Free oscillations of a thin luid layer of finite electric conductivity under the action of an external magnetic field

The problem of free oscillations of a thin layer of a heavy, incompressible, inviscid fluid of finite electrical conductivity in a horizontal magnetic field is reduced to a system of integrodifferential Fredholm equations with variable coefficients. A numerical analysis is performed over a broad range of input parameters, and the results obtained are supplemented with asymptotic formulas with large and small magnetic Reynolds numbers. A classification of the resulting wave modes is proposed. It is shown that certain conditions can lead to the occurrence of unstable oscillations of the fluid layer that grow in time.