Source-tract interaction with prescribed vocal fold motion

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.

Aerodynamic sound of a body in arbitrary, deformable motion, with application to phonation

“…The subglottal tract is also regarded here as rigid and uniform with cross-section A L < A and terminating in the lung complex, where wave energy is absorbed without reflection, or reflected in accordance with a suitable impedance condition [17 – 21]. Similarly, the glottis is modelled by a ‘necked’ duct of v rectangular cross-section and streamwise length ${\ell }_{\text{g}}\ll \sqrt{A}$, which opens and closes at a nominal ‘free’ vocal fold frequency f o ( ~ 125 Hz for an adult male) [22, 23].…”

Voicing produced by a constant velocity lung source

“…locally incompressible, and where the inertia of the fluid column in motion through the glottis is balanced against an aggregate of forces consisting of a constant subglottal overpressure p I , back-pressure associated with turbulence losses and interactions with the upper tract, and viscous forces at the walls. Subsequent analyses involving increasingly sophisticated applications of this basic equation have been discussed extensively in the literature and reviewed by McGowan and Howe (2012). Howe and McGowan (2011) derived the general form of Fant's equation from the equations of aerodynamic sound.…”

“…1 we can put F ¼ F g þ F f , where F g , F f , respectively, denote the force components produced respectively by jet interactions with the glottis and the false folds. The glottis component F g can be evaluated approximately by use of a simple quasi-static, "free-streamline" model of the jet (Howe and McGowan, 2011;McGowan and Howe, 2012). The jet in the vicinity of the idealized glottis is modeled as in Fig.…”

“…1. It would actually have been more satisfactory to extend the treatment given by McGowan and Howe (2012) which uses experimental data from measurements of the acoustic impedance on either side of the glottis. But, because the desired modification that properly accounts for steady lung contraction is contentious, it was decided to proceed first with a model that is simple enough to permit an exact comparison to be made of predictions of Q(t) and the subglottal pressure for a prescribed lung contraction rate Q o with corresponding predictions derived from the constant-pressure-driven Fant equation.…”

“…The constant pressure hypothesis was used in previous work by McGowan and Howe (2012) on Level I source-tract interactions [for which the motion of the vocal folds is prescribed; Titze (2008)], the assumption being that the pressure of air flowing from the lungs was approximately constant at the glottis, at least prior to any back-reaction from the subglottal system. In this paper we show that this assumption is unnecessary, and that an arguably more natural and rigorous boundary condition is the specification of the flow rate Q o at the lungs without regard to the nature of the resulting pressure behind the glottis.…”