Sound propagation in slowly varying lined flow ducts of arbitrary 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 wave length and a typical duct radius) in axial direction is a multiple-scales solution. This was previously done for the simpler problem of circular ducts with homentropic irrotational flow. In the present paper, this solution is generalised 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 type of 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.