Acoustic energy flux in nonhomogeneous ducts

Abstract

A study of the scattering matrix S of sound waves propagating in an inhomogeneous duct which consists of two semi-infinite ducts, connected somehow, is undertaken. The scattering matrix contains all reflection and transmission coefficients from the nth into the mth propagating mode for upstream and downstream propagation. A mean flow u0 is included. From energy conservation one finds, that S is unitary for potential flow, SS+=E, if the mode normalization is performed to unity energy flux. Invariance considerations with respect to time reversal reveal that S* (ω, −u0), the conjugate complex of the reverse flow scattering matrix, obeys the relation S* (ω, −u0) S (ω, u0) =E. From these two results, the reciprocity relation ST(ω, −u0) =s (ω, u0) is obtained, where ST denotes the transpose of S. Some implications of these results for sound waves propagating in potential flow are discussed. The modifications which occur in rotational flow are also described.