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Abstract: Starting from a multiple-scales approach to model sound transmission through a slowly varying duct of arbitrary cross section, an explicit analytical solution is derived for a mode undergoing cut-on cut-off transition. The solution is a composite one, removing the singularity that appears in the WKB approximation by encompassing the inner boundary-layer solution with the behaviour far upstream and downstream. The solution should not only prove to be a useful benchmark for computational aeroacoustic models, but also enable a designer to continue to use multiple-scales theory to examine sound transmission even when cut-on cut-off transitional modes are present. [Copyright &y& Elsevier]