High rates of development of construction equipment create the necessary prerequisites for the design and construction of buildings and other structures of elements that have significant strength and stability with low weight and small thickness [1–8]. At the same time, the development of technology leads to the emergence of more powerful machines and to an increasing number of vehicles, which leads to an increase in noise in populated areas and civil and industrial buildings. Acoustic improvement of premises becomes an actual problem in each design and construction of each building [9–13]. In the framework of solving this problem, the problem of the sound- insulating ability of the enclosing and supporting structures is first of all solved. To solve the problem, it is necessary to analyze the sound effect on the structure. In the framework of this paper, we analyzed sound- insulating enclosing and load-bearing building constructions. We will analyze this effect using the example of transverse oscillations of a multilayer construction under the influence of a plane sound wave perpendicular to the interface between the layers of the plate construction. The qualitative structure of the considered construction is presented in Fig. 1.

The solution of the following wave equation has determined oscillations in the considered multilayer construction.

where E (z) is the modulus of elasticity;  (z) is the density of materials of the considered structure;  (z) is the Poisson ratio, u (x,y,z,t) is the displacement of the points of the structure during oscillations; F (x,y,z,t) is the external processing (knock, sound wave, etc.); Lx, Ly and Lz are the dimensions of the considered structure in the directions indicated in the indices; x, y and z are spatial coordinates; t is the current time. Let us consider the case when the edges of the structure are rigidly fixed, and there is no effect on it at the time of the beginning of the considered processing. Then the boundary and initial conditions for equation (1) could be written in the following form.

, , ,

Now, let us solve Eq. (1) using the method of averaging functional corrections [14]. In the framework of the method to obtain the first-order approximation of the desired function u (x,y,z,t) one shall replace on the not yet known average value 1 in the right-hand side of the Eq. (1). Then the equation for the first-order approximation of the function u (x,y,z,t) takes the form.

Integration of both sides of Eq. (3) in time leads to the following result:

The not yet known average value 1 of the function u (x,y,z,t) should be determined using the standard relation.

where  is the continuance of observation of the oscillation in the considered structure. Substitution of the Relation (3.a) into Relation (4) leads to the following result:

The second-order approximation of the function u (x,y,z,t) was calculated by replacing of the considered function in the right-hand side of the Eq. (1) on the sum of the approximation of the previous order and the average value of the considered approximation 2, i.e. by the sum 2+u1(x,y,z,t). Then the equation for the second-order approximation of the function u (x,y,z,t) takes the form

The solution of Eq. (5) could be written as

The average value 2 of the second-order approximation of the considered function u (x,y,z,t) should be calculated by using the following standard relation [14].

where n is the order of the required approximation, substitution of Relation (3.a) and Relation (5.a) into Relation (6) leads to the following result.

Analysis of the spatio-temporal distribution of the displacement of points of the considered structures during oscillation has been done analytically in the framework of the second-order approximation by the method of averaging the functional corrections. The approximation is usually enough for qualitative analysis and obtaining some quantitative results. The results of the analytical calculations were verified by comparison with the results of the direct numerical simulations.

In this section we analyzed the spatio-temporal distribution of the displacement of the points of the considered multilayer building construction during their oscillations under the influence of a plane wave F (x,y,z,t) = Aexp( t-kzz), where A is the amplitude of the wave, kz is the projection of the wave number on the Oz axis, and  is the wave frequency. Fig. 1 shows the qualitative spatial distribution of the displacement of points of the considered structure as a function of the coordinates x and y at a fixed value of time. Fig. 2 shows the qualitative spatio-temporal distribution of the displacement of the considered structure points as a function of the coordinate z and time t for fixed values of the x and y coordinates.

Analysis of the spatio-temporal distribution of the considered displacement shows, that compromise between increasing of sound-insulation of by multilayer constructions and increasing of their complications are using two layer structures, which are presented on Fig. 4. These structures consists of load-bearing layer for reflection of received signal (see layer 1 on Fig. 4) and facing layer for absorption of the received signal (see layer 2 on Fig. 4).

In this paper, we introduce a model for the prognosis of sound insulation by a myltilayer building construction. To analyze the model, we introduce an analytical approach for the analysis of oscillations of points of the considered structure during sound processing. We formulate recommendations to increase the sound insulation of the considered constructions.