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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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