  We consider a problem of stability of a membrane of an infinite span and a
finite chord length that is submerged in a uniform flow of finite depth with
free surface. In the shallow water approximation, Nemtsov (1985) has shown that
an infinite-chord membrane is susceptible to flutter instability due to
excitation of long gravity waves on the free surface if the velocity of the
flow exceeds the phase velocity of the waves and placed this phenomenon into
the general physical context of the anomalous Doppler effect. In the present
work we derive a full nonlinear eigenvalue problem for an integro-differential
equation in the case of the finite-chord Nemtsov membrane in the finite-depth
flow. In the shallow- and deep water limits we develop a perturbation theory in
the small added mass ratio parameter acting as an effective dissipation
parameter in the system, to find explicit analytical expressions for the
frequencies and the growth rates of the membrane modes coupled to the surface
waves. This result reveals a new intricate pattern of instability pockets in
the parameter space and allows for its analytical description. The case of an
arbitrary depth flow with free surface requires numerical solution of a new
non-polynomial nonlinear eigenvalue problem. We propose an original approach
combining methods of complex analysis and residue calculus, Galerkin
discretization, Newton method and parallelization techniques implemented in
MATLAB to produce high-accuracy stability diagrams within an unprecedentedly
wide range of system's parameters. We believe that the Nemtsov membrane appears
to play the same paradigmatic role for understanding radiation-induced
instabilities as the famous Lamb oscillator coupled to a string has played for
understanding radiation damping.
