Within the framework of the linearized theory of elasticity, as exemplified by the problem of the shear harmonic oscillations of the pre-stressed functionally graded layer, the influence of the type of inhomogeneity and of the character of the initial stressed state on the distribution of the displacements with depth is investigated. The initially deformed state is assumed to be homogeneous, the inhomogeneity of the initial stresses is caused by the change of the material properties. The transformation of the displacements with depth for different oscillation frequencies of the layer under the conditions of the equal and arbitrary intensity of the change of properties is demonstrated. The possibility to dampen the amplitude of displacements at the definite frequencies by means of changing the initial actions is established.