Modal analysis of a time-varying vocal tract with losses: A comparison of left and right eigenfunctions

Sound propagation during speech can be modeled by a linear wave operator describing the spatiotemporal evolution of acoustic pressure and velocity distributions on a bounded domain representing the interior of the vocal tract. The eigenvalues of the wave operator determine the formant frequencies and bandwidths. The right eigenfunctions describe the standing wave patterns associated with each formant, whereas the left eigenfunctions describe how an arbitrary source distribution projects onto each formant. In speech research, it has always been implicitly assumed that left and right eigenfunctions are identical. This only holds if the wave operator is self-adjoint, which cannot be true when losses due to viscous damping, radiation effects, and wall vibration are taken into account. To examine this issue, a time-domain finite-volume simulation of acoustic wave propagation in the vocal tract was developed, including the above loss mechanisms. Eigenvalues and eigenfunctions were calculated from the resulting state-space recursion, for time-varying geometries representing VCV sequences containing fricatives and stops. Left and right eigenfunctions are similar in shape, but there are differences in the zero-crossing points and the amplitudes of the different lobes. Predictions based on right eigenfunctions alone will be incorrect. [Work supported by NIH.]