Resonances of a bent vocal tract

Analyses of wave propagation in a vocal tract invariably assume the tract to be a variable area tube with a straight axis. No estimates appear to be available of the effect of curvature of the tract on its resonances (or, equivalently, on its transfer function). For arbitrarily varying cross section, the computation is difficult. However, one can choose an idealized bent vocal tract with a uniform cross section for which the wave equation is separable. The resonance frequencies for such a bent tract are computed and compared to those of the corresponding straight tract. The comparison shows that for typical dimensions of the tract the shift in the resonance frequencies below 4 kHz is in the range of 2%-8%.