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.

A Reverse Flow Theorem and Acoustic Reciprocity in Compressible Potential Flows in Ducts

“…By essentially using this starting point, Eversman [2] has demonstrated and numerically veri"ed reciprocal properties of the scattering matrix for acoustic modes incident, re#ected, and transmitted in a non-uniform duct in the absence of mean #ow. Moehring [3], by using an approach based on energy conservation, has arrived at the same result when di!erences in de"nitions of the normalization of acoustic modes are considered. Moehring's approach depends on a suitable de"nition of acoustic energy density and acoustic energy #ux, which are well known in the case of propagation in a stationary medium.…”

“…On this basis, Moehring [3] extended his observations on properties of the scattering matrix to include non-uniform ducts with rigid walls and potential mean #ow with the result that the reciprocal properties also depend on #ow reversal. Godin [5] has studied extensively issues of acoustic energy, acoustic reciprocity, and #ow reversal theorems in a highly generalized sense directly from the "eld equations.…”

“…Moehring's approach depends on a suitable de"nition of acoustic energy density and acoustic energy #ux, which are well known in the case of propagation in a stationary medium. Since the result of Moehring [3] depends on the normal derivative of the acoustic energy #ux vanishing on the walls of the duct, it would appear to exclude dissipative walls. However, the classical result [1] concludes that reciprocal relations still hold, provided the duct walls have a locally reacting impedance model (whether dissipative or not) where for harmonic disturbances, acoustic particle velocity is proportional to acoustic pressure.…”

“…Godin [5] has studied extensively issues of acoustic energy, acoustic reciprocity, and #ow reversal theorems in a highly generalized sense directly from the "eld equations. He has not speci"cally addressed the simpler case of propagation in non-uniform ducts with compressible irrotational #ow, citing Moehring [3] as having demonstrated reverse #ow reciprocity in this case [6]. Godin's citations to the literature can be consulted for an extensive survey of the "eld.…”

Reciprocity principle in duct acoustics

“…8 A more fundamental view of the reciprocity principle in duct acoustics was offered by l'6hring. 9 His approach is not, however, applicable to a lossy system like a lined duct, because the scattering matrix fails to be unitary in such a system.…”