An analysis of acoustic wave propagation in a waveguide carrying an incompressible mean flow is presented. The radius of the waveguide is taken to vary slowly as a function of axial location. It is shown that the dynamic behavior of the enclosed fluid can be parametrized by the small parameter ε, where ε is the ratio of the typical duct radiusR0and the wall wavelengthL0. An analytical solution for the pressure field in the duct is given in terms of a regular perturbation expansion in ε. The method of matched asymptotic expansions is used to evaluate the refractive effect of a thin mean‐flow boundary layer on the acoustic pressure field. It is shown that in the case where the duct geometry conforms to that of a circular cosh duct the effect of higher‐order turning points in the wave equation can be effectively handled by a closed‐form solution that approximately solves the governing equations. The results of analysis are compared to those obtained using numerical methods.