The problem of diffraction of a plane elastic wave by a gradient transversely isotropic layer is considered. Using the method of overdetermined boundary value problem in combination with the Fourier transform method, the system of ordinary differential equations of the second order with boundary conditions of the third type is obtained which is solved by the grid method. Results of calculations obtained using the above-mentioned technique for the case of piecewise linear profiles for the Young modulus of the layer are given.

In nature, many of the geological formations form layered structures with elastic properties differing in various directions. Of all the formations and media, the special interest is often given to transversely isotropic media in which elastic modula of the media are the same in the plane normal to the axis of symmetry but differ from those of the direction along the axis of symmetry. Studies show that many sedimentary rocks indeed are transversely isotropic [

Furthermore, transversely isotropic structures are normally used at production of composites. If fibers packed in parallel are used as a reinforcing agent, then the composite possesses a unidirectional structure and is treated as a transversely isotropic material in the planes normal to the direction of reinforcement [

A number of works have been dedicated to studying processes of propagation of sound waves through anisotropic elastic layers. For example, in [

Differential equations governing the diffraction problem are considered separately for half-planes and for a plate. The problems for half-planes are overdetermined allowing establishing a relationship between traces of desired functions at the media’s interface [

Using the above-mentioned technique, dependence of energy of transmitted wave on angle and frequency of incidence is studied numerically. Differences in behavior of energy of transmitted wave at diffraction by uniformly anisotropic and nonuniformly anisotropic layer are outlined.

Let an elastic harmonic wave of type

Geometry of the problem.

We seek a solution to the plane harmonic problem from the elasticity theory at

General equations of two-dimensional oscillations are written in the form

We assume that rotational components of the forces can not result in stretching of the body. Then some of the components become equal zero:

At the media’s interface, the following conjugation conditions are to be fulfilled:

Of all the possible solutions to the system (

Desired functions for the Lame system (

It is worth noting here that the unknowns

For the upper half-plane

In equalities (

We perform transition from traces of functions of medium 1 to traces of functions of the layer in equalities (

We eliminate Fourier transforms of the stresses from the obtained conditions using (

On the lower half-plane

Using (

Physical meanings of solutions of (

Let us consider three-dimensional oscillations of an elastic transversely isotropic medium. To describe deformations of the medium, the following model will be used [

Equations (

We will assume that the field does not depend on the

The system (

The system (

Thus, the problem of diffraction of an elastic harmonic wave by a transversely isotropic layer reduces to the boundary value problem (

Before discussions of the numerical results, we will give some notes regarding dependence of solution of the problem (

Therefore, in the case of Fourier transforms of traces of the incident field being singular distributions, for example, in the case of the incident wave being a plane wave, the solutions of the problems will also be singular distributions with the same carrier. From this it follows that diffraction of one plane wave results in two reflected waves: longitudinal and transverse, and excitation of waveguide waves in the layer does not occur. It is obvious that the last statement is true under condition of uniqueness of the diffraction problem (homogeneous conditions (

The desired problem can be solved using many approximation methods. A uniform, finite-difference grid with the mesh size

After carrying out the numerical solution, it is necessary to reconstruct the fields in the half-planes

Taking into account the conditions at infinity, displacements for the reflected field will have the following form:

The unknown coefficients

We will consider the case of diffraction by a plane longitudinal wave with displacements of the following kind:

We apply Fourier transformation to components of the incident field and arrive at the result that all the components of the field are singular distributions with the multiplier

For carrying out the numerical experiments, we will consider the case when the layer of thickness ^{3},

In the present work, two sets of studies were carried out. The first set of studies is dedicated to searching for dependence of normalized energy of the transmitted longitudinal wave on angular frequency

Dependence of normalized energy of the transmitted longitudinal wave on angular frequency

Dependence of normalized energy of the transmitted longitudinal wave on angular frequency

In Figure

The second set of studies is dedicated to searching for dependence of normalized energy of the transmitted longitudinal wave on the angle of incidence

In Figure

Dependence of normalized energy of the transmitted longitudinal wave having the angular frequency

Dependence of normalized energy of the transmitted longitudinal wave having the angular frequency

The method of overdetermined boundary value problem, used in the present work, when combined with and the Fourier transform method is shown to be efficient, especially, for the cases the Fourier transforms of traces of the incident field are singular distributions. Then the approximation problem is solved just at the value

Results obtained with respect to propagation of elastic waves through anisotropic layers can be used in geophysics for the initial analysis of structure of the layers of rock strata. Also results of propagation of elastic waves through nonuniform anisotropic structures can be used in industries in which anisotropic materials are applied as well as at designing protective layers for various processes and apparatuses.

This work was supported by RFBR 12-01-97012-r-povolzh’e-a.