On the boundary value problem for the vibration and wave processes in two-dimensional environs

Abstract

In the article research of the third boundary value problem for the two-dimensional equation of free membrane vibrations is presented. The solution of the original differential operator is found in the form of a combination of the linearly independent system of orthonormal eigenfunctions on the given interval. Using the spectral decomposition for sufficiently smooth function, one can obtain the exact analytical representation of the deflection function for investigated problem in two-dimensional environs. The deflection function characterizes a membrane deviation from the equilibrium position.