Equations of higher order of accuracy describing the vibrations of thin plates

Methods developed in the mathematical theory of the averaging of processes in periodic media are used to derive two-dimensional equations describing the propagation of waves in non-uniform anisotropic plates with a periodic structure. Equations of higher order of accuracy in a small parameter - the ratio of the typical thickness of the plate to the typical wavelength, are derived. The case of uniform isotropic thin plates is considered in detail. Equations of different order of accuracy, derived in this paper, are analysed and compared with the equations proposed by others. Some corrections for the coefficients in Timoshenko-type equations, which increase the accuracy of these equations, are proposed.