Radiation-induced instability of a membrane interacting with a uniformly moving free surface flow of finite depth

In a recent study, we investigate the motion of waves emitted due to flutter of an elastic membrane being at the bottom of a uniform horizontal potential flow of an inviscid and incompressible fluid of finite depth with free surface. The membrane is rigidly fixed at both extremities and placed at the center of a solid floor. The membrane has an infinite span in the direction perpendicular tothe direction of the flow and a finite width along the flow axis. In the case where the length of the membrane is supposed infinite, we managed to derive a full dispersion relation that is valid for arbitrary depth of the fluid layer and find conditions for the flutter of the membrane due to emission of surface gravity waves. We describe this radiation-induced instability by means of the wave energy of the flow related to the concept of negative energy waves and study its relation to the anomalous Doppler effect. An extension to the case of a membrane with finite chord is also subject of a discussion in terms of stability, despite the more complex settings and analysis involved