Resonance tuning in vocal tract acoustics from modal perturbation analysis instead of nonlinear radiation pressure

The analysis of planar wave resonances in ducts of variable cross-section is of importance in many areas of acoustics such as horn theory, the design of musical instruments or the numerical generation of voice. An interesting problem is that of modifying the tube geometry and physical parameters to move its resonances to desired target values. In vowel production, for instance, one can properly adjust the length, cross-sectional area and wall admittance of the vocal tract (VT) to change the frequency of a resonance or to bring together or set apart a group of them for achieving some natural voice effects. An optimization strategy can be implemented to that purpose, which induces small sequential variations of the VT parameters according to sensitivity functions, until one obtains the desired resonance values. The sensitivity functions are usually derived from the work done by variations in the non-linear radiation pressure within the duct. In this paper we will show that, elegant as it may be, there is in fact no need of non-linear phenomena to determine appropriate sensitivity functions. These can readily be obtained from first order modal perturbation analysis. To demonstrate that, we transform the three-dimensional (3D) problem of voice generation into a one-dimensional one (1D), driven by a Webster-type equation. Despite of the latter being 1D, we discretize it using the finite element method (FEM) to benefit from its weak formulation to deduce expressions for the sensitivity functions. The latter are recovered from a perturbation modal analysis of the discrete Webster equation for VT area, length and wall admittance variations. Several examples concerning vowel to vowel transformations are presented to illustrate the potential and limitations of the method. The herein proposed 1D approach for resonance tuning could be easily coupled to 3D FEM codes for expressive vowel production in the future.