Wave propagation in strongly curved ducts

Abstract

A theoretical and experimental investigation has been made of wave propagation in the long-wavelength limit in curved ducts of both rectangular and circular cross section. The two-dimensional solution of the wave equation for propagation in curved rectangular bends is adapted for the case of a circular cross section. The wave admittance and phase velocity are computed in terms of integrals over the cross section of the duct, in which there is a radial variation in flow. Experiments to measure the wave admittance and phase velocity are carried out with semicircular segments from a baritone horn musical instrument tuning slide. Three air column arrangements are used; the curved duct assembly is placed in the center of the air column between segments of straight tubing, or antisymmetrically placed near the open or closed ends of the air column. The first-mode resonance frequencies of the air column are measured, and the frequency shifts relative to a straight duct are used to compute the wave-admittance and phase-velocity shifts in the curved duct assembly. For a toroidal duct with a radius of curvature of 12.7 mm and whose circular cross section has a radius of 9.25 mm, the measured increase in the wave admittance is 6.3% and in the phase velocity is 4.7% relative to their straight duct values. Dissipationless theory predicts equal shifts of 8.9%. The viscous energy losses within the interior of the curved duct due to shear are computed, and these losses are not negligible relative to the wall loss. The presence of the additional shearing losses lessen the admittance and phase velocity shifts by equal amounts. The observed inequality between the wave-admittance and phase-velocity shifts thus remains unaccounted for.