Time harmonic acoustic scattering in presence of a shear flow and a Myers impedance condition

We study theoretically and numerically the acoustic radiation of a source in a general flow. The Goldstein equations are chosen to describe the propagation of the acoustic perturbations. When the flow is potential, the perturbations are potential and the Goldstein equations reduce to a scalar wave equation. Introducing appropriate radiation conditions, the potential problem is shown to be well-posed for a subsonic flow. In a vortical flow, an additional vorticity unknown must be added to the velocity potential. This vorticity satisfies a non-standard harmonic transport equation and we show that resonant streamlines may exist. On a simpler toy model, we determine the spectrum of resonance frequencies and we show that the problem is well-posed outside these resonances. A numerical simulation outside a resonance is displayed for a complex 2D flow.