NUMERICAL EXPERIMENTS ON ACOUSTIC RECIPROCITY IN COMPRESSIBLE POTENTIAL FLOWS IN DUCTS

A reciprocity theorem for the scattering matrix for the propagation of acoustic modes in a duct with acoustically hard walls or with acoustically absorbing walls has been given in a companion publication. It was found that for a source at a specified end of the duct, suitably scaled reflection matrices in direct and reverse flow have a reciprocal relationship. Scaled transmission matrices obtained for direct flow and reversed flow with simultaneous switching of source location from one end to the other also have a reciprocal relationship. A reverse flow theorem for the equivalent one-dimensional propagation model, which is a good approximation to the three-dimensional model at low frequencies, was also obtained. In this case, using reciprocity and acoustic power conservation arguments it is additionally found that the acoustic power transmission coefficient is the same for a source at either end of the duct for a given flow direction. This result leads to an invariance theorem which relates acoustic power propagated due to sources of equal pressure amplitude at the two ends of the duct. A numerical verification of these reciprocal relationships is given here for propagation in axially symmetric (circular and annular) ducts with multi-modal propagation and at low frequencies when a one-dimensional model is appropriate.