Reflection of an acoustic line source by an impedance surface with uniform flow

An exact analytic solution is derived for the 2D acoustic pressure field generated by a time-harmonic line mass source located above an impedance surface with uniform grazing flow. Closed-form asymptotic solutions in the far-field are also provided. The analysis is valid for both locally-reacting and nonlocally-reacting impedances, as is demonstrated by analysing a nonlocally reacting effective impedance representing the presence of a thin boundary layer over the surface. The analytic solution may be written in a form suggesting a generalization of the method of images to account for the impedance surface. The line source is found to excite surface waves on the impedance surface, some of which may be leaky waves which contradict the assumption of decay away from the surface predicted in previous analyses of surface waves with flow. The surface waves may be treated either (correctly) as unstable waves or (artificially) as stable waves, enabling comparison with previous numerical or mathematical studies which make either of these assumptions. The computer code for evaluating the analytic solution and far-field asymptotics is provided in the supplementary material. It is hoped this work will provide a useful benchmark solution for validating 2D numerical acoustic codes.