Diffraction of a Plane Elastic Wave by a Gradient Transversely Isotropic Layer

Abstract

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.

1. Introduction

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 [1–3]. Besides, a thin-layered packet of parallel beds each of which is isotropic but properties of which differ from properties of the other beds within the packet behaves as a transversely isotropic medium at presence of deformations.

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 [4]. Most often sheet metals are not isotropic and possess normal anisotropy (transversely isotropic). Ferroconcrete containing cracks is considered a transversely isotropic material with the plane of isotropy parallel to the plane of the crack [5]. Transversely isotropic structures also occur at production of laminated wood [6].

A number of works have been dedicated to studying processes of propagation of sound waves through anisotropic elastic layers. For example, in [7, 8], an elastic layer was considered as uniform and anisotropic whereas [9] dealt with the problem of propagation of the sound wave through a transversely isotropic nonuniform layer. A simpler case of the problem was considered by the authors of present paper earlier [10]. In the present work considered is the problem of diffraction of an elastic wave by a nonuniform transversely isotropic plate with constant elasticity characteristics along the axis of the layer and a continuous distribution of elasticity parameters in the section.

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 [11]. Thus, the original problem reduces to the boundary value problem for the Lame system with the boundary conditions of the third type. Then the Fourier transform with respect to the variable for which uniformity is preserved is applied to the boundary value problem. The obtained system of ordinary differential equations is solved using the grid method.

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.

2. Statement of the Problem

Let an elastic harmonic wave of type 𝐮_𝟎(𝑥,𝑦)exp{𝑖𝜔𝑡} fall on a nonuniform in the transverse direction anisotropic layer of thickness 𝐿 (medium 2 {0<𝑦<𝐿}) with continuous density 𝜌₂(𝑦) and tensor (3×3) of elasticity modulus 𝐊(𝑦) from medium 1 {𝑦>𝐿} under the angle 𝜃 with respect to the axis 𝑦 (see Figure 1). The diffraction results in a wave 𝐮_𝟏(𝑥,𝑦) reflected toward medium 1, a wave 𝐮_𝟑(𝑥,𝑦) propagating toward medium 3 {𝑦<0}, and field 𝐮_𝟐(𝑥,𝑦) in the layer. Desired is the full diffracted field. Media 1 and 3 are assumed to be uniform and isotropic.

Figure 1 

Geometry of the problem.

We seek a solution to the plane harmonic problem from the elasticity theory at 𝑦<0 and 𝑦>𝐿 in the form 𝜕𝜎_𝑥𝑛+𝜕𝑥𝜕𝜏_𝑛𝜕𝑦+𝜌_𝑛𝜔²𝑢_𝑥𝑛=0,𝜕𝜏_𝑛+𝜕𝑥𝜕𝜎_𝑦𝑛𝜕𝑦+𝜌_𝑛𝜔²𝑢_𝑦𝑛𝜎=0,_𝑥𝑛=𝜆_𝑛+2𝜇_𝑛𝜕𝑢_𝑥𝑛𝜕𝑥+𝜆_𝑛𝜕𝑢_𝑦𝑛,𝜎𝜕𝑦_𝑦𝑛=𝜆_𝑛𝜕𝑢_𝑥𝑛+𝜆𝜕𝑥^𝑛+2𝜇_𝑛𝜕𝑢_𝑦𝑛,𝜏𝜕𝑦_𝑛=𝜇_𝑛𝜕𝑢_𝑥𝑛+𝜕𝑦𝜕𝑢_𝑦𝑛𝜕𝑥 (1)(2) for 𝑛=1,3 with the constant Lame coefficients 𝜆_𝑛, 𝜇_𝑛 and density 𝜌_𝑛.

General equations of two-dimensional oscillations are written in the form −𝜌₂𝜔²𝑢_𝑥2=𝜕𝜎_𝑥2+𝜕𝑥𝜕𝜏₂,𝜕𝑦−𝜌₂𝜔²𝑢_𝑦2=𝜕𝜏₂+𝜕𝑥𝜕𝜎_𝑦2,𝜎𝜕𝑦_𝑥2=𝑘_{𝑥𝑥𝑥𝑥}𝜕𝑢_𝑥2𝜕𝑥+𝑘_{𝑥𝑥𝑥𝑦}𝜕𝑢_𝑥2+𝜕𝑦𝜕𝑢_𝑦2𝜕𝑥+𝑘_{𝑥𝑥𝑦𝑦}𝜕𝑢_𝑦2,𝜎𝜕𝑦_𝑦2=𝑘_{𝑦𝑦𝑥𝑥}𝜕𝑢_𝑥2𝜕𝑥+𝑘_{𝑦𝑦𝑥𝑦}𝜕𝑢_𝑥2+𝜕𝑦𝜕𝑢_𝑦2𝜕𝑥+𝑘_{𝑦𝑦𝑦𝑦}𝜕𝑢_𝑦2,𝜏𝜕𝑦₂=𝑘_{𝑥𝑦𝑥𝑥}𝜕𝑢_𝑥2𝜕𝑥+𝑘_{𝑥𝑦𝑥𝑦}𝜕𝑢_𝑥2+𝜕𝑦𝜕𝑢_𝑦2𝜕𝑥+𝑘_{𝑥𝑦𝑦𝑦}𝜕𝑢_𝑦2,𝜕𝑦 (3)(4) where 𝑘_∗∗∗∗ are components of the elasticity modulus tensor. Let us introduce a standard notation for indices 𝑘_∗∗∗∗: {𝑥𝑥}→{1}, {𝑦𝑦}→{2} and {𝑥𝑦}→{3} and substitute (4) into (3) to obtain −𝜌₂𝜔²𝑢_𝑥2=𝜕𝑘𝜕𝑥¹¹𝜕𝑢_𝑥2𝜕𝑥+𝑘₁₃𝜕𝑢_𝑥2+𝜕𝑦𝜕𝑢_𝑦2𝜕𝑥+𝑘₁₂𝜕𝑢_𝑦2+𝜕𝜕𝑦𝑘𝜕𝑦¹³𝜕𝑢_𝑥2𝜕𝑥+𝑘₃₃𝜕𝑢_𝑥2+𝜕𝑦𝜕𝑢_𝑦2𝜕𝑥+𝑘₂₃𝜕𝑢_𝑦2,𝜕𝑦−𝜌₂𝜔²𝑢_𝑦2=𝜕𝑘𝜕𝑥¹³𝜕𝑢_𝑥2𝜕𝑥+𝑘₃₃𝜕𝑢_𝑥2+𝜕𝑦𝜕𝑢_𝑦2𝜕𝑥+𝑘₂₃𝜕𝑢_𝑦2+𝜕𝜕𝑦𝑘𝜕𝑦¹²𝜕𝑢_𝑥2𝜕𝑥+𝑘₂₃𝜕𝑢_𝑥2+𝜕𝑦𝜕𝑢_𝑦2𝜕𝑥+𝑘₂₂𝜕𝑢_𝑦2.𝜕𝑦 (5)

We assume that rotational components of the forces can not result in stretching of the body. Then some of the components become equal zero: 𝑘₁₃=𝑘₂₃=0. Note that, for the case of isotropic body, the elasticity modulus tensor takes the following form: 𝐊=𝜆+2𝜇𝜆0𝜆𝜆+2𝜇000𝜇. (6)

At the media’s interface, the following conjugation conditions are to be fulfilled: 𝑢_𝑥1(𝑥,𝐿+0)+𝑢_𝑥0(𝑥,𝐿+0)=𝑢_𝑥2(),𝑢𝑥,𝐿−0_𝑦1(𝑥,𝐿+0)+𝑢_𝑦0(𝑥,𝐿+0)=𝑢_𝑦2(),𝜏𝑥,𝐿−0₁(𝑥,𝐿+0)+𝜏₀(𝑥,𝐿+0)=𝜏₂(),𝜎𝑥,𝐿−0_𝑦1(𝑥,𝐿+0)+𝜎_𝑦0(𝑥,𝐿+0)=𝜎_𝑦2(𝑥,𝐿−0), (7) at 𝑦=𝐿 and 𝑢_𝑥3(𝑥,0−0)=𝑢_𝑥2(),𝑢𝑥,0+0_𝑦3(𝑥,0−0)=𝑢_𝑦2(),𝜏𝑥,0+0₃(𝑥,0−0)=𝜏₂(),𝜎𝑥,0+0_𝑦3(𝑥,0−0)=𝜎_𝑦2(𝑥,0+0), (8) at 𝑦=0.

Of all the possible solutions to the system (1), (2), (5), (7), and (8), we pick solutions corresponding to the waves going to infinity.

3. Boundary Value Problem for the System of Ordinary Differential Equations Describing a Field in the Gradient Layer

Desired functions for the Lame system (5) for any fixed 𝑦 from the interval (0,𝐿) will be considered in the class 𝐿_1,loc, and it will be assumed that the functions undergo a slow growth at infinity in the 𝑥-direction. This allows applying the Fourier transform with respect to the 𝑥-direction permitting both waves decaying at infinity and propagating waves. Thus, we perform the change of variables from variable 𝑥 to variable 𝜉 and obtain the system of equations at 𝑦∈(0,𝐿): 𝑘₃₃𝑢^′_𝑥2^′+𝜌₂𝜔²−𝑘₁₁𝜉²𝑢_𝑥2−𝑖𝜉𝑘₁₂+𝑘₃₃𝑢^′_𝑦2−𝑖𝜉𝑘^′₃₃𝑢_𝑦2𝑘=0,₂₂𝑢^′_𝑦2^′+𝜌₂𝜔²−𝑘₃₃𝜉²𝑢_𝑦2−𝑖𝜉𝑘₁₂+𝑘₃₃𝑢^′_𝑥2−𝑖𝜉𝑘^′₁₂𝑢_𝑥2=0 (9) with respect to the Fourier transform for displacements 𝑢_𝑥2(𝜉,𝑦) and 𝑢_𝑦2(𝜉,𝑦).

It is worth noting here that the unknowns 𝑢_𝑥2(𝜉,𝑦) and 𝑢_𝑦2(𝜉,𝑦) with respect to 𝑦 are ordinary functions, and, therefore, all the derivatives are to be understood in the classical sense. This allows discretization of the problem with respect to 𝑦. For any fixed 𝑦, desired functions with respect to 𝜉 are distributions of the slow growth.

For the upper half-plane {𝑦>𝐿}, it will be assumed that the solutions (1), (2) belong to 𝐿_1,loc(𝑅) and their traces 𝜏₁(𝑥,𝐿+0), 𝜎_𝑦1(𝑥,𝐿+0), 𝑢_𝑥1(𝑥,𝐿+0) and 𝑢_𝑦1(𝑥,𝐿+0) are correctly specified. We will consider that the desired functions are distributions of the slow growth at infinity, and, moreover, their traces are also distributions of the slow growth at infinity. In the work [11], it was shown that solutions corresponding to the wave moving in the positive 𝑦-direction satisfy the equalities linking to each other Fourier transforms of traces of components of the field 𝜉𝜏₁(𝜉,𝐿)+𝛾_1,1(𝜉)𝜎_𝑦1(𝜉,𝐿)−2𝑖𝜇₁𝜉𝛾_1,1(𝜉)𝑢_𝑥1()𝜌𝜉,𝐿−𝑖₁𝜔²−2𝜇₁𝜉²𝑢_𝑦1(𝜉,𝐿)=0,−𝛾_2,1(𝜉)𝜏₁(𝜉,𝐿)+𝜉𝜎_𝑦1()𝜌𝜉,𝐿+𝑖₁𝜔²−2𝜇₁𝜉²𝑢_𝑥1(𝜉,𝐿)−2𝑖𝜇₁𝜉𝛾_2,1(𝜉)𝑢_𝑦1(𝜉,𝐿)=0, (10) where 𝑘²₁₁=𝜌₁𝜔²/(𝜆₁+2𝜇₁), 𝑘²₂₁=𝜌₁𝜔²/𝜇₁ and branches of roots of the functions 𝛾_1,1=𝑘²₁₁−𝜉², 𝛾_2,1=𝑘²₂₁−𝜉² are chosen such that the real part is positive, and in the case of the real part being zero, positive imaginary roots are chosen.

In equalities (10), traces of all desired functions are considered at 𝑦=𝐿, but since the considered functions are continuous in the whole domain, the limit will be considered as the value at 𝑦=𝐿. We will proceed with the other traces of the desired functions in the same manner.

We perform transition from traces of functions of medium 1 to traces of functions of the layer in equalities (10). For doing this, we express the traces via conditions (7) and substitute the obtained expressions into (10). Thus, we obtain the following boundary conditions for the Fourier transforms of components of the field defined in the layer 𝜉𝜏₂(𝜉,𝐿)+𝛾_1,1(𝜉)𝜎_𝑦2(𝜉,𝐿)−2𝑖𝜇₁𝜉𝛾_1,1(𝜉)𝑢_𝑥2()𝜌𝜉,𝐿−𝑖₁𝜔²−2𝜇₁𝜉²𝑢_𝑦2(𝜉,𝐿)=𝑓₁(𝜉),−𝛾_2,1(𝜉)𝜏₂(𝜉,𝐿)+𝜉𝜎_𝑦2()𝜌𝜉,𝐿+𝑖₁𝜔²−2𝜇₁𝜉²𝑢_𝑥2(𝜉,𝐿)−2𝑖𝜇₁𝜉𝛾_2,1(𝜉)𝑢_𝑦2(𝜉,𝐿)=𝑓₂(𝜉), (11) where 𝑓₁(𝜉)=𝜉𝜏₀(𝜉,𝐿)+𝛾_1,1(𝜉)𝜎_𝑦0(𝜉,𝐿)−2𝑖𝜇₁𝜉𝛾_1,1(𝜉)𝑢_𝑥0()𝜌𝜉,𝐿−𝑖₁𝜔²−2𝜇₁𝜉²𝑢_𝑦0(),𝑓𝜉,𝐿₂(𝜉)=−𝛾_2,1(𝜉)𝜏₀(𝜉,𝐿)+𝜉𝜎_𝑦0(𝜉,𝐿)+𝑖𝜌₁𝜔²−2𝜇₁𝜉²𝑢_𝑥0(𝜉,𝐿)−2𝑖𝜇₁𝜉𝛾_2,1(𝜉)𝑢_𝑦0(𝜉,𝐿). (12)

We eliminate Fourier transforms of the stresses from the obtained conditions using (4). Thus, we obtain relations between traces of Fourier transforms of displacements in the layer 𝑎₁(𝜉)𝑢^′_𝑥2(𝜉,𝐿)+𝑎₂(𝜉)𝑢_𝑥2(𝜉,𝐿)+𝑎₃(𝜉)𝑢^′_𝑦2(𝜉,𝐿)+𝑎₄(𝜉)𝑢_𝑦2(𝜉,𝐿)=𝑓₁(𝜉),𝑎₅(𝜉)𝑢^′_𝑥2(𝜉,𝐿)+𝑎₆(𝜉)𝑢_𝑥2(𝜉,𝐿)+𝑎₇(𝜉)𝑢^′_𝑦2(𝜉,𝐿)+𝑎₈(𝜉)𝑢_𝑦2(𝜉,𝐿)=𝑓₂(𝜉), (13) where 𝑎₁(𝜉)=𝑘₃₃(𝐿),𝑎₂(𝜉)=−𝑘₁₂(𝐿)+2𝜇₁𝑖𝜉𝛾_1,1(𝜉),𝑎₃(𝜉)=𝑘₂₂(𝐿)𝛾_1,1(𝜉),𝑎₄(𝜉)=−𝑖𝜌₁𝜔²−2𝜇₁−𝑘₃₃(𝐿)𝜉²,𝑎₅(𝜉)=−𝑘₃₃(𝐿)𝛾_2,1(𝜉),𝑎₆(𝜉)=𝑖𝜌₁𝜔²−𝑘₁₂(𝐿)+2𝜇₁𝜉²,𝑎₇(𝜉)=𝑘₂₂(𝐿)𝑎𝜉,₈(𝜉)=−𝑖2𝜇₁−𝑘₃₃(𝐿)𝜉𝛾_2,1(𝜉). (14)

On the lower half-plane {𝑦<0}, solutions (1) will be sought in the class 𝐿_1,loc with the slow growth at infinity taking into account that the traces 𝜏₃(𝑥,0),  𝜎_𝑦3(𝑥,0),   𝑢_𝑥3(𝑥,0), and 𝑢_𝑦3(𝑥,0) are correctly determined and they also belong to 𝐿_1,loc. Then solutions from the class of distributions of the slow growth corresponding to waves moving in the negative 𝑦-direction satisfy equations establishing relations between Fourier transforms of components of the field [11] 𝜉𝜏₃(𝜉,0)−𝛾_1,3(𝜉)𝜎_𝑦3(𝜉,0)+2𝑖𝜇₃𝜉𝛾_1,3(𝜉,0)𝑢_𝑥3(𝜉)−𝑖𝜌₃𝜔²−2𝜇₃𝜉²𝑢_𝑦3()𝛾𝜉,0=0,_2,3(𝜉)𝜏₃(𝜉,0)+𝜉𝜎_𝑦3()𝜌𝜉,0+𝑖₃𝜔²−2𝜇₃𝜉²𝑢_𝑥3(𝜉,0)+2𝑖𝜇₃𝜉𝛾_2,3(𝜉)𝑢_𝑦3(𝜉,0)=0, (15) which are equivalent to conditions related to traces of the field components at the lower boundary 𝜉𝜏₂(𝜉,0)−𝛾_1,3(𝜉)𝜎_𝑦2(𝜉,0)+2𝑖𝜇₃𝜉𝛾_1,3(𝜉)𝑢_𝑥2()𝜌𝜉,0−𝑖₃𝜔²−2𝜇₃𝜉²𝑢_𝑦2()𝛾𝜉,0=0,_2,3(𝜉)𝜏₂(𝜉,0)+𝜉𝜎_𝑦2()𝜌𝜉,0+𝑖₃𝜔²−2𝜇₃𝜉²𝑢_𝑥2(𝜉,0)+2𝑖𝜇₃𝜉𝛾_2,3(𝜉)𝑢_𝑦2(𝜉,0)=0, (16) where 𝑘²₁₃=𝜌₃𝜔²/(𝜆₃+2𝜇₃), 𝑘²₂₃=𝜌₃𝜔²/𝜇₃ and branches of roots of the functions 𝛾_1,3=𝑘²₁₃−𝜉², 𝛾_2,3=𝑘²₂₃−𝜉² are chosen in the same way as branches of roots of the functions of the upper half-plane.

Using (4), we obtain 𝑏₁(𝜉)𝑢^′_𝑥2(𝜉,0)+𝑏₂(𝜉)𝑢_𝑥2(𝜉,0)+𝑏₃(𝜉)𝑢^′_𝑦2(𝜉,0)+𝑏₄(𝜉)𝑢_𝑦2()𝑏𝜉,0=0,₅(𝜉)𝑢^′_𝑥2(𝜉,0)+𝑏₆(𝜉)𝑢_𝑥2(𝜉,0)+𝑏₇(𝜉)𝑢^′_𝑦2(𝜉,0)+𝑏₈(𝜉)𝑢_𝑦2(𝜉,0)=0, (17) where 𝑏₁(𝜉)=𝑘₃₃(0)𝑏𝜉,₂(𝜉)=𝑖𝑘₁₂(0)+2𝜇₃𝜉𝛾_1,3(𝜉),𝑏₃(𝜉)=−𝑘₂₂(0)𝛾_1,3(𝜉),𝑏₄(𝜉)=−𝑖𝜌₃𝜔²−2𝜇₃−𝑘₃₃(0)𝜉²,𝑏₅(𝜉)=𝑘₃₃(0)𝛾_2,3(𝜉),𝑏₆(𝜉)=𝑖𝜌₃𝜔²−𝑘₁₂(0)+2𝜇₃𝜉²,𝑏₇(𝜉)=𝑘₂₂(0)𝑏𝜉,₈(𝜉)=𝑖2𝜇₃−𝑘₃₃(0)𝜉𝛾_2,3(𝜉). (18)

Physical meanings of solutions of (9) with the boundary conditions (13) and (17) are displacements (𝑢_𝑥2,𝑢_𝑦2) which describe the field at 0<𝑦<𝐿 in the problem of diffraction in the elastic layer.

4. Elastic Oscillations of a Transversely Isotropic Body

Let us consider three-dimensional oscillations of an elastic transversely isotropic medium. To describe deformations of the medium, the following model will be used [12]: 𝜀_𝑥𝑥=1𝐸𝜎_𝑥𝑥−𝜈𝜎_𝑧𝑧−𝜈^′𝐸^′𝜎_𝑦𝑦,𝜀_𝑧𝑧=1𝐸𝜎_𝑧𝑧−𝜈𝜎_𝑥𝑥−𝜈^′𝐸^′𝜎_𝑦𝑦,𝜀_𝑦𝑦𝜈=−^′𝐸^′𝜎_𝑥𝑥+𝜎_𝑧𝑧+1𝐸^′𝜎_𝑦𝑦,𝜀_𝑥𝑦=12𝐺^′𝜎_𝑥𝑦,𝜀_𝑥𝑧=1𝜎2𝐺^𝑥𝑧,𝜀_𝑦𝑧=12𝐺^′𝜎_𝑦𝑧. (19) Here the plane 𝑥𝑧 is the plane of isotropy and the planes 𝑥𝑦 and 𝑦𝑧 are the planes of elastic symmetry. The parameters 𝐸, 𝐸^′ are the Young modula, 𝜈 and 𝜈^′ are the Poisson coefficients, 𝐺=𝐸/(2(1+𝜈)) and 𝐺^′ are the displacement modula. Parameters without the prime sign correspond to deformations in the plane of isotropy whereas parameters with the prime sign correspond to deformations in the plane of elastic symmetry.

Equations (19) could be transformed to the form [13] 𝜎_𝑥𝑥=𝜆+2𝜇𝜀_𝑥𝑥+𝜆𝜀_𝑧𝑧+𝜆^′𝜀_𝑦𝑦,𝜎_𝑧𝑧=𝜆𝜀_𝑥𝑥+𝜆+2𝜇𝜀_𝑧𝑧+𝜆^′𝜀_𝑦𝑦,𝜎_𝑦𝑦=𝜆^′𝜀_𝑥𝑥+𝜀_𝑧𝑧+𝜆^′+2𝜇^′𝜀_𝑦𝑦,𝜎_𝑥𝑦=2𝐺^′𝜀_𝑥𝑦,𝜎_𝑥𝑧=2𝜇𝜀_𝑥𝑧,𝜎_𝑦𝑧=2𝐺^′𝜀_𝑦𝑧, (20)(21) in which the used notations imply the following: 𝐸𝜆+2𝜇=()𝑑𝜈1+𝜈1−^′2𝐸𝐸^′,𝐸𝜆=()𝑑𝜈1+𝜈𝜈+^′2𝐸𝐸^′,𝜆^′+2𝜇^′=𝐸^′(1−𝜈)𝑑,𝜆^′=𝐸𝜈^′𝑑,𝑑=1−𝜈−2𝜈^′2𝐸𝐸^′. (22)

We will assume that the field does not depend on the 𝑧 coordinate: 𝜕/𝜕𝑧≡0. Then we have 𝜀_𝑧𝑧=0, and the system of (20) falls into two independent subsystems. The first subsystem describes oscillations in the plane 𝑥𝑦: 𝜎_𝑥𝑥=𝜆+2𝜇𝜀_𝑥𝑥+𝜆^′𝜀_𝑦𝑦,𝜎_𝑦𝑦=𝜆^′𝜀_𝑥𝑥+𝜆^′+2𝜇^′𝜀_𝑦𝑦,𝜎_𝑥𝑦=2𝐺^′𝜀_𝑥𝑦, (23) whereas the second subsystem describes oscillations in the 𝑧 direction: 𝜎_𝑧𝑧=𝜆𝜀_𝑥𝑥+𝜆^′𝜀_𝑦𝑦,𝜎_𝑥𝑧=2𝜇𝜀_𝑥𝑧,𝜎_𝑦𝑧=2𝐺^′𝜀_𝑦𝑧. (24)

The system (24) with the use of equations of motion transforms to the following: 𝜇𝜕²𝑢_𝑧𝜕𝑥²+𝜕𝐺𝜕𝑦^′𝜕𝑢_𝑧𝜕𝑦+𝜌₂𝜔²𝑢_𝑧=0. (25)

The system (23) corresponds to (4) with the following notations: 𝜎_𝑥2=𝜎_𝑥𝑥, 𝜎_𝑦2=𝜎_𝑦, and 𝜏₂=𝜎_𝑥𝑦. Under the conditions 𝑘₁₃=𝑘₂₃=0 considered in Section 1, the elasticity tensor 𝐊 linking stress and deformations to each other takes the following form: 𝐊=𝜆+2𝜇𝜆^′0𝜆^′𝜆^′+2𝜇^′000𝐺^′. (26)

Thus, the problem of diffraction of an elastic harmonic wave by a transversely isotropic layer reduces to the boundary value problem (9), (13), and (17) with the elasticity tensor defined in (26).

5. Numerical Results

Before discussions of the numerical results, we will give some notes regarding dependence of solution of the problem (9), (13), and (17) on parameter 𝜉. All the coefficients of the boundary value problem are continuous functions of 𝜉. Then if right-hand sides of (13) are regular distributions on 𝜉, then solutions will also be considered as regular with respect to 𝜉. However, if 𝑓₁ and 𝑓₂ are singular distributions on 𝜉, then the solutions themselves will also be considered as singular. For example, if 𝑓₁=𝐶₁𝛿(𝜉−𝜉₀) and 𝑓₂=𝐶₂𝛿(𝜉−𝜉₀), then 𝑢_𝑥(𝑦)=𝛿(𝜉−𝜉₀)𝑤_𝑥(𝑦;𝜉₀) and 𝑢_𝑦(𝑦)=𝛿(𝜉−𝜉₀)𝑤_𝑦(𝑦;𝜉₀). In this case, it is convenient to “normalize” the boundary value problem by 𝛿(𝜉−𝜉₀). For doing that, we perform the change of variables from variable 𝜉 to variable 𝜉₀ all over and solve (9), (13), and (17) with respect to 𝑤_𝑥(𝑦;𝜉₀) and 𝑤_𝑦(𝑦;𝜉₀).

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 (13) result in a trivial solution to the problem (9), (13), and (17)) and under condition the eigenvalues of the waveguide formed by the layer which differ from 𝜉₀.

The desired problem can be solved using many approximation methods. A uniform, finite-difference grid with the mesh size ℎ was chosen to approximate the boundary value problem (9), (13), and (17) with the accuracy on the order of 𝑂(ℎ). When choosing the mesh size it is taken into account that the finite difference analogs of elastic profiles of the layer describe adequately the original continuous models. On the other hand, the mesh size ℎ must be smaller than the wavelength in the layer and, consequently, inversely proportional to the frequency 𝜔.

After carrying out the numerical solution, it is necessary to reconstruct the fields in the half-planes 𝑢₁ and 𝑢₃. For doing it, we consider displacements in a homogeneous isotropic 𝑛th medium which can be written in the general form in the following way [14]: 𝑢_𝑥𝑛𝑦=𝜉𝐴_𝑛𝑒^{−𝑖𝛾1𝑛𝑦}−𝜉𝐵_𝑛𝑒^{𝑖𝛾1𝑛𝑦}+𝛾_2𝑛𝐶_𝑛𝑒^{−𝑖𝛾2𝑛𝑦}+𝛾_2𝑛𝐷_𝑛𝑒^{𝑖𝛾2𝑛𝑦},𝑢_𝑦𝑛𝑦=𝛾_1𝑛𝐴_𝑛𝑒^{−𝑖𝛾1𝑛𝑦}+𝛾_1𝑛𝐵_𝑛𝑒^{𝑖𝛾1𝑛𝑦}−𝜉𝐶_𝑛𝑒^{−𝑖𝛾2𝑛𝑦}+𝜉𝐷_𝑛𝑒^{𝑖𝛾2𝑛𝑦}. (27)

Taking into account the conditions at infinity, displacements for the reflected field will have the following form: 𝑢_𝑥1𝑦=−𝜉₀𝐵₁𝑒^{𝑖𝛾11(𝑦−𝐿)}+𝛾₂₁𝐷₁𝑒^{𝑖𝛾21(𝑦−𝐿)},𝑢_𝑦1𝑦=𝛾₁₁𝐵₁𝑒^{𝑖𝛾11(𝑦−𝐿)}+𝜉₀𝐷₁𝑒^{𝑖𝛾21(𝑦−𝐿)}, (28) and for the transmitted wave 𝑢_𝑥3𝑦=𝜉₀𝐴₃𝑒^{−𝑖𝛾13𝑦}+𝛾₂₃𝐶₃𝑒^{−𝑖𝛾23𝑦},𝑢_𝑦3𝑦=𝛾₁₃𝐴₃𝑒^{−𝑖𝛾13𝑦}−𝜉₀𝐶₃𝑒^{−𝑖𝛾23𝑦}. (29)

The unknown coefficients 𝐵₁, 𝐷₁, 𝐴₃, and 𝐶₃ are found via the following expressions: 𝐵₁=𝑢_𝑦1𝛾₂₁−𝑢_𝑥1𝜉₀𝛾₁₁𝛾₂₁+𝜉²₀,𝐷₁=𝑢_𝑥1𝛾₁₁+𝑢_𝑦1𝜉₀𝛾₁₁𝛾₂₁+𝜉²₀,𝐴₃=𝑢_𝑦3𝛾₂₃+𝑢_𝑥3𝜉₀𝛾₁₃𝛾₂₃+𝜉²₀,𝐶₃=𝑢_𝑥3𝛾₁₃−𝑢_𝑦3𝜉₀𝛾₁₃𝛾₂₃+𝜉²₀, (30) where 𝑢_𝑥𝑛 and 𝑢_𝑦𝑛 are traces of displacements of the 𝑛th medium which are expressed using (7) and (8).

We will consider the case of diffraction by a plane longitudinal wave with displacements of the following kind: 𝑢_𝑥0𝑥,𝑦=𝐴₀𝑘_1,1sin𝜃×exp−𝑖𝑘_1,1sin𝜃𝑥−𝑖𝑘_1,1𝑢cos𝜃𝑦−𝐿_𝑦0𝑥,𝑦=𝐴₀𝑘_1,1cos𝜃×exp−𝑖𝑘_1,1sin𝜃𝑥−𝑖𝑘_1,1cos𝜃𝑦−𝐿. (31)

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 𝛿(𝜉−𝜉₀), 𝜉₀=𝑘_1,1sin𝜃. For example, the Fourier transform of the trace 𝑢_𝑥0(𝑥,𝑦) at 𝑦=𝐿 takes the following form: 𝑢_𝑥0(𝜉,𝐿)=𝐴₀𝑘_1,1sin𝜃𝛿𝜉−𝑘_1,1sin𝜃. (32) Since the right-hand sides of (13) are singular distributions, then it is sufficient to solve the problem (9), (13), and (17) just at the value 𝜉=𝜉₀.

For carrying out the numerical experiments, we will consider the case when the layer of thickness 𝐿=10m made of siltstone is located in sandstone. Parameters of sandstone filling in medium 1 and medium 3 are 𝜌=2400kg/m³, 𝑣_𝑝=3300m/sec, and 𝑣_𝑠=2000m/sec. The layer is considered anisotropic; we will consider three cases of distributions of elasticity parameters in the layer. A uniform layer has the following parameters [15]: 𝐸=56.8GPa, 𝐸^′=62.1GPa, 𝜈=0.29, 𝜈^′=0.26, and 𝐺^′=22.9GPa. In the case of “compressed siltstone,” all the parameters of the medium remain constant except for the Young modulus 𝐸(𝑦) which, in the plane of isotropy, grows linearly from 𝐸(0)=56.8GPa to 𝐸(𝐿/2)=85.0GPa and then reduces linearly to the value 𝐸(𝐿)=56.8GPa. In the case of “relaxed siltstone,” the Young modulus 𝐸(𝑦) reduces linearly from 𝐸(0)=56.8GPa to 𝐸(𝐿/2)=28.4GPa and grows linearly to 𝐸(𝐿)=56.8GPa.

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 𝜔. As a result, the conclusion is made so that the transmitted energy grows as the value of the Young modulus 𝐸(𝑦) reduces in the middle of the layer. The difference between various structures increases with the increase of angle of incidence 𝜃. The dependences are given in Figures 2 and 3.

Figure 2 

Dependence of normalized energy of the transmitted longitudinal wave on angular frequency 𝜔 at its incidence under the angle 𝜃=20∘upon a siltstone layer of thickness 𝐿=10m placed into sandstone. The solid curve corresponds to a uniform layer of siltstone, the dotted curve “compressed siltstone,” and the dashed curve “relaxed siltstone.”

Figure 3 

Dependence of normalized energy of the transmitted longitudinal wave on angular frequency 𝜔 at its incidence under the angle 𝜃=40∘upon a siltstone layer of thickness 𝐿=10m placed into sandstone. The solid curve corresponds to a uniform layer of siltstone, the dotted curve “compressed siltstone,” and the dashed curve “relaxed siltstone.”

In Figure 2, dependence of normalized energy of the transmitted longitudinal wave on angular frequency 𝜔 at its incidence under the angle 𝜃=20^∘upon a siltstone layer of thickness 𝐿=10m placed into sandstone is shown. The solid curve corresponds to a uniform layer of siltstone, the dotted curve “compressed siltstone,” and the dashed curve “relaxed siltstone.” In Figure 3, the same type of dependence is shown but for the angle 𝜃=40^∘.

The second set of studies is dedicated to searching for dependence of normalized energy of the transmitted longitudinal wave on the angle of incidence 𝜃. Just as in the previous set of studies, confirmed is the conclusion that the transmitted energy increases with the decrease of the Young modulus 𝐸(𝑦) in the middle of the layer. It is worth noting here that the difference in the transmitted energy for different structures decreases with the increase of angular frequency 𝜔.

In Figure 4, dependence of normalized energy of the transmitted longitudinal wave on 𝜃 at the angular frequency of the wave 𝜔=2⋅10³ radian per second upon a siltstone layer of thickness 𝐿=10m placed into sandstone is shown. The solid curve corresponds to a uniform layer of siltstone, the dotted curve “compressed siltstone,” and the dashed curve “relaxed siltstone.” In Figure 5, the same type of dependence is shown but for 𝜔=4⋅10³ radian per second.

Figure 4 

Dependence of normalized energy of the transmitted longitudinal wave having the angular frequency 𝜔=2000rad/sec on its angle of incidence 𝜃 upon a siltstone layer of thickness 𝐿=10m placed into sandstone. The solid curve corresponds to a uniform layer of siltstone, the dotted curve “compressed siltstone,” and the dashed curve “relaxed siltstone.”

Figure 5 

Dependence of normalized energy of the transmitted longitudinal wave having the angular frequency 𝜔=4000rad/sec on its angle of incidence 𝜃  upon a siltstone layer of thickness L = 10 m placed into sandstone. The solid curve corresponds to a uniform layer of siltstone, the dotted curve “compressed siltstone,” and the dashed curve “relaxed siltstone.”

6. Conclusions

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 𝜉=𝜉₀. In the case of the Fourier transforms being regular distributions (e.g., at diffraction of a Gauss beam by a plate), the problem (9), (13), and (17) is solved for several values of the parameter 𝜉.

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.

Acknowledgment

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