Phonation threshold pressure: comparison of calculations and measurements taken with physical models of the vocal fold mucosa

In an important paper on the physics of small amplitude oscillations, Titze showed that the essence of the vertical phase difference, which allows energy to be transferred from the flowing air to the motion of the vocal folds, could be captured in a surface wave model, and he derived a formula for the phonation threshold pressure with an explicit dependence on the geometrical and biomechanical properties of the vocal folds. The formula inspired a series of experiments e.g., R. Chan and I. Titze, J. Acoust. Soc. Am 119, 2351-2362 (2006)]. Although the experiments support many aspects of Titze's formula, including a linear dependence on the glottal half-width, the behavior of the experiments at the smallest values of this parameter is not consistent with the formula. It is shown that a key element for removing this discrepancy lies in a careful examination of the properties of the entrance loss coefficient. In particular, measurements of the entrance loss coefficient at small widths done with a physical model of the glottis (M5) show that this coefficient varies inversely with the glottal width. A numerical solution of the time-dependent equations of the surface wave model shows that adding a supraglottal vocal tract lowers the phonation threshold pressure by an amount approximately consistent with Chan and Titze's experiments.