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

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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.

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