Using the method of A. L. Gol'denveizer we reduce the three-dimensional problem of high-frequency steady-state vibrations of a thin isotropic plate to a series of two-dimensional problems. We construct two recursive processes that describe the basic stress-strain state of the plate. It is established that the first coefficient of the asymptotic expansion of the natural frequency of vibrations is determined by the boundary conditions on the plane faces of the plate and is independent of the conditions on its lateral surface. Therefore to solve the problem it is necessary to take account of at least two approximations of the recursive processes.