Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2013 Article ID 262067 8 pageshttpdxdoiorg1011552013262067

Research ArticleDiffraction of a Plane Elastic Wave by a GradientTransversely Isotropic Layer

Anastasiia Anufrieva and Dmitrii Tumakov

Institute of Computer Science and Information Technology Kazan Federal University Kazan 420008 Russia

Correspondence should be addressed to Dmitrii Tumakov dtumakovksuru

Received 24 February 2013 Accepted 5 September 2013

Academic Editor Akira Ikuta

Copyright copy 2013 A Anufrieva and D Tumakov This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

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

1 Introduction

In nature many of the geological formations form layeredstructures with elastic properties differing in various direc-tions Of all the formations and media the special interest isoften given to transversely isotropic media in which elasticmodula of the media are the same in the plane normal to theaxis of symmetry but differ from those of the direction alongthe axis of symmetry Studies show that many sedimentaryrocks indeed are transversely isotropic [1ndash3] Besides a thin-layered packet of parallel beds each of which is isotropic butproperties of which differ from properties of the other bedswithin the packet behaves as a transversely isotropic mediumat presence of deformations

Furthermore transversely isotropic structures are nor-mally used at production of composites If fibers packed inparallel are used as a reinforcing agent then the compositepossesses a unidirectional structure and is treated as atransversely isotropic material in the planes normal to thedirection of reinforcement [4] Most often sheet metals arenot isotropic and possess normal anisotropy (transverselyisotropic) Ferroconcrete containing cracks is considered atransversely isotropic material with the plane of isotropyparallel to the plane of the crack [5] Transversely isotropicstructures also occur at production of laminated wood [6]

A number of works have been dedicated to studyingprocesses of propagation of sound waves through anisotropicelastic layers For example in [7 8] an elastic layer wasconsidered as uniform and anisotropic whereas [9] dealt withthe problem of propagation of the sound wave through atransversely isotropic nonuniform layer A simpler case ofthe problem was considered by the authors of present paperearlier [10] In the present work considered is the problem ofdiffraction of an elastic wave by a nonuniform transverselyisotropic plate with constant elasticity characteristics alongthe axis of the layer and a continuous distribution of elasticityparameters in the section

Differential equations governing the diffraction problemare considered separately for half-planes and for a plate Theproblems for half-planes are overdetermined allowing estab-lishing a relationship between traces of desired functions atthe mediarsquos interface [11] Thus the original problem reducesto the boundary value problem for the Lame system withthe boundary conditions of the third type Then the Fouriertransform with respect to the variable for which uniformityis preserved is applied to the boundary value problem Theobtained system of ordinary differential equations is solvedusing the grid method

Using the above-mentioned technique dependence ofenergy of transmitted wave on angle and frequency of

2 Advances in Acoustics and Vibration

0

L

y

2

1

3

u1 u0

u2

u3

1205791205881 1205821 1205831

1205883 1205823 1205833

1205882(y) K(y)

x

Figure 1 Geometry of the problem

incidence is studied numerically Differences in behaviorof energy of transmitted wave at diffraction by uniformlyanisotropic and nonuniformly anisotropic layer are outlined

2 Statement of the Problem

Let an elastic harmonic wave of type u0(119909 119910) exp119894120596119905 fall ona nonuniform in the transverse direction anisotropic layerof thickness 119871 (medium 2 0 lt 119910 lt 119871) with continuousdensity 120588

2(119910) and tensor (3 times 3) of elasticity modulus K(119910)

from medium 1 119910 gt 119871 under the angle 120579 with respect to theaxis 119910 (see Figure 1)The diffraction results in a wave u1(119909 119910)reflected toward medium 1 a wave u3(119909 119910) propagatingtoward medium 3 119910 lt 0 and field u2(119909 119910) in the layerDesired is the full diffracted field Media 1 and 3 are assumedto be uniform and isotropic

We seek a solution to the plane harmonic problem fromthe elasticity theory at 119910 lt 0 and 119910 gt 119871 in the form

120597120590119909119899

120597119909+120597120591119899

120597119910+ 1205881198991205962119906119909119899

= 0

120597120591119899

120597119909+120597120590119910119899

120597119910+ 1205881198991205962119906119910119899

= 0

(1)

120590119909119899

= (120582119899+ 2120583119899)120597119906119909119899

120597119909+ 120582119899

120597119906119910119899

120597119910

120590119910119899

= 120582119899

120597119906119909119899

120597119909+ (120582119899+ 2120583119899)120597119906119910119899

120597119910

120591119899= 120583119899(120597119906119909119899

120597119910+120597119906119910119899

120597119909)

(2)

for 119899 = 1 3 with the constant Lame coefficients 120582119899 120583119899and

density 120588119899

General equations of two-dimensional oscillations arewritten in the form

minus120588212059621199061199092

=1205971205901199092

120597119909+1205971205912

120597119910

minus120588212059621199061199102

=1205971205912

120597119909+1205971205901199102

120597119910

(3)

1205901199092

= 119896119909119909119909119909

1205971199061199092

120597119909+ 119896119909119909119909119910

(1205971199061199092

120597119910+1205971199061199102

120597119909) + 119896119909119909119910119910

1205971199061199102

120597119910

1205901199102

= 119896119910119910119909119909

1205971199061199092

120597119909+ 119896119910119910119909119910

(1205971199061199092

120597119910+1205971199061199102

120597119909) + 119896119910119910119910119910

1205971199061199102

120597119910

1205912= 119896119909119910119909119909

1205971199061199092

120597119909+ 119896119909119910119909119910

(1205971199061199092

120597119910+1205971199061199102

120597119909) + 119896119909119910119910119910

1205971199061199102

120597119910

(4)

where 119896lowastlowastlowastlowast

are components of the elasticity modulus tensorLet us introduce a standard notation for indices 119896

lowastlowastlowastlowast

119909119909 rarr 1 119910119910 rarr 2 and 119909119910 rarr 3 and substitute(4) into (3) to obtain

minus120588212059621199061199092

=120597

120597119909(11989611

1205971199061199092

120597119909+ 11989613(1205971199061199092

120597119910+1205971199061199102

120597119909)

+11989612

1205971199061199102

120597119910)

+120597

120597119910(11989613

1205971199061199092

120597119909+ 11989633(1205971199061199092

120597119910+1205971199061199102

120597119909)

+11989623

1205971199061199102

120597119910)

minus120588212059621199061199102

=120597

120597119909(11989613

1205971199061199092

120597119909+ 11989633(1205971199061199092

120597119910+1205971199061199102

120597119909)

+11989623

1205971199061199102

120597119910)

+120597

120597119910(11989612

1205971199061199092

120597119909+ 11989623(1205971199061199092

120597119910+1205971199061199102

120597119909)

+11989622

1205971199061199102

120597119910)

(5)

We assume that rotational components of the forcescan not result in stretching of the body Then some of thecomponents become equal zero 119896

13= 11989623

= 0 Note thatfor the case of isotropic body the elasticity modulus tensortakes the following form

K = (

120582 + 2120583 120582 0

120582 120582 + 2120583 0

0 0 120583

) (6)

At the mediarsquos interface the following conjugation condi-tions are to be fulfilled

1199061199091(119909 119871 + 0) + 119906

1199090(119909 119871 + 0) = 119906

1199092(119909 119871 minus 0)

1199061199101(119909 119871 + 0) + 119906

1199100(119909 119871 + 0) = 119906

1199102(119909 119871 minus 0)

1205911(119909 119871 + 0) + 120591

0(119909 119871 + 0) = 120591

2(119909 119871 minus 0)

1205901199101(119909 119871 + 0) + 120590

1199100(119909 119871 + 0) = 120590

1199102(119909 119871 minus 0)

(7)

Advances in Acoustics and Vibration 3

at 119910 = 119871 and

1199061199093(119909 0 minus 0) = 119906

1199092(119909 0 + 0)

1199061199103(119909 0 minus 0) = 119906

1199102(119909 0 + 0)

1205913(119909 0 minus 0) = 120591

2(119909 0 + 0)

1205901199103(119909 0 minus 0) = 120590

1199102(119909 0 + 0)

(8)

at 119910 = 0Of all the possible solutions to the system (1) (2) (5) (7)

and (8) we pick solutions corresponding to the waves goingto infinity

3 Boundary Value Problem for the System ofOrdinary Differential EquationsDescribing a Field in the Gradient Layer

Desired functions for the Lame system (5) for any fixed 119910

from the interval (0 119871) will be considered in the class 1198711loc

and it will be assumed that the functions undergo a slowgrowth at infinity in the 119909-directionThis allows applying theFourier transform with respect to the 119909-direction permittingboth waves decaying at infinity and propagating wavesThuswe perform the change of variables from variable119909 to variable120585 and obtain the system of equations at 119910 isin (0 119871)

(119896331199061015840

1199092)1015840

+ [12058821205962minus 119896111205852] 1199061199092

minus 119894120585 (11989612+ 11989633) 1199061015840

1199102minus 1198941205851198961015840

331199061199102

= 0

(119896221199061015840

1199102)1015840

+ [12058821205962minus 119896331205852] 1199061199102

minus 119894120585 (11989612+ 11989633) 1199061015840

1199092minus 1198941205851198961015840

121199061199092

= 0

(9)

with respect to the Fourier transform for displacements1199061199092(120585 119910) and 119906

1199102(120585 119910)

It is worth noting here that the unknowns 1199061199092(120585 119910)

and 1199061199102(120585 119910) with respect to 119910 are ordinary functions and

therefore all the derivatives are to be understood in theclassical sense This allows discretization of the problem withrespect to 119910 For any fixed 119910 desired functions with respectto 120585 are distributions of the slow growth

For the upper half-plane 119910 gt 119871 it will be assumedthat the solutions (1) (2) belong to 119871

1loc(119877) and their traces1205911(119909 119871 + 0) 120590

1199101(119909 119871 + 0) 119906

1199091(119909 119871 + 0) and 119906

1199101(119909 119871 + 0)

are correctly specified We will consider that the desiredfunctions are distributions of the slow growth at infinityand moreover their traces are also distributions of the slowgrowth at infinity In thework [11] it was shown that solutionscorresponding to the wavemoving in the positive119910-directionsatisfy the equalities linking to each other Fourier transformsof traces of components of the field

1205851205911(120585 119871) + 120574

11(120585) 1205901199101(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199091(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199101(120585 119871) = 0

minus 12057421

(120585) 1205911(120585 119871) + 120585120590

1199101(120585 119871)

+ 119894 (12058811205962minus 212058311205852) 1199061199091(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199101(120585 119871) = 0

(10)

where 119896211

= 12058811205962(1205821+ 21205831) 119896221

= 120588112059621205831and branches of

roots of the functions 12057411

= radic119896211minus 1205852 120574

21= radic1198962

21minus 1205852 are

chosen such that the real part is positive and in the case ofthe real part being zero positive imaginary roots are chosen

In equalities (10) traces of all desired functions areconsidered at 119910 = 119871 but since the considered functions arecontinuous in the whole domain the limit will be consideredas the value at 119910 = 119871 We will proceed with the other traces ofthe desired functions in the same manner

We perform transition from traces of functions ofmedium 1 to traces of functions of the layer in equalities (10)For doing this we express the traces via conditions (7) andsubstitute the obtained expressions into (10) Thus we obtainthe following boundary conditions for the Fourier transformsof components of the field defined in the layer

1205851205912(120585 119871) + 120574

11(120585) 1205901199102(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199092(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199102(120585 119871) = 119891

1(120585)

minus 12057421

(120585) 1205912(120585 119871) + 120585120590

1199102(120585 119871)

+ 119894 (12058811205962minus 212058311205852) 1199061199092(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199102(120585 119871) = 119891

2(120585)

(11)

where1198911(120585) = 120585120591

0(120585 119871) + 120574

11(120585) 1205901199100(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199090(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199100(120585 119871)

1198912(120585) = minus120574

21(120585) 1205910(120585 119871)

+ 1205851205901199100(120585 119871) + 119894 (120588

11205962minus 212058311205852) 1199061199090(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199100(120585 119871)

(12)

We eliminate Fourier transforms of the stresses fromthe obtained conditions using (4) Thus we obtain relationsbetween traces of Fourier transforms of displacements in thelayer

1198861(120585) 1199061015840

1199092(120585 119871) + 119886

2(120585) 1199061199092(120585 119871)

+ 1198863(120585) 1199061015840

1199102(120585 119871)

+ 1198864(120585) 1199061199102(120585 119871) = 119891

1(120585)

1198865(120585) 1199061015840

1199092(120585 119871) + 119886

6(120585) 1199061199092(120585 119871)

+ 1198867(120585) 1199061015840

1199102(120585 119871)

+ 1198868(120585) 1199061199102(120585 119871) = 119891

2(120585)

(13)

4 Advances in Acoustics and Vibration

where

1198861(120585) = 119896

33(119871)

1198862(120585) = minus (119896

12(119871) + 2120583

1) 11989412058512057411

(120585)

1198863(120585) = 119896

22(119871) 12057411

(120585)

1198864(120585) = minus119894 (120588

11205962minus (21205831minus 11989633(119871)) 120585

2)

1198865(120585) = minus119896

33(119871) 12057421

(120585)

1198866(120585) = 119894 (120588

11205962minus (11989612(119871) + 2120583

1) 1205852)

1198867(120585) = 119896

22(119871) 120585

1198868(120585) = minus119894 (2120583

1minus 11989633(119871)) 120585120574

21(120585)

(14)

On the lower half-plane 119910 lt 0 solutions (1) will besought in the class 119871

1loc with the slow growth at infinity tak-ing into account that the traces 120591

3(119909 0) 120590

1199103(119909 0) 119906

1199093(119909 0)

and 1199061199103(119909 0) are correctly determined and they also belong

to 1198711loc Then solutions from the class of distributions of the

slow growth corresponding to waves moving in the negative119910-direction satisfy equations establishing relations betweenFourier transforms of components of the field [11]

1205851205913(120585 0) minus 120574

13(120585) 1205901199103(120585 0)

+ 2119894120583312058512057413

(120585 0) 1199061199093(120585)

minus 119894 (12058831205962minus 212058331205852) 1199061199103(120585 0) = 0

12057423

(120585) 1205913(120585 0) + 120585120590

1199103(120585 0)

+ 119894 (12058831205962minus 212058331205852) 1199061199093(120585 0)

+ 2119894120583312058512057423

(120585) 1199061199103(120585 0) = 0

(15)

which are equivalent to conditions related to traces of the fieldcomponents at the lower boundary

1205851205912(120585 0) minus 120574

13(120585) 1205901199102(120585 0)

+ 2119894120583312058512057413

(120585) 1199061199092(120585 0)

minus 119894 (12058831205962minus 212058331205852) 1199061199102(120585 0) = 0

12057423

(120585) 1205912(120585 0) + 120585120590

1199102(120585 0)

+ 119894 (12058831205962minus 212058331205852) 1199061199092(120585 0)

+ 2119894120583312058512057423

(120585) 1199061199102(120585 0) = 0

(16)

where 119896213

= 12058831205962(120582

3+ 21205833) 119896223

= 120588312059621205833and branches of

roots of the functions 12057413

= radic119896213minus 1205852 120574

23= radic1198962

23minus 1205852 are

chosen in the same way as branches of roots of the functionsof the upper half-plane

Using (4) we obtain

1198871(120585) 1199061015840

1199092(120585 0) + 119887

2(120585) 1199061199092(120585 0)

+ 1198873(120585) 1199061015840

1199102(120585 0)

+ 1198874(120585) 1199061199102(120585 0) = 0

1198875(120585) 1199061015840

1199092(120585 0) + 119887

6(120585) 1199061199092(120585 0)

+ 1198877(120585) 1199061015840

1199102(120585 0)

+ 1198878(120585) 1199061199102(120585 0) = 0

(17)

where

1198871(120585) = 119896

33(0) 120585

1198872(120585) = 119894 (119896

12(0) + 2120583

3) 12058512057413

(120585)

1198873(120585) = minus119896

22(0) 12057413

(120585)

1198874(120585) = minus119894 (120588

31205962minus (21205833minus 11989633(0)) 1205852)

1198875(120585) = 119896

33(0) 12057423

(120585)

1198876(120585) = 119894 (120588

31205962minus (11989612(0) + 2120583

3) 1205852)

1198877(120585) = 119896

22(0) 120585

1198878(120585) = 119894 (2120583

3minus 11989633(0)) 120585120574

23(120585)

(18)

Physical meanings of solutions of (9) with the boundaryconditions (13) and (17) are displacements (119906

1199092 1199061199102) which

describe the field at 0 lt 119910 lt 119871 in the problem of diffractionin the elastic layer

4 Elastic Oscillations of a TransverselyIsotropic Body

Let us consider three-dimensional oscillations of an elastictransversely isotropic medium To describe deformations ofthe medium the following model will be used [12]

120576119909119909

=1

119864(120590119909119909

minus ]120590119911119911) minus

]1015840

1198641015840120590119910119910

120576119911119911

=1

119864(120590119911119911minus ]120590119909119909) minus

]1015840

1198641015840120590119910119910

120576119910119910

= minus]1015840

1198641015840(120590119909119909

+ 120590119911119911) +

1

1198641015840120590119910119910

120576119909119910

=1

21198661015840120590119909119910 120576

119909119911=

1

2119866120590119909119911

120576119910119911

=1

21198661015840120590119910119911

(19)

Here the plane 119909119911 is the plane of isotropy and the planes119909119910 and 119910119911 are the planes of elastic symmetryThe parameters119864 1198641015840 are the Young modula ] and ]1015840 are the Poissoncoefficients 119866 = 119864(2(1 + ])) and 1198661015840 are the displacement

Advances in Acoustics and Vibration 5

modula Parameters without the prime sign correspond todeformations in the plane of isotropy whereas parameterswith the prime sign correspond to deformations in the planeof elastic symmetry

Equations (19) could be transformed to the form [13]

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 120582120576119911119911+ 1205821015840120576119910119910

120590119911119911

= 120582120576119909119909

+ (120582 + 2120583) 120576119911119911+ 1205821015840120576119910119910

120590119910119910

= 1205821015840(120576119909119909

+ 120576119911119911) + (120582

1015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910 120590

119909119911= 2120583120576

119909119911

(20)

120590119910119911

= 21198661015840120576119910119911 (21)

in which the used notations imply the following

120582 + 2120583 =119864

(1 + ]) 119889(1 minus (]1015840)

2 119864

1198641015840)

120582 =119864

(1 + ]) 119889(] + (]1015840)

2 119864

1198641015840)

1205821015840+ 21205831015840=1198641015840 (1 minus ])

119889

1205821015840=119864]1015840

119889 119889 = 1 minus ] minus 2(]1015840)

2 119864

1198641015840

(22)

We will assume that the field does not depend on the119911 coordinate 120597120597119911 equiv 0 Then we have 120576

119911119911= 0 and the

system of (20) falls into two independent subsystems Thefirst subsystem describes oscillations in the plane 119909119910

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 1205821015840120576119910119910

120590119910119910

= 1205821015840120576119909119909

+ (1205821015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910

(23)

whereas the second subsystem describes oscillations in the 119911direction

120590119911119911

= 120582120576119909119909

+ 1205821015840120576119910119910 120590

119909119911= 2120583120576

119909119911

120590119910119911

= 21198661015840120576119910119911

(24)

The system (24) with the use of equations of motiontransforms to the following

1205831205972119906119911

1205971199092+

120597

120597119910(1198661015840 120597119906119911

120597119910) + 12058821205962119906119911= 0 (25)

The system (23) corresponds to (4) with the followingnotations 120590

1199092= 120590119909119909 1205901199102

= 120590119910 and 120591

2= 120590119909119910 Under the

conditions 11989613

= 11989623

= 0 considered in Section 1 the elasticitytensor K linking stress and deformations to each other takesthe following form

K = (

120582 + 2120583 1205821015840 0

1205821015840 1205821015840 + 21205831015840 0

0 0 1198661015840

) (26)

Thus the problem of diffraction of an elastic harmonicwave by a transversely isotropic layer reduces to the boundaryvalue problem (9) (13) and (17) with the elasticity tensordefined in (26)

5 Numerical Results

Before discussions of the numerical results we will give somenotes regarding dependence of solution of the problem (9)(13) and (17) on parameter 120585 All the coefficients of theboundary value problem are continuous functions of 120585 Thenif right-hand sides of (13) are regular distributions on 120585 thensolutions will also be considered as regular with respect to120585 However if 119891

1and 119891

2are singular distributions on 120585 then

the solutions themselves will also be considered as singularFor example if 119891

1= 1198621120575(120585 minus 120585

0) and 119891

2= 1198622120575(120585 minus 120585

0) then

119906119909(119910) = 120575(120585 minus 120585

0)119908119909(119910 1205850) and 119906

119910(119910) = 120575(120585 minus 120585

0)119908119910(119910 1205850) In

this case it is convenient to ldquonormalizerdquo the boundary valueproblem by 120575(120585 minus 120585

0) For doing that we perform the change

of variables from variable 120585 to variable 1205850all over and solve

(9) (13) and (17) with respect to 119908119909(119910 1205850) and 119908

119910(119910 1205850)

Therefore in the case of Fourier transforms of traces ofthe incident field being singular distributions for examplein the case of the incident wave being a plane wave thesolutions of the problems will also be singular distributionswith the same carrier From this it follows that diffraction ofone plane wave results in two reflected waves longitudinaland transverse and excitation of waveguide waves in the layerdoes not occur It is obvious that the last statement is trueunder condition of uniqueness of the diffraction problem(homogeneous conditions (13) result in a trivial solution tothe problem (9) (13) and (17)) and under condition theeigenvalues of the waveguide formed by the layer which differfrom 120585

0

The desired problem can be solved using many approx-imation methods A uniform finite-difference grid with themesh size ℎ was chosen to approximate the boundary valueproblem (9) (13) and (17) with the accuracy on the order of119874(ℎ) When choosing the mesh size it is taken into accountthat the finite difference analogs of elastic profiles of thelayer describe adequately the original continuous modelsOn the other hand the mesh size ℎ must be smaller thanthe wavelength in the layer and consequently inverselyproportional to the frequency 120596

After carrying out the numerical solution it is necessaryto reconstruct the fields in the half-planes 119906

1and 119906

3 For

doing it we consider displacements in a homogeneousisotropic 119899th medium which can be written in the generalform in the following way [14]

119906119909119899(119910) = 120585119860

119899119890minus1198941205741119899119910minus 1205851198611198991198901198941205741119899119910

+ 1205742119899119862119899119890minus1198941205742119899119910+ 12057421198991198631198991198901198941205742119899119910

119906119910119899(119910) = 120574

1119899119860119899119890minus1198941205741119899119910+ 12057411198991198611198991198901198941205741119899119910

minus 120585119862119899119890minus1198941205742119899119910+ 1205851198631198991198901198941205742119899119910

(27)

6 Advances in Acoustics and Vibration

Taking into account the conditions at infinity displace-ments for the reflected field will have the following form

1199061199091(119910) = minus120585

0119861111989011989412057411(119910minus119871)

+ 12057421119863111989011989412057421(119910minus119871)

1199061199101(119910) = 120574

11119861111989011989412057411(119910minus119871)

+ 1205850119863111989011989412057421(119910minus119871)

(28)

and for the transmitted wave

1199061199093(119910) = 120585

01198603119890minus11989412057413119910+ 120574231198623119890minus11989412057423119910

1199061199103(119910) = 120574

131198603119890minus11989412057413119910minus 12058501198623119890minus11989412057423119910

(29)

The unknown coefficients 1198611 1198631 1198603 and 119862

3are found

via the following expressions

1198611=119906119910112057421minus 11990611990911205850

1205741112057421+ 12058520

1198631=119906119909112057411+ 11990611991011205850

1205741112057421+ 12058520

1198603=119906119910312057423+ 11990611990931205850

1205741312057423+ 12058520

1198623=119906119909312057413minus 11990611991031205850

1205741312057423+ 12058520

(30)

where 119906119909119899

and 119906119910119899

are traces of displacements of the 119899thmedium which are expressed using (7) and (8)

We will consider the case of diffraction by a planelongitudinal wave with displacements of the following kind

1199061199090(119909 119910)

= 119860011989611

sin 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

1199061199100(119909 119910)

= 119860011989611

cos 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

(31)

We apply Fourier transformation to components of theincident field and arrive at the result that all the componentsof the field are singular distributions with themultiplier 120575(120585minus1205850) 1205850= 11989611

sin 120579 For example the Fourier transform of thetrace 119906

1199090(119909 119910) at 119910 = 119871 takes the following form

1199061199090(120585 119871) = 119860

011989611

sin 120579120575 (120585 minus 11989611

sin 120579) (32)

Since the right-hand sides of (13) are singular distribu-tions then it is sufficient to solve the problem (9) (13) and(17) just at the value 120585 = 120585

0

For carrying out the numerical experiments we willconsider the case when the layer of thickness 119871 = 10mmadeof siltstone is located in sandstone Parameters of sandstonefilling in medium 1 and medium 3 are 120588 = 2400 kgm3 V

119901=

3300msec and V119904= 2000msec The layer is considered

anisotropic we will consider three cases of distributions ofelasticity parameters in the layer A uniform layer has thefollowing parameters [15] 119864 = 568GPa 1198641015840 = 621GPa] = 029 ]1015840 = 026 and 1198661015840 = 229GPa In the caseof ldquocompressed siltstonerdquo all the parameters of the mediumremain constant except for the Young modulus 119864(119910) which

00102030405060708091

0 1000 2000 3000 4000120596

Figure 2 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 20

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

in the plane of isotropy grows linearly from 119864(0) = 568GPato 119864(1198712) = 850GPa and then reduces linearly to the value119864(119871) = 568GPa In the case of ldquorelaxed siltstonerdquo the Youngmodulus 119864(119910) reduces linearly from 119864(0) = 568GPa to119864(1198712) = 284GPa and grows linearly to 119864(119871) = 568GPa

In the present work two sets of studies were carried outThe first set of studies is dedicated to searching for depen-dence of normalized energy of the transmitted longitudinalwave on angular frequency 120596 As a result the conclusion ismade so that the transmitted energy grows as the value ofthe Young modulus 119864(119910) reduces in the middle of the layerThe difference between various structures increases with theincrease of angle of incidence 120579 The dependences are givenin Figures 2 and 3

In Figure 2 dependence of normalized energy of thetransmitted longitudinal wave on angular frequency 120596 at itsincidence under the angle 120579 = 20

∘upon a siltstone layer ofthickness 119871 = 10m placed into sandstone is shownThe solidcurve corresponds to a uniform layer of siltstone the dottedcurve ldquocompressed siltstonerdquo and the dashed curve ldquorelaxedsiltstonerdquo In Figure 3 the same type of dependence is shownbut for the angle 120579 = 40

∘

The second set of studies is dedicated to searchingfor dependence of normalized energy of the transmittedlongitudinal wave on the angle of incidence 120579 Just as in theprevious set of studies confirmed is the conclusion that thetransmitted energy increases with the decrease of the Youngmodulus 119864(119910) in the middle of the layer It is worth notinghere that the difference in the transmitted energy for differentstructures decreases with the increase of angular frequency120596

In Figure 4 dependence of normalized energy of thetransmitted longitudinal wave on 120579 at the angular frequencyof the wave 120596 = 2 sdot 10

3 radian per second upon a siltstonelayer of thickness 119871 = 10m placed into sandstone is shownThe solid curve corresponds to a uniform layer of siltstonethe dotted curve ldquocompressed siltstonerdquo and the dashed curveldquorelaxed siltstonerdquo In Figure 5 the same type of dependenceis shown but for 120596 = 4 sdot 10

3 radian per second

Advances in Acoustics and Vibration 7

00102030405060708091

0 1000 2000 3000 4000120596

Figure 3 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 40

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 4 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 2000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness 119871 = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 5 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 4000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness L = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

6 Conclusions

The method of overdetermined boundary value problemused in the present work when combined with and theFourier transformmethod is shown to be efficient especiallyfor the cases the Fourier transforms of traces of the incidentfield are singular distributions Then the approximationproblem is solved just at the value 120585 = 120585

0 In the case of

the Fourier transforms being regular distributions (eg atdiffraction of a Gauss beam by a plate) the problem (9) (13)and (17) is solved for several values of the parameter 120585

Results obtained with respect to propagation of elasticwaves through anisotropic layers can be used in geophysicsfor the initial analysis of structure of the layers of rock strataAlso results of propagation of elastic waves through nonuni-form anisotropic structures can be used in industries inwhich anisotropicmaterials are applied as well as at designingprotective layers for various processes and apparatuses

Acknowledgment

Thiswork was supported by RFBR 12-01-97012-r-povolzhrsquoe-a

References

[1] S Ryan-Grigor ldquoEmpirical relationships between transverseisotropy parameters and VplVS implications for AVOrdquo Geo-physics vol 62 no 5 pp 1359ndash1364 1997

[2] T Alkhalifah ldquoVelocity analysis using nonhyperbolic moveoutin transversely isotropic mediardquo Geophysics vol 62 no 6 pp1839ndash1854 1997

[3] CM Sayers ldquoSimplified anisotropy parameters for transverselyisotropic sedimentary rocksrdquoGeophysics vol 60 pp 1933ndash19351995

[4] V J Nemirovskii and A P Jankovskii ldquoDetermining effectivephysical and mechanical characteristics of hybrid compositescrisscross reinforced by transversely isotropic fibers and com-parisons of computed characteristics versus experimental datardquoMekhanIka Kompozicionnykh MaterIalov i Konstrukcii vol 13no 1 pp 3ndash32 2007

[5] GAGeniev VN Kissjuk andGA TjupinTheory of Plasticityof Concrete and Ferroconcrete Strojizdat Moscow Russia 1974

[6] B V Labudin ldquoJustifying a computational model treating lam-inated wood as an orthogonal transversely isotropic materialrdquoIzvestiya VUZov Lesnoj Zhurnal no 6 pp 136ndash139 2006

[7] M P Lonkevich ldquoPropagation of sound through a layer of atransversely isotropic material of finite thicknessrdquo AkusticheskijZhurnal vol 17 no 1 pp 85ndash92 1971

[8] E L Shenderov ldquoPropagation of sound through a layer of atransversely isotropic platerdquo Akusticheskii Zhurnal vol 30 no1 pp 122ndash129 1984

[9] S A Skobelitsyn and L A Tolokonnikov ldquoPropagation ofsound through a transversely isotropic nonuniform flat layerrdquoAkusticheskii Zhurnal vol 36 no 4 pp 740ndash744 1990

[10] A V Anufrieva D N Tumakov and V L Kipot ldquoElastic wavepropagation through a layer with graded-index distribution ofdensityrdquo in Proceedings of the Days on Diffraction (DD rsquo12) pp21ndash26 2012

[11] I E Pleshchinskaya and N B Pleshchinskii ldquoOver-determinedboundary value problems for linear equations of elastodynam-ics and their applications to elastic wave diffraction theoryrdquo in

8 Advances in Acoustics and Vibration

Advances inMathematics Research A R Baswell Ed vol 17 pp102ndash138 Nova Science New York NY USA 2012

[12] S G Lekhnickii Theory of Elasticity of the Anisotropic BodyNauka Moscow Russia 1977

[13] B D Annin ldquoTransversely isotropic elastic model of geologicalmaterialsrdquo Sibirskiı Zhurnal Industrial noı Matematiki vol 12no 3 pp 5ndash14 2009

[14] K N Vdovina N B Pleshchinskii and D N TumakovldquoConcerning orthogonality of proper waves of a half-openedelastic waveguiderdquo Izvestiya VUZov Matematika no 9 pp 69ndash75 2008

[15] S A Batugin and R K Nirenburg ldquoApproximate rela-tion between the elastic constants of anisotropic rocks andthe anisotropy parametersrdquo Fiziko-Tekhnicheskie ProblemyRazrabotki Poleznykh Iskopaemykh vol 7 no 1 pp 7ndash12 1972

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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International Journal of

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DistributedSensor Networks

International Journal of

2 Advances in Acoustics and Vibration

0

L

y

2

1

3

u1 u0

u2

u3

1205791205881 1205821 1205831

1205883 1205823 1205833

1205882(y) K(y)

x

Figure 1 Geometry of the problem

incidence is studied numerically Differences in behaviorof energy of transmitted wave at diffraction by uniformlyanisotropic and nonuniformly anisotropic layer are outlined

2 Statement of the Problem

Let an elastic harmonic wave of type u0(119909 119910) exp119894120596119905 fall ona nonuniform in the transverse direction anisotropic layerof thickness 119871 (medium 2 0 lt 119910 lt 119871) with continuousdensity 120588

2(119910) and tensor (3 times 3) of elasticity modulus K(119910)

from medium 1 119910 gt 119871 under the angle 120579 with respect to theaxis 119910 (see Figure 1)The diffraction results in a wave u1(119909 119910)reflected toward medium 1 a wave u3(119909 119910) propagatingtoward medium 3 119910 lt 0 and field u2(119909 119910) in the layerDesired is the full diffracted field Media 1 and 3 are assumedto be uniform and isotropic

We seek a solution to the plane harmonic problem fromthe elasticity theory at 119910 lt 0 and 119910 gt 119871 in the form

120597120590119909119899

120597119909+120597120591119899

120597119910+ 1205881198991205962119906119909119899

= 0

120597120591119899

120597119909+120597120590119910119899

120597119910+ 1205881198991205962119906119910119899

= 0

(1)

120590119909119899

= (120582119899+ 2120583119899)120597119906119909119899

120597119909+ 120582119899

120597119906119910119899

120597119910

120590119910119899

= 120582119899

120597119906119909119899

120597119909+ (120582119899+ 2120583119899)120597119906119910119899

120597119910

120591119899= 120583119899(120597119906119909119899

120597119910+120597119906119910119899

120597119909)

(2)

for 119899 = 1 3 with the constant Lame coefficients 120582119899 120583119899and

density 120588119899

General equations of two-dimensional oscillations arewritten in the form

minus120588212059621199061199092

=1205971205901199092

120597119909+1205971205912

120597119910

minus120588212059621199061199102

=1205971205912

120597119909+1205971205901199102

120597119910

(3)

1205901199092

= 119896119909119909119909119909

1205971199061199092

120597119909+ 119896119909119909119909119910

(1205971199061199092

120597119910+1205971199061199102

120597119909) + 119896119909119909119910119910

1205971199061199102

120597119910

1205901199102

= 119896119910119910119909119909

1205971199061199092

120597119909+ 119896119910119910119909119910

(1205971199061199092

120597119910+1205971199061199102

120597119909) + 119896119910119910119910119910

1205971199061199102

120597119910

1205912= 119896119909119910119909119909

1205971199061199092

120597119909+ 119896119909119910119909119910

(1205971199061199092

120597119910+1205971199061199102

120597119909) + 119896119909119910119910119910

1205971199061199102

120597119910

(4)

where 119896lowastlowastlowastlowast

are components of the elasticity modulus tensorLet us introduce a standard notation for indices 119896

lowastlowastlowastlowast

119909119909 rarr 1 119910119910 rarr 2 and 119909119910 rarr 3 and substitute(4) into (3) to obtain

minus120588212059621199061199092

=120597

120597119909(11989611

1205971199061199092

120597119909+ 11989613(1205971199061199092

120597119910+1205971199061199102

120597119909)

+11989612

1205971199061199102

120597119910)

+120597

120597119910(11989613

1205971199061199092

120597119909+ 11989633(1205971199061199092

120597119910+1205971199061199102

120597119909)

+11989623

1205971199061199102

120597119910)

minus120588212059621199061199102

=120597

120597119909(11989613

1205971199061199092

120597119909+ 11989633(1205971199061199092

120597119910+1205971199061199102

120597119909)

+11989623

1205971199061199102

120597119910)

+120597

120597119910(11989612

1205971199061199092

120597119909+ 11989623(1205971199061199092

120597119910+1205971199061199102

120597119909)

+11989622

1205971199061199102

120597119910)

(5)

We assume that rotational components of the forcescan not result in stretching of the body Then some of thecomponents become equal zero 119896

13= 11989623

= 0 Note thatfor the case of isotropic body the elasticity modulus tensortakes the following form

K = (

120582 + 2120583 120582 0

120582 120582 + 2120583 0

0 0 120583

) (6)

At the mediarsquos interface the following conjugation condi-tions are to be fulfilled

1199061199091(119909 119871 + 0) + 119906

1199090(119909 119871 + 0) = 119906

1199092(119909 119871 minus 0)

1199061199101(119909 119871 + 0) + 119906

1199100(119909 119871 + 0) = 119906

1199102(119909 119871 minus 0)

1205911(119909 119871 + 0) + 120591

0(119909 119871 + 0) = 120591

2(119909 119871 minus 0)

1205901199101(119909 119871 + 0) + 120590

1199100(119909 119871 + 0) = 120590

1199102(119909 119871 minus 0)

(7)

Advances in Acoustics and Vibration 3

at 119910 = 119871 and

1199061199093(119909 0 minus 0) = 119906

1199092(119909 0 + 0)

1199061199103(119909 0 minus 0) = 119906

1199102(119909 0 + 0)

1205913(119909 0 minus 0) = 120591

2(119909 0 + 0)

1205901199103(119909 0 minus 0) = 120590

1199102(119909 0 + 0)

(8)

at 119910 = 0Of all the possible solutions to the system (1) (2) (5) (7)

and (8) we pick solutions corresponding to the waves goingto infinity

3 Boundary Value Problem for the System ofOrdinary Differential EquationsDescribing a Field in the Gradient Layer

Desired functions for the Lame system (5) for any fixed 119910

from the interval (0 119871) will be considered in the class 1198711loc

and it will be assumed that the functions undergo a slowgrowth at infinity in the 119909-directionThis allows applying theFourier transform with respect to the 119909-direction permittingboth waves decaying at infinity and propagating wavesThuswe perform the change of variables from variable119909 to variable120585 and obtain the system of equations at 119910 isin (0 119871)

(119896331199061015840

1199092)1015840

+ [12058821205962minus 119896111205852] 1199061199092

minus 119894120585 (11989612+ 11989633) 1199061015840

1199102minus 1198941205851198961015840

331199061199102

= 0

(119896221199061015840

1199102)1015840

+ [12058821205962minus 119896331205852] 1199061199102

minus 119894120585 (11989612+ 11989633) 1199061015840

1199092minus 1198941205851198961015840

121199061199092

= 0

(9)

with respect to the Fourier transform for displacements1199061199092(120585 119910) and 119906

1199102(120585 119910)

It is worth noting here that the unknowns 1199061199092(120585 119910)

and 1199061199102(120585 119910) with respect to 119910 are ordinary functions and

therefore all the derivatives are to be understood in theclassical sense This allows discretization of the problem withrespect to 119910 For any fixed 119910 desired functions with respectto 120585 are distributions of the slow growth

For the upper half-plane 119910 gt 119871 it will be assumedthat the solutions (1) (2) belong to 119871

1loc(119877) and their traces1205911(119909 119871 + 0) 120590

1199101(119909 119871 + 0) 119906

1199091(119909 119871 + 0) and 119906

1199101(119909 119871 + 0)

are correctly specified We will consider that the desiredfunctions are distributions of the slow growth at infinityand moreover their traces are also distributions of the slowgrowth at infinity In thework [11] it was shown that solutionscorresponding to the wavemoving in the positive119910-directionsatisfy the equalities linking to each other Fourier transformsof traces of components of the field

1205851205911(120585 119871) + 120574

11(120585) 1205901199101(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199091(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199101(120585 119871) = 0

minus 12057421

(120585) 1205911(120585 119871) + 120585120590

1199101(120585 119871)

+ 119894 (12058811205962minus 212058311205852) 1199061199091(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199101(120585 119871) = 0

(10)

where 119896211

= 12058811205962(1205821+ 21205831) 119896221

= 120588112059621205831and branches of

roots of the functions 12057411

= radic119896211minus 1205852 120574

21= radic1198962

21minus 1205852 are

chosen such that the real part is positive and in the case ofthe real part being zero positive imaginary roots are chosen

In equalities (10) traces of all desired functions areconsidered at 119910 = 119871 but since the considered functions arecontinuous in the whole domain the limit will be consideredas the value at 119910 = 119871 We will proceed with the other traces ofthe desired functions in the same manner

We perform transition from traces of functions ofmedium 1 to traces of functions of the layer in equalities (10)For doing this we express the traces via conditions (7) andsubstitute the obtained expressions into (10) Thus we obtainthe following boundary conditions for the Fourier transformsof components of the field defined in the layer

1205851205912(120585 119871) + 120574

11(120585) 1205901199102(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199092(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199102(120585 119871) = 119891

1(120585)

minus 12057421

(120585) 1205912(120585 119871) + 120585120590

1199102(120585 119871)

+ 119894 (12058811205962minus 212058311205852) 1199061199092(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199102(120585 119871) = 119891

2(120585)

(11)

where1198911(120585) = 120585120591

0(120585 119871) + 120574

11(120585) 1205901199100(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199090(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199100(120585 119871)

1198912(120585) = minus120574

21(120585) 1205910(120585 119871)

+ 1205851205901199100(120585 119871) + 119894 (120588

11205962minus 212058311205852) 1199061199090(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199100(120585 119871)

(12)

We eliminate Fourier transforms of the stresses fromthe obtained conditions using (4) Thus we obtain relationsbetween traces of Fourier transforms of displacements in thelayer

1198861(120585) 1199061015840

1199092(120585 119871) + 119886

2(120585) 1199061199092(120585 119871)

+ 1198863(120585) 1199061015840

1199102(120585 119871)

+ 1198864(120585) 1199061199102(120585 119871) = 119891

1(120585)

1198865(120585) 1199061015840

1199092(120585 119871) + 119886

6(120585) 1199061199092(120585 119871)

+ 1198867(120585) 1199061015840

1199102(120585 119871)

+ 1198868(120585) 1199061199102(120585 119871) = 119891

2(120585)

(13)

4 Advances in Acoustics and Vibration

where

1198861(120585) = 119896

33(119871)

1198862(120585) = minus (119896

12(119871) + 2120583

1) 11989412058512057411

(120585)

1198863(120585) = 119896

22(119871) 12057411

(120585)

1198864(120585) = minus119894 (120588

11205962minus (21205831minus 11989633(119871)) 120585

2)

1198865(120585) = minus119896

33(119871) 12057421

(120585)

1198866(120585) = 119894 (120588

11205962minus (11989612(119871) + 2120583

1) 1205852)

1198867(120585) = 119896

22(119871) 120585

1198868(120585) = minus119894 (2120583

1minus 11989633(119871)) 120585120574

21(120585)

(14)

On the lower half-plane 119910 lt 0 solutions (1) will besought in the class 119871

1loc with the slow growth at infinity tak-ing into account that the traces 120591

3(119909 0) 120590

1199103(119909 0) 119906

1199093(119909 0)

and 1199061199103(119909 0) are correctly determined and they also belong

to 1198711loc Then solutions from the class of distributions of the

slow growth corresponding to waves moving in the negative119910-direction satisfy equations establishing relations betweenFourier transforms of components of the field [11]

1205851205913(120585 0) minus 120574

13(120585) 1205901199103(120585 0)

+ 2119894120583312058512057413

(120585 0) 1199061199093(120585)

minus 119894 (12058831205962minus 212058331205852) 1199061199103(120585 0) = 0

12057423

(120585) 1205913(120585 0) + 120585120590

1199103(120585 0)

+ 119894 (12058831205962minus 212058331205852) 1199061199093(120585 0)

+ 2119894120583312058512057423

(120585) 1199061199103(120585 0) = 0

(15)

which are equivalent to conditions related to traces of the fieldcomponents at the lower boundary

1205851205912(120585 0) minus 120574

13(120585) 1205901199102(120585 0)

+ 2119894120583312058512057413

(120585) 1199061199092(120585 0)

minus 119894 (12058831205962minus 212058331205852) 1199061199102(120585 0) = 0

12057423

(120585) 1205912(120585 0) + 120585120590

1199102(120585 0)

+ 119894 (12058831205962minus 212058331205852) 1199061199092(120585 0)

+ 2119894120583312058512057423

(120585) 1199061199102(120585 0) = 0

(16)

where 119896213

= 12058831205962(120582

3+ 21205833) 119896223

= 120588312059621205833and branches of

roots of the functions 12057413

= radic119896213minus 1205852 120574

23= radic1198962

23minus 1205852 are

chosen in the same way as branches of roots of the functionsof the upper half-plane

Using (4) we obtain

1198871(120585) 1199061015840

1199092(120585 0) + 119887

2(120585) 1199061199092(120585 0)

+ 1198873(120585) 1199061015840

1199102(120585 0)

+ 1198874(120585) 1199061199102(120585 0) = 0

1198875(120585) 1199061015840

1199092(120585 0) + 119887

6(120585) 1199061199092(120585 0)

+ 1198877(120585) 1199061015840

1199102(120585 0)

+ 1198878(120585) 1199061199102(120585 0) = 0

(17)

where

1198871(120585) = 119896

33(0) 120585

1198872(120585) = 119894 (119896

12(0) + 2120583

3) 12058512057413

(120585)

1198873(120585) = minus119896

22(0) 12057413

(120585)

1198874(120585) = minus119894 (120588

31205962minus (21205833minus 11989633(0)) 1205852)

1198875(120585) = 119896

33(0) 12057423

(120585)

1198876(120585) = 119894 (120588

31205962minus (11989612(0) + 2120583

3) 1205852)

1198877(120585) = 119896

22(0) 120585

1198878(120585) = 119894 (2120583

3minus 11989633(0)) 120585120574

23(120585)

(18)

Physical meanings of solutions of (9) with the boundaryconditions (13) and (17) are displacements (119906

1199092 1199061199102) which

describe the field at 0 lt 119910 lt 119871 in the problem of diffractionin the elastic layer

4 Elastic Oscillations of a TransverselyIsotropic Body

Let us consider three-dimensional oscillations of an elastictransversely isotropic medium To describe deformations ofthe medium the following model will be used [12]

120576119909119909

=1

119864(120590119909119909

minus ]120590119911119911) minus

]1015840

1198641015840120590119910119910

120576119911119911

=1

119864(120590119911119911minus ]120590119909119909) minus

]1015840

1198641015840120590119910119910

120576119910119910

= minus]1015840

1198641015840(120590119909119909

+ 120590119911119911) +

1

1198641015840120590119910119910

120576119909119910

=1

21198661015840120590119909119910 120576

119909119911=

1

2119866120590119909119911

120576119910119911

=1

21198661015840120590119910119911

(19)

Here the plane 119909119911 is the plane of isotropy and the planes119909119910 and 119910119911 are the planes of elastic symmetryThe parameters119864 1198641015840 are the Young modula ] and ]1015840 are the Poissoncoefficients 119866 = 119864(2(1 + ])) and 1198661015840 are the displacement

Advances in Acoustics and Vibration 5

modula Parameters without the prime sign correspond todeformations in the plane of isotropy whereas parameterswith the prime sign correspond to deformations in the planeof elastic symmetry

Equations (19) could be transformed to the form [13]

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 120582120576119911119911+ 1205821015840120576119910119910

120590119911119911

= 120582120576119909119909

+ (120582 + 2120583) 120576119911119911+ 1205821015840120576119910119910

120590119910119910

= 1205821015840(120576119909119909

+ 120576119911119911) + (120582

1015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910 120590

119909119911= 2120583120576

119909119911

(20)

120590119910119911

= 21198661015840120576119910119911 (21)

in which the used notations imply the following

120582 + 2120583 =119864

(1 + ]) 119889(1 minus (]1015840)

2 119864

1198641015840)

120582 =119864

(1 + ]) 119889(] + (]1015840)

2 119864

1198641015840)

1205821015840+ 21205831015840=1198641015840 (1 minus ])

119889

1205821015840=119864]1015840

119889 119889 = 1 minus ] minus 2(]1015840)

2 119864

1198641015840

(22)

We will assume that the field does not depend on the119911 coordinate 120597120597119911 equiv 0 Then we have 120576

119911119911= 0 and the

system of (20) falls into two independent subsystems Thefirst subsystem describes oscillations in the plane 119909119910

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 1205821015840120576119910119910

120590119910119910

= 1205821015840120576119909119909

+ (1205821015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910

(23)

whereas the second subsystem describes oscillations in the 119911direction

120590119911119911

= 120582120576119909119909

+ 1205821015840120576119910119910 120590

119909119911= 2120583120576

119909119911

120590119910119911

= 21198661015840120576119910119911

(24)

The system (24) with the use of equations of motiontransforms to the following

1205831205972119906119911

1205971199092+

120597

120597119910(1198661015840 120597119906119911

120597119910) + 12058821205962119906119911= 0 (25)

The system (23) corresponds to (4) with the followingnotations 120590

1199092= 120590119909119909 1205901199102

= 120590119910 and 120591

2= 120590119909119910 Under the

conditions 11989613

= 11989623

= 0 considered in Section 1 the elasticitytensor K linking stress and deformations to each other takesthe following form

K = (

120582 + 2120583 1205821015840 0

1205821015840 1205821015840 + 21205831015840 0

0 0 1198661015840

) (26)

Thus the problem of diffraction of an elastic harmonicwave by a transversely isotropic layer reduces to the boundaryvalue problem (9) (13) and (17) with the elasticity tensordefined in (26)

5 Numerical Results

Before discussions of the numerical results we will give somenotes regarding dependence of solution of the problem (9)(13) and (17) on parameter 120585 All the coefficients of theboundary value problem are continuous functions of 120585 Thenif right-hand sides of (13) are regular distributions on 120585 thensolutions will also be considered as regular with respect to120585 However if 119891

1and 119891

2are singular distributions on 120585 then

the solutions themselves will also be considered as singularFor example if 119891

1= 1198621120575(120585 minus 120585

0) and 119891

2= 1198622120575(120585 minus 120585

0) then

119906119909(119910) = 120575(120585 minus 120585

0)119908119909(119910 1205850) and 119906

119910(119910) = 120575(120585 minus 120585

0)119908119910(119910 1205850) In

this case it is convenient to ldquonormalizerdquo the boundary valueproblem by 120575(120585 minus 120585

0) For doing that we perform the change

of variables from variable 120585 to variable 1205850all over and solve

(9) (13) and (17) with respect to 119908119909(119910 1205850) and 119908

119910(119910 1205850)

Therefore in the case of Fourier transforms of traces ofthe incident field being singular distributions for examplein the case of the incident wave being a plane wave thesolutions of the problems will also be singular distributionswith the same carrier From this it follows that diffraction ofone plane wave results in two reflected waves longitudinaland transverse and excitation of waveguide waves in the layerdoes not occur It is obvious that the last statement is trueunder condition of uniqueness of the diffraction problem(homogeneous conditions (13) result in a trivial solution tothe problem (9) (13) and (17)) and under condition theeigenvalues of the waveguide formed by the layer which differfrom 120585

0

The desired problem can be solved using many approx-imation methods A uniform finite-difference grid with themesh size ℎ was chosen to approximate the boundary valueproblem (9) (13) and (17) with the accuracy on the order of119874(ℎ) When choosing the mesh size it is taken into accountthat the finite difference analogs of elastic profiles of thelayer describe adequately the original continuous modelsOn the other hand the mesh size ℎ must be smaller thanthe wavelength in the layer and consequently inverselyproportional to the frequency 120596

After carrying out the numerical solution it is necessaryto reconstruct the fields in the half-planes 119906

1and 119906

3 For

doing it we consider displacements in a homogeneousisotropic 119899th medium which can be written in the generalform in the following way [14]

119906119909119899(119910) = 120585119860

119899119890minus1198941205741119899119910minus 1205851198611198991198901198941205741119899119910

+ 1205742119899119862119899119890minus1198941205742119899119910+ 12057421198991198631198991198901198941205742119899119910

119906119910119899(119910) = 120574

1119899119860119899119890minus1198941205741119899119910+ 12057411198991198611198991198901198941205741119899119910

minus 120585119862119899119890minus1198941205742119899119910+ 1205851198631198991198901198941205742119899119910

(27)

6 Advances in Acoustics and Vibration

Taking into account the conditions at infinity displace-ments for the reflected field will have the following form

1199061199091(119910) = minus120585

0119861111989011989412057411(119910minus119871)

+ 12057421119863111989011989412057421(119910minus119871)

1199061199101(119910) = 120574

11119861111989011989412057411(119910minus119871)

+ 1205850119863111989011989412057421(119910minus119871)

(28)

and for the transmitted wave

1199061199093(119910) = 120585

01198603119890minus11989412057413119910+ 120574231198623119890minus11989412057423119910

1199061199103(119910) = 120574

131198603119890minus11989412057413119910minus 12058501198623119890minus11989412057423119910

(29)

The unknown coefficients 1198611 1198631 1198603 and 119862

3are found

via the following expressions

1198611=119906119910112057421minus 11990611990911205850

1205741112057421+ 12058520

1198631=119906119909112057411+ 11990611991011205850

1205741112057421+ 12058520

1198603=119906119910312057423+ 11990611990931205850

1205741312057423+ 12058520

1198623=119906119909312057413minus 11990611991031205850

1205741312057423+ 12058520

(30)

where 119906119909119899

and 119906119910119899

are traces of displacements of the 119899thmedium which are expressed using (7) and (8)

We will consider the case of diffraction by a planelongitudinal wave with displacements of the following kind

1199061199090(119909 119910)

= 119860011989611

sin 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

1199061199100(119909 119910)

= 119860011989611

cos 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

(31)

We apply Fourier transformation to components of theincident field and arrive at the result that all the componentsof the field are singular distributions with themultiplier 120575(120585minus1205850) 1205850= 11989611

sin 120579 For example the Fourier transform of thetrace 119906

1199090(119909 119910) at 119910 = 119871 takes the following form

1199061199090(120585 119871) = 119860

011989611

sin 120579120575 (120585 minus 11989611

sin 120579) (32)

Since the right-hand sides of (13) are singular distribu-tions then it is sufficient to solve the problem (9) (13) and(17) just at the value 120585 = 120585

0

For carrying out the numerical experiments we willconsider the case when the layer of thickness 119871 = 10mmadeof siltstone is located in sandstone Parameters of sandstonefilling in medium 1 and medium 3 are 120588 = 2400 kgm3 V

119901=

3300msec and V119904= 2000msec The layer is considered

anisotropic we will consider three cases of distributions ofelasticity parameters in the layer A uniform layer has thefollowing parameters [15] 119864 = 568GPa 1198641015840 = 621GPa] = 029 ]1015840 = 026 and 1198661015840 = 229GPa In the caseof ldquocompressed siltstonerdquo all the parameters of the mediumremain constant except for the Young modulus 119864(119910) which

00102030405060708091

0 1000 2000 3000 4000120596

Figure 2 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 20

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

in the plane of isotropy grows linearly from 119864(0) = 568GPato 119864(1198712) = 850GPa and then reduces linearly to the value119864(119871) = 568GPa In the case of ldquorelaxed siltstonerdquo the Youngmodulus 119864(119910) reduces linearly from 119864(0) = 568GPa to119864(1198712) = 284GPa and grows linearly to 119864(119871) = 568GPa

In the present work two sets of studies were carried outThe first set of studies is dedicated to searching for depen-dence of normalized energy of the transmitted longitudinalwave on angular frequency 120596 As a result the conclusion ismade so that the transmitted energy grows as the value ofthe Young modulus 119864(119910) reduces in the middle of the layerThe difference between various structures increases with theincrease of angle of incidence 120579 The dependences are givenin Figures 2 and 3

In Figure 2 dependence of normalized energy of thetransmitted longitudinal wave on angular frequency 120596 at itsincidence under the angle 120579 = 20

∘upon a siltstone layer ofthickness 119871 = 10m placed into sandstone is shownThe solidcurve corresponds to a uniform layer of siltstone the dottedcurve ldquocompressed siltstonerdquo and the dashed curve ldquorelaxedsiltstonerdquo In Figure 3 the same type of dependence is shownbut for the angle 120579 = 40

∘

The second set of studies is dedicated to searchingfor dependence of normalized energy of the transmittedlongitudinal wave on the angle of incidence 120579 Just as in theprevious set of studies confirmed is the conclusion that thetransmitted energy increases with the decrease of the Youngmodulus 119864(119910) in the middle of the layer It is worth notinghere that the difference in the transmitted energy for differentstructures decreases with the increase of angular frequency120596

In Figure 4 dependence of normalized energy of thetransmitted longitudinal wave on 120579 at the angular frequencyof the wave 120596 = 2 sdot 10

3 radian per second upon a siltstonelayer of thickness 119871 = 10m placed into sandstone is shownThe solid curve corresponds to a uniform layer of siltstonethe dotted curve ldquocompressed siltstonerdquo and the dashed curveldquorelaxed siltstonerdquo In Figure 5 the same type of dependenceis shown but for 120596 = 4 sdot 10

3 radian per second

Advances in Acoustics and Vibration 7

00102030405060708091

0 1000 2000 3000 4000120596

Figure 3 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 40

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 4 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 2000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness 119871 = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 5 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 4000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness L = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

6 Conclusions

The method of overdetermined boundary value problemused in the present work when combined with and theFourier transformmethod is shown to be efficient especiallyfor the cases the Fourier transforms of traces of the incidentfield are singular distributions Then the approximationproblem is solved just at the value 120585 = 120585

0 In the case of

the Fourier transforms being regular distributions (eg atdiffraction of a Gauss beam by a plate) the problem (9) (13)and (17) is solved for several values of the parameter 120585

Results obtained with respect to propagation of elasticwaves through anisotropic layers can be used in geophysicsfor the initial analysis of structure of the layers of rock strataAlso results of propagation of elastic waves through nonuni-form anisotropic structures can be used in industries inwhich anisotropicmaterials are applied as well as at designingprotective layers for various processes and apparatuses

Acknowledgment

Thiswork was supported by RFBR 12-01-97012-r-povolzhrsquoe-a

References

[1] S Ryan-Grigor ldquoEmpirical relationships between transverseisotropy parameters and VplVS implications for AVOrdquo Geo-physics vol 62 no 5 pp 1359ndash1364 1997

[2] T Alkhalifah ldquoVelocity analysis using nonhyperbolic moveoutin transversely isotropic mediardquo Geophysics vol 62 no 6 pp1839ndash1854 1997

[3] CM Sayers ldquoSimplified anisotropy parameters for transverselyisotropic sedimentary rocksrdquoGeophysics vol 60 pp 1933ndash19351995

[4] V J Nemirovskii and A P Jankovskii ldquoDetermining effectivephysical and mechanical characteristics of hybrid compositescrisscross reinforced by transversely isotropic fibers and com-parisons of computed characteristics versus experimental datardquoMekhanIka Kompozicionnykh MaterIalov i Konstrukcii vol 13no 1 pp 3ndash32 2007

[5] GAGeniev VN Kissjuk andGA TjupinTheory of Plasticityof Concrete and Ferroconcrete Strojizdat Moscow Russia 1974

[6] B V Labudin ldquoJustifying a computational model treating lam-inated wood as an orthogonal transversely isotropic materialrdquoIzvestiya VUZov Lesnoj Zhurnal no 6 pp 136ndash139 2006

[7] M P Lonkevich ldquoPropagation of sound through a layer of atransversely isotropic material of finite thicknessrdquo AkusticheskijZhurnal vol 17 no 1 pp 85ndash92 1971

[8] E L Shenderov ldquoPropagation of sound through a layer of atransversely isotropic platerdquo Akusticheskii Zhurnal vol 30 no1 pp 122ndash129 1984

[9] S A Skobelitsyn and L A Tolokonnikov ldquoPropagation ofsound through a transversely isotropic nonuniform flat layerrdquoAkusticheskii Zhurnal vol 36 no 4 pp 740ndash744 1990

[10] A V Anufrieva D N Tumakov and V L Kipot ldquoElastic wavepropagation through a layer with graded-index distribution ofdensityrdquo in Proceedings of the Days on Diffraction (DD rsquo12) pp21ndash26 2012

[11] I E Pleshchinskaya and N B Pleshchinskii ldquoOver-determinedboundary value problems for linear equations of elastodynam-ics and their applications to elastic wave diffraction theoryrdquo in

8 Advances in Acoustics and Vibration

Advances inMathematics Research A R Baswell Ed vol 17 pp102ndash138 Nova Science New York NY USA 2012

[12] S G Lekhnickii Theory of Elasticity of the Anisotropic BodyNauka Moscow Russia 1977

[13] B D Annin ldquoTransversely isotropic elastic model of geologicalmaterialsrdquo Sibirskiı Zhurnal Industrial noı Matematiki vol 12no 3 pp 5ndash14 2009

[14] K N Vdovina N B Pleshchinskii and D N TumakovldquoConcerning orthogonality of proper waves of a half-openedelastic waveguiderdquo Izvestiya VUZov Matematika no 9 pp 69ndash75 2008

[15] S A Batugin and R K Nirenburg ldquoApproximate rela-tion between the elastic constants of anisotropic rocks andthe anisotropy parametersrdquo Fiziko-Tekhnicheskie ProblemyRazrabotki Poleznykh Iskopaemykh vol 7 no 1 pp 7ndash12 1972

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International Journal of

Advances in Acoustics and Vibration 3

at 119910 = 119871 and

1199061199093(119909 0 minus 0) = 119906

1199092(119909 0 + 0)

1199061199103(119909 0 minus 0) = 119906

1199102(119909 0 + 0)

1205913(119909 0 minus 0) = 120591

2(119909 0 + 0)

1205901199103(119909 0 minus 0) = 120590

1199102(119909 0 + 0)

(8)

at 119910 = 0Of all the possible solutions to the system (1) (2) (5) (7)

and (8) we pick solutions corresponding to the waves goingto infinity

3 Boundary Value Problem for the System ofOrdinary Differential EquationsDescribing a Field in the Gradient Layer

Desired functions for the Lame system (5) for any fixed 119910

from the interval (0 119871) will be considered in the class 1198711loc

and it will be assumed that the functions undergo a slowgrowth at infinity in the 119909-directionThis allows applying theFourier transform with respect to the 119909-direction permittingboth waves decaying at infinity and propagating wavesThuswe perform the change of variables from variable119909 to variable120585 and obtain the system of equations at 119910 isin (0 119871)

(119896331199061015840

1199092)1015840

+ [12058821205962minus 119896111205852] 1199061199092

minus 119894120585 (11989612+ 11989633) 1199061015840

1199102minus 1198941205851198961015840

331199061199102

= 0

(119896221199061015840

1199102)1015840

+ [12058821205962minus 119896331205852] 1199061199102

minus 119894120585 (11989612+ 11989633) 1199061015840

1199092minus 1198941205851198961015840

121199061199092

= 0

(9)

with respect to the Fourier transform for displacements1199061199092(120585 119910) and 119906

1199102(120585 119910)

It is worth noting here that the unknowns 1199061199092(120585 119910)

and 1199061199102(120585 119910) with respect to 119910 are ordinary functions and

therefore all the derivatives are to be understood in theclassical sense This allows discretization of the problem withrespect to 119910 For any fixed 119910 desired functions with respectto 120585 are distributions of the slow growth

For the upper half-plane 119910 gt 119871 it will be assumedthat the solutions (1) (2) belong to 119871

1loc(119877) and their traces1205911(119909 119871 + 0) 120590

1199101(119909 119871 + 0) 119906

1199091(119909 119871 + 0) and 119906

1199101(119909 119871 + 0)

are correctly specified We will consider that the desiredfunctions are distributions of the slow growth at infinityand moreover their traces are also distributions of the slowgrowth at infinity In thework [11] it was shown that solutionscorresponding to the wavemoving in the positive119910-directionsatisfy the equalities linking to each other Fourier transformsof traces of components of the field

1205851205911(120585 119871) + 120574

11(120585) 1205901199101(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199091(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199101(120585 119871) = 0

minus 12057421

(120585) 1205911(120585 119871) + 120585120590

1199101(120585 119871)

+ 119894 (12058811205962minus 212058311205852) 1199061199091(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199101(120585 119871) = 0

(10)

where 119896211

= 12058811205962(1205821+ 21205831) 119896221

= 120588112059621205831and branches of

roots of the functions 12057411

= radic119896211minus 1205852 120574

21= radic1198962

21minus 1205852 are

chosen such that the real part is positive and in the case ofthe real part being zero positive imaginary roots are chosen

In equalities (10) traces of all desired functions areconsidered at 119910 = 119871 but since the considered functions arecontinuous in the whole domain the limit will be consideredas the value at 119910 = 119871 We will proceed with the other traces ofthe desired functions in the same manner

We perform transition from traces of functions ofmedium 1 to traces of functions of the layer in equalities (10)For doing this we express the traces via conditions (7) andsubstitute the obtained expressions into (10) Thus we obtainthe following boundary conditions for the Fourier transformsof components of the field defined in the layer

1205851205912(120585 119871) + 120574

11(120585) 1205901199102(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199092(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199102(120585 119871) = 119891

1(120585)

minus 12057421

(120585) 1205912(120585 119871) + 120585120590

1199102(120585 119871)

+ 119894 (12058811205962minus 212058311205852) 1199061199092(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199102(120585 119871) = 119891

2(120585)

(11)

where1198911(120585) = 120585120591

0(120585 119871) + 120574

11(120585) 1205901199100(120585 119871)

minus 2119894120583112058512057411

(120585) 1199061199090(120585 119871)

minus 119894 (12058811205962minus 212058311205852) 1199061199100(120585 119871)

1198912(120585) = minus120574

21(120585) 1205910(120585 119871)

+ 1205851205901199100(120585 119871) + 119894 (120588

11205962minus 212058311205852) 1199061199090(120585 119871)

minus 2119894120583112058512057421

(120585) 1199061199100(120585 119871)

(12)

We eliminate Fourier transforms of the stresses fromthe obtained conditions using (4) Thus we obtain relationsbetween traces of Fourier transforms of displacements in thelayer

1198861(120585) 1199061015840

1199092(120585 119871) + 119886

2(120585) 1199061199092(120585 119871)

+ 1198863(120585) 1199061015840

1199102(120585 119871)

+ 1198864(120585) 1199061199102(120585 119871) = 119891

1(120585)

1198865(120585) 1199061015840

1199092(120585 119871) + 119886

6(120585) 1199061199092(120585 119871)

+ 1198867(120585) 1199061015840

1199102(120585 119871)

+ 1198868(120585) 1199061199102(120585 119871) = 119891

2(120585)

(13)

4 Advances in Acoustics and Vibration

where

1198861(120585) = 119896

33(119871)

1198862(120585) = minus (119896

12(119871) + 2120583

1) 11989412058512057411

(120585)

1198863(120585) = 119896

22(119871) 12057411

(120585)

1198864(120585) = minus119894 (120588

11205962minus (21205831minus 11989633(119871)) 120585

2)

1198865(120585) = minus119896

33(119871) 12057421

(120585)

1198866(120585) = 119894 (120588

11205962minus (11989612(119871) + 2120583

1) 1205852)

1198867(120585) = 119896

22(119871) 120585

1198868(120585) = minus119894 (2120583

1minus 11989633(119871)) 120585120574

21(120585)

(14)

On the lower half-plane 119910 lt 0 solutions (1) will besought in the class 119871

1loc with the slow growth at infinity tak-ing into account that the traces 120591

3(119909 0) 120590

1199103(119909 0) 119906

1199093(119909 0)

and 1199061199103(119909 0) are correctly determined and they also belong

to 1198711loc Then solutions from the class of distributions of the

slow growth corresponding to waves moving in the negative119910-direction satisfy equations establishing relations betweenFourier transforms of components of the field [11]

1205851205913(120585 0) minus 120574

13(120585) 1205901199103(120585 0)

+ 2119894120583312058512057413

(120585 0) 1199061199093(120585)

minus 119894 (12058831205962minus 212058331205852) 1199061199103(120585 0) = 0

12057423

(120585) 1205913(120585 0) + 120585120590

1199103(120585 0)

+ 119894 (12058831205962minus 212058331205852) 1199061199093(120585 0)

+ 2119894120583312058512057423

(120585) 1199061199103(120585 0) = 0

(15)

which are equivalent to conditions related to traces of the fieldcomponents at the lower boundary

1205851205912(120585 0) minus 120574

13(120585) 1205901199102(120585 0)

+ 2119894120583312058512057413

(120585) 1199061199092(120585 0)

minus 119894 (12058831205962minus 212058331205852) 1199061199102(120585 0) = 0

12057423

(120585) 1205912(120585 0) + 120585120590

1199102(120585 0)

+ 119894 (12058831205962minus 212058331205852) 1199061199092(120585 0)

+ 2119894120583312058512057423

(120585) 1199061199102(120585 0) = 0

(16)

where 119896213

= 12058831205962(120582

3+ 21205833) 119896223

= 120588312059621205833and branches of

roots of the functions 12057413

= radic119896213minus 1205852 120574

23= radic1198962

23minus 1205852 are

chosen in the same way as branches of roots of the functionsof the upper half-plane

Using (4) we obtain

1198871(120585) 1199061015840

1199092(120585 0) + 119887

2(120585) 1199061199092(120585 0)

+ 1198873(120585) 1199061015840

1199102(120585 0)

+ 1198874(120585) 1199061199102(120585 0) = 0

1198875(120585) 1199061015840

1199092(120585 0) + 119887

6(120585) 1199061199092(120585 0)

+ 1198877(120585) 1199061015840

1199102(120585 0)

+ 1198878(120585) 1199061199102(120585 0) = 0

(17)

where

1198871(120585) = 119896

33(0) 120585

1198872(120585) = 119894 (119896

12(0) + 2120583

3) 12058512057413

(120585)

1198873(120585) = minus119896

22(0) 12057413

(120585)

1198874(120585) = minus119894 (120588

31205962minus (21205833minus 11989633(0)) 1205852)

1198875(120585) = 119896

33(0) 12057423

(120585)

1198876(120585) = 119894 (120588

31205962minus (11989612(0) + 2120583

3) 1205852)

1198877(120585) = 119896

22(0) 120585

1198878(120585) = 119894 (2120583

3minus 11989633(0)) 120585120574

23(120585)

(18)

Physical meanings of solutions of (9) with the boundaryconditions (13) and (17) are displacements (119906

1199092 1199061199102) which

describe the field at 0 lt 119910 lt 119871 in the problem of diffractionin the elastic layer

4 Elastic Oscillations of a TransverselyIsotropic Body

Let us consider three-dimensional oscillations of an elastictransversely isotropic medium To describe deformations ofthe medium the following model will be used [12]

120576119909119909

=1

119864(120590119909119909

minus ]120590119911119911) minus

]1015840

1198641015840120590119910119910

120576119911119911

=1

119864(120590119911119911minus ]120590119909119909) minus

]1015840

1198641015840120590119910119910

120576119910119910

= minus]1015840

1198641015840(120590119909119909

+ 120590119911119911) +

1

1198641015840120590119910119910

120576119909119910

=1

21198661015840120590119909119910 120576

119909119911=

1

2119866120590119909119911

120576119910119911

=1

21198661015840120590119910119911

(19)

Here the plane 119909119911 is the plane of isotropy and the planes119909119910 and 119910119911 are the planes of elastic symmetryThe parameters119864 1198641015840 are the Young modula ] and ]1015840 are the Poissoncoefficients 119866 = 119864(2(1 + ])) and 1198661015840 are the displacement

Advances in Acoustics and Vibration 5

modula Parameters without the prime sign correspond todeformations in the plane of isotropy whereas parameterswith the prime sign correspond to deformations in the planeof elastic symmetry

Equations (19) could be transformed to the form [13]

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 120582120576119911119911+ 1205821015840120576119910119910

120590119911119911

= 120582120576119909119909

+ (120582 + 2120583) 120576119911119911+ 1205821015840120576119910119910

120590119910119910

= 1205821015840(120576119909119909

+ 120576119911119911) + (120582

1015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910 120590

119909119911= 2120583120576

119909119911

(20)

120590119910119911

= 21198661015840120576119910119911 (21)

in which the used notations imply the following

120582 + 2120583 =119864

(1 + ]) 119889(1 minus (]1015840)

2 119864

1198641015840)

120582 =119864

(1 + ]) 119889(] + (]1015840)

2 119864

1198641015840)

1205821015840+ 21205831015840=1198641015840 (1 minus ])

119889

1205821015840=119864]1015840

119889 119889 = 1 minus ] minus 2(]1015840)

2 119864

1198641015840

(22)

We will assume that the field does not depend on the119911 coordinate 120597120597119911 equiv 0 Then we have 120576

119911119911= 0 and the

system of (20) falls into two independent subsystems Thefirst subsystem describes oscillations in the plane 119909119910

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 1205821015840120576119910119910

120590119910119910

= 1205821015840120576119909119909

+ (1205821015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910

(23)

whereas the second subsystem describes oscillations in the 119911direction

120590119911119911

= 120582120576119909119909

+ 1205821015840120576119910119910 120590

119909119911= 2120583120576

119909119911

120590119910119911

= 21198661015840120576119910119911

(24)

The system (24) with the use of equations of motiontransforms to the following

1205831205972119906119911

1205971199092+

120597

120597119910(1198661015840 120597119906119911

120597119910) + 12058821205962119906119911= 0 (25)

The system (23) corresponds to (4) with the followingnotations 120590

1199092= 120590119909119909 1205901199102

= 120590119910 and 120591

2= 120590119909119910 Under the

conditions 11989613

= 11989623

= 0 considered in Section 1 the elasticitytensor K linking stress and deformations to each other takesthe following form

K = (

120582 + 2120583 1205821015840 0

1205821015840 1205821015840 + 21205831015840 0

0 0 1198661015840

) (26)

Thus the problem of diffraction of an elastic harmonicwave by a transversely isotropic layer reduces to the boundaryvalue problem (9) (13) and (17) with the elasticity tensordefined in (26)

5 Numerical Results

Before discussions of the numerical results we will give somenotes regarding dependence of solution of the problem (9)(13) and (17) on parameter 120585 All the coefficients of theboundary value problem are continuous functions of 120585 Thenif right-hand sides of (13) are regular distributions on 120585 thensolutions will also be considered as regular with respect to120585 However if 119891

1and 119891

2are singular distributions on 120585 then

the solutions themselves will also be considered as singularFor example if 119891

1= 1198621120575(120585 minus 120585

0) and 119891

2= 1198622120575(120585 minus 120585

0) then

119906119909(119910) = 120575(120585 minus 120585

0)119908119909(119910 1205850) and 119906

119910(119910) = 120575(120585 minus 120585

0)119908119910(119910 1205850) In

this case it is convenient to ldquonormalizerdquo the boundary valueproblem by 120575(120585 minus 120585

0) For doing that we perform the change

of variables from variable 120585 to variable 1205850all over and solve

(9) (13) and (17) with respect to 119908119909(119910 1205850) and 119908

119910(119910 1205850)

Therefore in the case of Fourier transforms of traces ofthe incident field being singular distributions for examplein the case of the incident wave being a plane wave thesolutions of the problems will also be singular distributionswith the same carrier From this it follows that diffraction ofone plane wave results in two reflected waves longitudinaland transverse and excitation of waveguide waves in the layerdoes not occur It is obvious that the last statement is trueunder condition of uniqueness of the diffraction problem(homogeneous conditions (13) result in a trivial solution tothe problem (9) (13) and (17)) and under condition theeigenvalues of the waveguide formed by the layer which differfrom 120585

0

The desired problem can be solved using many approx-imation methods A uniform finite-difference grid with themesh size ℎ was chosen to approximate the boundary valueproblem (9) (13) and (17) with the accuracy on the order of119874(ℎ) When choosing the mesh size it is taken into accountthat the finite difference analogs of elastic profiles of thelayer describe adequately the original continuous modelsOn the other hand the mesh size ℎ must be smaller thanthe wavelength in the layer and consequently inverselyproportional to the frequency 120596

After carrying out the numerical solution it is necessaryto reconstruct the fields in the half-planes 119906

1and 119906

3 For

doing it we consider displacements in a homogeneousisotropic 119899th medium which can be written in the generalform in the following way [14]

119906119909119899(119910) = 120585119860

119899119890minus1198941205741119899119910minus 1205851198611198991198901198941205741119899119910

+ 1205742119899119862119899119890minus1198941205742119899119910+ 12057421198991198631198991198901198941205742119899119910

119906119910119899(119910) = 120574

1119899119860119899119890minus1198941205741119899119910+ 12057411198991198611198991198901198941205741119899119910

minus 120585119862119899119890minus1198941205742119899119910+ 1205851198631198991198901198941205742119899119910

(27)

6 Advances in Acoustics and Vibration

Taking into account the conditions at infinity displace-ments for the reflected field will have the following form

1199061199091(119910) = minus120585

0119861111989011989412057411(119910minus119871)

+ 12057421119863111989011989412057421(119910minus119871)

1199061199101(119910) = 120574

11119861111989011989412057411(119910minus119871)

+ 1205850119863111989011989412057421(119910minus119871)

(28)

and for the transmitted wave

1199061199093(119910) = 120585

01198603119890minus11989412057413119910+ 120574231198623119890minus11989412057423119910

1199061199103(119910) = 120574

131198603119890minus11989412057413119910minus 12058501198623119890minus11989412057423119910

(29)

The unknown coefficients 1198611 1198631 1198603 and 119862

3are found

via the following expressions

1198611=119906119910112057421minus 11990611990911205850

1205741112057421+ 12058520

1198631=119906119909112057411+ 11990611991011205850

1205741112057421+ 12058520

1198603=119906119910312057423+ 11990611990931205850

1205741312057423+ 12058520

1198623=119906119909312057413minus 11990611991031205850

1205741312057423+ 12058520

(30)

where 119906119909119899

and 119906119910119899

are traces of displacements of the 119899thmedium which are expressed using (7) and (8)

We will consider the case of diffraction by a planelongitudinal wave with displacements of the following kind

1199061199090(119909 119910)

= 119860011989611

sin 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

1199061199100(119909 119910)

= 119860011989611

cos 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

(31)

We apply Fourier transformation to components of theincident field and arrive at the result that all the componentsof the field are singular distributions with themultiplier 120575(120585minus1205850) 1205850= 11989611

sin 120579 For example the Fourier transform of thetrace 119906

1199090(119909 119910) at 119910 = 119871 takes the following form

1199061199090(120585 119871) = 119860

011989611

sin 120579120575 (120585 minus 11989611

sin 120579) (32)

Since the right-hand sides of (13) are singular distribu-tions then it is sufficient to solve the problem (9) (13) and(17) just at the value 120585 = 120585

0

For carrying out the numerical experiments we willconsider the case when the layer of thickness 119871 = 10mmadeof siltstone is located in sandstone Parameters of sandstonefilling in medium 1 and medium 3 are 120588 = 2400 kgm3 V

119901=

3300msec and V119904= 2000msec The layer is considered

anisotropic we will consider three cases of distributions ofelasticity parameters in the layer A uniform layer has thefollowing parameters [15] 119864 = 568GPa 1198641015840 = 621GPa] = 029 ]1015840 = 026 and 1198661015840 = 229GPa In the caseof ldquocompressed siltstonerdquo all the parameters of the mediumremain constant except for the Young modulus 119864(119910) which

00102030405060708091

0 1000 2000 3000 4000120596

Figure 2 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 20

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

in the plane of isotropy grows linearly from 119864(0) = 568GPato 119864(1198712) = 850GPa and then reduces linearly to the value119864(119871) = 568GPa In the case of ldquorelaxed siltstonerdquo the Youngmodulus 119864(119910) reduces linearly from 119864(0) = 568GPa to119864(1198712) = 284GPa and grows linearly to 119864(119871) = 568GPa

In the present work two sets of studies were carried outThe first set of studies is dedicated to searching for depen-dence of normalized energy of the transmitted longitudinalwave on angular frequency 120596 As a result the conclusion ismade so that the transmitted energy grows as the value ofthe Young modulus 119864(119910) reduces in the middle of the layerThe difference between various structures increases with theincrease of angle of incidence 120579 The dependences are givenin Figures 2 and 3

In Figure 2 dependence of normalized energy of thetransmitted longitudinal wave on angular frequency 120596 at itsincidence under the angle 120579 = 20

∘upon a siltstone layer ofthickness 119871 = 10m placed into sandstone is shownThe solidcurve corresponds to a uniform layer of siltstone the dottedcurve ldquocompressed siltstonerdquo and the dashed curve ldquorelaxedsiltstonerdquo In Figure 3 the same type of dependence is shownbut for the angle 120579 = 40

∘

The second set of studies is dedicated to searchingfor dependence of normalized energy of the transmittedlongitudinal wave on the angle of incidence 120579 Just as in theprevious set of studies confirmed is the conclusion that thetransmitted energy increases with the decrease of the Youngmodulus 119864(119910) in the middle of the layer It is worth notinghere that the difference in the transmitted energy for differentstructures decreases with the increase of angular frequency120596

In Figure 4 dependence of normalized energy of thetransmitted longitudinal wave on 120579 at the angular frequencyof the wave 120596 = 2 sdot 10

3 radian per second upon a siltstonelayer of thickness 119871 = 10m placed into sandstone is shownThe solid curve corresponds to a uniform layer of siltstonethe dotted curve ldquocompressed siltstonerdquo and the dashed curveldquorelaxed siltstonerdquo In Figure 5 the same type of dependenceis shown but for 120596 = 4 sdot 10

3 radian per second

Advances in Acoustics and Vibration 7

00102030405060708091

0 1000 2000 3000 4000120596

Figure 3 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 40

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 4 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 2000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness 119871 = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 5 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 4000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness L = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

6 Conclusions

The method of overdetermined boundary value problemused in the present work when combined with and theFourier transformmethod is shown to be efficient especiallyfor the cases the Fourier transforms of traces of the incidentfield are singular distributions Then the approximationproblem is solved just at the value 120585 = 120585

0 In the case of

the Fourier transforms being regular distributions (eg atdiffraction of a Gauss beam by a plate) the problem (9) (13)and (17) is solved for several values of the parameter 120585

Results obtained with respect to propagation of elasticwaves through anisotropic layers can be used in geophysicsfor the initial analysis of structure of the layers of rock strataAlso results of propagation of elastic waves through nonuni-form anisotropic structures can be used in industries inwhich anisotropicmaterials are applied as well as at designingprotective layers for various processes and apparatuses

Acknowledgment

Thiswork was supported by RFBR 12-01-97012-r-povolzhrsquoe-a

References

[1] S Ryan-Grigor ldquoEmpirical relationships between transverseisotropy parameters and VplVS implications for AVOrdquo Geo-physics vol 62 no 5 pp 1359ndash1364 1997

[2] T Alkhalifah ldquoVelocity analysis using nonhyperbolic moveoutin transversely isotropic mediardquo Geophysics vol 62 no 6 pp1839ndash1854 1997

[3] CM Sayers ldquoSimplified anisotropy parameters for transverselyisotropic sedimentary rocksrdquoGeophysics vol 60 pp 1933ndash19351995

[4] V J Nemirovskii and A P Jankovskii ldquoDetermining effectivephysical and mechanical characteristics of hybrid compositescrisscross reinforced by transversely isotropic fibers and com-parisons of computed characteristics versus experimental datardquoMekhanIka Kompozicionnykh MaterIalov i Konstrukcii vol 13no 1 pp 3ndash32 2007

[5] GAGeniev VN Kissjuk andGA TjupinTheory of Plasticityof Concrete and Ferroconcrete Strojizdat Moscow Russia 1974

[6] B V Labudin ldquoJustifying a computational model treating lam-inated wood as an orthogonal transversely isotropic materialrdquoIzvestiya VUZov Lesnoj Zhurnal no 6 pp 136ndash139 2006

[7] M P Lonkevich ldquoPropagation of sound through a layer of atransversely isotropic material of finite thicknessrdquo AkusticheskijZhurnal vol 17 no 1 pp 85ndash92 1971

[8] E L Shenderov ldquoPropagation of sound through a layer of atransversely isotropic platerdquo Akusticheskii Zhurnal vol 30 no1 pp 122ndash129 1984

[9] S A Skobelitsyn and L A Tolokonnikov ldquoPropagation ofsound through a transversely isotropic nonuniform flat layerrdquoAkusticheskii Zhurnal vol 36 no 4 pp 740ndash744 1990

[10] A V Anufrieva D N Tumakov and V L Kipot ldquoElastic wavepropagation through a layer with graded-index distribution ofdensityrdquo in Proceedings of the Days on Diffraction (DD rsquo12) pp21ndash26 2012

[11] I E Pleshchinskaya and N B Pleshchinskii ldquoOver-determinedboundary value problems for linear equations of elastodynam-ics and their applications to elastic wave diffraction theoryrdquo in

8 Advances in Acoustics and Vibration

Advances inMathematics Research A R Baswell Ed vol 17 pp102ndash138 Nova Science New York NY USA 2012

[12] S G Lekhnickii Theory of Elasticity of the Anisotropic BodyNauka Moscow Russia 1977

[13] B D Annin ldquoTransversely isotropic elastic model of geologicalmaterialsrdquo Sibirskiı Zhurnal Industrial noı Matematiki vol 12no 3 pp 5ndash14 2009

[14] K N Vdovina N B Pleshchinskii and D N TumakovldquoConcerning orthogonality of proper waves of a half-openedelastic waveguiderdquo Izvestiya VUZov Matematika no 9 pp 69ndash75 2008

[15] S A Batugin and R K Nirenburg ldquoApproximate rela-tion between the elastic constants of anisotropic rocks andthe anisotropy parametersrdquo Fiziko-Tekhnicheskie ProblemyRazrabotki Poleznykh Iskopaemykh vol 7 no 1 pp 7ndash12 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Advances in Acoustics and Vibration

where

1198861(120585) = 119896

33(119871)

1198862(120585) = minus (119896

12(119871) + 2120583

1) 11989412058512057411

(120585)

1198863(120585) = 119896

22(119871) 12057411

(120585)

1198864(120585) = minus119894 (120588

11205962minus (21205831minus 11989633(119871)) 120585

2)

1198865(120585) = minus119896

33(119871) 12057421

(120585)

1198866(120585) = 119894 (120588

11205962minus (11989612(119871) + 2120583

1) 1205852)

1198867(120585) = 119896

22(119871) 120585

1198868(120585) = minus119894 (2120583

1minus 11989633(119871)) 120585120574

21(120585)

(14)

On the lower half-plane 119910 lt 0 solutions (1) will besought in the class 119871

1loc with the slow growth at infinity tak-ing into account that the traces 120591

3(119909 0) 120590

1199103(119909 0) 119906

1199093(119909 0)

and 1199061199103(119909 0) are correctly determined and they also belong

to 1198711loc Then solutions from the class of distributions of the

slow growth corresponding to waves moving in the negative119910-direction satisfy equations establishing relations betweenFourier transforms of components of the field [11]

1205851205913(120585 0) minus 120574

13(120585) 1205901199103(120585 0)

+ 2119894120583312058512057413

(120585 0) 1199061199093(120585)

minus 119894 (12058831205962minus 212058331205852) 1199061199103(120585 0) = 0

12057423

(120585) 1205913(120585 0) + 120585120590

1199103(120585 0)

+ 119894 (12058831205962minus 212058331205852) 1199061199093(120585 0)

+ 2119894120583312058512057423

(120585) 1199061199103(120585 0) = 0

(15)

which are equivalent to conditions related to traces of the fieldcomponents at the lower boundary

1205851205912(120585 0) minus 120574

13(120585) 1205901199102(120585 0)

+ 2119894120583312058512057413

(120585) 1199061199092(120585 0)

minus 119894 (12058831205962minus 212058331205852) 1199061199102(120585 0) = 0

12057423

(120585) 1205912(120585 0) + 120585120590

1199102(120585 0)

+ 119894 (12058831205962minus 212058331205852) 1199061199092(120585 0)

+ 2119894120583312058512057423

(120585) 1199061199102(120585 0) = 0

(16)

where 119896213

= 12058831205962(120582

3+ 21205833) 119896223

= 120588312059621205833and branches of

roots of the functions 12057413

= radic119896213minus 1205852 120574

23= radic1198962

23minus 1205852 are

chosen in the same way as branches of roots of the functionsof the upper half-plane

Using (4) we obtain

1198871(120585) 1199061015840

1199092(120585 0) + 119887

2(120585) 1199061199092(120585 0)

+ 1198873(120585) 1199061015840

1199102(120585 0)

+ 1198874(120585) 1199061199102(120585 0) = 0

1198875(120585) 1199061015840

1199092(120585 0) + 119887

6(120585) 1199061199092(120585 0)

+ 1198877(120585) 1199061015840

1199102(120585 0)

+ 1198878(120585) 1199061199102(120585 0) = 0

(17)

where

1198871(120585) = 119896

33(0) 120585

1198872(120585) = 119894 (119896

12(0) + 2120583

3) 12058512057413

(120585)

1198873(120585) = minus119896

22(0) 12057413

(120585)

1198874(120585) = minus119894 (120588

31205962minus (21205833minus 11989633(0)) 1205852)

1198875(120585) = 119896

33(0) 12057423

(120585)

1198876(120585) = 119894 (120588

31205962minus (11989612(0) + 2120583

3) 1205852)

1198877(120585) = 119896

22(0) 120585

1198878(120585) = 119894 (2120583

3minus 11989633(0)) 120585120574

23(120585)

(18)

Physical meanings of solutions of (9) with the boundaryconditions (13) and (17) are displacements (119906

1199092 1199061199102) which

describe the field at 0 lt 119910 lt 119871 in the problem of diffractionin the elastic layer

4 Elastic Oscillations of a TransverselyIsotropic Body

Let us consider three-dimensional oscillations of an elastictransversely isotropic medium To describe deformations ofthe medium the following model will be used [12]

120576119909119909

=1

119864(120590119909119909

minus ]120590119911119911) minus

]1015840

1198641015840120590119910119910

120576119911119911

=1

119864(120590119911119911minus ]120590119909119909) minus

]1015840

1198641015840120590119910119910

120576119910119910

= minus]1015840

1198641015840(120590119909119909

+ 120590119911119911) +

1

1198641015840120590119910119910

120576119909119910

=1

21198661015840120590119909119910 120576

119909119911=

1

2119866120590119909119911

120576119910119911

=1

21198661015840120590119910119911

(19)

Here the plane 119909119911 is the plane of isotropy and the planes119909119910 and 119910119911 are the planes of elastic symmetryThe parameters119864 1198641015840 are the Young modula ] and ]1015840 are the Poissoncoefficients 119866 = 119864(2(1 + ])) and 1198661015840 are the displacement

Advances in Acoustics and Vibration 5

modula Parameters without the prime sign correspond todeformations in the plane of isotropy whereas parameterswith the prime sign correspond to deformations in the planeof elastic symmetry

Equations (19) could be transformed to the form [13]

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 120582120576119911119911+ 1205821015840120576119910119910

120590119911119911

= 120582120576119909119909

+ (120582 + 2120583) 120576119911119911+ 1205821015840120576119910119910

120590119910119910

= 1205821015840(120576119909119909

+ 120576119911119911) + (120582

1015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910 120590

119909119911= 2120583120576

119909119911

(20)

120590119910119911

= 21198661015840120576119910119911 (21)

in which the used notations imply the following

120582 + 2120583 =119864

(1 + ]) 119889(1 minus (]1015840)

2 119864

1198641015840)

120582 =119864

(1 + ]) 119889(] + (]1015840)

2 119864

1198641015840)

1205821015840+ 21205831015840=1198641015840 (1 minus ])

119889

1205821015840=119864]1015840

119889 119889 = 1 minus ] minus 2(]1015840)

2 119864

1198641015840

(22)

We will assume that the field does not depend on the119911 coordinate 120597120597119911 equiv 0 Then we have 120576

119911119911= 0 and the

system of (20) falls into two independent subsystems Thefirst subsystem describes oscillations in the plane 119909119910

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 1205821015840120576119910119910

120590119910119910

= 1205821015840120576119909119909

+ (1205821015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910

(23)

whereas the second subsystem describes oscillations in the 119911direction

120590119911119911

= 120582120576119909119909

+ 1205821015840120576119910119910 120590

119909119911= 2120583120576

119909119911

120590119910119911

= 21198661015840120576119910119911

(24)

The system (24) with the use of equations of motiontransforms to the following

1205831205972119906119911

1205971199092+

120597

120597119910(1198661015840 120597119906119911

120597119910) + 12058821205962119906119911= 0 (25)

The system (23) corresponds to (4) with the followingnotations 120590

1199092= 120590119909119909 1205901199102

= 120590119910 and 120591

2= 120590119909119910 Under the

conditions 11989613

= 11989623

= 0 considered in Section 1 the elasticitytensor K linking stress and deformations to each other takesthe following form

K = (

120582 + 2120583 1205821015840 0

1205821015840 1205821015840 + 21205831015840 0

0 0 1198661015840

) (26)

Thus the problem of diffraction of an elastic harmonicwave by a transversely isotropic layer reduces to the boundaryvalue problem (9) (13) and (17) with the elasticity tensordefined in (26)

5 Numerical Results

Before discussions of the numerical results we will give somenotes regarding dependence of solution of the problem (9)(13) and (17) on parameter 120585 All the coefficients of theboundary value problem are continuous functions of 120585 Thenif right-hand sides of (13) are regular distributions on 120585 thensolutions will also be considered as regular with respect to120585 However if 119891

1and 119891

2are singular distributions on 120585 then

the solutions themselves will also be considered as singularFor example if 119891

1= 1198621120575(120585 minus 120585

0) and 119891

2= 1198622120575(120585 minus 120585

0) then

119906119909(119910) = 120575(120585 minus 120585

0)119908119909(119910 1205850) and 119906

119910(119910) = 120575(120585 minus 120585

0)119908119910(119910 1205850) In

this case it is convenient to ldquonormalizerdquo the boundary valueproblem by 120575(120585 minus 120585

0) For doing that we perform the change

of variables from variable 120585 to variable 1205850all over and solve

(9) (13) and (17) with respect to 119908119909(119910 1205850) and 119908

119910(119910 1205850)

Therefore in the case of Fourier transforms of traces ofthe incident field being singular distributions for examplein the case of the incident wave being a plane wave thesolutions of the problems will also be singular distributionswith the same carrier From this it follows that diffraction ofone plane wave results in two reflected waves longitudinaland transverse and excitation of waveguide waves in the layerdoes not occur It is obvious that the last statement is trueunder condition of uniqueness of the diffraction problem(homogeneous conditions (13) result in a trivial solution tothe problem (9) (13) and (17)) and under condition theeigenvalues of the waveguide formed by the layer which differfrom 120585

0

The desired problem can be solved using many approx-imation methods A uniform finite-difference grid with themesh size ℎ was chosen to approximate the boundary valueproblem (9) (13) and (17) with the accuracy on the order of119874(ℎ) When choosing the mesh size it is taken into accountthat the finite difference analogs of elastic profiles of thelayer describe adequately the original continuous modelsOn the other hand the mesh size ℎ must be smaller thanthe wavelength in the layer and consequently inverselyproportional to the frequency 120596

After carrying out the numerical solution it is necessaryto reconstruct the fields in the half-planes 119906

1and 119906

3 For

doing it we consider displacements in a homogeneousisotropic 119899th medium which can be written in the generalform in the following way [14]

119906119909119899(119910) = 120585119860

119899119890minus1198941205741119899119910minus 1205851198611198991198901198941205741119899119910

+ 1205742119899119862119899119890minus1198941205742119899119910+ 12057421198991198631198991198901198941205742119899119910

119906119910119899(119910) = 120574

1119899119860119899119890minus1198941205741119899119910+ 12057411198991198611198991198901198941205741119899119910

minus 120585119862119899119890minus1198941205742119899119910+ 1205851198631198991198901198941205742119899119910

(27)

6 Advances in Acoustics and Vibration

Taking into account the conditions at infinity displace-ments for the reflected field will have the following form

1199061199091(119910) = minus120585

0119861111989011989412057411(119910minus119871)

+ 12057421119863111989011989412057421(119910minus119871)

1199061199101(119910) = 120574

11119861111989011989412057411(119910minus119871)

+ 1205850119863111989011989412057421(119910minus119871)

(28)

and for the transmitted wave

1199061199093(119910) = 120585

01198603119890minus11989412057413119910+ 120574231198623119890minus11989412057423119910

1199061199103(119910) = 120574

131198603119890minus11989412057413119910minus 12058501198623119890minus11989412057423119910

(29)

The unknown coefficients 1198611 1198631 1198603 and 119862

3are found

via the following expressions

1198611=119906119910112057421minus 11990611990911205850

1205741112057421+ 12058520

1198631=119906119909112057411+ 11990611991011205850

1205741112057421+ 12058520

1198603=119906119910312057423+ 11990611990931205850

1205741312057423+ 12058520

1198623=119906119909312057413minus 11990611991031205850

1205741312057423+ 12058520

(30)

where 119906119909119899

and 119906119910119899

are traces of displacements of the 119899thmedium which are expressed using (7) and (8)

We will consider the case of diffraction by a planelongitudinal wave with displacements of the following kind

1199061199090(119909 119910)

= 119860011989611

sin 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

1199061199100(119909 119910)

= 119860011989611

cos 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

(31)

We apply Fourier transformation to components of theincident field and arrive at the result that all the componentsof the field are singular distributions with themultiplier 120575(120585minus1205850) 1205850= 11989611

sin 120579 For example the Fourier transform of thetrace 119906

1199090(119909 119910) at 119910 = 119871 takes the following form

1199061199090(120585 119871) = 119860

011989611

sin 120579120575 (120585 minus 11989611

sin 120579) (32)

Since the right-hand sides of (13) are singular distribu-tions then it is sufficient to solve the problem (9) (13) and(17) just at the value 120585 = 120585

0

For carrying out the numerical experiments we willconsider the case when the layer of thickness 119871 = 10mmadeof siltstone is located in sandstone Parameters of sandstonefilling in medium 1 and medium 3 are 120588 = 2400 kgm3 V

119901=

3300msec and V119904= 2000msec The layer is considered

anisotropic we will consider three cases of distributions ofelasticity parameters in the layer A uniform layer has thefollowing parameters [15] 119864 = 568GPa 1198641015840 = 621GPa] = 029 ]1015840 = 026 and 1198661015840 = 229GPa In the caseof ldquocompressed siltstonerdquo all the parameters of the mediumremain constant except for the Young modulus 119864(119910) which

00102030405060708091

0 1000 2000 3000 4000120596

Figure 2 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 20

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

in the plane of isotropy grows linearly from 119864(0) = 568GPato 119864(1198712) = 850GPa and then reduces linearly to the value119864(119871) = 568GPa In the case of ldquorelaxed siltstonerdquo the Youngmodulus 119864(119910) reduces linearly from 119864(0) = 568GPa to119864(1198712) = 284GPa and grows linearly to 119864(119871) = 568GPa

In the present work two sets of studies were carried outThe first set of studies is dedicated to searching for depen-dence of normalized energy of the transmitted longitudinalwave on angular frequency 120596 As a result the conclusion ismade so that the transmitted energy grows as the value ofthe Young modulus 119864(119910) reduces in the middle of the layerThe difference between various structures increases with theincrease of angle of incidence 120579 The dependences are givenin Figures 2 and 3

In Figure 2 dependence of normalized energy of thetransmitted longitudinal wave on angular frequency 120596 at itsincidence under the angle 120579 = 20

∘upon a siltstone layer ofthickness 119871 = 10m placed into sandstone is shownThe solidcurve corresponds to a uniform layer of siltstone the dottedcurve ldquocompressed siltstonerdquo and the dashed curve ldquorelaxedsiltstonerdquo In Figure 3 the same type of dependence is shownbut for the angle 120579 = 40

∘

The second set of studies is dedicated to searchingfor dependence of normalized energy of the transmittedlongitudinal wave on the angle of incidence 120579 Just as in theprevious set of studies confirmed is the conclusion that thetransmitted energy increases with the decrease of the Youngmodulus 119864(119910) in the middle of the layer It is worth notinghere that the difference in the transmitted energy for differentstructures decreases with the increase of angular frequency120596

In Figure 4 dependence of normalized energy of thetransmitted longitudinal wave on 120579 at the angular frequencyof the wave 120596 = 2 sdot 10

3 radian per second upon a siltstonelayer of thickness 119871 = 10m placed into sandstone is shownThe solid curve corresponds to a uniform layer of siltstonethe dotted curve ldquocompressed siltstonerdquo and the dashed curveldquorelaxed siltstonerdquo In Figure 5 the same type of dependenceis shown but for 120596 = 4 sdot 10

3 radian per second

Advances in Acoustics and Vibration 7

00102030405060708091

0 1000 2000 3000 4000120596

Figure 3 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 40

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 4 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 2000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness 119871 = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 5 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 4000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness L = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

6 Conclusions

The method of overdetermined boundary value problemused in the present work when combined with and theFourier transformmethod is shown to be efficient especiallyfor the cases the Fourier transforms of traces of the incidentfield are singular distributions Then the approximationproblem is solved just at the value 120585 = 120585

0 In the case of

the Fourier transforms being regular distributions (eg atdiffraction of a Gauss beam by a plate) the problem (9) (13)and (17) is solved for several values of the parameter 120585

Results obtained with respect to propagation of elasticwaves through anisotropic layers can be used in geophysicsfor the initial analysis of structure of the layers of rock strataAlso results of propagation of elastic waves through nonuni-form anisotropic structures can be used in industries inwhich anisotropicmaterials are applied as well as at designingprotective layers for various processes and apparatuses

Acknowledgment

Thiswork was supported by RFBR 12-01-97012-r-povolzhrsquoe-a

References

[1] S Ryan-Grigor ldquoEmpirical relationships between transverseisotropy parameters and VplVS implications for AVOrdquo Geo-physics vol 62 no 5 pp 1359ndash1364 1997

[2] T Alkhalifah ldquoVelocity analysis using nonhyperbolic moveoutin transversely isotropic mediardquo Geophysics vol 62 no 6 pp1839ndash1854 1997

[3] CM Sayers ldquoSimplified anisotropy parameters for transverselyisotropic sedimentary rocksrdquoGeophysics vol 60 pp 1933ndash19351995

[4] V J Nemirovskii and A P Jankovskii ldquoDetermining effectivephysical and mechanical characteristics of hybrid compositescrisscross reinforced by transversely isotropic fibers and com-parisons of computed characteristics versus experimental datardquoMekhanIka Kompozicionnykh MaterIalov i Konstrukcii vol 13no 1 pp 3ndash32 2007

[5] GAGeniev VN Kissjuk andGA TjupinTheory of Plasticityof Concrete and Ferroconcrete Strojizdat Moscow Russia 1974

[6] B V Labudin ldquoJustifying a computational model treating lam-inated wood as an orthogonal transversely isotropic materialrdquoIzvestiya VUZov Lesnoj Zhurnal no 6 pp 136ndash139 2006

[7] M P Lonkevich ldquoPropagation of sound through a layer of atransversely isotropic material of finite thicknessrdquo AkusticheskijZhurnal vol 17 no 1 pp 85ndash92 1971

[8] E L Shenderov ldquoPropagation of sound through a layer of atransversely isotropic platerdquo Akusticheskii Zhurnal vol 30 no1 pp 122ndash129 1984

[9] S A Skobelitsyn and L A Tolokonnikov ldquoPropagation ofsound through a transversely isotropic nonuniform flat layerrdquoAkusticheskii Zhurnal vol 36 no 4 pp 740ndash744 1990

[10] A V Anufrieva D N Tumakov and V L Kipot ldquoElastic wavepropagation through a layer with graded-index distribution ofdensityrdquo in Proceedings of the Days on Diffraction (DD rsquo12) pp21ndash26 2012

[11] I E Pleshchinskaya and N B Pleshchinskii ldquoOver-determinedboundary value problems for linear equations of elastodynam-ics and their applications to elastic wave diffraction theoryrdquo in

8 Advances in Acoustics and Vibration

Advances inMathematics Research A R Baswell Ed vol 17 pp102ndash138 Nova Science New York NY USA 2012

[12] S G Lekhnickii Theory of Elasticity of the Anisotropic BodyNauka Moscow Russia 1977

[13] B D Annin ldquoTransversely isotropic elastic model of geologicalmaterialsrdquo Sibirskiı Zhurnal Industrial noı Matematiki vol 12no 3 pp 5ndash14 2009

[14] K N Vdovina N B Pleshchinskii and D N TumakovldquoConcerning orthogonality of proper waves of a half-openedelastic waveguiderdquo Izvestiya VUZov Matematika no 9 pp 69ndash75 2008

[15] S A Batugin and R K Nirenburg ldquoApproximate rela-tion between the elastic constants of anisotropic rocks andthe anisotropy parametersrdquo Fiziko-Tekhnicheskie ProblemyRazrabotki Poleznykh Iskopaemykh vol 7 no 1 pp 7ndash12 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 5

modula Parameters without the prime sign correspond todeformations in the plane of isotropy whereas parameterswith the prime sign correspond to deformations in the planeof elastic symmetry

Equations (19) could be transformed to the form [13]

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 120582120576119911119911+ 1205821015840120576119910119910

120590119911119911

= 120582120576119909119909

+ (120582 + 2120583) 120576119911119911+ 1205821015840120576119910119910

120590119910119910

= 1205821015840(120576119909119909

+ 120576119911119911) + (120582

1015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910 120590

119909119911= 2120583120576

119909119911

(20)

120590119910119911

= 21198661015840120576119910119911 (21)

in which the used notations imply the following

120582 + 2120583 =119864

(1 + ]) 119889(1 minus (]1015840)

2 119864

1198641015840)

120582 =119864

(1 + ]) 119889(] + (]1015840)

2 119864

1198641015840)

1205821015840+ 21205831015840=1198641015840 (1 minus ])

119889

1205821015840=119864]1015840

119889 119889 = 1 minus ] minus 2(]1015840)

2 119864

1198641015840

(22)

We will assume that the field does not depend on the119911 coordinate 120597120597119911 equiv 0 Then we have 120576

119911119911= 0 and the

system of (20) falls into two independent subsystems Thefirst subsystem describes oscillations in the plane 119909119910

120590119909119909

= (120582 + 2120583) 120576119909119909

+ 1205821015840120576119910119910

120590119910119910

= 1205821015840120576119909119909

+ (1205821015840+ 21205831015840) 120576119910119910

120590119909119910

= 21198661015840120576119909119910

(23)

whereas the second subsystem describes oscillations in the 119911direction

120590119911119911

= 120582120576119909119909

+ 1205821015840120576119910119910 120590

119909119911= 2120583120576

119909119911

120590119910119911

= 21198661015840120576119910119911

(24)

The system (24) with the use of equations of motiontransforms to the following

1205831205972119906119911

1205971199092+

120597

120597119910(1198661015840 120597119906119911

120597119910) + 12058821205962119906119911= 0 (25)

The system (23) corresponds to (4) with the followingnotations 120590

1199092= 120590119909119909 1205901199102

= 120590119910 and 120591

2= 120590119909119910 Under the

conditions 11989613

= 11989623

= 0 considered in Section 1 the elasticitytensor K linking stress and deformations to each other takesthe following form

K = (

120582 + 2120583 1205821015840 0

1205821015840 1205821015840 + 21205831015840 0

0 0 1198661015840

) (26)

Thus the problem of diffraction of an elastic harmonicwave by a transversely isotropic layer reduces to the boundaryvalue problem (9) (13) and (17) with the elasticity tensordefined in (26)

5 Numerical Results

Before discussions of the numerical results we will give somenotes regarding dependence of solution of the problem (9)(13) and (17) on parameter 120585 All the coefficients of theboundary value problem are continuous functions of 120585 Thenif right-hand sides of (13) are regular distributions on 120585 thensolutions will also be considered as regular with respect to120585 However if 119891

1and 119891

2are singular distributions on 120585 then

the solutions themselves will also be considered as singularFor example if 119891

1= 1198621120575(120585 minus 120585

0) and 119891

2= 1198622120575(120585 minus 120585

0) then

119906119909(119910) = 120575(120585 minus 120585

0)119908119909(119910 1205850) and 119906

119910(119910) = 120575(120585 minus 120585

0)119908119910(119910 1205850) In

this case it is convenient to ldquonormalizerdquo the boundary valueproblem by 120575(120585 minus 120585

0) For doing that we perform the change

of variables from variable 120585 to variable 1205850all over and solve

(9) (13) and (17) with respect to 119908119909(119910 1205850) and 119908

119910(119910 1205850)

Therefore in the case of Fourier transforms of traces ofthe incident field being singular distributions for examplein the case of the incident wave being a plane wave thesolutions of the problems will also be singular distributionswith the same carrier From this it follows that diffraction ofone plane wave results in two reflected waves longitudinaland transverse and excitation of waveguide waves in the layerdoes not occur It is obvious that the last statement is trueunder condition of uniqueness of the diffraction problem(homogeneous conditions (13) result in a trivial solution tothe problem (9) (13) and (17)) and under condition theeigenvalues of the waveguide formed by the layer which differfrom 120585

0

The desired problem can be solved using many approx-imation methods A uniform finite-difference grid with themesh size ℎ was chosen to approximate the boundary valueproblem (9) (13) and (17) with the accuracy on the order of119874(ℎ) When choosing the mesh size it is taken into accountthat the finite difference analogs of elastic profiles of thelayer describe adequately the original continuous modelsOn the other hand the mesh size ℎ must be smaller thanthe wavelength in the layer and consequently inverselyproportional to the frequency 120596

After carrying out the numerical solution it is necessaryto reconstruct the fields in the half-planes 119906

1and 119906

3 For

doing it we consider displacements in a homogeneousisotropic 119899th medium which can be written in the generalform in the following way [14]

119906119909119899(119910) = 120585119860

119899119890minus1198941205741119899119910minus 1205851198611198991198901198941205741119899119910

+ 1205742119899119862119899119890minus1198941205742119899119910+ 12057421198991198631198991198901198941205742119899119910

119906119910119899(119910) = 120574

1119899119860119899119890minus1198941205741119899119910+ 12057411198991198611198991198901198941205741119899119910

minus 120585119862119899119890minus1198941205742119899119910+ 1205851198631198991198901198941205742119899119910

(27)

6 Advances in Acoustics and Vibration

Taking into account the conditions at infinity displace-ments for the reflected field will have the following form

1199061199091(119910) = minus120585

0119861111989011989412057411(119910minus119871)

+ 12057421119863111989011989412057421(119910minus119871)

1199061199101(119910) = 120574

11119861111989011989412057411(119910minus119871)

+ 1205850119863111989011989412057421(119910minus119871)

(28)

and for the transmitted wave

1199061199093(119910) = 120585

01198603119890minus11989412057413119910+ 120574231198623119890minus11989412057423119910

1199061199103(119910) = 120574

131198603119890minus11989412057413119910minus 12058501198623119890minus11989412057423119910

(29)

The unknown coefficients 1198611 1198631 1198603 and 119862

3are found

via the following expressions

1198611=119906119910112057421minus 11990611990911205850

1205741112057421+ 12058520

1198631=119906119909112057411+ 11990611991011205850

1205741112057421+ 12058520

1198603=119906119910312057423+ 11990611990931205850

1205741312057423+ 12058520

1198623=119906119909312057413minus 11990611991031205850

1205741312057423+ 12058520

(30)

where 119906119909119899

and 119906119910119899

are traces of displacements of the 119899thmedium which are expressed using (7) and (8)

We will consider the case of diffraction by a planelongitudinal wave with displacements of the following kind

1199061199090(119909 119910)

= 119860011989611

sin 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

1199061199100(119909 119910)

= 119860011989611

cos 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

(31)

We apply Fourier transformation to components of theincident field and arrive at the result that all the componentsof the field are singular distributions with themultiplier 120575(120585minus1205850) 1205850= 11989611

sin 120579 For example the Fourier transform of thetrace 119906

1199090(119909 119910) at 119910 = 119871 takes the following form

1199061199090(120585 119871) = 119860

011989611

sin 120579120575 (120585 minus 11989611

sin 120579) (32)

Since the right-hand sides of (13) are singular distribu-tions then it is sufficient to solve the problem (9) (13) and(17) just at the value 120585 = 120585

0

For carrying out the numerical experiments we willconsider the case when the layer of thickness 119871 = 10mmadeof siltstone is located in sandstone Parameters of sandstonefilling in medium 1 and medium 3 are 120588 = 2400 kgm3 V

119901=

3300msec and V119904= 2000msec The layer is considered

anisotropic we will consider three cases of distributions ofelasticity parameters in the layer A uniform layer has thefollowing parameters [15] 119864 = 568GPa 1198641015840 = 621GPa] = 029 ]1015840 = 026 and 1198661015840 = 229GPa In the caseof ldquocompressed siltstonerdquo all the parameters of the mediumremain constant except for the Young modulus 119864(119910) which

00102030405060708091

0 1000 2000 3000 4000120596

Figure 2 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 20

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

in the plane of isotropy grows linearly from 119864(0) = 568GPato 119864(1198712) = 850GPa and then reduces linearly to the value119864(119871) = 568GPa In the case of ldquorelaxed siltstonerdquo the Youngmodulus 119864(119910) reduces linearly from 119864(0) = 568GPa to119864(1198712) = 284GPa and grows linearly to 119864(119871) = 568GPa

In the present work two sets of studies were carried outThe first set of studies is dedicated to searching for depen-dence of normalized energy of the transmitted longitudinalwave on angular frequency 120596 As a result the conclusion ismade so that the transmitted energy grows as the value ofthe Young modulus 119864(119910) reduces in the middle of the layerThe difference between various structures increases with theincrease of angle of incidence 120579 The dependences are givenin Figures 2 and 3

In Figure 2 dependence of normalized energy of thetransmitted longitudinal wave on angular frequency 120596 at itsincidence under the angle 120579 = 20

∘upon a siltstone layer ofthickness 119871 = 10m placed into sandstone is shownThe solidcurve corresponds to a uniform layer of siltstone the dottedcurve ldquocompressed siltstonerdquo and the dashed curve ldquorelaxedsiltstonerdquo In Figure 3 the same type of dependence is shownbut for the angle 120579 = 40

∘

The second set of studies is dedicated to searchingfor dependence of normalized energy of the transmittedlongitudinal wave on the angle of incidence 120579 Just as in theprevious set of studies confirmed is the conclusion that thetransmitted energy increases with the decrease of the Youngmodulus 119864(119910) in the middle of the layer It is worth notinghere that the difference in the transmitted energy for differentstructures decreases with the increase of angular frequency120596

In Figure 4 dependence of normalized energy of thetransmitted longitudinal wave on 120579 at the angular frequencyof the wave 120596 = 2 sdot 10

3 radian per second upon a siltstonelayer of thickness 119871 = 10m placed into sandstone is shownThe solid curve corresponds to a uniform layer of siltstonethe dotted curve ldquocompressed siltstonerdquo and the dashed curveldquorelaxed siltstonerdquo In Figure 5 the same type of dependenceis shown but for 120596 = 4 sdot 10

3 radian per second

Advances in Acoustics and Vibration 7

00102030405060708091

0 1000 2000 3000 4000120596

Figure 3 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 40

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 4 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 2000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness 119871 = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 5 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 4000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness L = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

6 Conclusions

The method of overdetermined boundary value problemused in the present work when combined with and theFourier transformmethod is shown to be efficient especiallyfor the cases the Fourier transforms of traces of the incidentfield are singular distributions Then the approximationproblem is solved just at the value 120585 = 120585

0 In the case of

the Fourier transforms being regular distributions (eg atdiffraction of a Gauss beam by a plate) the problem (9) (13)and (17) is solved for several values of the parameter 120585

Results obtained with respect to propagation of elasticwaves through anisotropic layers can be used in geophysicsfor the initial analysis of structure of the layers of rock strataAlso results of propagation of elastic waves through nonuni-form anisotropic structures can be used in industries inwhich anisotropicmaterials are applied as well as at designingprotective layers for various processes and apparatuses

Acknowledgment

Thiswork was supported by RFBR 12-01-97012-r-povolzhrsquoe-a

References

[1] S Ryan-Grigor ldquoEmpirical relationships between transverseisotropy parameters and VplVS implications for AVOrdquo Geo-physics vol 62 no 5 pp 1359ndash1364 1997

[2] T Alkhalifah ldquoVelocity analysis using nonhyperbolic moveoutin transversely isotropic mediardquo Geophysics vol 62 no 6 pp1839ndash1854 1997

[3] CM Sayers ldquoSimplified anisotropy parameters for transverselyisotropic sedimentary rocksrdquoGeophysics vol 60 pp 1933ndash19351995

[4] V J Nemirovskii and A P Jankovskii ldquoDetermining effectivephysical and mechanical characteristics of hybrid compositescrisscross reinforced by transversely isotropic fibers and com-parisons of computed characteristics versus experimental datardquoMekhanIka Kompozicionnykh MaterIalov i Konstrukcii vol 13no 1 pp 3ndash32 2007

[5] GAGeniev VN Kissjuk andGA TjupinTheory of Plasticityof Concrete and Ferroconcrete Strojizdat Moscow Russia 1974

[6] B V Labudin ldquoJustifying a computational model treating lam-inated wood as an orthogonal transversely isotropic materialrdquoIzvestiya VUZov Lesnoj Zhurnal no 6 pp 136ndash139 2006

[7] M P Lonkevich ldquoPropagation of sound through a layer of atransversely isotropic material of finite thicknessrdquo AkusticheskijZhurnal vol 17 no 1 pp 85ndash92 1971

[8] E L Shenderov ldquoPropagation of sound through a layer of atransversely isotropic platerdquo Akusticheskii Zhurnal vol 30 no1 pp 122ndash129 1984

[9] S A Skobelitsyn and L A Tolokonnikov ldquoPropagation ofsound through a transversely isotropic nonuniform flat layerrdquoAkusticheskii Zhurnal vol 36 no 4 pp 740ndash744 1990

[10] A V Anufrieva D N Tumakov and V L Kipot ldquoElastic wavepropagation through a layer with graded-index distribution ofdensityrdquo in Proceedings of the Days on Diffraction (DD rsquo12) pp21ndash26 2012

[11] I E Pleshchinskaya and N B Pleshchinskii ldquoOver-determinedboundary value problems for linear equations of elastodynam-ics and their applications to elastic wave diffraction theoryrdquo in

8 Advances in Acoustics and Vibration

Advances inMathematics Research A R Baswell Ed vol 17 pp102ndash138 Nova Science New York NY USA 2012

[12] S G Lekhnickii Theory of Elasticity of the Anisotropic BodyNauka Moscow Russia 1977

[13] B D Annin ldquoTransversely isotropic elastic model of geologicalmaterialsrdquo Sibirskiı Zhurnal Industrial noı Matematiki vol 12no 3 pp 5ndash14 2009

[14] K N Vdovina N B Pleshchinskii and D N TumakovldquoConcerning orthogonality of proper waves of a half-openedelastic waveguiderdquo Izvestiya VUZov Matematika no 9 pp 69ndash75 2008

[15] S A Batugin and R K Nirenburg ldquoApproximate rela-tion between the elastic constants of anisotropic rocks andthe anisotropy parametersrdquo Fiziko-Tekhnicheskie ProblemyRazrabotki Poleznykh Iskopaemykh vol 7 no 1 pp 7ndash12 1972

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Control Scienceand Engineering

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Advances in Acoustics and Vibration

Taking into account the conditions at infinity displace-ments for the reflected field will have the following form

1199061199091(119910) = minus120585

0119861111989011989412057411(119910minus119871)

+ 12057421119863111989011989412057421(119910minus119871)

1199061199101(119910) = 120574

11119861111989011989412057411(119910minus119871)

+ 1205850119863111989011989412057421(119910minus119871)

(28)

and for the transmitted wave

1199061199093(119910) = 120585

01198603119890minus11989412057413119910+ 120574231198623119890minus11989412057423119910

1199061199103(119910) = 120574

131198603119890minus11989412057413119910minus 12058501198623119890minus11989412057423119910

(29)

The unknown coefficients 1198611 1198631 1198603 and 119862

3are found

via the following expressions

1198611=119906119910112057421minus 11990611990911205850

1205741112057421+ 12058520

1198631=119906119909112057411+ 11990611991011205850

1205741112057421+ 12058520

1198603=119906119910312057423+ 11990611990931205850

1205741312057423+ 12058520

1198623=119906119909312057413minus 11990611991031205850

1205741312057423+ 12058520

(30)

where 119906119909119899

and 119906119910119899

are traces of displacements of the 119899thmedium which are expressed using (7) and (8)

We will consider the case of diffraction by a planelongitudinal wave with displacements of the following kind

1199061199090(119909 119910)

= 119860011989611

sin 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

1199061199100(119909 119910)

= 119860011989611

cos 120579

times exp minus11989411989611

sin 120579119909 minus 11989411989611

cos 120579 (119910 minus 119871)

(31)

We apply Fourier transformation to components of theincident field and arrive at the result that all the componentsof the field are singular distributions with themultiplier 120575(120585minus1205850) 1205850= 11989611

sin 120579 For example the Fourier transform of thetrace 119906

1199090(119909 119910) at 119910 = 119871 takes the following form

1199061199090(120585 119871) = 119860

011989611

sin 120579120575 (120585 minus 11989611

sin 120579) (32)

Since the right-hand sides of (13) are singular distribu-tions then it is sufficient to solve the problem (9) (13) and(17) just at the value 120585 = 120585

0

For carrying out the numerical experiments we willconsider the case when the layer of thickness 119871 = 10mmadeof siltstone is located in sandstone Parameters of sandstonefilling in medium 1 and medium 3 are 120588 = 2400 kgm3 V

119901=

3300msec and V119904= 2000msec The layer is considered

anisotropic we will consider three cases of distributions ofelasticity parameters in the layer A uniform layer has thefollowing parameters [15] 119864 = 568GPa 1198641015840 = 621GPa] = 029 ]1015840 = 026 and 1198661015840 = 229GPa In the caseof ldquocompressed siltstonerdquo all the parameters of the mediumremain constant except for the Young modulus 119864(119910) which

00102030405060708091

0 1000 2000 3000 4000120596

Figure 2 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 20

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

in the plane of isotropy grows linearly from 119864(0) = 568GPato 119864(1198712) = 850GPa and then reduces linearly to the value119864(119871) = 568GPa In the case of ldquorelaxed siltstonerdquo the Youngmodulus 119864(119910) reduces linearly from 119864(0) = 568GPa to119864(1198712) = 284GPa and grows linearly to 119864(119871) = 568GPa

In the present work two sets of studies were carried outThe first set of studies is dedicated to searching for depen-dence of normalized energy of the transmitted longitudinalwave on angular frequency 120596 As a result the conclusion ismade so that the transmitted energy grows as the value ofthe Young modulus 119864(119910) reduces in the middle of the layerThe difference between various structures increases with theincrease of angle of incidence 120579 The dependences are givenin Figures 2 and 3

In Figure 2 dependence of normalized energy of thetransmitted longitudinal wave on angular frequency 120596 at itsincidence under the angle 120579 = 20

∘upon a siltstone layer ofthickness 119871 = 10m placed into sandstone is shownThe solidcurve corresponds to a uniform layer of siltstone the dottedcurve ldquocompressed siltstonerdquo and the dashed curve ldquorelaxedsiltstonerdquo In Figure 3 the same type of dependence is shownbut for the angle 120579 = 40

∘

The second set of studies is dedicated to searchingfor dependence of normalized energy of the transmittedlongitudinal wave on the angle of incidence 120579 Just as in theprevious set of studies confirmed is the conclusion that thetransmitted energy increases with the decrease of the Youngmodulus 119864(119910) in the middle of the layer It is worth notinghere that the difference in the transmitted energy for differentstructures decreases with the increase of angular frequency120596

In Figure 4 dependence of normalized energy of thetransmitted longitudinal wave on 120579 at the angular frequencyof the wave 120596 = 2 sdot 10

3 radian per second upon a siltstonelayer of thickness 119871 = 10m placed into sandstone is shownThe solid curve corresponds to a uniform layer of siltstonethe dotted curve ldquocompressed siltstonerdquo and the dashed curveldquorelaxed siltstonerdquo In Figure 5 the same type of dependenceis shown but for 120596 = 4 sdot 10

3 radian per second

Advances in Acoustics and Vibration 7

00102030405060708091

0 1000 2000 3000 4000120596

Figure 3 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 40

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 4 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 2000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness 119871 = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 5 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 4000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness L = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

6 Conclusions

The method of overdetermined boundary value problemused in the present work when combined with and theFourier transformmethod is shown to be efficient especiallyfor the cases the Fourier transforms of traces of the incidentfield are singular distributions Then the approximationproblem is solved just at the value 120585 = 120585

0 In the case of

the Fourier transforms being regular distributions (eg atdiffraction of a Gauss beam by a plate) the problem (9) (13)and (17) is solved for several values of the parameter 120585

Results obtained with respect to propagation of elasticwaves through anisotropic layers can be used in geophysicsfor the initial analysis of structure of the layers of rock strataAlso results of propagation of elastic waves through nonuni-form anisotropic structures can be used in industries inwhich anisotropicmaterials are applied as well as at designingprotective layers for various processes and apparatuses

Acknowledgment

Thiswork was supported by RFBR 12-01-97012-r-povolzhrsquoe-a

References

[1] S Ryan-Grigor ldquoEmpirical relationships between transverseisotropy parameters and VplVS implications for AVOrdquo Geo-physics vol 62 no 5 pp 1359ndash1364 1997

[2] T Alkhalifah ldquoVelocity analysis using nonhyperbolic moveoutin transversely isotropic mediardquo Geophysics vol 62 no 6 pp1839ndash1854 1997

[3] CM Sayers ldquoSimplified anisotropy parameters for transverselyisotropic sedimentary rocksrdquoGeophysics vol 60 pp 1933ndash19351995

[4] V J Nemirovskii and A P Jankovskii ldquoDetermining effectivephysical and mechanical characteristics of hybrid compositescrisscross reinforced by transversely isotropic fibers and com-parisons of computed characteristics versus experimental datardquoMekhanIka Kompozicionnykh MaterIalov i Konstrukcii vol 13no 1 pp 3ndash32 2007

[5] GAGeniev VN Kissjuk andGA TjupinTheory of Plasticityof Concrete and Ferroconcrete Strojizdat Moscow Russia 1974

[6] B V Labudin ldquoJustifying a computational model treating lam-inated wood as an orthogonal transversely isotropic materialrdquoIzvestiya VUZov Lesnoj Zhurnal no 6 pp 136ndash139 2006

[7] M P Lonkevich ldquoPropagation of sound through a layer of atransversely isotropic material of finite thicknessrdquo AkusticheskijZhurnal vol 17 no 1 pp 85ndash92 1971

[8] E L Shenderov ldquoPropagation of sound through a layer of atransversely isotropic platerdquo Akusticheskii Zhurnal vol 30 no1 pp 122ndash129 1984

[9] S A Skobelitsyn and L A Tolokonnikov ldquoPropagation ofsound through a transversely isotropic nonuniform flat layerrdquoAkusticheskii Zhurnal vol 36 no 4 pp 740ndash744 1990

[10] A V Anufrieva D N Tumakov and V L Kipot ldquoElastic wavepropagation through a layer with graded-index distribution ofdensityrdquo in Proceedings of the Days on Diffraction (DD rsquo12) pp21ndash26 2012

[11] I E Pleshchinskaya and N B Pleshchinskii ldquoOver-determinedboundary value problems for linear equations of elastodynam-ics and their applications to elastic wave diffraction theoryrdquo in

8 Advances in Acoustics and Vibration

Advances inMathematics Research A R Baswell Ed vol 17 pp102ndash138 Nova Science New York NY USA 2012

[12] S G Lekhnickii Theory of Elasticity of the Anisotropic BodyNauka Moscow Russia 1977

[13] B D Annin ldquoTransversely isotropic elastic model of geologicalmaterialsrdquo Sibirskiı Zhurnal Industrial noı Matematiki vol 12no 3 pp 5ndash14 2009

[14] K N Vdovina N B Pleshchinskii and D N TumakovldquoConcerning orthogonality of proper waves of a half-openedelastic waveguiderdquo Izvestiya VUZov Matematika no 9 pp 69ndash75 2008

[15] S A Batugin and R K Nirenburg ldquoApproximate rela-tion between the elastic constants of anisotropic rocks andthe anisotropy parametersrdquo Fiziko-Tekhnicheskie ProblemyRazrabotki Poleznykh Iskopaemykh vol 7 no 1 pp 7ndash12 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Advances in Acoustics and Vibration 7

00102030405060708091

0 1000 2000 3000 4000120596

Figure 3 Dependence of normalized energy of the transmittedlongitudinal wave on angular frequency 120596 at its incidence under theangle 120579 = 40

∘upon a siltstone layer of thickness 119871 = 10m placedinto sandstone The solid curve corresponds to a uniform layer ofsiltstone the dotted curve ldquocompressed siltstonerdquo and the dashedcurve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 4 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 2000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness 119871 = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

00102030405060708091

0 10 20 30 40 50 60 70 80 90120579

Figure 5 Dependence of normalized energy of the transmittedlongitudinal wave having the angular frequency120596 = 4000 radsec onits angle of incidence 120579 upon a siltstone layer of thickness L = 10mplaced into sandstone The solid curve corresponds to a uniformlayer of siltstone the dotted curve ldquocompressed siltstonerdquo and thedashed curve ldquorelaxed siltstonerdquo

6 Conclusions

The method of overdetermined boundary value problemused in the present work when combined with and theFourier transformmethod is shown to be efficient especiallyfor the cases the Fourier transforms of traces of the incidentfield are singular distributions Then the approximationproblem is solved just at the value 120585 = 120585

0 In the case of

the Fourier transforms being regular distributions (eg atdiffraction of a Gauss beam by a plate) the problem (9) (13)and (17) is solved for several values of the parameter 120585

Results obtained with respect to propagation of elasticwaves through anisotropic layers can be used in geophysicsfor the initial analysis of structure of the layers of rock strataAlso results of propagation of elastic waves through nonuni-form anisotropic structures can be used in industries inwhich anisotropicmaterials are applied as well as at designingprotective layers for various processes and apparatuses

Acknowledgment

Thiswork was supported by RFBR 12-01-97012-r-povolzhrsquoe-a

References

[1] S Ryan-Grigor ldquoEmpirical relationships between transverseisotropy parameters and VplVS implications for AVOrdquo Geo-physics vol 62 no 5 pp 1359ndash1364 1997

[2] T Alkhalifah ldquoVelocity analysis using nonhyperbolic moveoutin transversely isotropic mediardquo Geophysics vol 62 no 6 pp1839ndash1854 1997

[3] CM Sayers ldquoSimplified anisotropy parameters for transverselyisotropic sedimentary rocksrdquoGeophysics vol 60 pp 1933ndash19351995

[4] V J Nemirovskii and A P Jankovskii ldquoDetermining effectivephysical and mechanical characteristics of hybrid compositescrisscross reinforced by transversely isotropic fibers and com-parisons of computed characteristics versus experimental datardquoMekhanIka Kompozicionnykh MaterIalov i Konstrukcii vol 13no 1 pp 3ndash32 2007

[5] GAGeniev VN Kissjuk andGA TjupinTheory of Plasticityof Concrete and Ferroconcrete Strojizdat Moscow Russia 1974

[6] B V Labudin ldquoJustifying a computational model treating lam-inated wood as an orthogonal transversely isotropic materialrdquoIzvestiya VUZov Lesnoj Zhurnal no 6 pp 136ndash139 2006

[7] M P Lonkevich ldquoPropagation of sound through a layer of atransversely isotropic material of finite thicknessrdquo AkusticheskijZhurnal vol 17 no 1 pp 85ndash92 1971

[8] E L Shenderov ldquoPropagation of sound through a layer of atransversely isotropic platerdquo Akusticheskii Zhurnal vol 30 no1 pp 122ndash129 1984

[9] S A Skobelitsyn and L A Tolokonnikov ldquoPropagation ofsound through a transversely isotropic nonuniform flat layerrdquoAkusticheskii Zhurnal vol 36 no 4 pp 740ndash744 1990

[10] A V Anufrieva D N Tumakov and V L Kipot ldquoElastic wavepropagation through a layer with graded-index distribution ofdensityrdquo in Proceedings of the Days on Diffraction (DD rsquo12) pp21ndash26 2012

[11] I E Pleshchinskaya and N B Pleshchinskii ldquoOver-determinedboundary value problems for linear equations of elastodynam-ics and their applications to elastic wave diffraction theoryrdquo in

8 Advances in Acoustics and Vibration

Advances inMathematics Research A R Baswell Ed vol 17 pp102ndash138 Nova Science New York NY USA 2012

[12] S G Lekhnickii Theory of Elasticity of the Anisotropic BodyNauka Moscow Russia 1977

[13] B D Annin ldquoTransversely isotropic elastic model of geologicalmaterialsrdquo Sibirskiı Zhurnal Industrial noı Matematiki vol 12no 3 pp 5ndash14 2009

[14] K N Vdovina N B Pleshchinskii and D N TumakovldquoConcerning orthogonality of proper waves of a half-openedelastic waveguiderdquo Izvestiya VUZov Matematika no 9 pp 69ndash75 2008

[15] S A Batugin and R K Nirenburg ldquoApproximate rela-tion between the elastic constants of anisotropic rocks andthe anisotropy parametersrdquo Fiziko-Tekhnicheskie ProblemyRazrabotki Poleznykh Iskopaemykh vol 7 no 1 pp 7ndash12 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Advances in Acoustics and Vibration

Advances inMathematics Research A R Baswell Ed vol 17 pp102ndash138 Nova Science New York NY USA 2012

[12] S G Lekhnickii Theory of Elasticity of the Anisotropic BodyNauka Moscow Russia 1977

[13] B D Annin ldquoTransversely isotropic elastic model of geologicalmaterialsrdquo Sibirskiı Zhurnal Industrial noı Matematiki vol 12no 3 pp 5ndash14 2009

[14] K N Vdovina N B Pleshchinskii and D N TumakovldquoConcerning orthogonality of proper waves of a half-openedelastic waveguiderdquo Izvestiya VUZov Matematika no 9 pp 69ndash75 2008

[15] S A Batugin and R K Nirenburg ldquoApproximate rela-tion between the elastic constants of anisotropic rocks andthe anisotropy parametersrdquo Fiziko-Tekhnicheskie ProblemyRazrabotki Poleznykh Iskopaemykh vol 7 no 1 pp 7ndash12 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of