Sound transmission through ducts of constant cross-sectionwith a uniform inviscid mean flow and a constant acoustic lining (impedance wall)is classically described by a modal expansion, where the modes are eigenfunctionsof the corresponding Laplace eigenvalue problem along a duct cross-section. Anatural extension for ducts with cross-section and wall impedance that arevarying slowly (compared to a typical acoustic wavelength and a typical ductradius) in the axial direction is a multiple-scales solution. This has been donefor the simpler problem of circular ducts with homentropic irrotational flow. Inthe present paper, this solution is generalized to the problem of ducts ofarbitrary cross-section. It is shown that the multiple-scales problem allows anexact solution, given the cross-sectional Laplace eigensolutions. The formulationincludes both hollow and annular geometries. In addition, the turning pointanalysis is given for a single hard-wall cut-on, cut-off transition. This appearsto yield the same reflection and transmission coefficients as in the circularduct problem.