Theoretical consideration of the flow behavior in oscillating vocal fold

ARTICLE IN PRESS Journal of Biomechanics 42 (2009) 824–829

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Theoretical consideration of the flow behavior in oscillating vocal fold Shinji Deguchi a,b,, Toru Hyakutake c a b c

Graduate School of Biomedical Engineering, Tohoku University, 6-6-11-1306-2 Aramaki-Aoba, Sendai 980-8579, Japan Graduate School of Engineering, Tohoku University, 6-6-11-1306-2 Aramaki-Aoba, Sendai 980-8579, Japan Graduate School of Natural Science and Technology, Okayama University, 3-1-1 Tsushima-Naka, Okayama 700-8530, Japan

a r t i c l e in f o

a b s t r a c t

Article history: Accepted 26 January 2009

Self-excited oscillation of the vocal folds produces a source sound of the human voiced speech. The mechanism of the self-excitation remains elusive partly because characteristics of the flow in rapidly oscillating vocal folds are unclear. This paper deals with theoretical considerations of the flow behavior in oscillating constriction based on general flow equations. The cause-and-effect relationships between time-varying glottal width and physical variables such as glottal pressure, velocity, and volume flow are analytically derived as functions of oscillatory frequency through perturbation analysis. The result shows that the unsteady effect due to convective acceleration of vocal fold wall-induced flow becomes comparable in magnitude to the Bernoulli effect at a high but physiological frequency of phonation. Consequently, a phase difference between the vocal fold motion and glottal pressure appears, enabling self-excited oscillation. The phase-lead of the pressure compared to wall motion is described as a monotonically increasing function of the Strouhal number. The above two effects essentially play the dominant role in the glottal flow. These explicit descriptions containing flow-related variables are useful for understanding of the glottal aerodynamics particularly at high frequency range of the falsetto voice register. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Vocal fold Flow-structure interaction Strouhal number Self-excited oscillation Phonation

1. Introduction Self-excited oscillation of the vocal folds is a major sound source of human voiced speech (Ishizaka and Flanagan, 1972; Titze, 1988; Pelorson et al., 1994; Ikeda et al., 2001; Tao and Jiang, 2007; Tao et al., 2007). Accurate estimation of aerodynamic forces, which deform the vocal folds and induce the self-excitation, is a critical issue for elucidation of the mechanism. Previous studies using rigid flow channels mimicking the glottis (i.e., passage between a pair of vocal folds) demonstrated that pressure drop along the constricted glottis was produced mainly by the Bernoulli effect and viscous resistance (e.g., Van den Berg et al., 1957). However, it is unclear that the findings obtained under such non-oscillatory conditions can be applicable to more general cases of oscillatory flow in the actual vocal folds having a fundamental frequency of 4100 Hz. Theoretical and experimental assessments of unsteady aerodynamic effects have recently been reported by Krane et al. (2006, 2007). They showed based on flow equations that the flow separation point motion is delayed in phase compared to glottal wall motion, thereby indicating that the shape of the

 Corresponding author at: Tohoku University, 6-6-11-1306-2 Aramaki-Aoba, Sendai 980-8579, Japan. Tel./fax: +81 22 795 3936. E-mail address: [email protected] (S. Deguchi).

0021-9290/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2009.01.027

glottal volume flow waveform could be influenced. We also analytically derived, from general flow equations, an unsteady aerodynamic pressure that could induce self-excitations (Deguchi, 2005; Deguchi et al., 2006). In addition, we investigated from experimental and numerical approaches with the lattice Boltzman method that the airflow behavior including flow separation is wall oscillation-dependent (Hyakutake et al., 2006). Nevertheless, the cause-and-effect relationships between time-varying glottal width and involved physical variables remain elusive. The aim of this paper is to clarify the relationships. Our theoretical considerations show that physical variables related to the flow behavior is clearly expressed as functions of glottal width and oscillatory frequency.

2. Mathematical model 2.1. Problem definition (1) Model geometry: The inferior half part of the glottis is modeled (Fig. 1A). Vocal fold shape and its motion are assumed symmetrical with respect to the mid-sagittal plane (Fig. 1B). The position along the glottis is given by a spatial coordinate x. The glottal half-width B(x,t) perpendicular to the x direction, where t is time, is assumed a constant value of b¯ u at the

ARTICLE IN PRESS S. Deguchi, T. Hyakutake / Journal of Biomechanics 42 (2009) 824–829

Nomenclature B(x,t) b(x,t) b¯(x) bc(t) b¯c b¯u f h lc lg Nw P(x,t) Pt p(x,t) p¯(x) pc(t) p¯c p¯u Q(x,t) q(x,t)

q¯(x) time average of Q(x, t) q(x ¼ lc, t) qc(t) Re modified Reynolds number St Strouhal number t time U(x,t) flow velocity in glottis u(x,t) infinitesimal component of U(x, t) % u(x) time average of U(x, t) u(x ¼ lc, t) uc(t) % ¼ l c) u% c u(x % ¼ 0) u% u u(x x spatial coordinate a, b, g, d, x, Z, z, f, and c coefficients describing effects of vocal fold geometry and/or time-averaged airflow properties on glottal pressure e factor related to vena contracta n air kinematic viscosity r air density k phase difference between velocity and wall motion with the addition of p y phase difference between pressure and wall motion

glottal half-width infinitesimal component of B(x, t) time average of B(x, t) b(x ¼ lc, t) b¯(x ¼ lc) b¯(x ¼ 0) frequency of wall oscillation amplitude of wall oscillation axial length of glottis depth of glottis Womersley number air static pressure upstream total pressure infinitesimal component of P(x, t) time average of P(x, t) p(x ¼ lc, t) p¯(x ¼ lc) p¯(x ¼ 0) glottal volume flow infinitesimal component of Q(x, t)

upstream inlet (x ¼ 0) and a time-varying value at the downstream outlet (x ¼ lc). Each of the cross-sections is modeled as having a rectangular geometry of a constant depth lg. It is assumed that flow is still attached at x ¼ lc, and flow separation as well as downstream pressure recovery are not considered. (2) Airflow model: A one-dimensional flow model used for collapsible-tube analyses (Cancelli and Pedley, 1985; Fung, 1990) is employed. The flow equations are originally derived from two-dimensional flow equations by assuming a flow profile along the glottal width, integrating the equations over the width, and then substituting energy conservation

Epiglottis (Vocal tract)

Flow Vocal folds

bu uu pu

B (x,t) U (x,t) P (x,t)

0

bc lc

False vocal folds

Frequency f Amplitude h

Flow

equation into them (Deguchi et al., 2007). The Poiseuille flow (i.e., a parabolic flow profile) is employed to estimate viscous effects. In addition, flow is assumed incompressible. Pressure and velocity at x ¼ 0 are assumed to have constant values p¯u and u% u, respectively. (3) Analysis: Each variable is decomposed into a time-averaged component and a perturbation component (i.e., a timevarying deviation). The perturbations are small, and terms of higher order than their quadratic are neglected.

2.2. Governing equations and perturbation analysis The mass and momentum conservation equations are described by

Inferior half part of glottis

Trachea (Subglottis)

825

@B @BU þ ¼ 0, @t @x

(1)

@U 1 @U 2 1 @P U þ þ  þ 3n 2 ¼ 0, r @x @t 2 @x B

(2)

respectively, where e is a factor that depends on vena contracta (Ishizaka and Flanagan, 1972), and r is the fluid density. Each variable is decomposed into a time-averaged component (written in small letter with an overbar) and a perturbation component (small letter) Pðx; tÞ ¼ pðxÞ þ pðx; tÞ, ¯

(3)

Uðx; tÞ ¼ uðxÞ þ uðx; tÞ, ¯

(4)

¯ Bðx; tÞ ¼ bðxÞ þ bðx; tÞ.

(5)

Medial axis uc(t), pc(t)

x

Subglottis Inferior half part of glottis (Constriction) Fig. 1. Schema of the vocal folds. (A) Coronal cross-section of the whole glottis. The inferior half part of the glottis (dashed rectangular region) is modeled. (B) Model of the glottal flow in oscillating constriction.

Substituting Eqs. (4) and (5) into Eq. (1) and integrating along x, u¼

1@ b¯ @t

Z 0

x

bdx 

u¯ b. b¯

(6)

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Similarly, substituting Eqs. (3)–(5) into Eq. (2) and integrating, pressure at the narrowest constriction is @ pc ¼ r @t

Z

lc

udx  ru¯ c uc þ 6rvb¯ u u¯ u

0

Z

lc

b 4 b¯

0

dx  3rv

Z

lc 0

u 2 b¯

dx. (7)

To integrate Eqs. (6) and (7), the geometry of the glottis is assumed Bðx; tÞ ¼ b¯ u þ ðb¯ c þ bc  b¯ u Þx=lc .

(8)

b¯ u 15 mm;

suggesting that the coefficients of Eq. (18) are all positive except that only a is negative. It is then rewritten by pc ¼ ab€ c þ fb_ c þ cbc ,

b ¼ bc x=lc ,

(9)

which is easily integrated and differentiated. 2.3. Glottal velocity, volume flow, and pressure

(21)

where ao0, f ¼ b+g+Z4 0, and c ¼ d+x+z4 0. To extract fundamental features, a sinusoidal oscillation is given bðlc ; tÞ ¼ bc ¼ h sin 2pft,

(22)

where f and h are frequency and amplitude, respectively. Perturbation volume flow is qc ¼ qðlc ; tÞ ¼ pflc lg h sin ð2pft  p=2Þ,

With Eq. (5),

(20)

(23)

showing that it is proportional to the frequency, and its phase is delayed from the wall by p/2 rad. The velocity is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h uc ¼ u¯ 2c þ ðplc f Þ2 sinð2pft þ k þ pÞ, (24) ¯bc where

k ¼ tan1 ðplc f =u¯ c Þ,

Velocity of Eq. (6) becomes

(25)

where a dot over a variable denotes its time-derivative. Volume flow is defined by

suggesting that the response precedes the wall motion by a phase difference of k+p rad that is dependent on the frequency f. From Eq. (21), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pc ¼ h fc  að2pf Þ2 g2 þ ð2pf fÞ2 sinð2pft þ yÞ, (26)

Q ðx; tÞ ¼ lg Bðx; tÞUðx; tÞ,

where

uðx; tÞ ¼ 

x2 _ b¯ u u¯ u x b  2 bc , ¯ c c 2bl b¯ l

(10)

c

(11)

and divided depending on whether time-dependent or not Q ðx; tÞ ¼ qðxÞ þ qðx; tÞ. ¯

tan y ¼ (12)

qðx; tÞ ¼ lg x2 b_ c =2lc .

(13) 2.4. Simplification by scale analysis

Then, Eq. (7) is integrated ½1st term of right side of Eq: ð7Þ ( ) 2 b¯ u b¯ c € rl2 3b¯ u  b¯ c ¼ c bc þ ln 2 2ðb¯ u  b¯ c Þ2 ðb¯ u  b¯ c Þ3 b¯ u ( ) 1 1 b¯ u _ bc . þ rb¯ u u¯ u lc þ ln b¯ c ðb¯ u  b¯ c Þb¯ c ðb¯ u  b¯ c Þ2

½3rd term of right side of Eq: ð7Þ ¼

According to literature (e.g., Sˇidlof et al., 2008), lc 10 mm:

(14)

rb¯ u u¯ u lc _ 2 2b¯ c

bc þ

rvu¯ u lc ðb¯ c þ 2b¯ u Þ 3 b¯ c b¯ u

2

rb¯ u u¯ 2u 3 b¯ c

bc .

bc .

3 2b¯ c b¯ u

bc .

(16)

The air viscosity at a standard condition (101.3 kPa, 20 1C) is n ¼ 1.5  105 m2/s. Fundamental frequency of voiced sound is 100–400 Hz. To assess its effect, let us consider a wider range of f 162500 Hz:

(29)

(30)

From these,

(17)

(18)

where a, b, g, d, x, Z, and z denote the corresponding coefficients. According to literature (e.g., Sˇidlof et al., 2008), the length scale describing the glottis is b¯ c 0:5 mm,

assumed unity. From literature (e.g., Deguchi et al., 2007), velocity in the subglottal channel is u¯ u 103 mm=s.

Taken together, pressure at the narrowest constriction is pc ¼ ab€ c þ bb_ c þ gb_ c þ dbc þ xbc þ Zb_ c þ zbc ¼ ab€ c þ ðb þ g þ ZÞb_ c þ ðd þ x þ zÞbc ,

e of 1.37 was used (Ishizaka and Flanagan, 1972). Here, e is

(15)

c

þ

(28)

In a constricted region where flow separation does not occur, an

½4th term of right side of Eq: ð7Þ ( ) 2 2 2 3rvlc b¯ c 2ðb¯ u  b¯ c Þ b¯ u  b¯ c _ ¼ bc ln þ  2 b¯ u b¯ c 2ðb¯ c  b¯ u Þ3 2b¯

rvu¯ u lc ðb¯ c þ 2b¯ u Þ

(27)

suggesting that the pressure response is delayed by y rad that is again dependent on the frequency.

Substituting Eqs. (4), (5) and (12) to Eq. (11),

½2nd term of right side of Eq: ð7Þ ¼

2pf f , c  að2pf Þ2

(19)

ab€ c   7:3  104 rbc  7:2  107 rbc ,

(31)

bb€ c   2:3  106 rbc  7:2  107 rbc ,

(32)

gb_ c   3:0  107 rbc  9:4  108 rbc ,

(33)

dbc 1:8  109 rbc ,

(34)

xbc 2:4  103 rbc ,

(35)

Zb_ c 2:9  10rbc 29:1  102 rbc ,

(36)

zbc 1:2  103 rbc .

(37)

ARTICLE IN PRESS S. Deguchi, T. Hyakutake / Journal of Biomechanics 42 (2009) 824–829

where Pt is the subglottal pressure (shown by the total pressure). The latter, obtained under the plausible treatment n ¼ 0, corresponds to the Bernoulli law for steady flow.

1011

Magnitude of each term of the glottal pressure (mm/s2)

Physiological range

827

·

bc

2.5. Descriptions using nondimensional parameters

109

·· bc ·

bc

bc

1010

Although some terms in Eq. (18) are small in magnitude and negligible (Fig. 2), let us here keep bb˙c, xbc, and zbc to explore the effects of the Strouhal and Reynolds numbers. In addition, although the Womersley number determining the velocity profile in the lateral direction (Fung, 1990) has an appropriate meaning only when two or three-dimensional flow is considered, we also include the parameter here to provide a measure of its effect. Here, the Strouhal number (i.e., the ratio describing which effect is stronger, wall motion-induced gb˙c or convection-induced dbc) is defined by

108 107

106

bc

104 102

105

··

bc

·

bc

·

bc ·

bc

bc

100 0

100

200

300

400

500

St ¼ 2pflc =u¯ c .

(43)

A modified Reynolds number (i.e., the ratio of convective inertial force dbc to viscous force xbc+zbc) is defined by

103

Re ¼ u¯ c L=n,

bc

bc

·

bc

(44)

in which L is a characteristic length defined by L ¼ b¯ c b¯ u =3lc .

101 0

1000 2000 3000 4000 5000 6000 7000 8000 Frequency f (Hz)

Fig. 2. Comparison of the magnitude of the terms of the glottal pressure Eq. (18) as a function of oscillatory frequency. Note that each value was divided by rbc that had been originally contained in all the terms. A realistic frequency range of voiced speech, o500 Hz, is magnified in the inset.

¨c (representing the force due to transient acceleraAt 16 Hz, ab tion of wall motion-induced flow) is only 0.24% of gb˙c (representing the force due to convective acceleration of wall motion-induced flow); yet, the former approaches the latter as frequency increases and becomes 7.7% at 500 Hz (Fig. 2). At the whole frequency range, bb˙c (representing transient inertial force) is 7.7% of gb˙c. At 16 Hz, gb˙c is 1.7% of dbc (representing convective inertial force or the Bernoulli effect); yet, the former becomes 52% of the latter at 500 Hz. The other terms related to n (xbc, Zb˙c, and zbc) are negligible compared to the formers, indicating that the viscous effect is small at the gradually narrowing channel. Taken together, in the realistic frequency range, Eq. (21) is reduced to pc ¼ gb_ c þ dbc .

(38)

Then, Eq. (26) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pc ¼ h d þ ð2pf gÞ2 sinð2pft þ yÞ.

(39)

The phase difference between the pressure and wall motion at the minimum constriction is thus

y ¼ tan1 ðplc f =u¯ c Þ.

(40)

Since y ¼ k from Eqs. (25) and (40), Eqs. (24) and (26) indicate that the pressure and velocity are in the opposite phase. Note again that this equality is satisfied, to be exact, in the physiological frequency range. Temporal averages of Eqs. (1) and (2) satisfy b¯ c u¯ c ¼ b¯ u u¯ u ¼ q=l ¯ g, 2 2

p¯ c ¼ Pt  rq¯ 2 =2b¯ c lg ¼ Pt  ru¯ 2c =2,

(41) (42)

(45)

The Womersley number (i.e., the square root of the ratio of transient inertial force bb˙c to viscous force xbc+zbc) is defined by qffiffiffiffiffiffiffiffiffiffiffiffiffi Nw ¼ b¯ c 2pf =n. (46) The pressure at the constriction is then rewritten under approximation of b¯ubb¯c  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ru¯ 2c 2h ½Amplitude of pc  ¼ ð1 þ 1=ReÞ2 þ ðSt=2 þ Nw2 =ReÞ2 , 2 b¯ c (47)

yðx ¼ lc Þ ¼ tan1

  St=2 þ Nw2 =Re , 1 þ 1=Re

(48)

indicating that the pressure amplitude is the product of the kinetic pressure there, the ratio of wall movement distance to the channel width, and a function containing the nondimensional parameters. It is noteworthy that the parameters-containing parts in Eqs. (47) and (48) are practically dependent only on St since the viscous effect is negligible. From (24),  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u¯ c 2h (49) 1 þ ðSt=2Þ2 , ½Amplitude of uc  ¼ 2 b¯ c indicating that the velocity amplitude is the product of a half of the mean velocity, the ratio of the wall movement distance to the channel width, and a Strouhal number-related function. In contrast, Eq. (23) indicates that the glottal volume flow varies as a function of flc but not St in reality because it does not contain u% c.

3. Discussion Starting from the general Eqs. (1) and (2), the airflow behavior in oscillating constriction is analytically derived. Flow-related variables are described as functions of time-varying glottal width, thereby illuminating their cause-and-effect relationships. The glottal pressure of Eq. (38) contains a term having a timederivative of the width. This term is comparable in magnitude to the second term (equivalent to the Bernoulli effect) at a high but realistic frequency range (Fig. 2). The time-derivative distinguishes the moving direction of the vocal fold, as its sign becomes

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the opposite depending on the direction. Such a flow can sustain oscillation because the flow gives the wall the energy required for overcoming viscous damping inside the oscillating object. More specifically, the velocity is represented as a reversed-signed sum of the displacement itself and its time-derivative according to Eq. (6). The pressure is represented similarly as a reversed-signed sum of the velocity itself and its time-derivative according to Eq. (7) if the practically small viscous effect is neglected by giving n ¼ 0. Fig. 3 illustrates the relationships between variables when a wall oscillates sinusoidally. Note that, to focus in this figure on the phase difference critical for self-excitation, the magnitude of the variable coefficients is assumed the same in each graph. The pressure precedes the wall by p/2 rad. The phase difference in more general case where the coefficient values differ from each other is presented in Eq. (27) for the whole frequency range or in Eq. (40) for the realistic range. Under respective cases, the phase difference has a positive value between 0 and p/2. Therefore a higher pressure, as a driving force, is applied to the wall during its opening phase than the closing phase (Fig. 3). Similar phase-lag of glottal volume flow (of approximately p/2), phase-lead of glottal

Opening phase

Closing phase

db / dt

-db / dt

b

t

-b Wall displacement -du / dt

Wall velocity -u

t

du / dt

u Flow velocity

Glottal pressure Vocal fold

p

Vocal fold

Low pressure

t

High pressure Fig. 3. The relationships between the wall displacement, velocity, and pressure perturbation components. It is noteworthy that a higher pressure is obtained at the opening phase than the closing phase as schematically shown in the bottom panel.

pressure (of approximately p/2), and phase reversal of velocity (with a further phase-lead of approximately p/2) compared to vocal fold motion without closure have been observed in numerical simulations incorporating unsteady terms in flow equations (Deguchi et al., 2007; Ikeda and Matsuzaki, 1999). In the above case, a wall oscillation can gradually develop over time while receiving net positive energy from the flow. In a case of the oscillation of living tissues, whether self-excitation occurs or not depends on the tissue mechanical properties (Titze, 1988; Tao and Jiang, 2008). In the present study, the glottal pressure that drives the vocal fold motion is expressed explicitly as a function of a channel displacement in Eq. (18) whose coefficients are determined from the vocal fold geometry and time-averaged flow-related parameters. The expression is therefore applicable to investigations of criteria regarding how an unsteady flow can cause a self-excitation of the vocal folds by employing a channel displacement-dependent spring-mass-damper model instead of the use of a forced-oscillation system. Analysis on this point, or the phonation onset, will be the subject of future investigation. Thus, even if one-dimensional wall motion perpendicular to the mainstream is permitted, without considering the mucosal wave motion (propagating in the mainstream direction) and/or air column oscillation in the axially long vocal tract or trachea (Titze, 1988), consideration of the time-dependent effects in flow equations yields a condition in which self-excitation can occur (Deguchi, 2005; Krane and Wei, 2006; Deguchi et al., 2006). This type of self-oscillation may be particularly significant to the falsetto voice register, where the vocal folds do not show remarkable mucosal waves nor collisions because of their small amplitude, and in addition the fundamental frequency is high enough to provoke the phase-altering effect. Looking at other variables, Eq. (24) shows that the velocity is in the opposite phase with wall motion, but precedes by a phase of k. This leading phase is the same in magnitude with the difference between the pressure and wall motion, y of Eq. (39), at a physiological frequency range. By contrast, Eq. (23) indicates that the volume flow lags behind wall motion by p/2 rad. The phaselag is independent of frequency, different from the characteristics of the velocity and pressure. The origin of this nature is explained as follows: the velocity Eq. (10) contains a term proportional to bc and another term having its time-derivative b˙c. The effect of frequency differs between these two; specifically, bc is unaware of frequency magnitude, whereas b˙c is frequency-dependent. Its relative scale therefore becomes different as frequency increases. The same holds true for the case of the pressure expressed by Eq. (38) having bc in addition to b˙c. Meanwhile, there is only one term having b˙c in the volume flow of Eq. (13), indicating that such a relative difference between terms never appears, and the phaselag is frequency-independent accordingly. In Section 2.5, some equations that are particularly important for considering the self-excitation mechanism are described using nondimensional parameters. These expressions may be helpful for interpretation of other experimental or numerical studies because they provide a universal form independent of the specific system. The Strouhal number St is defined as the product of the frequency of oscillatory flow and a characteristic channel length divided by the mean velocity. When the axial constriction length lc is presumed the characteristic length, the phase difference between the wall and pressure, Eq. (48), is expressed as a monotonically increasing function of St. In this study, the flow-structure interaction was described using one-dimensional flow equations, highlighting many important aspects of the phenomenon. To include the unsteady flow behavior in flow separation region (i.e., the superior half part of the glottis) within one-dimensional descriptions, however, a certain assumption (such as an assumption on flow streamline

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geometry) must be introduced (Ikeda and Matsuzaki, 1999). The Borda–Carnot equation, a one-dimensional description of pressure loss due to a flow expansion, is often used in phonation modeling (e.g., Ishizaka and Flanagan, 1972); however, its effectiveness at a high frequency range is unclear. Thus, we did not include the flow separation region to develop our theory without loss of generality. Yet, the explicit form of the phase difference k of Eq. (25) will be informative in future study on the role of flow separation in the oscillation mechanism, because velocity influences time-varying flow separation (Pelorson et al., 1994; Ikeda and Matsuzaki, 1999; Krane and Wei, 2006). In addition, the use of the coefficient e of the convective term may be useful in those analyses because the coefficient has been used to estimate the loss due to flow separation in collapsible-tube studies (Cancelli and Pedley, 1985). Finally, we touch on our choice of the forced sinusoidal oscillator instead of using a self-excitation model. The glottal wall shape during self-excited oscillation is determined essentially by the vocal fold material properties in addition to driving aerodynamic forces. For the flow itself, its behavior is independent of whether the wall movement is induced by self-excitation or forced oscillation (Hyakutake et al., 2006). In order to isolate the intrinsic characteristics of flow from such elusive phenomena of flow-structure interaction, it is therefore a good approach to examine the flow behavior under given and simple wall displacements and to finally build a flow model, as done in the current study, that is applicable to more general cases.

Conflict of interest statement The authors report no conflicts of interest.

Acknowledgement This work was supported in part by grant-in-aid for Scientific Research from the Japan Ministry of Education, Culture, Sports, Science, and Technology (19700382).

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