آرام سازی برای جریان یک بعدی مربوط به دهانه نای در یک مدل تار صوتی

Postglottal flow in low-order dynamical systems modeling vocal fold motion is customarily considered one-dimensional. A relaxation distance is however mandatory before the flow effectively complies with this approximation. A continuous vocal fold model is used to show that this relaxation distance can impact voice simulation through the coupling strength between source and tract. The degree of interaction raises if relaxation occurs closer to the glottis, introducing complexity in the response of the system.

This work addresses an issue that has plagued speech modeling for a number of years, that is, coupling a flow solution that requires a finite distance for the flow and acoustics to reach one-dimensionality, with commonly-employed acoustic solvers that assume that one-dimensionality occurs instantly at the glottis. Voice production can be modeled with different degrees of complexity. The essentials of the fluid–structure-acoustics interaction process can be captured by simple ordinary differential equation systems (ODEs), where the folds are represented by a mass-spring system, the fluid is represented by a quasi-parallel (1D) flow, and the acoustic source is represented by a plane wave emitter at the glottis (Sciamarella and Artana, 2009). The mucosal-wave model (Titze, 1988) is an example of the low-order modeling approach, in which the flapping motion of the vocal folds is condensed in one second-order ODE. This model, initially conceived for small amplitude oscillations, was later extended to account for large amplitude oscillations (Laje et al., 2001). In the extended version, an ad hoc nonlinear damping term was added in the ODE to account for an ensemble of effects ranging from the formation of the glottal jet to the saturation mechanism responsible for stopping the folds and interrupting the flow during vocal fold collision. The extended model has the particular advantage of being continuous: the returning points of the oscillation are included without resorting to piecewise functions. The approach, shown to produce vocal fold oscillation with physiologically realistic values for the parameters (Lucero, 2005) and also applied to labial oscillation modeling in birdsong (Laje and Mindlin, 2008), was employed to study the effect of source-tract coupling in phonation, i.e. of delayed feedback on vocal fold dynamics. Feedback arises when the glottal system is coupled to the vocal tract and pressure reverberations are allowed to perturb vocal fold motion after a time delay given by sound speed and vocal tract length. The inclusion of this delay transforms the single ODE system into a DDE system (delay differential equation), endowing the simple oscillator with a complexity that can lead to subharmonic and non-periodic solutions (Laje et al., 2001). In an application of the DDE system to source-tract interaction in birdsong (Laje and Mindlin, 2008), the transition zone between the avian source and the base of the tract is modeled in terms of characteristic distances which are redefined in this work for application to the case of human voice. A transition or relaxation distance separates the glottal outlet from the region where postglottal flow can be effectively considered 1-D. This distance is incorporated into the continuous vocal fold model, leading to an expression for the pressure perturbations that depends on this length scale. This study considers the role of the relaxation distance in human voice production. Unlike many of the parameters involved in low-order vocal fold models, the finite distance required for the flow and acoustics to reach one-dimensionality has a direct physical correlate in the development of the glottal jet. It corresponds to the distance it takes the flow exiting the glottis to regain a unidirectional profile across the vocal tract section. Different values of this parameter are to be expected depending on the spreading rate of the jet and on the geometry of the jet-developing region – epilarynx tube and vocal tract (Titze, 2008). The spreading rate of a jet is known to depend on numerous parameters (Gutmark and Grinstein, 1999), such as Reynolds number, nozzle geometry and aspect-ratio. The pulsating nature of the glottal jet makes the scenario still more complex, because most of these parameters are time-varying. Moreover, the elongated geometry of the glottal outlet leads to spreading rates with initially opposed tendencies in the coronal and sagittal planes, that result in axis switching (Sciamarella et al., 2012). Recent in vitro studies (Krebs et al., 2012) are addressing the quantification of the full flow field in the proximity of the glottis, and therefore on the problem on which this work focuses, with simple modeling tools. Correlations will be proposed in this work with experimental data, in order to show how the solution is affected by measured variations in the development length of the flow. The paper is organized as follows. Section 2 presents the derivation of the equation system modeling human voice with the relaxation length as an additional parameter, together with an analysis of the involved scales. Section 3 contains numerical examples showing how the model produces qualitatively different behavior for different values of the parameter. It also shows how solutions are affected if the relaxation length is time dependent. Conclusions are provided in Section 4.

Simple mathematical models are particularly useful to study in isolation the different sources of complexity that intervene in vocal fold behavior and hence in sound production. In studies devoted to the problem of source-tract (Laje and Mindlin, 2008) and source-source interaction (Laje et al., 2008) in oscine birds, coupling is treated with an approach that is more general than the traditional impedance approach. This rationale is applied back to speech communication in this work. Following recent velocimetry measurements for the glottal jet (Krebs et al., 2012), it is postulated that glottal flow regains a quasi-one-dimensional profile across the tract at a finite distance from the source. Combining ingredients from human (Laje et al., 2001) and birdsong (Laje and Mindlin, 2008) toy models, a delayed differential equation system is obtained and used as a voice simulator to evaluate the influence of this relaxation distance. Numerical examples show that a significant sensitivity to this magnitude exists through the coupling strength. The assumption of 1D postglottal flow is in fact an unphysical hypothesis. Our study shows that when feedback is taken into account, this common assumption is not deprived of consequences. For low values of the relaxation distance (i.e. comparable to the size of the vocal folds), the degree of coupling is shown to reach levels where subharmonic responses are possible. Delayed feedback is one of the possible mechanisms underlying subharmonicity, also reported to occur in the case of asymmetrical vocal fold motion ( Steinecke and Herzel, 1995 and Svec et al., 1996) or in the Kargyraa style of harmonic chant ( Levin and Edgerton, 1999). Larger values of the relaxation distance, instead, are shown to undermine the enhancement of delayed feedback, moving the system back to normal vocal fold behavior – where subharmonic response is untypical. Moreover, an intra-cycle time-variation of the relaxation length (in phase with glottal flow, as indicated by experimental studies Krebs et al., 2012) can drive the response from the system back from a period-2 state to a period-1 solution, as shown by our numerical simulations. In short, to the parameter set that controls the degree of source-tract interaction in phonation (including geometrical factors, such as the exact dimensions of the epilarynx), this work incorporates the role of commonly disregarded fluid dynamical factors, such as the non-planar characteristics and intra-cycle dynamics of postglottal flow: both are shown to have a moderating influence over coupling strength and therefore, over dynamical complexity. In other words, jet relaxation is found to provide a physical mechanism inducing a regularization of vocal fold behavior, there where the assumption of one-dimensionality occurring instantly at the glottis produces subharmonicity. There is still a paucity of information regarding the three-dimensional fluid dynamics of postglottal flow in phonation, which this work encourages to fulfill. An analysis of the manner in which the relaxation distance changes with subglottal pressure, stress–strain vocal fold characteristics, and other parameters controlling vocal fold motion would provide further insight on the effective role of this physical mechanism in real phonation.