On a Long-Wave Instability of Steady-State Plane-Parallel Flows of an Ideal Fluid with Free Boundary in the Gravity Field

We consider the problem of linear stability of steady-state plane-parallel flows of inviscid incompressible fluid of uniform density with free surface in the gravity field. Using the method of coupling the integrals of motion, we prove that sufficient conditions for the stability of these flows to small plane long-wave perturbations are absent. We construct an analytic example of a steady-state plane-parallel flow with small plane long-wave perturbations superimposed as normal modes