A REVERSE FLOW THEOREM AND ACOUSTIC RECIPROCITY IN COMPRESSIBLE POTENTIAL FLOWS IN DUCTS

A reverse flow theorem for acoustic propagation in compressible potential flow has been obtained directly from the field equations without recourse to energy conservation arguments. A reciprocity theorem for the scattering matrix for the propagation of acoustic modes in a duct with either acoustically rigid walls or acoustically absorbing walls follows. It is found that for a source at a specific 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 related reverse flow theorem specialized to one-dimensional acoustic propagation has also been obtained. Reciprocity relationships for the scattering coefficients for propagation are derived, and are found to be similar though much simpler than in the case of multi-mode propagation. In one-dimensional flow, reciprocal relations and power conservation arguments are used to show that scaled power reflection and transmission coefficients are invariant to flow reversal and switching of source location from one end of the duct to the other. Numerical verification of the reciprocal relationships is given in a companion paper.