Near cut-on/cut-off transition in lined ducts with flow

Using multiple-scales techniques, transition-type phenomena of acoustic eigenmodes in slowly-varying cylindrical ducts are investigated. In hard-walled ducts, it is known that singularities in the solution can appear where a cut-on mode is completely reflected into an opposite-running mode of equal amplitude but with a change in phase. In this paper, we will attempt to show that a similar but partial reflection of an eigenmode is also possible in soft-wall ducts. The magnitude and phase of the reflected mode depend both on the irrotational mean flow and on the magnitude and phase of the wall impedance. The result remains consistent with hard-wall transition in the limit of infinite impedance