VARIATIONAL METHODS FOR ACOUSTIC RADIATION IN A DUCT WITH A SHEAR FLOW AND AN ABSORBING BOUNDARY

The well-posedness of the acoustic radiation in a 2D duct in the presence of both a fluid in flow and an absorbing wall described by the Myers boundary condition is studied using variational methods. Without flow the problem is found to be well-posed for any impedance value. The presence of a flow complicates the results. With a uniform flow the problem is proven to be always of the Fredholm type but is found to be well-posed only when considering a dissipative radiation problem. With a general shear flow, the Fredholm property is recovered for a weak enough shear and the dissipative radiation problem requires the introduction of extra conditions to be well-posed: enough dissipation, a large enough frequency, and nonintuitive conditions on the impedance value.