A modified reciprocal equation to predict the complex transmission coefficient of a flow duct

Reciprocity is a basic property existing in wave phenomena characterized by complex transmission coefficients. Reciprocity in a duct describes the relationship between the direct response at the duct outlet to a source applied at the inlet and the reverse response at the inlet to a source at the outlet. The reciprocity has been adopted to simplify the methodology for predicting the reverse transmission coefficient from the direct coefficient and vice versa. The existing methods, however, cannot predict the complex form of the coefficient because they only consider the reciprocity in the moduli but neglect the reciprocity in the phases. The study on the reciprocity in a duct with a temperature gradient has hardly been seen. The present work derived a reciprocal equation with a time domain method to characterize the reciprocity in a duct with spatially varying temperature. In addition to considering the reciprocity in the moduli of the direct and reverse transmission coefficients, the proposed method invented a correction term to consider the reciprocity in their phases as well. The correction term was developed from the time delay between the incident and the transmitted pulses in the duct solved with the time domain method. The modified reciprocal equation using the correction term was derived and used to predict the reverse transmission coefficient for calculating exhaust noise of an engine connecting to the duct. The result has shown that the proposed time domain method is an effective approach to accurately describe the phase relationship of waves in flow ducts.