# Transverse Wave Propagation in a Thin Isotropic Plate Part I
## 1 Introduction  
The issue of plate vibration has a history of more than 200 years. A number of plate oscillation studies and monographs have been published that address different issues and analyze different influences, but this issue is still not satisfactorily solved. [PERSON] was the first scientist who dealt with the mathematical expression of the problem of plates oscillation. In 1776, he analyzed the free oscillations of plates. The existence of various types of free oscillations of thin plates was discovered and experimentally confirmed by [PERSON] [1] (the experiment was performed on a fixed horizontal plate on which he applied a thin layer of powder, which formed regular patterns after vibrating the plate). [PERSON] tried to theoretically justify the results of these experiments. [PERSON]'s solution was based on [PERSON]'s previous work, and the result was [PERSON]'s theory of beam bending. [PERSON] presents the slab as a system of belts, where each belt is considered a beam [2]. [PERSON] presented his experiments to [PERSON], who in 1808 financially supported the announcement of a competition (in 1808) by the French Academy of Sciences for the prize for the best solution for plate vibrations. [PERSON] [3] derived a differential equation for a plate, but did not include the deformation of the slab. This part was completed by [PERSON]. In 1829, this solution was supplemented by a static load (Poisson), in which the plate bending stiffness was assumed constant. A satisfactory theory of plate bending solutions was presented by [PERSON], who already considered the plate thickness as a function of stiffness in the general equation of the plate. In 1850, [PERSON] developed the theory of thin elastic isotropic plates, which only respected the vertical displacement of the plate element [4]. [PERSON]'s theory, or more precisely, [PERSON]'s basic theory, assumes only vertical displacements and corresponding inertial effects. This theory was extended byRayleigh by influence of the moments of inertia due to the rotation of the cross sections [5]. Further refinement of the theory was made by [PERSON] ([PERSON]), who extended the basic [PERSON] model by the effect of shear strain and inclination on the rotation of the normal to the cross section [6]. However, they did not consider the effect of inertia moments due to the rotation of the cross sections. [PERSON] and, independently of him, [PERSON] corrected the basic [PERSON] model involving both [PERSON]'s effect of cross-section rotation and [PERSON]'s effect of shear [7; 8]. With the development of information technology and the use of thin shell structures in the second half of the 20 th century, the mathematical solution of thin plates has also been developed. Initially, it was a solution of plates made of isotropic materials, which gradually expanded to anisotropic and heterogeneous materials. Various geometric models of the plates were solved, which, however, were based on [PERSON]'s basic theory and used only [PERSON]'s material model. New mathematical methods and especially the use of computers have made it possible to extend the original solutions to composite plates (since the 1980s). At present, attention is paid mainly to the solution of wave propagation in both isotropic and orthotropic plates for various material models and boundary conditions. These solutions for the propagation of stress waves and deformations in materials allow us to develop and use gradually thin structures in mechanical engineering (e.g., automobiles, aircraft, ships), as well as in civil engineering (bridges, buildings) and other fields. [PERSON] [9] dealt with the analysis of oscillation propagation in isotropic and orthotropic plates under different boundary conditions. Using the DQM (Differential Quadrature Method), he gradually supported these plates from different sides, first from one side, then from two sides, three sides, and finally from four sides. He compared the obtained results for both types of boards.