Mode-analysis in curved and ring-shaped ducts. Part 1: Curved ducts

Sound fields in ducts with ring-shaped cross-sections or in flat ducts with a circular bend of their axis can be represented as mode sums of modes in the cylindrical co-ordinate system in which the duct wall surfaces are co-ordinate surfcaces. Ring-shaped lined ducts are a common form of silencers; bent ducts with absorbing walls are favorite constructions as turning vanes for the flow in the corners of wind tunnels. The modal analysis of the sound fields in both types of ducts proceeds in parallel over wide parts. The most significant differences are found in the determination of the mode wave numbers. Whereas in the ring-shaped duct the arguments of the cylindrical functions are sought as radial wave numbers for given (mostly integer) orders of these functions, their (complex) orders are to be determined as azimuthal wave numbers in the curved duct, in which the arguments of the cylindrical functions can be supposed as known quantities, because their determination then is a rather simple standard task. If more than just the first-order approximation is required to the attenuation in a duct system containing a ring-shaped or a bent duct section (for that approximation the knowledge of the least attenuated mode would be sufficient) a set of mode wave numbers must be evaluated. This difficult sub-task of the modal analysis in both types of ducts is the principal topic of the present paper. The present part I of the paper is mainly concerned with curved ducts.