We study the time-harmonic acoustic scattering in a duct in presence of a flow and of adiscontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuousone leads to still open modeling questions, as in particular the singularity of thesolution at the abrupt transition and the choice of the right unknown to formulate thescattering problem. To address these questions we propose a mathematical approach based onvariational formulations set in weighted Sobolev spaces. Considering the discontinuousimpedance as the limit of a continuous boundary condition, we prove that only the problemformulated in terms of the velocity potential converges to a well-posed problem. Moreoverwe identify the limit problem and determine some Kutta-like condition satisfied by thevelocity: its convective derivative must vanish at the ends of the impedance area. Finallywe justify why it is not possible to define limit problems for the pressure and thedisplacement. Numerical examples illustrate the convergence process.
