  We consider stability of an elastic membrane being on the bottom of a uniform
horizontal flow of an inviscid and incompressible fluid of finite depth with
free surface. The membrane is simply supported at the leading and the trailing
edges which attach it to the two parts of the horizontal rigid floor. The
membrane has an infinite span in the direction perpendicular to the direction
of the flow and a finite length in the direction of the flow. For the membrane
of infinite length we 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 perturbation theory of the roots
of the dispersion relation and the concept of negative energy waves and discuss
its relation to the anomalous Doppler effect.
