the analytical form of the dispersion equation of...

Research ArticleThe Analytical Form of the Dispersion Equation ofElastic Waves in Periodically Inhomogeneous Medium ofDifferent Classes of Crystals

Nurlybek A Ispulov1 Abdul Qadir2 Marat K Zhukenov1

Talgat S Dossanov1 and Tanat G Kissikov3

1S Toraighyrov Pavlodar State University Pavlodar 140008 Kazakhstan2Sukkur Institute of Business Administration Sindh Pakistan3University of California Davis CA 95616 USA

Correspondence should be addressed to Abdul Qadir aqadiriba-sukedupk

Received 28 March 2016 Revised 28 June 2016 Accepted 16 November 2016 Published 29 January 2017

Academic Editor Andre Nicolet

Copyright copy 2017 Nurlybek A Ispulov et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The investigation of thermoelastic wave propagation in elastic media is bound to have much influence in the fields of materialscience geophysics seismology and so on The heat conduction equations and bound equations of motions differ by the difficultylevel and presence ofmany physical andmechanical parameters in themTherefore thermoelasticity is being extensively studied anddeveloped In this paper by using analytical matrizant method set of equation of motions in elastic media are reduced to equivalentset of first-order differential equations Moreover for given set of equations the structure of fundamental solutions for the generalcase has been derived and also dispersion relations are obtained

1 Introduction

The theory of thermoelasticity deals with the study of mutualinteractions of thermal andmechanical fields in elastic bodies[1 2] It has vast applications in the various branches ofPhysics as well as in engineering like materials engineeringmechanical engineering nuclear engineering and so forthTheory of thermoelasticity is based on assumption thattemperature distribution in an elastic object is governedby hyperbolic type parabolic-type partial differential equa-tion as described by Fourier law of heat conduction [3ndash5] According to Fourier law any thermal impulse is felteverywhere instantly in an object Obviously it raised someserious concerns due to its unrealistic point of view Inorder to circumvent this problem and to make it realistic ageneralized theory of thermoelasticity was proposed whichtakes into account a finite thermal relaxation time In this the-ory the temperature distribution is governed by hyperbolictype equations which results in heat propagation in solids

being considered as wave phenomenon instead of diffusionphenomenon

In order to investigate the wave propagation in aniso-tropic inhomogeneous medium a new method of matrizantwas developed This method allows investigation of wavepropagation in anisotropicmediumwith various physical andmechanical properties [6ndash8]

In 1950 Thompson [9] proposed a matrix method inorder to investigate the elastic wave propagation in isotropicstratified media Haskell also enhanced the method in 1953[10] After that major work was carried out by Stroh andothers [11 12] He analytically investigated the dislocationsin anisotropic medium by expressing first-order motionequations using (6times6)matrix In order to investigate the insu-lators made up of piezoelectric materials six-dimensionalframework was enhanced to eight-dimensional formalismMatrixmethod also paved the way for carrying out numericalsimulation in anisotropic media [13 14] Various researchershave investigated the ordered structures and layered medium

HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 5236898 8 pageshttpsdoiorg10115520175236898

2 Advances in Mathematical Physics

by using the matrix method In this connection followingpapers are of particular interest Wave propagation has beeninvestigated bymatrix algebramethod [15 16]WKBmethodand ray method [17ndash19] Some investigations have tried toemploy matricant in which infinite product of truncatedexponentials of the matrix of system coefficients and alsoPeano expansion are satisfied [7 8] However in case ofperiodic structure Peano expansion cannot be fully solvedTherefore development of analytical techniques will opennew dimensions to understand wave propagation in periodicstructure

In case of layered and periodic medium the dispersionequations have been obtained and also the matricant struc-ture was formed for nonhomogenous isotropic medium [3]In [4] the matricant was obtained employing Chebyshev-Gegenbauerrsquos polynomial form for the case of finite periodicinhomogeneous layer For such structures the modifiedconditions in determining the dispersion relationship havingmutual transformation of transverse and longitudinal wavesare obtained In [20 21] these results have been generalizedin case of anisotropic inhomogeneous media

The applications of matrizant method for nondestructivetesting and wave propagation in thermoelastic media areconsidered [22]

Periodically heterogeneous media have lot of importancefrom applied and theoretical perspective Wave propagationin discrete periodic structures has been extensively studiedWhile in case of continuous medium layered homogeneousisotropic periodic structures are well studied However theinvestigations of wave propagation in more complex period-ically heterogeneous medium are carried out using variousnumerical methods or approximate analytical method Oneof them is matrizant method it was initially developed inlate 70s in the Kazakh scientific school of Jakhan SuleimenulyErzhanov to investigate the dynamics of inhomogeneousmedium

The method aims at reducing original equations ofmotion by using method of separation of variables to anequivalent system of ordinary differential equations of firstorder with variable coefficients After that the resultingsystem of equations defines the structure of matrizant

2 Elastic Waves

The motion equations in elastic medium and generalizedHookersquos law describe wave propagation in elastic media asfollows [21] 120597120590119894119895120597119909119895 = 12058812059721199061198941205971199052120590119894119895 = 119888119894119895119896119897120576119896119897 (1)

where 120576119896119897 = (12)(120597119906119894120597119909119895 + 120597119906119895120597119909119894) represents the de-formation tensor components 119906119894 denotes the mechanicaldisplacement vector 120590119894119895 are the stress tensor 119888119894119895119896119897 representsthe elastic parameters of nonisotropic media and density ofmedium is represented by 120588

The medium is assumed to be stratified that is param-eters employed to describe the material depend on spacevariable along z-axis

Using the representation of the solution119891 (119909 119910 119911 119905) = 119891 (119911) exp (119894120596119905 minus 119894119896119909119909 minus 119894119896119910119910) (2)

where 120596 is radial frequency 119896119894 are a projection of wavenumber The multiplier exp(119894120596119905 minus 119894119896119909119909 minus 119894119896119910119910) is omitted inthe following equations for clarity

Taking into consideration propagation direction deriva-tive of anisotropic medium on z-axis and using (2) then (1)are reduced to a system of first-order ODEs having variablecoefficients 119889W119889119911 = BW

W = (119906119911 120590119911119911 119906119909 120590119909119911 119906119910 120590119910119911)119905 (3)

the transposition operator is denoted by 119905The coefficient matrix B for the case of triclinic

anisotropic medium takes the following form

B =((((((

0 11988712 11988713 11988714 11988715 1198871611988721 0 0 11988724 0 1198872611988724 11988714 11988733 11988734 11988735 119887360 11988713 11988743 11988733 11988745 1198874611988726 11988716 11988746 11988736 11988755 119887560 11988715 11988745 11988735 11988765 11988755))))))

(4)

For the orthorhombic anisotropy

B =((((((

0 11988712 11988713 0 11988715 011988721 0 0 11988724 0 1198872611988724 0 0 11988734 0 119887360 11988713 11988743 0 11988745 011988726 0 0 11988736 0 119887560 11988715 11988745 0 11988765 0))))))

(5)

If the vector representing the direction of wave propaga-tion is in (119909119911) plane of anisotropy orthorhombic mediumthe coefficient matrix given in (5) is divided into 4

If the wave propagation direction vector lies in the mediamatrix (5) splits into (4 times 4) and (2 times 2) matrices

B1015840 =( 0 11988712 11988713 011988721 0 0 1198872411988724 0 0 119887340 11988713 11988743 0 )W = (119906119911 120590119911119911 119906119909 120590119909119911)119905 B10158401015840 = ( 0 1198875611988765 0 ) W = (119906119910 120590119910119911)119905

(6)

Advances in Mathematical Physics 3

3 Matricant Structure and Its Implications

31 Matricant Method In order to describe the wave prop-agation in elastic medium various analytical approaches areused like the formalism proposed by StrohndashBarnettndashLothe[23] for piezoelasticity the state vector ldquo119882rdquo method [13]The matricant approach is different from others analyticaltechniques used to investigate wave propagation in elasticmedium In the matricant approach the vector 119882 of coeffi-cient matrix 119861 is chosen in (119906119894 120590119894119895) (120579 119902119894) pairs for instanceThe selection of pairs depends on the type of wave to beinvestigated Coupled waves in the general case and alongmain crystal axes are suitably described by the use of suchnotations

Solutions of (5) are written as

W (119911) = T (119911 1199110) W (1199110) (7)

Here T(119911 1199110) is the matricant that is the normalizedfundamental solution matrix of the systems of ODEs ForT(119911 1199110) and Tminus1(119911 1199110) there are representations in the formof the infinite matrix integral series of exponential type asfollows [9 10]

T (119911 1199110) = I + int1199111199110

B (1199111) d1199111+ int1199111199110

int11991111199110

B (1199111)B (1199112) d1199111d1199112 + sdot sdot sdot (8)

Tminus1 (119911 1199110) = I minus int1199110B1 (1199111) d1199111+ int119911

0int11991110

B2 (1199112)B1 (1199111) d1199111d119911 minus sdot sdot sdot (9)

the identity matrix is denoted by IThe expansion in (9) is the alternating-sign series with

reverse argument ordering of the integrated product of B(119911119894)Note that the matrix B(119911119894) does not commute As the initialsystem of equations are satisfied by the matricant so thesuccessive approximation methods can be used to obtain (8)

dTd119911 = BT (10)

It follows from substitution of (7) into (10)Similar to (10) inversematricantTminus1 is the solution of the

equation

dTminus1

d119911 = minusTminus1B (11)

It follows from differentiation that the identity

TTminus1 equiv Tminus1T equiv I (12)

The solutions of (2 times 2) order matrix are well known

B = ( 0 1198871211988721 0 ) (13)

the matricants have the structure

T = (11990511 1199051211990521 11990522) Tminus1 = ( 11990522 minus11990512minus11990521 11990511 ) (14)

For unimodular matrices the above result can be carriedout

In case of (4times4) coefficientmatrix in (6) its inversematrixis given by

Tminus1 = [[[[[[11990522 minus11990512 minus11990542 11990532minus11990521 11990511 11990541 minus11990531minus11990524 11990514 11990544 minus1199053411990523 minus11990513 minus11990543 11990533

]]]]]] (15)

The result given in (15) defines the properties of solutionsof systems of first-order ODEs with variable coefficientsThishas been obtained with term to term comparison of elementsof (8) and (9) and with the help of mathematical induction[5 24]

In the case of the coefficient matrix (5) the (6 times 6)matricant structure is obtained in the following form

Tminus1 =[[[[[[[[[[[[

11990522 minus11990512 minus11990542 11990532 minus11990562 11990552minus11990521 11990511 11990541 minus11990531 11990561 minus11990551minus11990524 11990514 11990544 minus11990534 11990564 minus1199055411990523 minus11990513 minus11990543 11990533 minus11990563 11990553minus11990526 11990516 11990546 minus11990536 11990566 minus1199055611990525 minus11990515 minus11990545 11990535 minus11990565 11990555]]]]]]]]]]]] (16)

It may be noted that elements of Tminus1 contain only ofelements from matricant T It is a one to one correspon-dence of elements of direct and its inverse matricant givenin (15) and (16) The conservation laws are contained byinvariant relationships these laws have to be satisfied inwave processes In 1D inhomogeneous isotropic mediumhaving various crystals symmetry is described by matricantstructureThematricant of coefficient matrix is of order (2119899times2119899) [5]32 Periodic Structures Suppose the variation in parametersthat describes medium is 119891119894(119911) The condition119891119894(119911 + ℎ) =119891119894(119911) is satisfied by periodic structure and also by parametersaltered by environment The period of inhomogeneity isdenoted by ℎ Monodromy matrix is matricant of singleperiod of inhomogeneity It is known for single periodinhomogeneity that W(ℎ) = T(0 ℎ)W0(0) and the FloquetndashBloch conditions W(119911 + ℎ) = W0(119911) exp(119894119896119911ℎ) From thisfollows the following equality(T minus I exp (119894119896119911ℎ)) W0 = 0 (17)


the equivalent condition are obtained bymultiplying (17)withTminus1119890119894119896ℎ (Tminus1 minus I exp (minus119894119896119911ℎ)) W0 = 0 (18)

Using auxiliary matrix and (17) and (18) the modifiedconditions as given below are derived as follows(p minus I cos 119896119911ℎ) W0 = 0

p = 12 (T + Tminus1) (19)

Matrix p for cases (6) is given by

p =( 11990111 0 11990113 119901140 11990111 11990123 11990124minus11990124 11990114 11990122 011990123 minus11990113 0 11990122) (20)

The dispersion in periodic structure is determined by charac-teristics of (19) 119904 det 1003816100381610038161003816p minus I cos 119896119911ℎ1003816100381610038161003816 = 0 (21)

The dispersion equation in case of (4 times 4) matrices is foundby

cos 1198961119911ℎ = 1cos 1198962119911ℎ = 2= 12 (11990111 + 11990122)plusmn 12radic(11990111 minus 11990122)2 + 4 (1199011411990123 minus 1199011311990124)

(22)

The equation of dispersion is obtained However theorder of characteristic equation is reduced by half when theconditions as laid down in (19) are imposed

Following recurrence relationship can be obtained from(19) as follows

T2 = 2pT minus I (23)

Matricant representing periodically inhomogeneous lay-ers can be obtained applying (24)

In the presence of 119899 periods of heterogeneity in form119867 =119899ℎ we can obtain

T (119867) = Tnm (ℎ) = Pn (p)Tm minus Pnminus1 (p) (24)

where Tm = T(ℎ) represents the monodromy matrix andPn(p)denotesmatrix polynomials of Chebyshev-Gegenbauer[5 25]The results of Brillouin andParodi [26] are generalizedby above equations

33 Structuring the Matrizant Structuring the matrizant(normalized matrix of fundamental solutions) is based on itsrepresentation in the form of the exponential matrix series[5 7] (8) and (9)

Thesematrix series converge absolutely and uniformly onany finite interval In this case the following relation is true(12)

For the matrizant the following expressions also hold

(i) 119879(1199110 119911) = 119879(1199111 119911)119879(1199110 1199111)(ii) ln |119879(1199110 119911)| = int1199111199110 119904119901119861(1199111)1198891199111(iii) If119861 = 1198610 ndash constantmatrix then1198790 = exp[1198610(119911minus1199110)](iv) 119889119879minus1119889119911 = minus119879minus1119861Matrix series (8) and (9) can be written in summation

form 119879 = infinsum119899=0

119879(119899)119879minus1 = infinsum

119899=0

119879minus1(119899) (25)

The index 119899 corresponds to the number of multipliedunder signs of integral matrices where 119861(119911119894) is the numberof integrals of the matrix in each term of the series as givenin (1) Moreover the terms of the series with even and oddvalues of 119899 are as follows119879even = infinsum

119899=0

119879(2119899)119879odd = infinsum

119899=0

119879(2119899+1)119879minus1even = infinsum

119899=0

119879minus1(2119899)119879minus1odd = infinsum

119899=0

119879minus1(2119899+1) (119899 = 0 1 2 3 ) (26)

In such case constructing the matrizant is basicallyexpressing the relationships between the elements of the 119879and119879minus1matrices it is based on the element-wise comparison

As a first approximation 119905(1)119894119895 = minus119905(1)119894119895 119905(minus1)119894119895 = int1199110119887119894119895(119911)119889119911

Elastic waves propagating in the orthorhombic syngonyof the classes mm2 and 222 hexagonal syngony (6 6 6226mm 6m2) tetragonal syngony (class 422) and matrizantstructure are built based on the structure of the coefficientmatrix based on the element-wise comparison of matrix 119879and 119879minus1 The structure of the coefficient matrix in the bulkcase is as follows [5]

119861 =[[[[[[[[[[[[

0 11988712 11988713 0 11988715 011988721 0 0 11988724 0 1198872611988724 0 0 11988734 0 00 11988713 11988743 0 11988745 011988726 0 0 0 0 119887560 11988715 11988745 0 11988765 0]]]]]]]]]]]] (27)


See (3)

W = (119906119911 120590119911119911 119906119909 120590119909119911 119906119910 120590119910119911)119905 (28)

The coefficients of the matrix 119887119894119895 (27) for the crystals oforthorhombic syngony are equal to

11988712 = 111988833 11988713 = 11988742 = 119894119896119909 1198881311988833 11988715 = 11988762 = 119894119896119910 1198883211988833 11988721 = minus120588120596211988724 = 11988731 = 11989411989611990911988726 = 11988751 = 11989411989611991011988734 = 111988855 11988743 = 119896211991011988866 minus 1205881205962 + 1198962119909 (11988811 minus 11988821311988833) 11988745 = 11988763 = (11988866 + 11988812 minus 119888131198883211988833 )11989611990911989611991011988756 = 111988844 11988765 = 119896211990911988866 minus 1205881205962 + (11988822 minus 11988822311988833)1198962119910

(29)

where 119888120572120573- are elastic parameters 119896119909 119896119910- are componentsof the wave vector 120588 is medium density and 120596 isangularfrequency

Coefficients 119887119894119895 for hexagonal syngony (classes 6 6 6226mm 6m2) have the form

11988712 = 111988833 11988713 = 119894119896119909 1198881311988833 11988715 = 119894119896119910 1198881311988833 11988721 = minus120588120596211988724 = 11989411989611990911988726 = 11989411989611991011988734 = 11988756 = 111988844

11988743 = 1198962119910 (11988811 minus 119888122 ) + 1198962119909 (11988811 minus 11988821311988833) minus 120588120596211988745 = (11988812 + 11988811 minus 119888122 minus 11988821311988833)11989611990911989611991011988765 = 1198962119909 11988811 minus 119888122 + (11988811 minus 11988821311988833)1198962119910 minus 1205881205962(30)

Coefficients 119887119894119895 for tetragonal syngony (class 422) have theform

11988712 = 111988833 11988713 = 119894119896119909 1198881311988833 11988715 = 119894119896119910 1198881311988833 11988721 = minus120588120596211988724 = 11989411989611990911988726 = 11989411989611991011988734 = 11988756 = 111988844 11988743 = 119896211991011988866 + 1198962119909 (11988811 minus 11988821311988833) minus 120588120596211988745 = (11988812 + 11988866 minus 11988821311988833)11989611990911989611991011988765 = 119896211990911988866 + (minus11988821311988833)1198962119910 minus 1205881205962

(31)

As it can be seen from the last relations coefficients 119887119894119895of crystals of high and average symmetry differ only in thevalues of the elastic constants 119888120572120573

Matrix of order 6 describes the propagation of boundelastic one longitudinal and two transverse waves

The second approximation matrizant has the form

119879(2) = int1199110int11991110119861 (1199111) 119861 (1199112) 11988911991111198891199112 (32)

Inverse matrizant in the second approximation takes theform

119879minus1(2) = int1199110int11991110119861 (1199112) 119861 (1199111) 11988911991111198891199112 (33)


Comparison of the terms of the second approximationgives the following relationship between the elements ofmatrizant 119879 and 119879minus1

119879minus1(2) =[[[[[[[[[[[[

11990522 0 0 11990532 0 119905520 11990511 11990541 0 11990561 00 11990514 11990544 0 11990564 011990523 0 0 11990533 0 119905530 11990516 11990546 0 11990566 011990525 0 0 11990535 0 11990555]]]]]]]]]]]](2)

(34)

The elements 119905119894119895 are the elements of the direct matrizant(27)

Similarly elements of the third approximation are com-pared 119879(3) =∭119861(1199111) 119861 (1199112) 119861 (1199113) 119889119911111988911991121198891199113

or 119879(3) = 119879(2)119861 (1199113)119879(3) =

[[[[[[[[[[[[

0 11990512 11990513 0 11990515 011990521 0 0 11990524 0 1199052611990531 0 0 11990534 0 119905360 11990542 11990543 0 11990545 011990551 0 0 11990554 0 119905560 11990562 11990563 0 11990565 0]]]]]]]]]]]](3)

(35)

Inversematrizant in the third approximation has the form119879minus1(3) =∭119861(1199113) 119861 (1199112) 119861 (1199111) 119889119911111988911991121198891199113or 119879minus1(3) = 119861 (1199113) 119879minus1(2) (36)

and has the structure

119879minus1(3) =[[[[[[[[[[[[

0 11990512 11990542 0 11990562 011990521 0 0 11990515 0 1199055111990524 0 0 11990534 0 119905540 11990513 11990543 0 11990563 011990526 0 0 11990536 0 119905560 11990515 11990545 0 11990565 0]]]]]]]]]]]](3)

(37)

The elements 119905119894119895 are the elements of the matrizant (35)Endless rows of the matrix can be written as follows [5]119879 = 119879even + 119879odd119879minus1 = 119879minus1even minus 119879minus1odd (38)

where 119879plusmn1 ndash corresponds to the sum of odd and even rows (910)

Mathematical induction proves that the structure of the119879minus1(119899) is preserved for any 119899

Structure (34) is valid for all even 119879minus1 and the structureof (37) is valid for all odd values of 119879minus1 Generalizing (34)and (37) according to (38) we obtain the structure of the 119879minus1matrizant as follows [5]

119879minus1 =[[[[[[[[[[[[


Elements 119905119894119895 of matrices 119879minus1even and 119879minus1odd are elements ofmatrices 119879even and 119879odd are the elements of direct matrix 119879respectively

Thus the structure of matrizant is a relationship betweenelements of the forward and inverse matrizant in the form(39) as well as the relationship between the elements 119879 and119879minus1 as follows from (12)

Analytical representation of matrizant of periodicallyinhomogeneous layer is derived from knowing the structure119879minus14 Dispersion Equations for the Elastic

Anisotropic Mediums

We introduce the following matrix [5] = 12 (119879 + 119879minus1) (40)

substituting values of 119879 and 119879minus1 in (40) we obtain

2 =((((((

1199011 0 11990113 11990114 11990115 119901160 1199011 11990123 11990124 11990125 11990126minus11990124 11990114 1199013 0 11990135 1199013611990123 minus11990113 0 1199013 11990145 11990146minus11990126 11990116 11990146 minus11990136 1199015 011990125 minus11990115 minus11990145 11990135 0 1199015))))))

(41)

The matrix 119901 as given in (40) which is important for theregular structures gives the recurrence relation as in (23)

Consistent application of (23) gives possibility of repre-senting 119879119899 in the form119879119899 = 119875119899 (119901) 119879 minus 119875119899minus1 (119901) (42)

where 119875119899(119901) is Chebyshev-Gegenbaur matrix polynomials ofthe second kind

Equation (42) allows obtaining 119879119899 in an explicit analyticform 119879119899 = 3sum

119894=1

119875119894 [119875119899 (119894) 119879 minus 119875119899 (119894) I] (43)


where 119875119894 = (1(119894 minus 119895)(119894 minus 119896))[ minus 119895I][ minus 119896I] 119894119895 119896-1 2 3 119894 = 119895 119895 = 119896 119894 = 1198961 21 31are the roots of the characteristic equationsatisfying the following condition (23)

Thematrixmethod ofmatrizant allows twice lowering thedegree of the characteristic equation which in the end has theform [1205823 + 1198861205822 + 119887120582 + 119888]2 = 01205823 + 1198861205822 + 119887120582 + 119888 = 0 (44)

where120582 = cos ℎ119894119886 = minus (1199011 + 1199012 + 1199015)119887 = 11990111199013 + 11990111199015 + 11990131199015 minus 1199011411990123 + 1199011311990124 minus 1199011611990125+ 1199011511990126 + 1199013511990146119888 = minus1199011 (11990131199015 + 1199013611990145 minus 1199013511990146)+ 1199013 (1199011611990125 minus 1199011511990126) + 1199015 (1199011411990123 minus 1199011311990124) minusminus 11990116 (1199012311990135 + 1199012411990145) + 11990113 (1199012611990135 minus 1199012511990136)+ 11990115 (1199012311990136 + 1199012411990146) + 11990114 (1199012611990145 minus 1199012511990146) (45)

The solution of the characteristic equation (44) givesthree roots which have the following form

cos ℎ1 = 16 (158 3radic120575 minus 2119886 + 252 (1198862 minus 3119887)3radic120575 )cos ℎ2 = 112 (minus317 3radic120575 minus 4119886 minus 4 3radicminus2 (1198862 minus 3119887)3radic120575 )cos ℎ3 = 112 (minus317 3radicminus1 3radic120575 minus 4119886 + 5 (1198862 minus 3119887)3radic120575 )

(46)

where 120575 = minus21198863 + 9119886119887 minus 27119888 +radic(21198863 minus 9119886119887 + 27119888)2 minus 4(1198862 minus 3119887)3Relations (46) determine the dispersion equations of

elastic waves in above-mentioned crystals The difference liesin the roots of the difference between the coefficients 119887119894119895 in thematrix (27)

5 Conclusion

In this paper we have developed the structure of matrizantand from it obtained invariant relations which reflects theinner symmetry of inner equations and contains conserva-tion laws Also we have derived an analytical representationof matrizant of periodically inhomogeneous layer usingChebyshev-Gegenbauer polynomials and obtained a separatedispersion equation Finally by using different crystals sys-tems the analytical solution of equations of motion for a wideclass of homogenous anisotropic medium has been obtained

Competing Interests

The authors declare that they have no competing interests

References

[1] W Nowacki Thermoelasticity Pergamon Press Oxford UK2nd edition 1986

[2] J P Nowacki V I Alshits and A Radowicz ldquo2D Electro-elasticfields in a piezoelectric layer-substrate structurerdquo InternationalJournal of Engineering Science vol 40 no 18 pp 2057ndash20762002

[3] S K Tleukenov ldquoCharacteristic matrix of a periodically inho-mogeneous layerrdquo Steklov Math Institute Leningrad vol 165pp 177ndash181 1987 (Translated to English by Springer Journal ofSoviet Mathematic vol 50 no 6 pp 2058ndash2062 1990)

[4] S K Tleukenov ldquoOn bending vibrations of a periodicallyinhomogeneous orthotropic plate vol 179 pp 179ndash181 SteklovMath Institute Leningrad (1989)rdquo (Translated to English bySpringer Journal of Mathematical Sciences 57 3181ndash3182 (1991)

[5] S K Tleukenov Matricant Method Psu Press Pavlodar Kaza-khstan 2004 (Russian)

[6] S K Tleukenov ldquoDisposition of roots of the dispersion equa-tion of a waveguide periodically inhomogeneous according todepthrdquo Journal ofMathematical Sciences vol 55 no 3 pp 1766ndash1770 1991

[7] F R Gantmacher The Theory of Matrices chapter 14 ChelseaNW USA 1964

[8] M C Pease Methods of Matrix Algebra chapter 7 AcademicPress New York NY USA 1965

[9] W T Thompson ldquoTransmission of elastic waves through astratified solid mediardquo Journal of Applied Physics vol 21 no 2pp 89ndash93 1950

[10] N A Haskell ldquoThe dispersion of surface waves on multilayeredmediardquo Bulletin of the Seismological Society of America vol 43pp 377ndash393 1953

[11] A N Stroh ldquoSteady state problems in anisotropic elasticityrdquoJournal of Mathematics and Physics vol 41 pp 77ndash103 1962

[12] G J Fryer and L N Frazer ldquoSeismic waves in stratified ani-sotropic mediardquo Geophysical Journal of the Royal AstronomicalSociety vol 78 no 3 pp 691ndash710 1984

[13] A H Fahmy and E L Adler ldquoPropagation of acoustic surfacewaves in multilayers a matrix descriptionrdquo Applied PhysicsLetters vol 22 no 10 pp 495ndash497 1973

[14] A H Nayfeh ldquoThe general problem of elastic wave propagationin multilayered anisotropic mediardquo Journal of the AcousticalSociety of America vol 89 no 4 pp 1521ndash1531 1991

[15] A L Shuvalov O Poncelet and M Deschamps ldquoGeneral for-malism for plane guided waves in transversely inhomogeneousanisotropic platesrdquo Wave Motion vol 40 no 4 pp 413ndash4262004

[16] B L N Kennet Seismic Wave Propagation in Stratified MediaCambridge University Press Cambridge UK 1983

[17] A Morro and G Caviglia ldquoWave propagation in a stratifiedlayer via the matricantrdquo Communications in Applied and Indus-trial Mathematics vol 2 no 1 2011

[18] V M Babich ldquoRay method of computation of intensity ofwave fronts in elastic inhomogeneous anisotropic mediardquo inProblems of Dynamic Theory of Propagation of Seismic WavesG I Petrashen Ed vol 5 pp 36ndash46 Leningrad University


Press 1961 (Translation to English Geophysical Journal Inter-national vol 118 pp 379ndash383 1994)

[19] V M Babich and V S Buldyrev Asymptotic Methods in Prob-lems of Diffraction of Shot Waves Translated to English bySpringer 1991 under the title Short-wavelength diffractiontheory Nauka Moscow Russia 1972 (Russian)

[20] S Tleukenov and K A Orynbasarov ldquoAbout matrices of funda-mental solutions of the equations of the dynamics of inhomoge-neous anisotropic mediardquo in Proceedings of KazSSR Physics andMathematics Series vol 5 pp 87ndash91 1991 (Russian)

[21] S Tleukenov ldquoThe structure of propagator matrix and itrsquosapplication in the case of the periodical inhomogeneousmediardquoin Proceedings of the Abstract for Seminar on EarthquakeProcesses and Their Consequences Seismological InvestigationsKurukshetra India 1989

[22] N A Ispulov A Qadir and M A Shah ldquoReflection of ther-moelastic wave on the interface of isotropic half-space andtetragonal syngony anisotropic medium of classes 4 4m withthermomechanical effectrdquo Chinese Physics B vol 25 no 3Article ID 038102 2016

[23] K Tanuma ldquoStroh formalism and Rayleigh wavesrdquo Journal ofElasticity vol 89 no 1 pp 5ndash154 2007

[24] O Baikonysov and S Tleukenov ldquoA method for solving someproblems of elastic wave propagation in the presence of periodicinhomogeneitiesrdquo inMathematical Questions ofTheory of WavePropagation 15 vol 148 of Notes of LOMI Scientific Seminars ofthe USSR pp 30ndash33 1985 (Russian)

[25] S K Tleukenov ldquoWave processes and method matricantrdquo Sci-entific Journal of the LGumilyov Eurasian National Universityno 4 pp 68ndash74 2011 (Russian)

[26] L Brillouin andM ParodiWave Propagation in Periodic Struc-tures Dover New York NY USA 1953

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


by using the matrix method In this connection followingpapers are of particular interest Wave propagation has beeninvestigated bymatrix algebramethod [15 16]WKBmethodand ray method [17ndash19] Some investigations have tried toemploy matricant in which infinite product of truncatedexponentials of the matrix of system coefficients and alsoPeano expansion are satisfied [7 8] However in case ofperiodic structure Peano expansion cannot be fully solvedTherefore development of analytical techniques will opennew dimensions to understand wave propagation in periodicstructure

In case of layered and periodic medium the dispersionequations have been obtained and also the matricant struc-ture was formed for nonhomogenous isotropic medium [3]In [4] the matricant was obtained employing Chebyshev-Gegenbauerrsquos polynomial form for the case of finite periodicinhomogeneous layer For such structures the modifiedconditions in determining the dispersion relationship havingmutual transformation of transverse and longitudinal wavesare obtained In [20 21] these results have been generalizedin case of anisotropic inhomogeneous media

The applications of matrizant method for nondestructivetesting and wave propagation in thermoelastic media areconsidered [22]

Periodically heterogeneous media have lot of importancefrom applied and theoretical perspective Wave propagationin discrete periodic structures has been extensively studiedWhile in case of continuous medium layered homogeneousisotropic periodic structures are well studied However theinvestigations of wave propagation in more complex period-ically heterogeneous medium are carried out using variousnumerical methods or approximate analytical method Oneof them is matrizant method it was initially developed inlate 70s in the Kazakh scientific school of Jakhan SuleimenulyErzhanov to investigate the dynamics of inhomogeneousmedium

The method aims at reducing original equations ofmotion by using method of separation of variables to anequivalent system of ordinary differential equations of firstorder with variable coefficients After that the resultingsystem of equations defines the structure of matrizant

2 Elastic Waves

The motion equations in elastic medium and generalizedHookersquos law describe wave propagation in elastic media asfollows [21] 120597120590119894119895120597119909119895 = 12058812059721199061198941205971199052120590119894119895 = 119888119894119895119896119897120576119896119897 (1)

where 120576119896119897 = (12)(120597119906119894120597119909119895 + 120597119906119895120597119909119894) represents the de-formation tensor components 119906119894 denotes the mechanicaldisplacement vector 120590119894119895 are the stress tensor 119888119894119895119896119897 representsthe elastic parameters of nonisotropic media and density ofmedium is represented by 120588

The medium is assumed to be stratified that is param-eters employed to describe the material depend on spacevariable along z-axis

Using the representation of the solution119891 (119909 119910 119911 119905) = 119891 (119911) exp (119894120596119905 minus 119894119896119909119909 minus 119894119896119910119910) (2)

where 120596 is radial frequency 119896119894 are a projection of wavenumber The multiplier exp(119894120596119905 minus 119894119896119909119909 minus 119894119896119910119910) is omitted inthe following equations for clarity

Taking into consideration propagation direction deriva-tive of anisotropic medium on z-axis and using (2) then (1)are reduced to a system of first-order ODEs having variablecoefficients 119889W119889119911 = BW

W = (119906119911 120590119911119911 119906119909 120590119909119911 119906119910 120590119910119911)119905 (3)

the transposition operator is denoted by 119905The coefficient matrix B for the case of triclinic

anisotropic medium takes the following form

B =((((((

0 11988712 11988713 11988714 11988715 1198871611988721 0 0 11988724 0 1198872611988724 11988714 11988733 11988734 11988735 119887360 11988713 11988743 11988733 11988745 1198874611988726 11988716 11988746 11988736 11988755 119887560 11988715 11988745 11988735 11988765 11988755))))))

(4)

For the orthorhombic anisotropy

B =((((((

0 11988712 11988713 0 11988715 011988721 0 0 11988724 0 1198872611988724 0 0 11988734 0 119887360 11988713 11988743 0 11988745 011988726 0 0 11988736 0 119887560 11988715 11988745 0 11988765 0))))))

(5)

If the vector representing the direction of wave propaga-tion is in (119909119911) plane of anisotropy orthorhombic mediumthe coefficient matrix given in (5) is divided into 4

If the wave propagation direction vector lies in the mediamatrix (5) splits into (4 times 4) and (2 times 2) matrices

B1015840 =( 0 11988712 11988713 011988721 0 0 1198872411988724 0 0 119887340 11988713 11988743 0 )W = (119906119911 120590119911119911 119906119909 120590119909119911)119905 B10158401015840 = ( 0 1198875611988765 0 ) W = (119906119910 120590119910119911)119905

(6)





W (119911) = T (119911 1199110) W (1199110) (7)


T (119911 1199110) = I + int1199111199110

B (1199111) d1199111+ int1199111199110

int11991111199110



0int11991110




dTd119911 = BT (10)


equation

dTminus1





B = ( 0 1198871211988721 0 ) (13)






]]]]]] (15)



Tminus1 =[[[[[[[[[[[[






p = 12 (T + Tminus1) (19)


p =( 11990111 0 11990113 119901140 11990111 11990123 11990124minus11990124 11990114 11990122 011990123 minus11990113 0 11990122) (20)




(22)













119879(119899)119879minus1 = infinsum

119899=0

119879minus1(119899) (25)


119899=0

119879(2119899)119879odd = infinsum

119899=0


119899=0


119899=0

119879minus1(2119899+1) (119899 = 0 1 2 3 ) (26)




119861 =[[[[[[[[[[[[

0 11988712 11988713 0 11988715 011988721 0 0 11988724 0 1198872611988724 0 0 11988734 0 00 11988713 11988743 0 11988745 011988726 0 0 0 0 119887560 11988715 11988745 0 11988765 0]]]]]]]]]]]] (27)


See (3)

W = (119906119911 120590119911119911 119906119909 120590119909119911 119906119910 120590119910119911)119905 (28)



(29)



11988712 = 111988833 11988713 = 119894119896119909 1198881311988833 11988715 = 119894119896119910 1198881311988833 11988721 = minus120588120596211988724 = 11989411989611990911988726 = 11989411989611991011988734 = 11988756 = 111988844




(31)




119879(2) = int1199110int11991110119861 (1199111) 119861 (1199112) 11988911991111198891199112 (32)


119879minus1(2) = int1199110int11991110119861 (1199112) 119861 (1199111) 11988911991111198891199112 (33)



119879minus1(2) =[[[[[[[[[[[[

11990522 0 0 11990532 0 119905520 11990511 11990541 0 11990561 00 11990514 11990544 0 11990564 011990523 0 0 11990533 0 119905530 11990516 11990546 0 11990566 011990525 0 0 11990535 0 11990555]]]]]]]]]]]](2)

(34)



or 119879(3) = 119879(2)119861 (1199113)119879(3) =

[[[[[[[[[[[[

0 11990512 11990513 0 11990515 011990521 0 0 11990524 0 1199052611990531 0 0 11990534 0 119905360 11990542 11990543 0 11990545 011990551 0 0 11990554 0 119905560 11990562 11990563 0 11990565 0]]]]]]]]]]]](3)

(35)



119879minus1(3) =[[[[[[[[[[[[

0 11990512 11990542 0 11990562 011990521 0 0 11990515 0 1199055111990524 0 0 11990534 0 119905540 11990513 11990543 0 11990563 011990526 0 0 11990536 0 119905560 11990515 11990545 0 11990565 0]]]]]]]]]]]](3)

(37)





119879minus1 =[[[[[[[[[[[[





Anisotropic Mediums



2 =((((((


(41)





119894=1

119875119894 [119875119899 (119894) 119879 minus 119875119899 (119894) I] (43)







(46)



5 Conclusion


Competing Interests


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













W (119911) = T (119911 1199110) W (1199110) (7)


T (119911 1199110) = I + int1199111199110

B (1199111) d1199111+ int1199111199110

int11991111199110



0int11991110




dTd119911 = BT (10)


equation

dTminus1





B = ( 0 1198871211988721 0 ) (13)






]]]]]] (15)



Tminus1 =[[[[[[[[[[[[






p = 12 (T + Tminus1) (19)


p =( 11990111 0 11990113 119901140 11990111 11990123 11990124minus11990124 11990114 11990122 011990123 minus11990113 0 11990122) (20)




(22)













119879(119899)119879minus1 = infinsum

119899=0

119879minus1(119899) (25)


119899=0

119879(2119899)119879odd = infinsum

119899=0


119899=0


119899=0

119879minus1(2119899+1) (119899 = 0 1 2 3 ) (26)




119861 =[[[[[[[[[[[[

0 11988712 11988713 0 11988715 011988721 0 0 11988724 0 1198872611988724 0 0 11988734 0 00 11988713 11988743 0 11988745 011988726 0 0 0 0 119887560 11988715 11988745 0 11988765 0]]]]]]]]]]]] (27)


See (3)

W = (119906119911 120590119911119911 119906119909 120590119909119911 119906119910 120590119910119911)119905 (28)



(29)



11988712 = 111988833 11988713 = 119894119896119909 1198881311988833 11988715 = 119894119896119910 1198881311988833 11988721 = minus120588120596211988724 = 11989411989611990911988726 = 11989411989611991011988734 = 11988756 = 111988844




(31)




119879(2) = int1199110int11991110119861 (1199111) 119861 (1199112) 11988911991111198891199112 (32)


119879minus1(2) = int1199110int11991110119861 (1199112) 119861 (1199111) 11988911991111198891199112 (33)



119879minus1(2) =[[[[[[[[[[[[

11990522 0 0 11990532 0 119905520 11990511 11990541 0 11990561 00 11990514 11990544 0 11990564 011990523 0 0 11990533 0 119905530 11990516 11990546 0 11990566 011990525 0 0 11990535 0 11990555]]]]]]]]]]]](2)

(34)



or 119879(3) = 119879(2)119861 (1199113)119879(3) =

[[[[[[[[[[[[

0 11990512 11990513 0 11990515 011990521 0 0 11990524 0 1199052611990531 0 0 11990534 0 119905360 11990542 11990543 0 11990545 011990551 0 0 11990554 0 119905560 11990562 11990563 0 11990565 0]]]]]]]]]]]](3)

(35)



119879minus1(3) =[[[[[[[[[[[[

0 11990512 11990542 0 11990562 011990521 0 0 11990515 0 1199055111990524 0 0 11990534 0 119905540 11990513 11990543 0 11990563 011990526 0 0 11990536 0 119905560 11990515 11990545 0 11990565 0]]]]]]]]]]]](3)

(37)





119879minus1 =[[[[[[[[[[[[





Anisotropic Mediums



2 =((((((


(41)





119894=1

119875119894 [119875119899 (119894) 119879 minus 119875119899 (119894) I] (43)







(46)



5 Conclusion


Competing Interests


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












p = 12 (T + Tminus1) (19)


p =( 11990111 0 11990113 119901140 11990111 11990123 11990124minus11990124 11990114 11990122 011990123 minus11990113 0 11990122) (20)




(22)













119879(119899)119879minus1 = infinsum

119899=0

119879minus1(119899) (25)


119899=0

119879(2119899)119879odd = infinsum

119899=0


119899=0


119899=0

119879minus1(2119899+1) (119899 = 0 1 2 3 ) (26)




119861 =[[[[[[[[[[[[

0 11988712 11988713 0 11988715 011988721 0 0 11988724 0 1198872611988724 0 0 11988734 0 00 11988713 11988743 0 11988745 011988726 0 0 0 0 119887560 11988715 11988745 0 11988765 0]]]]]]]]]]]] (27)


See (3)

W = (119906119911 120590119911119911 119906119909 120590119909119911 119906119910 120590119910119911)119905 (28)



(29)



11988712 = 111988833 11988713 = 119894119896119909 1198881311988833 11988715 = 119894119896119910 1198881311988833 11988721 = minus120588120596211988724 = 11989411989611990911988726 = 11989411989611991011988734 = 11988756 = 111988844




(31)




119879(2) = int1199110int11991110119861 (1199111) 119861 (1199112) 11988911991111198891199112 (32)


119879minus1(2) = int1199110int11991110119861 (1199112) 119861 (1199111) 11988911991111198891199112 (33)



119879minus1(2) =[[[[[[[[[[[[

11990522 0 0 11990532 0 119905520 11990511 11990541 0 11990561 00 11990514 11990544 0 11990564 011990523 0 0 11990533 0 119905530 11990516 11990546 0 11990566 011990525 0 0 11990535 0 11990555]]]]]]]]]]]](2)

(34)



or 119879(3) = 119879(2)119861 (1199113)119879(3) =

[[[[[[[[[[[[

0 11990512 11990513 0 11990515 011990521 0 0 11990524 0 1199052611990531 0 0 11990534 0 119905360 11990542 11990543 0 11990545 011990551 0 0 11990554 0 119905560 11990562 11990563 0 11990565 0]]]]]]]]]]]](3)

(35)



119879minus1(3) =[[[[[[[[[[[[

0 11990512 11990542 0 11990562 011990521 0 0 11990515 0 1199055111990524 0 0 11990534 0 119905540 11990513 11990543 0 11990563 011990526 0 0 11990536 0 119905560 11990515 11990545 0 11990565 0]]]]]]]]]]]](3)

(37)





119879minus1 =[[[[[[[[[[[[





Anisotropic Mediums



2 =((((((


(41)





119894=1

119875119894 [119875119899 (119894) 119879 minus 119875119899 (119894) I] (43)







(46)



5 Conclusion


Competing Interests


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










See (3)

W = (119906119911 120590119911119911 119906119909 120590119909119911 119906119910 120590119910119911)119905 (28)



(29)



11988712 = 111988833 11988713 = 119894119896119909 1198881311988833 11988715 = 119894119896119910 1198881311988833 11988721 = minus120588120596211988724 = 11989411989611990911988726 = 11989411989611991011988734 = 11988756 = 111988844




(31)




119879(2) = int1199110int11991110119861 (1199111) 119861 (1199112) 11988911991111198891199112 (32)


119879minus1(2) = int1199110int11991110119861 (1199112) 119861 (1199111) 11988911991111198891199112 (33)



119879minus1(2) =[[[[[[[[[[[[

11990522 0 0 11990532 0 119905520 11990511 11990541 0 11990561 00 11990514 11990544 0 11990564 011990523 0 0 11990533 0 119905530 11990516 11990546 0 11990566 011990525 0 0 11990535 0 11990555]]]]]]]]]]]](2)

(34)



or 119879(3) = 119879(2)119861 (1199113)119879(3) =

[[[[[[[[[[[[

0 11990512 11990513 0 11990515 011990521 0 0 11990524 0 1199052611990531 0 0 11990534 0 119905360 11990542 11990543 0 11990545 011990551 0 0 11990554 0 119905560 11990562 11990563 0 11990565 0]]]]]]]]]]]](3)

(35)



119879minus1(3) =[[[[[[[[[[[[

0 11990512 11990542 0 11990562 011990521 0 0 11990515 0 1199055111990524 0 0 11990534 0 119905540 11990513 11990543 0 11990563 011990526 0 0 11990536 0 119905560 11990515 11990545 0 11990565 0]]]]]]]]]]]](3)

(37)





119879minus1 =[[[[[[[[[[[[





Anisotropic Mediums



2 =((((((


(41)





119894=1

119875119894 [119875119899 (119894) 119879 minus 119875119899 (119894) I] (43)







(46)



5 Conclusion


Competing Interests


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879minus1(2) =[[[[[[[[[[[[

11990522 0 0 11990532 0 119905520 11990511 11990541 0 11990561 00 11990514 11990544 0 11990564 011990523 0 0 11990533 0 119905530 11990516 11990546 0 11990566 011990525 0 0 11990535 0 11990555]]]]]]]]]]]](2)

(34)



or 119879(3) = 119879(2)119861 (1199113)119879(3) =

[[[[[[[[[[[[

0 11990512 11990513 0 11990515 011990521 0 0 11990524 0 1199052611990531 0 0 11990534 0 119905360 11990542 11990543 0 11990545 011990551 0 0 11990554 0 119905560 11990562 11990563 0 11990565 0]]]]]]]]]]]](3)

(35)



119879minus1(3) =[[[[[[[[[[[[

0 11990512 11990542 0 11990562 011990521 0 0 11990515 0 1199055111990524 0 0 11990534 0 119905540 11990513 11990543 0 11990563 011990526 0 0 11990536 0 119905560 11990515 11990545 0 11990565 0]]]]]]]]]]]](3)

(37)





119879minus1 =[[[[[[[[[[[[





Anisotropic Mediums



2 =((((((


(41)





119894=1

119875119894 [119875119899 (119894) 119879 minus 119875119899 (119894) I] (43)







(46)



5 Conclusion


Competing Interests


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















(46)



5 Conclusion


Competing Interests


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









the analytical form of the dispersion equation of...

Documents