Quasilinear Theory of Resonant Interaction of Bending Vibrations of a Thin Plate with a Shear Hydrodynamic Flow

Abstract

A system of equations describing resonant interaction of bending waves of a thin plate with a liquid or gas flow around it leading to the evolution of wind instability is derived in a quasilinear approximation. It is demonstrated that as a result of the reverse effect of waves on the flow, a quasilinear relaxation of liquid particle distribution function occurs to the state with a plateau that leads to a slow smoothing of the velocity profile in the liquid in the resonant region and thereby to elimination of the reason causing the growth of waves in the linear stage of instability evolution. The resultant energy transferred from the flow to the waves in the course of quasilinear relaxation is calculated. The wavelength of the instability that most quickly grows due to evolution of the wind instability is obtained.