airway entrance. The symbol “*” indicates a complex conjugate operation, meaning that the sign of the reactance is reversed. If the supraglottal reactance is inertive (positive), then the subglottal reactance should be compliant (negative) for the complex conjugate operation (Kong, 1995). In terms of airway shapes, this translates into a wide trachea combined with a narrow epilaryngeal airway. This concept is beautifully demonstrated in trumpet playing. Trumpeters often widen the oral cavity behind the lips while playing into a narrow tube attached to the cup, the mouthpiece (Giordano, 2019).

Let the source impedance and vocal tract input impedance be written as

$$Z_s=R_s+i{X}_s,$$ (16)

$$Z_e=R_e+i{X}_e,$$ (17)

where the epilaryngeal input impedance Z_e now represents the entire supraglottal vocal tract impedance up to (and including) the radiation impedance at the mouth. The source impedance Z_s includes the tracheal impedance and time-varying glottal resistance R_g. There are now two powers transferred to the vocal tract, the resistive (dissipated) power W_r and the reactive (stored) power W_x (Kong, 1995). The resistive power to the load is

$$W_r=\hspace{0.17em}{P}_L^2{R}_e/\left[{\left(R_S+R_e\right)}^2+{\left(X_S+\hspace{0.17em}{X}_e\right)}^2\right],$$ (18)

and the reactive (stored) power to the load is

$$W_x=\hspace{0.17em}{P}_L^2{X}_e/\left[{\left(R_S+R_e\right)}^2+{\left(X_S+X_e\right)}^2\right].$$ (19)

For the maximum power transfer, X_s = −X_e and R_s = R_e. These relations can be obtained by differentiating the powers W_r and W_x with respect to X_e or X_s and setting the derivative to zero for maximization (Kong, 1995). Without repeating the mathematics here, visual inspection of Eqs. (18) and (19) shows that the denominators are minimized by cancelling the reactances, which maximizes the power functions. The total power delivered to the vocal tract is the square root of the sum of the squares,

$$W={(W_r^2+W_x^2)}^{1/2},$$ (20)

because the two powers are 90° out of phase. This total power delivered to the vocal tract was calculated here for every frequency from 0 to 5 kHz.

B. Methods

The methods for the impedance calculation have been reported in a previous work (Story et al., 2000). It involves a cascade 2 × 2 matrix computation of the pressure-flow relations in serial sections of the airway with complex numbers. For the supraglottal airway, it begins with the radiation impedance, which is composed of a parallel inertance-resistance circuit, followed by the impedance transfer through multiple serial tubelets (small sections of circular tubes of specified diameters) to obtain an impedance (pressure/flow ratio) at the input of the vocal tract, yielding both the resistance R_e and reactance X_e. The process is repeated for the subglottal (tracheal) system to obtain R_t and X_t. The glottal resistance R_g is then added to R_t to obtain the total source resistance R_s, and the source reactance X_s is equated to the tracheal reactance X_t.

C. Results

Two vocal tract shapes were considered. The first was a uniform tube of 3.0 cm² cross-sectional area and 17.5 cm length. The second shape was also a uniform tube, but a narrow epilaryngeal airway (0.5 cm² cross section) replaced the first 3.17 cm length above the source. The area function for the subglottal system was taken from Story et al. (1996) for a male subject. Figure 7 shows the impedance calculations X_e and R_e across a range of frequencies 1–5 kHz for the two vocal tract shapes. Figures 7(a) and 7(b) show reactance and resistance for the uniform tube, respectively, whereas Figs. 7(c) and 7(d) show the same for the tube with a narrow epilaryngeal airway. The resistance peaks occur exactly at the resonances of the tubes, whereas the positive (inertive) reactance peaks occur slightly below the resonaces and the compliant (negative) reactance peaks occur slightly above the resonances. Note the 10:1 vertical scale change between the upper and lower graphs. The reactances and resistances are between 0 and 10 kPa per L/s for the uniform tube (henceforth called low impedance tube) and between 0 and 100 kPa per L/s for the tube with a narrow epilarynx (henceforth called high impedance tube). The subglottal impedances (dashed lines) are comparable to the supraglottal impedances for the low impedance tube but negligible for the high impedance tube. Interestingly, for the high impedance tube, the impedance peaks are maximal around 2.5–3.5 kHz, a region often identified as the singers formant cluster region (Sundberg, 1977). For the low impedance tube, the impedance peaks decrease gradually with a higher frequency.

FIG. 7.

Impedance calculations for the supraglottal (solid lines) and subglottal (dashed lines). (a) Reactance for a uniform tube, (b) resistance for a uniform tube, (c) reactance for a uniform tube with a narrow epilaryngeal airway, and (d) resistance for uniform tune with a narrow epilaryngeal airway are shown.

In the acoustics of waves confined to a tube, a characteristic wave impedance is defined as ρc/A, where A is the cross-sectional area at the input of the tube (Story et al., 2000). This would be the impedance for an infinitely long tube or one in which all refletions are cancelled. Note that it shows an inverse relation with the tube entry area. For the two cases of interest, the characteristic wave impedance for the supraglottal tube ρc/A_e is 1.33 kPa per L/s for the low impdance tube and 7.98 kPa per L/s for the high impedance tube. The frequency-dependent impedances fluctuate around this characteristiv impedance.

Figures 8 and 9 show the power calculations, W, according to Eq. (20), for the two airways. The power in dB re 1.0 W is plotted for any single frequency sinusoid across the 0–5 kHz range. The four separate panels are for different glottal resistances, 1.0, 3.0, 5.0, and 7.0 kPa per L/s. In the vocal fold vibrations, these resistances are time varying. Here, the numbers are considered average glottal resistances. Two results are obvious. First, the power delivered to the vocal tract is highly variable with the frequency because the vocal tract resonates at specific frequencies. Second, the power transferred is inversely related to the glottal resistance (compare the upper left to lower right panels). This is best quantified by comparing the dashed horizontal lines, which show the mean power across all frequencies. This mean power was computed by averaging the power in W across the frequency before converting to dB. For the same lung pressure (1.5 kPa), the lowest glottal resistance (1.0 kPa per L/s in the upper left panel) delivers about 9 dB more power to the vocal tract across frequencies than the highest glottal resistance (7.0 kPa per L/s in lower right panel). There is a caveat, however. Low glottal resistance may not be conducive to maximum power production given that the vocal folds encounter less collision and therewith produce less high-frequency energy.

FIG. 8.

The acoustic power in dB re 1 W transferred to the uniform vocal tract for four different glottal resistances. The dashed lines depict the mean power across all frequencies.

FIG. 9.

The acoustic power transfer in dB re 1.0 W for a uniform vocal tract with a narrow epilaryngeal airway for four different glottal resistances. The dashed lines depict the mean power across all frequencies.

Figure 9 shows the power transfer for the vocal tract that includes a narrow epilaryngeal airway. The most obvious result is the greater power delivered at high frequencies (3.5–5 kHz). For example, with the largest glottal resistance [Fig. 9(d)], the mean power is −14 dB instead of −17.5 dB as in Fig. 8. In general, a higher vocal tract input impedance is a better match to the 1–7 kPa per L/s range of glottal resistances.

Whereas the input impedance varies greatly with the frequency, the characteristic wave impedance at the vocal tract input (ρc/A_e) is a good measure of the impedance match. In Fig. 9, A_e = 0.5 cm², which results in the above calculated characteristic input impedance of 7.98 kPa per L/s, nearly a match to the highest glottal resistance of 7.0 kPa per L/s. In contrast, in Fig. 8, for which A_e = 3.0 cm², the characteristic wave impedance is 1.33 kPa per L/s, nearly a match to the lowest 1.0 value of R_g chosen in Figs. 8 and 9. In both cases, however, the lowest glottal resistance produced the greatest power transfer across all frequencies. This is simply due to the fact that with the lung pressure held constant, more overall airflow is possible with a lower glottal resistance and, therefore, more power is delivered.

The results in this section are in agreement with earlier results (Titze, 2002) in which time waveforms for the glottal airflow and radiated pressure were generated with a low-dimensional self-oscillating model of the vocal folds. Two separate vowels were considered, /a/ and /i/. In both cases, the radiated power was the maximum when the glottal resistance was balanced with the vocal tract input impedance. Given that both steady airflow resistance and acoustic input impedance of the vocal tract vary inversely with the cross-sectional area of the epilaryngal airway, it appears that a better zero-frequency (aerodynamic) impedance match also produces a better high-frequency (acoustic) match. The small fraction of the transferred power radiated to free-space from the lips is a separate analysis that has been treated previously (Titze, 2018).

IV. DISCUSSION

It has been shown that airway resistances for steady airflow can be adjusted for the maximum aerodynamic power transfer with glottal widening and epilaryngeal airway narrowing. Although aerodynamic power transferred from the lungs to lips is not a necessary criterion for the acoustic power transfer, the covariance has been shown to be strong. The acoustic power transfer is much more variable because it involves the frequency-depenent impedance of a resonating tube and efficiency of sound radiation from the lips. Both acoustic and aerodynamic power transfers have a common requirement, however, namely, a vocal tract input impedance that can match the relatively high glottal impedance when self-sustained vocal fold oscillation occurs.

The questions addressed were (1) can a controlled airway resistance facilitate a balance between the resistances of the glottis and the airway resistance for maximum aerodynamic (non-oscillatory) power transfer from the source to the vocal tract?, and (2) does the resulting aerodynamic impedance match provide a better acoustic match between the source and airway? It has been shown that the vocal tract input impedance (for both steady and oscillatory airflow) is raised with a narrowed epilaryngeal airway, providing a better match to the typical glottal resistances.

For application to voice training, it has been shown that a lip tube resistance can serve as a prosthetic device to lower the source impedance while raising the vocal tract impedance as was shown in Fig. 3. This may give a vocalist the internal “feel” of impedance balance and maximum power transfer. A further hypothesis is that it may be a motoric recall from neonatal vocalization. In voice training of Parkinson's patients, “think loud” brings about both improved repiratory and pnonatory function (Ramig and Dromey, 1996), suggesting that the two functions are synergistic. The maximum aerodynamic power transfer is also invoked in blowing out a candle. One, generally, purses the lips to produce a powerful stream of air. The supraglottal resistance is raised while the glottal resistance is lowered automatically by the steady supraglottal pressure, which has been shown to cause vocal fold separation in this study and a previous one (Titze et al., 2021). The concept of flow phonation has also been shown to be efficacious for optimizing the glottal resistance (Sundberg, 1995, 2020), avoiding so-called “pressed voice.”

V. CONCLUSIONS

The maximum transfer of aerodynamic and acoustic power from the source to the vocal tract is likely an organizing principle for efficient vocalization. It seems to be innate with some indication that it occurs in infant cries. A narrow epilaryngeal airway, evolved for swallowing and airway protection, can be used as an impedance matcher between the source and supraglottal airway. If the impedance balance is less than ideal for a given individual, voice training with a semi-occlusion at the lips can begin to facilitate the balance by offerering a high vocal tract resistance. When the mouth is opened, a narrowed epilaryngeal airway can take over from the high lip tube resistance to balance the source resistance with the vocal tract resistance as Fig. 5 in this study has shown. The critical resistance match shifts from the distal end to the proximal end of the vocal tract. The details of various airway expansions and contractions with steady supraglottal pressures have been described in an earlier paper (Titze et al., 2021).

Although the results here suggest that lowering the glottal resistance with less adduction is categorically beneficial for the aerodynamic power transfer to the vocal tract, this unidirectional increase has only a limited benefit for the acoustic power generation and transfer. Increasing the glottal width, and therewith the airflow, has a point of diminishing the return because it ultimately eliminates all vocal fold contact. Rather, an optimal glottal width is one in which a balance is reached between the harmonic energy generation with vocal fold contact and the transfer of this energy to the vocal tract with reduced glottal resistance and increased airway impedance. An ideal ratio of the airway impedance to the source impedance depends on the type of vocalization desired. The maximum power transmitted to the listener is less of a requirement for conversational speech than it is for unamplfied singing, calling, or shouting.

A limitation of this study is that cylindrical tube shapes were used to quantify the airway resistances. Most of the segments of the airway are more complicated in geometry. Future studies might be directed toward detailed imaging and three-dimensional (3-D) model construction for a more precise resistance measurement or calculation. Future studies could also include a posterior glottal gap as an addition to the glottal resistance. With interarytenoid muscle activation, vocalists do have some control over this gap.