Sound propagation in slowly varying lined flow ducts ofarbitrary cross-section

Sound transmission through ducts of constant cross-section with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion, where the modes are eigenfunctions of the corresponding Laplace eigenvalue problem along a duct cross-section. A natural extension for ducts with cross-section and wall impedance that are varying slowly (compared to a typical acoustic wavelength and a typical duct radius) in the axial direction is a multiple-scales solution. This has been done for the simpler problem of circular ducts with homentropic irrotational flow. In the present paper, this solution is generalized to the problem of ducts of arbitrary cross-section. It is shown that the multiple-scales problem allows an exact solution, given the cross-sectional Laplace eigensolutions. The formulation includes both hollow and annular geometries. In addition, the turning point analysis is given for a single hard-wall cut-on, cut-off transition. This appears to yield the same reflection and transmission coefficients as in the circular duct problem.