^{1,a)}and Michael S. Howe

^{2}

### Abstract

An equation describing the time-evolution of glottal volume velocity with specified vocal fold motion is derived when the sub- and supra-glottal vocal tracts are present. The derivation of this Fant equation employs a property explicated in Howe and McGowan [(**2011**) J. Fluid Mech. **672**, 428–450] that the Fant equation is the adjoint to the equation characterizing the matching conditions of sub- and supra-glottal Green’s functions segments with the glottal segment. The present aeroacoustic development shows that measurable quantities such as input impedances at the glottis, provide the coefficients for the Fant equation when source-tract interaction is included in the development. Explicit expressions for the Green’s function are not required. With the poles and zeros of the input impedance functions specified, the Fant equation can be solved. After the general derivation of the Fant equation, the specific cases where plane wave acoustic propagation is described either by a Sturm-Liouville problem or concatenated cylindrical tubes is considered. Simulations show the expected skewing of the glottal volume velocity pulses depending on whether the fundamental frequency is below or above a sub- or supra-glottal formant. More complex glottal wave forms result when both the first supra-glottal fundamental frequencies are high and close to the first sub-glottal formant.

This work was supported by a subaward of Grant No. 1R01 DC009229 from the National Institute on Deafness and other Communication Disorders to the University of California, Los Angeles. We thank two anonymous reviewers for helping to improve this paper.

I. INTRODUCTION

II. THEORY

A. Preliminaries

B. The Fant equation

1. Green’s function matching conditions

2. Green’s function decomposition

3. Input impedances and relations to Green’s function segments

4. Combining results for the Fant equation

5. Aeroacoustic sources

6. Final form of the Fant equation

C. Low frequency impedance models

III. NUMERICAL SIMULATIONS

A. Definitions

B. Numerical results

IV. DISCUSSION AND CONCLUSION

### Key Topics

- Phonetic segments
- 81.0
- Vocal tract
- 46.0
- Speech analysis
- 41.0
- Green's function methods
- 40.0
- Solid surfaces
- 21.0

## Figures

Schematic of the vocal tract. The control volume is bounded by the solid walls of the vocal tract and the surfaces between the lips and through the bronchial tubes, denoted S_{L}. The surfaces S_{sub} and S_{sup} are used to segment the control volume into three portions: the sub-glottal tract, glottal region, and supra-glottal tract.

Schematic of the vocal tract. The control volume is bounded by the solid walls of the vocal tract and the surfaces between the lips and through the bronchial tubes, denoted S_{L}. The surfaces S_{sub} and S_{sup} are used to segment the control volume into three portions: the sub-glottal tract, glottal region, and supra-glottal tract.

Normalized glottal volume velocity, , versus normalized time, *f* _{0} *t*, at four different fundamental frequencies. The supra-glottal impedance is that for a semi-infinite tube, (*ρ* _{0} *c* _{0})/*A* _{sup}. The dark curves have sub-glottal input impedance specified in the text according to ^{ Ishizaka et al. (1976) } . Sub-glottal formant frequencies, with bandwidths in brackets are 640 Hz (256 Hz), 1400 Hz (156 Hz), and 2100 Hz (175 Hz). The dashed curve for comparison is when the sub-glottal tract is a semi-infinite tube with input impedance (*ρ* _{0} *c* _{0})/*A* _{sub}. (a) *f* _{0} = 100 Hz, (b) *f* _{0} =280 Hz, (c) *f* _{0} = 400 Hz, and (d) *f* _{0} = 800 Hz.

Normalized glottal volume velocity, , versus normalized time, *f* _{0} *t*, at four different fundamental frequencies. The supra-glottal impedance is that for a semi-infinite tube, (*ρ* _{0} *c* _{0})/*A* _{sup}. The dark curves have sub-glottal input impedance specified in the text according to ^{ Ishizaka et al. (1976) } . Sub-glottal formant frequencies, with bandwidths in brackets are 640 Hz (256 Hz), 1400 Hz (156 Hz), and 2100 Hz (175 Hz). The dashed curve for comparison is when the sub-glottal tract is a semi-infinite tube with input impedance (*ρ* _{0} *c* _{0})/*A* _{sub}. (a) *f* _{0} = 100 Hz, (b) *f* _{0} =280 Hz, (c) *f* _{0} = 400 Hz, and (d) *f* _{0} = 800 Hz.

Normalized glottal volume velocity, *Q _{norm} * = (

*Q*(

*t*)

*ρ*

_{0}

*c*

_{0})/(

*p*

_{sub}

*A*

_{sub}), versus normalized time,

*f*

_{0}

*t*at four different fundamental frequencies. Sub-glottal resonances are as specified for the dark curve in Fig. 2 . The dark curves are for supra-glottal formants appropriate for /i/, which are, with bandwidths in brackets, 280 Hz (65 Hz), 2250 Hz (98 Hz), and 2750 Hz (113 Hz). The dashed curves are when supra-glottal impedance is (

*ρ*

_{0}

*c*

_{0})/

*A*

_{sup}for a semi-infinite tube. (a)

*f*

_{0}= 100 Hz, (b)

*f*

_{0}=260 Hz, (c)

*f*

_{0}= 275 Hz, and (d)

*f*

_{0}= 290 Hz.

Normalized glottal volume velocity, *Q _{norm} * = (

*Q*(

*t*)

*ρ*

_{0}

*c*

_{0})/(

*p*

_{sub}

*A*

_{sub}), versus normalized time,

*f*

_{0}

*t*at four different fundamental frequencies. Sub-glottal resonances are as specified for the dark curve in Fig. 2 . The dark curves are for supra-glottal formants appropriate for /i/, which are, with bandwidths in brackets, 280 Hz (65 Hz), 2250 Hz (98 Hz), and 2750 Hz (113 Hz). The dashed curves are when supra-glottal impedance is (

*ρ*

_{0}

*c*

_{0})/

*A*

_{sup}for a semi-infinite tube. (a)

*f*

_{0}= 100 Hz, (b)

*f*

_{0}=260 Hz, (c)

*f*

_{0}= 275 Hz, and (d)

*f*

_{0}= 290 Hz.

Normalized glottal volume velocity, *Q _{norm} * = (

*Q*(

*t*)

*ρ*

_{0}

*c*

_{0})/(

*p*

_{sub}

*A*

_{sub}), versus normalized time,

*f*

_{0}

*t*at four different fundamental frequencies. Sub-glottal resonances are as specified for the dark curve in Fig. 2 . The dark curves are for supra-glottal formants appropriate for /r/, which are, with bandwidths in brackets, 250 Hz (69 Hz), 700 Hz (54 Hz), and 1400 Hz (73 Hz). The dashed curves are when supra-glottal impedance is (

*ρ*

_{0}

*c*

_{0})/

*A*

_{sup}for a semi-infinite tube. (a)

*f*

_{0}= 100 Hz, (b)

*f*

_{0}=240 Hz, (c)

*f*

_{0}= 245 Hz, and (d)

*f*

_{0}= 260 Hz.

Normalized glottal volume velocity, *Q _{norm} * = (

*Q*(

*t*)

*ρ*

_{0}

*c*

_{0})/(

*p*

_{sub}

*A*

_{sub}), versus normalized time,

*f*

_{0}

*t*at four different fundamental frequencies. Sub-glottal resonances are as specified for the dark curve in Fig. 2 . The dark curves are for supra-glottal formants appropriate for /r/, which are, with bandwidths in brackets, 250 Hz (69 Hz), 700 Hz (54 Hz), and 1400 Hz (73 Hz). The dashed curves are when supra-glottal impedance is (

*ρ*

_{0}

*c*

_{0})/

*A*

_{sup}for a semi-infinite tube. (a)

*f*

_{0}= 100 Hz, (b)

*f*

_{0}=240 Hz, (c)

*f*

_{0}= 245 Hz, and (d)

*f*

_{0}= 260 Hz.

Peak amplitudes for /i/ versus fundamental frequency, *f* _{0} at a distance *r* = 400 cm from the lips. (a) Normalized far field pressure, 10log_{10}( /*p* _{sub}) versus *f* _{0}, and (b) normalized volume velocity 10log_{10}(*Q _{max}ρ*

_{0}

*c*

_{0}/

*p*

_{sub}

*A*

_{sub}) versus

*f*

_{0}.

Peak amplitudes for /i/ versus fundamental frequency, *f* _{0} at a distance *r* = 400 cm from the lips. (a) Normalized far field pressure, 10log_{10}( /*p* _{sub}) versus *f* _{0}, and (b) normalized volume velocity 10log_{10}(*Q _{max}ρ*

_{0}

*c*

_{0}/

*p*

_{sub}

*A*

_{sub}) versus

*f*

_{0}.

Peak amplitudes for /r/ versus fundamental frequency, *f* _{0} at a distance *r* = 400 cm from the lips. (a) Normalized far field pressure, 10log_{10}( /*p* _{sub}) versus *f* _{0}, and (b) normalized volume velocity 10log_{10}(*Q _{max}ρ*

_{0}

*c*

_{0}/

*p*

_{sub}

*A*

_{sub}) versus

*f*

_{0}.

Peak amplitudes for /r/ versus fundamental frequency, *f* _{0} at a distance *r* = 400 cm from the lips. (a) Normalized far field pressure, 10log_{10}( /*p* _{sub}) versus *f* _{0}, and (b) normalized volume velocity 10log_{10}(*Q _{max}ρ*

_{0}

*c*

_{0}/

*p*

_{sub}

*A*

_{sub}) versus

*f*

_{0}.

Normalized glottal volume velocity, *Q _{norm} * = (

*Q*(

*t*)

*ρ*

_{0}

*c*

_{0})/(

*p*), versus normalized time,

_{sub}A_{sub}*f*

_{0}

*t*at four different fundamental frequencies. Sub-glottal resonances are as specified for the dark curve in Fig. 1 . The dark curves are for supra-glottal formants appropriate for /

*a*/, which are, with bandwidths in brackets, 750 Hz (54 Hz), 1100 Hz (64 Hz), and 2600 Hz (108 Hz). The dashed curves are when supra-glottal impedance is (

*ρ*

_{0}

*c*

_{0})/

*A*for a semi-infinite tube. (a)

_{sup}*f*

_{0}= 100 Hz, (b)

*f*

_{0}=650 Hz, (c)

*f*

_{0}= 745 Hz, and (d)

*f*

_{0}= 800 Hz.

Normalized glottal volume velocity, *Q _{norm} * = (

*Q*(

*t*)

*ρ*

_{0}

*c*

_{0})/(

*p*), versus normalized time,

_{sub}A_{sub}*f*

_{0}

*t*at four different fundamental frequencies. Sub-glottal resonances are as specified for the dark curve in Fig. 1 . The dark curves are for supra-glottal formants appropriate for /

*a*/, which are, with bandwidths in brackets, 750 Hz (54 Hz), 1100 Hz (64 Hz), and 2600 Hz (108 Hz). The dashed curves are when supra-glottal impedance is (

*ρ*

_{0}

*c*

_{0})/

*A*for a semi-infinite tube. (a)

_{sup}*f*

_{0}= 100 Hz, (b)

*f*

_{0}=650 Hz, (c)

*f*

_{0}= 745 Hz, and (d)

*f*

_{0}= 800 Hz.

Peak amplitudes for /*a*/ versus fundamental frequency, *f* _{0} at a distance *r* = 400 cm from the lips. (a) Normalized far field pressure, 10log_{10}( /*p* _{sub}) versus *f* _{0}, and (b) normalized volume velocity 10log_{10}(*Q _{max}ρ*

_{0}

*c*

_{0}/

*p*

_{sub}

*A*

_{sub}) versus

*f*

_{0}.

Peak amplitudes for /*a*/ versus fundamental frequency, *f* _{0} at a distance *r* = 400 cm from the lips. (a) Normalized far field pressure, 10log_{10}( /*p* _{sub}) versus *f* _{0}, and (b) normalized volume velocity 10log_{10}(*Q _{max}ρ*

_{0}

*c*

_{0}/

*p*

_{sub}

*A*

_{sub}) versus

*f*

_{0}.

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