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Production of sound by unsteady throttling of flow into a resonant cavity, with application to voiced speech

Published online by Cambridge University Press:  14 February 2011

M. S. HOWE  and
R. S. McGOWAN
M. S. HOWE*
Affiliation:
Boston University, College of Engineering, 110 Cummington Street, Boston, MA 02215, USA
R. S. McGOWAN
Affiliation:
CReSS LLC, 1 Seaborn Place, Lexington, MA 02420, USA
*
†Email address for correspondence: [email protected]
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Abstract

An analysis is made of the sound generated by the time-dependent throttling of a nominally steady stream of air through a small orifice into a flow-through resonant cavity. This is exemplified by the production of voiced speech, where air from the lungs enters the vocal tract through the glottis at a time-variable volume flow rate Q(t) controlled by oscillations of the glottis cross-section. Voicing theory has hitherto determined Q from a heuristic, reduced complexity ‘Fant’ differential equation. A new self-consistent, integro-differential form of this equation is derived in this paper using the theory of aerodynamic sound, with full account taken of the back-reaction of the resonant tract on the glottal flux Q. The theory involves an aeroacoustic Green's function (G) for flow–surface interactions in a time-dependent glottis, so making the problem non-self-adjoint. In complex problems of this type, it is not usually possible to obtain G in an explicit analytic form. The principal objective of this paper is to show how the Fant equation can still be derived in such cases from a consideration of the equation of aerodynamic sound and from the adjoint of the equation governing G in the neighbourhood of the ‘throttle’. The theory is illustrated by application to the canonical problem of throttled flow into a Helmholtz resonator.

JFM classification

Acoustics: Aeroacoustics Mathematical Foundations: General fluid mechanics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 428 - 450
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1970 (ed.) Handbook of Mathematical Functions (9th corrected printing), US Department of Commerce, National Bureau of Standards Applied Mathematics Series No. 55.Google Scholar
Austin, S. F. & Titze, I. R. 1997 The effect of subglottal resonance upon the vocal fold vibration. J. Voice 11, 391–402.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, Wakes and Cavities. Academic Press.Google Scholar
Crighton, D. G., Dowling, A. P., Ffowcs Williams, J. E., Heckl, M. & Leppington, F. G. 1992 Modern Methods in Analytical Acoustics (Lecture Notes). Springer.CrossRefGoogle Scholar
Cummings, A. 1984 Acoustic nonlinearities and power losses at orifices. AIAA J. 22, 786–792.CrossRefGoogle Scholar
Cummings, A. 1986 Transient and multiple frequency sound transmission through perforated plates at high amplitude. J. Acoust. Soc. Am. 79, 942–951.CrossRefGoogle Scholar
Duncan, C., Zhai, G. & Scherer, R. 2006 Modeling coupled aerodynamics and vocal fold dynamics using immersed boundary methods. J. Acoust. Soc. Am. 120, 2859–2871.CrossRefGoogle ScholarPubMed
Fant, G. 1960 Acoustic Theory of Speech Production. Mouton.Google Scholar
Flanagan, J. L. 1972 Speech Analysis Synthesis and Perception, 2nd edn. Springer.CrossRefGoogle Scholar
Flanagan, J. L. & Landgraf, L. L. 1968 Self-oscillating sources for vocal-tract synthesizer. IEEE Trans. Audio Electroacoust. AU-16, 57–64.CrossRefGoogle Scholar
Fulcher, L. P., Scherer, R. C., Melnykov, A., Gateva, V. & Limes, M. E. 2006 Negative Coulomb damping, limit cycles, and self-oscillation of the vocal folds. Am. J. Phys. 74, 386–393.CrossRefGoogle Scholar
Gupta, V., Wilson, T. A. & Beavers, G. S. 1973 A model for vocal cord excitation. J. Acoust. Soc. Am. 54, 1607–1617.CrossRefGoogle Scholar
Gurevich, M. I. 1965 Theory of Jets in Ideal Fluids. Academic Press.Google Scholar
Hofmans, G. C. J., Groot, G., Ranucci, M., Graziani, G. & Hirschberg, A. 2003 Unsteady flow through in-vitro models of the glottis. J. Acoust. Soc. Am. 113, 1658–1675.CrossRefGoogle ScholarPubMed
Howe, M. S. 1998 Acoustics of Fluid–Structure Interactions. Cambridge University Press.CrossRefGoogle Scholar
Howe, M. S. 2002 Theory of Vortex Sound. Cambridge University Press.CrossRefGoogle Scholar
Howe, M. S. 2008 Rayleigh Lecture 2007: flow–surface interaction noise. J. Sound Vib. 314, 113–146.CrossRefGoogle Scholar
Howe, M. S. & McGowan, R. S. 2007 Sound generated by aerodynamic sources near a deformable body, with application to voiced speech. J. Fluid Mech. 592, 367–392.CrossRefGoogle Scholar
Howe, M. S. & McGowan, R. S. 2009 Analysis of flow–structure coupling in a mechanical model of the vocal folds and the subglottal system. J. Fluids Struct. 25, 1299–1317.CrossRefGoogle Scholar
Howe, M. S. & McGowan, R. S. 2010 On the single-mass model of the vocal folds. Fluid Dyn. Res. 42, 015001.CrossRefGoogle ScholarPubMed
Howe, M. S. & McGowan, R. S. 2011 On the generalised Fant equation. J. Sound Vib. (in press).CrossRefGoogle Scholar
Ishizaka, K. & Flanagan, J. L. 1972 Synthesis of voiced sounds from a two-mass model of the vocal cords. Bell Syst. Tech. J. 51, 1233–1267.CrossRefGoogle Scholar
Joliveau, E., Smith, J. & Wolfe, J. 2004 Vocal tract resonances in singing: the soprano voice. J. Acoust. Soc. Am. 116, 2434–2439.CrossRefGoogle ScholarPubMed
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lighthill, M. J. 1952 On sound generated aerodynamically. Part I. General theory. Proc. R. Soc. Lond. A 211, 564–587.Google Scholar
Luong, T., Howe, M. S. & McGowan, R. S. 2005 On the Rayleigh conductivity of a bias-flow aperture. J. Fluids Struct. 21, 769–778.CrossRefGoogle Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, 5th edn. Macmillan.CrossRefGoogle Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, vol. 1. McGraw-Hill.Google Scholar
Morse, P. M. & Ingard, K. U. 1968 Theoretical Acoustics. McGraw-Hill.Google Scholar
Park, J. B. & Mongeau, L. 2007 Instantaneous orifice discharge coefficient of a physical, driven model of the human larynx. J. Acoust. Soc. Am. 121, 442–455.CrossRefGoogle ScholarPubMed
Pelorson, X., Hirschberg, A., van Hassel, R. R., Wijnands, A. P. J. & Auregan, Y. 1994 Theoretical and experimental study of quasisteady flow separation within the glottis during phonation. J. Acoust. Soc. Am. 96, 3416–3431.CrossRefGoogle Scholar
Poinsot, T. & Veynante, D. 2005 Theoretical and Numerical Combustion, 2nd edn. R. T. Edwards.Google Scholar
Rayleigh, Lord 1945 The Theory of Sound, vol. 2, chap. 16. Dover.Google Scholar
Rothenberg, M. 1981. Acoustic interaction between the glottal source and the vocal tract. In Vocal Fold Physiology (ed. Stevens, K. N. & Hirano, M.), pp. 305–328. University of Tokyo Press.Google Scholar
Stevens, K. N. 1998 Acoustic Phonetics. MIT Press.Google Scholar
Strahle, W. C. 1971 On combustion generated noise. J. Fluid Mech. 49, 399–414.CrossRefGoogle Scholar
Strahle, W. C. 1978 Combustion noise. Prog. Energy Combust. Sci. 4, 157–176.CrossRefGoogle Scholar
Thomson, S. L., Mongeau, L. & Frankel, S. H. 2005 Aerodynamic transfer of energy to the vocal folds. J. Acoust. Soc. Am. 118, 1689–1700.CrossRefGoogle Scholar
Titze, I. R. 1988 The physics of small-amplitude oscillation of the vocal folds. J. Acoust. Soc. Am. 83, 1536–1552.CrossRefGoogle ScholarPubMed
Titze, I. R. 2008 Nonlinear source-filter coupling in phonation: theory. J. Acoust. Soc. Am. 123, 2733–2749.CrossRefGoogle ScholarPubMed
Titze, I. R. & Story, B. H. 1995 Acoustic interactions of the voice source with the lower vocal tract. J. Acoust. Soc. Am. 101, 2234–2243.CrossRefGoogle Scholar
Zanartu, M., Mongeau, L. & Wodicka, G. R. 2007 Influence of acoustic loading on an effective single mass model of the vocal folds. J. Acoust. Soc. Am. 121, 1119–1129.CrossRefGoogle Scholar
Zhang, C., Zhao, W., Frankel, S. H. & Mongeau, L. 2002 Computational aeroacoustics of phonation. Part II. Effects of flow parameters and ventricular folds. J. Acoust. Soc. Am. 112, 2147–2154.CrossRefGoogle ScholarPubMed
Zhao, W., Zhang, C., Frankel, S. H. & Mongeau, L. 2002 Computational aeroacoustics of phonation. Part I. Computational methods and sound generation mechanisms. J. Acoust. Soc. Am. 112, 2134–2146.CrossRefGoogle ScholarPubMed