KURBANALIYEV L. T.

Candidate of the Physical and Mathematical Sciences,

Yassawi International Kazakh-Turkish University

 

 DUYSENOVA G. A.

Master-Lecturer,

Yassawi International Kazakh-Turkish University

 

USE OF THE MATRIX METHOD IN PROBLEMS OF DISTRIBUTION OF ELASTIC WAVES IN LAYERED ENVIRONMENTS

 

A number of new results of waves concerning distributions allows to gain a matrix method in layered environments [1]. In particular, it was succeeded to prove that the medium with final number of the periods in the field of low frequencies of W can be replaced is transversal a homogeneous environment. The error of this replacement appears generally proportional W2/n. In the elastic periodic environment this fact was proved earlier on the basis of a number of hypotheses [2]. The specified approach is applied to more difficult periodic environments in which between periodic packs of layers there is a contact, with slipping of record of fluctuations the lands received by means of seismographs, contain information as about the nature of the seismic source which has generated this movement, and about properties of the environment through which indignation (fig. 1) extended.


 


Figure 1. Schematic image of a task

 

Great practical value is represented by settlement methods which at a stage of preliminary estimates and final could give the necessary values of own frequencies of fluctuation of soil, quantitative amendments at the expense of soil conditions.

The analysis of the matrixes describing layers at distribution of elastic waves, gives the chance to write down at small curvature of layers approximate expression [1]:

                                          

for a matrix of any poorly bent elastic layer in which two-dimensional waves extend. Here k - the wave number, a matrix of C0 describes a plane-parallel layer, and r1 and r2 - radiuses of curvature of borders respectively in the direction of distribution of waves and along a layer, perpendicular to the first direction. The second in the third members is characterized by amendments on curvature of a surface of layers, correction matrixes ΔC1  and ΔC2 represent in the form of meeting ranks on degrees of the relation of thickness of a layer to length of a wave and have final ideas.

 Let's consider distribution of flat harmonious waves in the environment consisting of n - layers with plane-parallel limits of the section (fig. 2), including each layer porous, containing one of fillers: viscous liquid, gazo-liquid mix, gas.

Let's consider two cases when the studied layered environment is concluded: (A) - between two half-spaces (with indexes m=0 and m = m+1) or (B) - between a free surface and a half-space.

The task consists in definition for a case (A) - amplitude and phase ranges, reflection and refraction coefficients for all set hades; (B) - amplitude-frequency phase ranges of horizontal and vertical shifts on a free surface and reflection coefficients from a package of layers. These characteristics are necessary and serve as the main sizes for definition of an increment of a ballnost of the territory. Let's assume that from the bottom half-space (Z£Î) the flat harmonious wave with W frequency at an angle θ and with a phase speed of C from a remote source falls.


Boundary statements of the problem between layers (in all points of a platform of contact) we set for both cases in the form of equalities: speeds of shifts of a skeleton, tension firm components, pressure of liquid and preservation of a stream of substances.

 

 


Figure 2. Models of environments and direction of vectors

 

on the lower bound these conditions look like:

             (2)

Besides, on the upper bound of the environment for a task (A) the condition (1), and for a case is fair (B) conditions on a free surface in shape have to be set:

                                               (3)

Decision method. Potentials and in a formula (3) taking into account the reflected and refracted waves can be presented in the form of [2]:

                                          (4)

where  and  - potentials of the reflected longitudinal and cross waves;  and  - potentials of the longitudinal and cross refracted waves.

The solution of the equation (4) meeting boundary conditions (1-3) we will write down so:

                                (5)

 

In expression (5) coefficients are complex, depending on frequency. Their return values it is representable in a look:

At the solution of seismological tasks it is supposed [2]:

Tension and pressure in any layer can be expressed through the generalized potentials:

                                        (6)

 

Further we will substitute (5) in (4) and passing from shifts to speeds taking into account (6), we will receive a ratio in a look:

     (7)

Let's place the beginning of coordinates on border between m-l and m layers (fig. 1), then speeds of shifts and tension on this border will be connected with potentials of these layers dependences:

                  (8)

Speed of shifts and t tension – a layer is expressed through potentials in a layer in a look:

 

                                         (9)

 

Let's exclude from (8) and (9) potentials, we will receive a formula connecting speeds of shifts and tension on the top and bottom borders of m  a layer:

                            (10)    

 

Using boundary conditions, we will find ratios for m and layer m-1:

        

Repeating mathematical transformations of n of times of this sort, we will receive communication between characteristics of the bottom and top environment in a look:

                                             (11)

Substituting in (11) instead of speeds of shifts and tension of their expression through potentials, in case of a task (A) we will have:

             (12)

In the top half-space there are no reflected waves therefore ratios are fair . In case of falling of longitudinal waves of the first type on the lower bound of the environment has to be . Let's assume that , then for the second type of longitudinal and cross waves expressions are fair:

Considering above stated, ratios (12) we can write down in shape:

                                         (13)

where   - matrix elements

For a task (B) system of the equations (12) looks like:

                              (14)

After some mathematical transformations taking into account falling of each type of waves we will receive:

                             (15)

where   - matrix elements

Actions over matrixes C, E and the solution of the equations (13) and (15) are carried out by means of the COMPUTER. Coefficients of the reflected and refracted waves are defined as the relation of amplitudes of the corresponding waves.

 

LITERATURE

 

1. Molotkov  L.A. Matrix method in the theory of distribution of waves in layered elastic and liquid environments. - L.: Science, 1984.

2. Thomson W.T. Transmission of elastic waves trough a strati-fief solid material. J. Appl. Phys., 1950, 21, N2, 89-93.