Chapter 3Bulk waves

Chapter 3

BULK WAVES 3.1

An overview

The fundamental mechanical behavior of anisotropic solids is best demonstrated through their response to propagating bulk waves. Bulk waves exist in infinite homogeneous bodies and propagate indefinitely without being interrupted by boundaries or interfaces. These waves can be decomposed into infinite plane waves propagating along an arbitrary direction g within the solid. The properties of these waves are determined by the relationships between the propagation direction and the constitutive properties of the medium. Generally speaking, three types of waves are possible. These are associated with the directions of the three particle displacement vectors u-'(k), k - 1, 2, 3 defined as acoustic polarizations. Typical polarizations are depicted together with the propagation direction in figure 3.1. The three polarization vectors are mutually orthogonal but, in general, none of them are necessarily parallel or normal to g. A bulk wave is termed "pure" if its polarization vector is directed either along or normal to the propagation direction. For isotropic materials, only pure modes are possible; one of these waves is known as longitudinal with polarization directed along the propagation direction g. The other two are known as shear waves with polarizations directed normal to the propagation direction. Thus ff x g - 0 and ft. g - 0 define purely longitudinal and shear waves, respectively. For an isotropic material with Lame elastic properties A and # and density p, it is known (and shall also be shown later on in this chapter) that longitudinal waves travel with speeds vl = x/(~ + 2 # ) / p , whereas shear waves travel with speeds vt = V/-~/p. In changing the propagation direction, the individual characteristics of these waves, and in particular their speeds, do not change. 31

C H A P T E R 3. B U L K W A V E S

32

X3

X2

XI

Figure 3.1: Schematic of typical polarizations in anisotropic media.

To further demonstrate the "clean" character of propagation in an infinite isotropic medium, we imagine the case of a spherical source in the medium. This specialized example also serves as a forum for introducing terminology of elastic waves in solids. Three spherical wave front surfaces emanate from the source. The fastest one travels with the longitudinal wave speed vl, while the two shear waves coincide and travel with the shear wave speed vt. These wave front surfaces are also normal to the propagation direction which, in this case, is the radial direction. More precisely, the propagation direction is normal to the tangents of these surfaces. Here the longitudinal wave is polarized along the radial direction, whereas the shear ones are polarized normal to it. Associated with wave front surfaces are two other kinds of surfaces: (i) slowness surfaces, defined by the inverse of the wave front speeds and (ii) energy flow surfaces, also known as group velocity surfaces, which define the direction of the flow of energy associated with the propagating waves. For isotropic media, energy flow surfaces coincide with the wave front and slowness surfaces. For anisotropic media however, as shall be demonstrated below, none of the three polarization vectors a priori satisfies either ~ • g - 0 or i f - g 0. If one of the polarization vectors satisfies the first condition, namely • g = 0, then both of the remaining two satisfy the second condition ~ . g = 0, thereby defining three pure modes. If on the other hand, one of the polarization vectors satisfies the second condition, there is no guarantee

3.2. THE C H R I S T O F F E L E Q U A T I O N

33

that the remaining two will satisfy either condition. Hence, for the later case we are assured of the existence of at least one single pure mode. Generally speaking, for anisotropic media, pure modes can occur for some propagation directions depending upon the degree of symmetry of the material under consideration. According to Auld [12], pure modes can also be defined as those associated with directions of propagation that are coincident with the flow of energy directions. These pure modes are not necessarily the same as the ones described above. Interesting situations can then arise when both sets of "pure" modes overlap. We conjecture that the intersection defines "true" pure modes. More fundamental differences exist between isotropic and anisotropic media. The propagation velocities of individual wave types in anisotropic media are intrinsically dependent upon the direction of propagation. Specific dependence is influenced by the level of the anisotropy of the medium. Furthermore, the energy and slowness surfaces are no longer coincident with each other or with the corresponding wave front surfaces. Also the two wave front surfaces associated with the shear motions are no longer coincident. Consequently, fundamental questions arise concerning the manner in which wave fronts and their associated energy surfaces travel. Answers to these questions are presented in the remainder of this chapter.

3.2

The Christoffel equation

Combining the momentum equations (2.31) with the general constitutive relations (2.32) and the strain-displacement relations (2.33) lead to

02ui 1 0 OUk Oul p Ot2 - -~Cijkl~xj(-~x l § ~Xk).

(3.1)

Taking into consideration the various symmetries of cijkl, by interchanging the indices k and l, we can reduce equation (3.1) to

02ui

02ul

P - - ~ - -- Cijkl OXjOXk"

(3.2)

For bulk waves, solutions of equation (3.2) are sought in the complex plane wave form Ui -- Ui ev/-~(~njxj-wt)

(3.3)

where r and nj represent the bulk wavenumber (scalar) and the propagation direction (unit vector) with the components nl, n2 and n3; ~ is the circular

34

CHAPTER3.

BULK WAVES

frequency and Ui is the displacement amplitude vector which also defines polarization. Substituting from equation (3.3) into equation (3.2) leads to the eigenvalue relation (3.4)

w2Ui _~ )~ijkl~2nknjUl

where we used (3.5)

)~ijkl : Cijkl/P.

Defining the phase velocity v = w/~ and using the Kronecker delta property Ui - UtSiz recast equation (3.4) into the characteristic equation

(3.6)

()~ijklnknj -- v25il)Ul -- O.

It is now convenient to introduce the second order tensor All, given by All = Aijklnjnk

(3.7)

and allowing equation (3.6) to be written in the more compact form (Air - v:(fit)Uz = 0.

(3.8)

Both equations (3.6) and (3.8) make up the well-known Christoffel equation which defines a set of three homogeneous linear equations for the displacement amplitudes Ut. Since the phase velocity v is still unknown, each of these equations constitutes an eigenvalue problem with its eigenvalues identified as v 2. Associated with each eigenvalue is an eigenvector Ut which also defines a polarization direction. Because Aij is symmetric with real elements, the eigenvalues are real and the associated eigenvectors are orthogonal. 3.2.1

General features of the

Christoffel equation

The transformation representation Now let us consider the solution of equation (3.8) in greater detail. To this end we rewrite the eigenvalue equation (3.8) in the expanded matrix form

(Aiiv A12 A13

A12 A13 A22 - v 2 A23 A23 A33 - v 2

U2 U3

= 0

(3.9)

3.2. THE CHRISTOFFELEQUATION

35

where the various elements Aij are given from combinations of equations (3.7) and (3.5) as

pAll --- 611n21+ 666 n2 + C55n2

+2C16nln2 + 2C15nln3 + 2C56n2n3 pAl2 --

C16n21-} C26rt 2 -I- C45 n2

nt- (C12 -~- C66)nln2

+(C14 + C56)nln3 + (C46 + C25)n2n3 pAl3 = C15n21+ C46n22+ C35n~ + (C14 + C56)nln2 +(C13 + C55)nln3 + (C36 + C45)n2n3 =

+

+

+2C26nln2 + 2C46nln3 + 2C24n2n3 ph23 = C56n~ + C24n~ + C34n~ + ((746 + C25)nln2 +(C36 + C45)nln3 + (C23 + C44)n2n3 pA33 - C55n21+ C44n2 + C33n2 +2C45nln2 + 2C35nln3 + 2C34n2n3.

(3.10)

In equation (3.10), the definitions of the elastic properties are used in accordance with the notation adopted in section 2.3. In principle, once the material properties and the direction of propaga2 k = 1 2,3. tion are chosen, equation (3.9) can be solved for its three roots Vk, For each of these roots, the same equations can be solved for the corresponding amplitude ratios:

U2(k) r(2k ) = U~k)

A23(A11 - v~) - A13A12

r~k)__ U(k)

A23(All - v~) - A12A13 A12(A33 - v~) - A13A23

k)

A13(A22 - v~) - A23A12

(3.11)

(3.12)

These amplitude ratios define the polarization directions with respect to the coordinate system xi. Specifically, the polarizations are directed along the three vectors (1, r~k), r~k)), k - 1, 2, 3. The normalized components with /-

to their respective lengths V/1 + r(2k)2+-r(3k)2 define their direction cosines with the coordinate axes.

respect

CHAPTER 3. BULK WAVES

36

Crystollographical r e p r e s e n t a t i o n An alternative representation of the Christoeffel equation can be given directly in terms of the crystollographical (reference) coordinate system. This representation would be identical to the transformation representation for r - 0 ~ As shall be demonstrated later on, each of the two representations has certain advantages over the other. The crystollographical representation will be advantageous when dealing with propagation along a priori known axis of symmetry. But this will require the existence of orthotropic or higher-symmetry situations. For propagation along an off axis of symmetry, the transformation representation will lead to significant algebraic simplifications. The crystollographical representation of the Christoffel equation can be written in a form that parallels the transformation representation (3.9)(3.12), leading to =o -

(3.13)

v

with t 2 t 2 P A ~ l -- C l l n l + C66 n2 -~- C 5 5 n 3

PA~2 -

, +2C16nln2 -~- 2C~5 n l n 3 + 2C~6n2n3 ' t 2 t 2 t 2 t C 1 6 n I + C 2 6 n 2 -k- C'45n 3 + (C12 + C~6)nln2 +(C~4 + C~6)n,n3 + (Ci6 + C;5)n2n3 t

2

t

2

t

2

t

t

PA~3 -- C15n1 + C'46n2 + C35n 3 + (C14 + C~6)nln2 +(C~3 + C~5)nln3 + (C~6 + C45)n2n3 pA~2 = C~6nl , 2 + C~2n2 , 2 + C44n32

+2C~6nln2 + 2C46nln3 + 2C&n2n3 pA~3

---- C 5,6 n l2 -+- C 2,4 n 22 +

C~4 n2 + (C46 -I- C ; 5 ) n l n 2

+(C~6 + C45)nln3 + (C~3 + C44)n2n3 PA~3 -- C55 , n 2i -~- C~14 , n22 + (733 , n32

+2C45nln2 + 2C~5nln3 + 2C~4n2n3

(3.14)

and t(k) r2

U2 (k) ----

U~(k)

A ~ 3 ( A ] 1 _ v~2) _ A13A12tt __

AIa(A~2 -- v~2) -- A~3A~2

r~3(k) = U~(k) = A23(A~1 - V~k2) - A~2A~3 U~(a) A~2(A~a- v~2) - AIaA~a

(3.16)

3.2. THE CHRISTOFFELEQUATION

37

Once again, these amplitude ratios define the polarization directions with respect to the coordinate axes x ii. Specifically, the polarizations are directed along the three vectors (1, r~2(k),r~3(k)),k = 1,2,3. The normalized components with respect to their respective lengths V/1 + r~(k)2+ r~3(k)2 define their direction cosines with respect to the primed axes. The above methods enable us to find both the velocity and the associated displacement vector (polarization) for any given propagation direction. Pure mode criteria As discussed earlier, a situation in which one of the polarization vectors coincides with the propagation direction defines three pure modes. The way to establish this is to take the dot product of each of the three polarization vectors with the propagation direction vector. For unit-length (normalized) polarization vectors, each of the three dot products will be less or equal to unity. The largest value is found to correspond to the largest eigenvalue (i.e., to the fastest wave), and is customarily associated with the quasilongitudinal wave (~(1) in figure 3.1). The two remaining waves are then termed quasi-shear ones. Accordingly, three pure modes are obtained only for the propagation directions where one of the dot products is unity. These modes can thus exist if combinations of material properties and propagation directions happen to satisfy the explicit condition described above. Consequently, the above description does not preclude the existence of one or more pure modes even in triclinic materials. If these dot products are not unity, any polarization corresponding to a zero value dot product will then define a single type pure mode. Generally, the above discussion reinforces the argument that solutions to equations (3.9) or (3.13) do not necessarily satisfy the relations ff • g = 0 or i f - g = 0, which define pure longitudinal or pure shear waves, respectively. 3.2.2

Limitations

of analytic solutions

For the general anisotropic medium, no simple analytical solutions exist for either the eigenvalues or their associated polarization ratios. Only numerical methods are useful for obtaining the required solutions. With the recent advances in computational and graphical illustration methods, solutions can be obtained and demonstrated with relative ease. In solving the cubic equation, one obtains three roots; generally, each root is associated with a sheet of the wave front surface. However, in order to determine the set of roots that belong to a specific surface branch, one has to either plot the com-

38

C H A P T E R 3. B U L K W A V E S

plete results or augment the computations with an a priori selected sorting subroutine. By virtue of considering all possible propagation directions, the loci of the computed velocities define the three complete wave front surfaces. Because the phase velocities depend upon the propagation direction, these surfaces are not necessarily uniformly shaped such as spheres. The degrees of deviation from spherical shapes depend upon the specific anisotropy. Of the three surfaces, the outermost surface (belonging to the largest velocity) defines the quasi-longitudinal surface, while the other two define quasi-shear ones. It follows that the innermost surface belongs to the slower quasi-shear wave. The dual slowness surfaces can be constructed from the wave surfaces by merely plotting the inverses of the phase velocities against the propagation directions.

3.3 3.3.1

Material s y m m e t r y Analytical

solutions

In some special cases requiring the existence of material symmetry, equations (3.9) and (3.13) can be factored out resulting in significant computational simplifications and leading to simple analytical solutions. As shall be shown in the sequel below, factorization is associated with the presence of pure modes. Potentially, the cubic equations in v 2 (3.9) and (3.13) can factor into two or three terms. We shall refer to two term factorization as partial, and to three term factorization as total. By inspection of these equations we see that factorization is associated with the vanishing of at least two of the three off-diagonal entries A12, A13, and A23; partial or total factorization is achieved depending upon whether two or all of these elements vanish. Let's now consider the various polarizations associated with individual factorizations. Inspection of the various elements in equation (3.10) or (3.14) shows no possibility of partial or total factorization for triclinic materials. Moving to materials with monoclinic symmetry (see equations (2.24) and (2.35)), concentrating on the transformation representation and implementing their material restrictions on the elements of equation (3.10), reduce them to pAll -- Clln21 nt- C66n22 nt- C55 n2 -1- 2C16nln2 pAl2 - C16n21 -k- C26 n2 -t- C45n32 -1- (612 nt- C66)nln2 pal3 = (613 -~- 655)nln3 -+- (636 -+- 645)n2n3 pA22 = C66n21 + C22n 2 + C44n 2 + 2C26nln2

3.3. M A T E R I A L S Y M M E T R Y

39

pA23 = (C36 + C45)nln3 + (C23 + C44)n2n3 pA33 = C55n21 + C44n~ + C33n 2 + 2645nln2

(3.17)

where the various elements Cij are as given in equation (2.35). Once again, inspecting the various elements (3.17) for possible factorization, we find two possibilities. Both involve partial factorization and are associated with the two specific propagation directions fi = (0,0,1) and fi = (nl,n2,0). For these two special cases, the required vanishing of the determinant in equation (3.9) reduces, respectively, to case (i)

C5~ - pv 2 C45 C4~ C44 - pv 2 O

0

0 0 C33 - pv 2

=0

(3.18)

case (ii) All - v 2 A12 0 A12 A22 - v 2 0 0 0 A33 - v 2

(3.19)

=0

with the reduced elements pAll = C l l n 2 + C66n 2 + 2C16nln2 pAl2 = C16n~ + C26n~ + (612 + C66)nln2 pA22 = C66n21 + C22n 2 + 2C26nln2

(3.20)

pA33 = C55n 2 + C44n 2 + 2C45nln2.

Due to the similarity in the algebraic structure of equations (3.18) and (3.19), we start with the analysis of case (ii). Results pertaining to case (i) can then be obtained by merely exchanging appropriate parameters. The characteristic equation (3.19) now factors out as [(All-

V2 ) ( A 2 2 - V2 ) - A 2 2 ] [ A 3 3 - v

2]=0

(3.21)

and thus admits the three solutions

Vl =

(3.22)

2 - ~[(All 1 v2,3 + A22)+ ~/(A22 _ All )2 + 4A22].

(3.23)

40

CHAPTER3.

BULK

WAVES

The eigenvectors corresponding to these three velocities are given by (0, 0, 1) and (1, r~k), 0), k = 2, 3, respectively, with rk -

-

Vk2 _ All _

A12

A12 - v k2 -- A22

(3.24)

By substituting for v k2 from equation (3.23) we can rewrite r k in the alternative form rk = d-(-1)kv/d 2 + 1

(3.25)

where d=

A22 -- All 2A12

(3.26)

Replacing p a l l , pA22, pA33, pal2 in the relations (3.21)-(3.26) with C55, C44, C33 and C45, respectively, we obtain a complete description pertaining to case (i). In both cases, we find that one mode is directed along the x3-(or equivalently x~-) direction while the other two are polarized in the xl - x 2 plane, at angles 7k measured from the xl-axis whose tangents are r~ k). Note the interesting relation 73 = 7r/2 + 72 which implies that the two inplane polarizations are orthogonal. This can be realized from the two algebraic identities t a n 7 tan(~/2 - 7) = - 1 (d-

v / d 2 + 1)(d + ~/d 2 + 1 ) = - 1

(3.27)

(3.28)

for any 7 and any d. Although results for both cases are algebraically similar, their physical significance and interpretation are different. To facilitate discussions, we depict the possible polarizations of cases (i) and (ii) in figures 3.23 and 3.2b, respectively. Upon closer inspection of the results pertaining to case (i), we see the existence of a pure mode situation. A pure longitudinal mode is found to propagate along the x3-direction with the wave speed Vl = x / C 3 3 / P . The other two define a pair of coupled quasi-shear modes propagating with the wave speeds v2 and v3 respectively. For case (ii), on the other hand, a pure shear mode is polarized along the x3-direction while the remaining two define a coupled quasi-longitudinal and quasi-shear waves, polarized in the Xl - x 2 plane.

3.3. M A T E R I A L S Y M M E T R Y

41

x 3

n

X3

~u~R U)

(a)

x2

x 1

x2

u 2) case (i)

case (ii)

Figure 3.2: Polarizations associated with cases (i) and (ii). 3.3.2

Higher symmetry

Total factorization will be possible only for orthotropic and higher-symmetry materials, such as transversely isotropic, cubic and isotropic. This is in virtue of the fact that these classes of materials possess three orthogonal axes of symmetry which are often identified with the reference (crystollographical) coordinate axes. The axes of symmetry are also known as the principal axes, and the three planes normal to them are termed principal planes. Except for the case of isotropy, complete factorization can only be achieved for propagation along directions that coincide with the axes of symmetry. For vanishing C[6 , C~6 , C~6 and C~5 , the form of equation (3.17) appropriate for orthotropic materials will not change as long as r r 0 ~ or r ~ 90 ~ and will thus be applicable to orthotropic media. For propagation along an axis of symmetry, we use the crystollographical representation of the Christoffel equation. Examination of the entries of equation (3.14) reveals that complete factorization is possible only for propagation along any of the principal axes, namely for n' = (1, 0, 0), (0, 1, 0) or (0, 0, 1) directions. In all three cases, pure, uncoupled modes exist with polarizations along the three principal axes. The modes can be identified once the propagation direction has been specified. To help facilitate our discussion, we specialize equation (3.14) to orthotropic materials and get pA~l = C l!l n l12 + ,-,! ~66n212 + C ~I n ~ 2

pA~2 = (C~2 + C~6)nlln~2

CHAPTER3.

42

BULK WAVES

pA~3 = (C[3 + C~5)nln3', pA~2 ~, ,2 ~, ,2 ,~, ,2 ~66nl -i- ~22n2 -[- ~44n3

pA~

(C~ + C~4)~

pA~3 --" ~, ~55 nl,2 -]- ,~, ~44 n2,2 -~- ,~, ~33 n3,2 9

(3.29)

For the propagation direction n' - (0, 1,0), for example, a longitudinal mode propagates with wave speed (C~2/p)89along the x~-axis, and the remaining two modes are of shear type, polarized along the x~-and x~1 1 directions and propagate with speeds (C~6/p)~and (C~4/p)~, respectively. For propagation along an arbitrary direction in any of the principal planes, only partial factorization is possible and solutions will follow the formal procedure used above for the monoc]inic materials presented in cases (i) and (ii). Except for the case of isotropic symmetry where pure modes exist for all propagation directions, the procedure and results obtained for orthotropic symmetry can be directly adapted and applied to the highersymmetry cases. This is done by merely imposing the appropriate restrictions on the properties as fully exploited in chapter 2. For this reason we shall now move to the interesting and well-studied case of cubic materials. 3.3.3

Cubic symmetry

In higher than orthotropic symmetry materials, other symmetry axes besides the principal ones can potentially exist. In particular, we note the case of cubic materials. To facilitate the discussion pertaining to this case, we further impose the cubic material restrictions on the elements of equation (3.29) and get pA~l -- C[1

+ C~6(n~ 2 -+-

:

I

I

__

!

!

)

ph'~ (C~: + C~)n,n'~ pA'~ = C~(nl ~ + n'~~) + C~n'~ ~ pA'~

pA~

= _..

(C'~: + C ~ )n2n ' '3

C~(nl ~ + n~~) + . i l n ~ . ~,~I

! 2

(3.30)

For the transformation representation of the Christoffel equation in the cubic material, we first impose the appropriate material restrictions on equation (2.34) and get Cll = C[1 - 2(C11 - C[ 2 - 2C~6)$2G 2 C22 ~ C11

3.3. M A T E R I A L S Y M M E T R Y

43

C12 = C~2 + 2(C~1 - C ~ 2 - 2C~6)$2G 2 C2a = C l a = C~2 c )sa C16 C26 --" - C 1 6 C33 = c h = C36 = C 4 5 = 0 = C44 = C66 = C~6 -F 2(C[1 - C [ 2 - 2C~6)$2G 2. =

-

-

2c

)(s

-

(3.31)

These specialized properties result in simplifications of the various elements

Aij of equation (3.10) in accordance with pAll - C l l n 2 + C66n2 + C55n 2 + 2C16nln2 pAl2 = C16(n 2 - n 2) + (C12 + C66)nIn2 pAl3 - (C12 + C55)nln3 pA22 = C66n 2 + C l l n 2 + C44n~ + 2C26nln2 pA23 - (C23 + C44)n2n3 pA33 = C55n2 + C44n 2 + C33n2.

(3.32)

Although these representations appear to be algebraically different, they are equivalent in that they represent alternative Christoffel equations for cubic materials. The utility of these two representations can be demonstrated through a comparison of the manner in which they lead to the desired solutions. There are several propagation directions in cubic materials which can be identified that lead to factorization and hence to pure modes. These include the principal axes xi,' the face face diagonals and the body face diagonals. We start with the simplest situation of propagation along one of the principal axes, say x~. For this choice n' = (1, 0, 0) and, in light of r = 0, also fi = (1, 0, 0). Under these conditions, both representations immediately lead to total factorization of their respective Christoffel equations and hence to three pure modes. The longitudinal mode travels with the speed J C ~ l / p along the x~-direction and the remaining two shear modes are polarized along the x~ and x~-directions with the equal speeds of ~/C~6/p. Face d i a g o n a l Next, we consider propagation along face face diagonals, namely along bisectors to the in-plane angles of the principal planes. As a representative

44

CHAPTER

3.

BULK

WAVES

example, we choose the x] - x ~ face diagonal for the propagation direction. For this case, however, we have to identify the components of the propagation direction for both representations. For the crystollographical representation, it is obvious that n-~ - (1 / v/2, 1 / V~, 0). For the transformation representation on the other hand, this direction is uniquely specified by a combination of r = 45 ~ and g - (1, 0, 0). Under these conditions, the vanishing of the corresponding Christoffel determinants are given, respectively as

C~I -~- 6~6 - 2pv 2 6~2 + 6~6 0 6~2 + 6~6 C~I + 6~6 - 2pv 2 0 0 0 C~6 - pv 2

= 0

(3.33)

and Cll

0

0 C66 - p v 2

0 0

0

0

C66 - p v 2

-

pv 2

= 0

(3.34)

which display partial and total factorization, respectively. Expanding the determinant in equation (3.33) yields [(C~I + C~6 - 2pv2) 2 - (C~2 + C~6)2][C~6 - p v 2] = 0.

(3.35)

Recognizing that the first factor in equation (3.35) is a difference between two perfect squares, we can rewrite it as the product of the two terms (C~1 + C~2 + 2C~6 - 2pv2)(C~1 - C~2 - 2pv 2) = 0.

(3.36)

This allows equation (3.35) to yield the three solutions

(ci, + Vl

--

v2 =

vii

v3 -- ~ C ~ 6

-

2p

2p

+

(3.37) (3.38) (3.39)

The polarizations associated with equation (3.33) can be obtained from consideration of equation (3.25). Since here A~2 = All , application of equation (3.26) to the case of crystollographical representation, i.e., for the primed quantities, shows the vanishing of d implying two polarizations along 4-45 ~ from the x]-direction. These two modes propagate with the respective

3.3.

MATERIAL

45

SYMMETRY

velocities (3.37) and (3.38). Since the direction at 45 ~ coincides with the propagation direction, we have a situation of pure modes. The remaininfi mode is of shear type and propagates with the velocity (3.39) normal to n ~. By permutation of the axes, we can see that all bisectors of the in-plane angles constitute axes of symmetry for cubic materials. We now consider the solutions of equation (3.34). Notice the relative simplicity of the expression (3.34) revealing total trivial factorization and yielding three pure modes propagating with the speeds vl = x/C'll/p, v2 = v/C66/p and v3 = x/C55/p. The first of these belongs to a longitudinal mode polarized along the xl-direction, i.e., along the x~ - x ~ face diagonal, while the third is of shear type and is polarized in the x3- (also x~-) direction. This leaves the second one as a shear mode which is polarized along the x2-direction, namely at 90 ~ from the face diagonal. Specializing these results for r = 45 ~ in equation (3.31) identically recovers the results of the crystollographical representation, equations (3.37)-(3.39). Body diagonal Next, we show that propagation along all directions described by the rela12 12 12 tion n 3 = n 2 -- n 1 = 1/3 in the crystollographical axes, or by the relations n 2 - 2/3, n 2 - 0, n 2 = 1/3 with r = 45 ~ in transformed axes, define axes of symmetry in cubic materials. These coincide with the cubic body face diagonals. There are four such directions. We start with the crystollographical representation (3.30). The choice of any body diagonal for the propagation direction yields the restrictions A~3 = A~2 = A~I (all designated with A~I ) and A~3 - A~3 = A~2 (all designated with A'). These restrictions factor the characteristic equation (3.13) as -

A' - v 2 ) 2

'

+ 2h'-

= 0

(3.40)

which admits, with the help of equation (3.30), the three possible velocities

Vl = ~/h~l -~- 2At= v/(C~I --b 26~2 + 4C~6)/3p =

A/-- v/(C~I

6~2 .-[-C~6)/3p.

(3.41)

(3.42)

Thus far, we can only expect that the first solution belongs to a pure longitudinal wave, whereas the second belongs to the two remaining pure shear ones. The two shear waves are degenerate since their velocities are equal. To demonstrate this, we must examine the corresponding polarizations. In this instance, solutions depend upon the specific diagonal chosen for the propagation direction. We can however, treat all cases simultaneously by

46

CHAPTER3.

BULK

WAVES

substituting the cubic material restrictions into the amplitude ratios (3.15) and (3.16) leading to i(k)

=

_

(Ai

-

-

A'

-- ( n ~ l _ vk2) _ A/ -- 1

(3.43)

except when Vk 2 = A~I - A' where the expression is ill-defined. Examination of the expression (3.43) shows that vl of equation (3.41) corresponds to a longitudinal mode polarized along the propagation direction and hence defines a pure mode. Alternately, the choice of the other velocities, namely Vk -- v2 or v3, leads to ill-defined ratios and thus implies that any two orthogonal directions normal to n' define the two shear polarization directions. We finally show that the same results can be obtained by using the transformation representation. To this end, specializing equation (3.32) to n 2 = 2/3, n2 - 0 and n] - 1/3 leads to the characteristic equation 2Cll + C55 - 3pv 2 0

0

x/r2(c13 -t- C55)

2C66 + C44 - 3 p v 2

0

~/r~(C13 -{- C55)

0

2C55 -J- C33 - 3 p v 2

= 0. (3.44)

If this is followed by specializing the various elements in equation (3.31) to r = 45 ~ after some algebraic manipulation, one can show that the resulting solutions are identical with the results (3.41) and (3.42) obtained using the crystollographical representation.

3.3.4

The isotropic case

Thus far, we have dealt with situations where we identified specific directions along which pure modes exist. The question now arises whether there exist planes which admit pure modes for all propagation directions. This will undoubtedly require higher-symmetry materials. Indeed, as shall be demonstrated below, all propagation directions in isotropic media support pure modes. Thus, it follows that planes of transverse isotropy have similar properties. In this section we illustrate application of the procedures described so far to the case of isotropic materials. Returning to relation (3.6), we arbitrarily choose the wave amplitude vector as unity, namely UiUi - 1. Multiplying both sides of equation (3.6) by Ui, we obtain the interesting relation V 2 -- )~ijklnknjU1Ui.

(3.45)

Although it appears from equation (3.45) that the velocity v is dependent upon the displacement polarization, whereas the relations in equation (3.8)

3.3.

MATERIAL

47

SYMMETRY

do not show such a dependence, not withstanding, both relations are consistent. Since in equation (3.45) we normalized the polarization vector to unity, this equation also includes implicit information about the eigenvectors. The velocity v in equation (3.45) will be independent of the polarizations only if certain restrictions are applied to such polarizations. These restrictions will essentially result in the characterization of the required polarizations as would be obtained by the direct solution of equation (3.6). The simple example of propagation in isotropic media illuminates in best fashion the implications of the above argument. To this end, we specialize cijkl to that of an isotropic material as Cijkl = s

(3.46)

+ #(hikhjZ + 5iZhjk).

Then equation (3.45) can be written as (3.47)

pv 2 = ()~ + # ) n k n j U k V j + #

since nini -- 1 and ULUz = 1. For v to be independent of U, we must have one of the following two situations: (i) If [7 is directed along g, then n j U j - 1 which reduces equation (3.47) to (3.48)

p v 2 -- )~ + 2#

defining a pure longitudinal wave. (ii) If U is directed normal to g, then n j U j - 0 leading to (3.49)

pv 2 - #

which defines a pure shear wave. The same conclusions may be drawn by direct application of the eigenvalue equation (3.9). Substituting from equation (3.46) directly into equation (3.9) yields the characteristic equation (n2+v nln2 nln3

nln2 nln3)(U1) n 2 + "7 n2n3 U2 n2n3 n2+ ~ U3

= 0

(3.50)

where, for convenience, we introduced # -- pv 2

= ~ . A+#

(3.51)

Equation (3.50) can also be obtained from equation (3.9) by specializing the elements (3.10) to isotropic materials in accordance with Cll = 6'22 = C33 - - / ~ -~- 2 p , C12 -~ C13 - - C23 -- ~, C44 -- C55 - C66 - ]_t and such that all

C H A P T E R 3. BULK WAVES

48

remaining Cij properties vanish. For nontrivial solutions, the determinant of equation (3.50) must vanish. By expanding the determinant, using the condition n 2 + n 2 + n32 = 1

(3.52)

and following simple algebraic manipulations, we get ~/2(i + ~/) = 0

(3.53)

which, once again, yields the results (3.48) and (3.49). Notice that the relation (3.53) is independent of the propagation direction ~ signifying that only pure waves can propagate in isotropic media. This means that any choice of the propagation direction in equation (3.50) suffices to arrive at equation (3.53). An obvious example is the choice = (I,0,0). For this situation, equation (3.50) trivially recovers equation (3.53). For this case, the solution "7 = 0, if utilized in equation (3.50) leads to UI - 0 leaving U2 and U3 arbitrary, defining a shear wave. For "7 = - i , on the other hand, U2 and U3 vanish while UI is arbitrary and hence defining a longitudinal wave. Returning to the general propagation case described in equation (3.50), we see that for "7 - 0, consistent solutions of this equation require the relation

nl U1 + n2U2 + n3U3 - 0

(3.54)

which can only hold if U is normal to g describing a shear wave polarization. For "7 - - 1 , on the contrary, application of equation (3.50) leads to the relation U1 nl

.

.

.

U2 n2

.

U3 n3

(3.55)

which can only hold if U is parallel to ~ and thus describing a longitudinal polarization.

3.4

Computer aided analysis

For the symmetry classes considered so far, it was easy to analytically solve the Christoffel characteristic equation only for the simplest, usually easily identified symmetry cases. For most of these cases, solutions are obtained by inspection of the equation assisted by elementary algebraic manipulations. For the general case, however, numerical computation is required to calculate wave surfaces. In this section, we describe computer aided computational techniques, capable of treating the general anisotropic case, and also

3.4.

COMPUTER

49

AIDED ANALYSIS

p

X3

X3

ig,

X'2

3

X1

p

1(1

2

(a)

(b)

Figure 3.3: Strategy of the choices of propagation directions" (a) g~ in the reference coordinate system and (b) g in the transformed system.

of providing insight into the acoustic properties of symmetry classes such as those described above. Generally, good computational strategies are found to be of utility in the solution process. As a consequence of dealing with the representation of three-dimensional wave surfaces, it is of fundamental importance to adopt a systematic strategy for taking into account all propagation directions and the subsequent solutions of phase velocities and associated polarizations. Two methods are found to be particularly attractive. The first uses the reference coordinate system and is based upon the construction of wave curves for propagations in planes that are perpendicular to one pre-chosen axis, say x~ as shown in figure 3.3a. By stepping the component n~ of the propagation unit vector direction from- 1 to + 1, the whole three-dimensional surface can be constructed. The other technique is based upon the versatile properties of the linear orthogonal transformation of chapter 2. In this instance, solutions are first obtained for propagation in a plane that makes an azimuthal angle r with the x~-direction. This situation is illustrated schematically in figure 3.3b. Without any loss in generality, we can identify this plane with the Xl - x 3 plane obtained from a linear transformation of the original axes x i to the global one xi through a counterclockwise rotation of an angle r about the x~ axis; thus x2 is normal to this plane. With this choice of the coordinate system, being vertical, the propagation plane is often called the sagittal plane. Polarization normal to this plane defines

50

CHAPTER3. BULKWAVES

the horizontally polarized (SH) wave. Thus, the polarization in the sagittal plane constitutes a coupling between the quasi-longitudinal and the vertically polarized quasi-shear waves. By incrementally varying r from 0 ~ to 360 ~ we span the whole space. We now show how the calculations can be conducted in the transformed system xi with relative ease. To this end, let us examine the form of the formal solution (3.3) for the special case where n2 is zero. This means that we are solving the Christoffel equation in the Xl - x3 plane. But such a choice implies that all displacements, velocities and stresses are independent of the x2-direction. Consequently we can conduct the computations in two, rather than three dimensions. Since all field variables as well as the stiffness coefficients are tensors, by consistently applying the tensor transformation, we can conduct the analysis in the transformed coordinates as well. It turns out that the computations can thus be carried out independent of the x2-axis. Solutions can be obtained by replacing the properties by their respective transformed ones as described in section 3.2.1. As was indicated in chapter 2, we shall limit our computational illustrations to monoclinic and highersymmetry materials. We recall that the plane of symmetry of the monoclinic material is always chosen in this book to coincide with the x: - x 2 plane. Numerical identification of the pure modes can also be achieved in a rather simple and straightforward manner. First, we recall from section 3.2.1 that all wave modes are pure for situations where the longitudinal wave is polarized along the propagation direction g. Second, the dot products of the three normalized polarization vectors define the cosines of the angles between these individual vectors and the propagation direction. By taking the inverse cosines of the dot products, we obtain the skewing (deviation) angles of the polarization directions from the propagation direction. Guided by the fact that the pure shear modes propagate normal to the propagation directions, it is more appropriate to take the inverse sines of the dot products associated with the shear components and thus find the corresponding skewing angles from their pure directions. Numerical illustrations are now presented for selected materials chosen from the material menu listed in Appendix A. The choices of propagation planes are intended to exhibit the general features of slowness curves and to graphically isolate and illustrate the properties of the simplified cases discussed above. Figures 3.4a, b are samples of polar slowness and polarization skewing angle curves that belong to the cubic material InAs computed for r - 0 ~ In these, and in the remaining figures of this section, solid lines are reserved for the quasi-longitudinal wave component and the broken ones are reserved for

3.4. COMPUTER AIDED ANALYSIS

51

n/v .

..- .....

n~

] .......

.

- .." ;____,. : ,;....::.:. . . .

.-/

~,....,:, ,,,.,

".~'. / ,'"

,, ', ~.-

:,, .". .'~-.~- - -.,. . . ~. . .-. - ' ~, - ~ , ~ - ~ - -' -,. - ... .. .. . ~- .". ." ,

. ~

(a)

.~o

,..,,

,

,,

.,

o

,, ,,..:-

/

:

: ",%

"

oo.,, ,, ,, ::

; :

nJv

o

(b)

Figure 3.4: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation in the r - 0 ~ plane in InAs cubic material. Scale is (a) 0.15 sec/km-per-division and (b) 5~

the quasi-shear components. We start with the result depicted in figure 3.4a. The situation in this figure is similar to the case where propagation takes place in the plane of s y m m e t r y x~ - x~ where ~' - (n~, 0, n~). The presence of the perfect circle signifies a case of partial factorization that decouples the horizontally polarized shear wave. This wave component is polarized in the x~-direction and travels with the speed (v/C~6/p). The other two modes are coupled and are polarized in the sagittal plane. The specific polarizations of these two waves are governed by the relation (3.15) or (3.16). This figure shows two unique propagation directions along which the two shear waves are degenerate, namely, coalesce into a single one. Further inspection of this figure reveals three pure propagation directions. Two are along the principal axes x~ and x~ and the third is along the face diagonal. Specific results for these propagation directions confirm the analytical results discussed above. Using visualization of figure 3.4b, we first notice two rather than three curves exist. The third, which is associated with the perfect circle slowness curve in figure 3.4a, corresponds to a zero value dot product implying the existence of a pure mode for all propagation directions in the x l - x3 plane. As far as the other remaining two modes are concerned, we can identify four directions for possible propagation of pure modes in this system. We also notice the relatively smaller skewing angles associated with the quasi-longitudinal wave component. In figures 3.5a,b, the results depicted

CHAPTER3.

52

n]v

BULK WAVES

n~,

.g--'"~,i~~il---~ .... :~.

,' , ".,,

,,.9

,

~

!

-;; ',,

~. ,

% ~,"... ', .i

.~*~

,

g,

~'..~...~"ilk~ .....;."',

i

,, ', .

nJv

',

/

, ,i

"'.. % "'---]_ __',. . . . - : .."

(8.)

(b)

Figure 3.5: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation along r = 45 ~ in InAs cubic material. Scale is (a) 0.10 seclkm-per-division and (b) 5~ Horizontal axis corresponds to the face diagonal of the (x~ - x~) plane. The added solid line corresponds to the body diagonal n~ = n~ - n ~ = 0.577.

in figure 3.4a,b are repeated for the azimuthal angles r = 45 ~ and 30 ~ respectively. Notice in both the appearance of the third curve in the skewing angle figures. The situation in figure 3.5ia,b is somewhat more involved in comparison with that of figure 3.4a,b. Here, the S H wave also uncouples but no longer defines a perfect circle. In fact its slowness curve defines an ellipse with major and minor a x e s (x/p/C66) and (X/p/C44), respectively. This figure also displays the results obtained for the propagation along all of the axes of symmetry discussed earlier for cubic materials. The vertical axis is still a principal axis of the unrotated (original) cube, the horizontal axis corresponds to a face diagonal direction and the direction shown in the added solid line arrow defines a body diagonal of the cube. Here there are three directions of degeneracy of the shear waves. Like in the case of propagation in the principal plane, four directions are identified as pure mode directions in the present system. Finally, figures 3.6a,b are examples of what might happen when propagation takes place in an arbitrary sagittal plane in a cubic material. Here no possible factorization exists and the three waves are completely coupled resulting in more complicated slowness curve textures. Since all propagation planes contain the original unrotated x3-axis, the figures, like all possible

3.4. COMPUTER AIDED ANALYSIS

53

n~v

n~v :

.-•

--,,...

~-" "; "," ,"9176 -;9

'

,,. . . . . . . ,

nJv

,

~ %-" %*.**

_% 9

o.4~

,~~

(a)

-

-..

"4

",

"

-.. 9 . o . O - - . . , , ~o 9176 ', ~

*'~

,.,.... ',,. 9. . . . ,

*,

',

', ,

',

,

,

,'

u ;-.~...v -

~

,,

,

,

(b)

Figure 3.6: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation along r = 30 ~ in InAs cubic material. Scale is (a) 0.15 sec/km-per-division and (b) 5~

n~

n~

/:i:.::":::-'"'",,":::-ii:".. ~k

k

',,

,'

(a)

.

' iJ

n,/v

i

i

I

:**

9

',

',

nJv

(b)

Figure 3.7: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation along r -- 0 ~ in graphite-epoxy. Scale is (a) 0.15 sec/km-per-division and (b) 10~

CHAPTER3.

54

BULK WAVES

others, have a principal propagation direction, namely the x3-direction. Notice that only two directions of shear degeneracy exist for this case. For the 30 ~ sagittal plane orientation, only one direction along the x3-direction is possible for three pure modes to exist. Skewing angles ranging from 0 ~ to 25 ~ are possible for this situation with the smallest deviations belonging to the quasi-longitudinal mode. Some of the numerical results presented in figures 3.4a,b-3.6a,b are repeated for the unidirectional graphite-epoxy material and shown in figures 3.7a,b and 3.8a,b. Here uncoupling of the S H wave occurs for propagation in the principal plane r = 0 with its slowness curve being a perfect circle. The fact that only two curves are present in the skewing angle figure 3.7b once again suggests that a pure shear mode exists for all directions in this plane. This, of course, corresponds to the perfect circle associated with the horizontally polarized S H shear wave component. Notice also that only propagations along the principal directions Xl and x3 define pure modes. For any other sagittal plane, like r = 30 ~ of figure 3.8a,b, the three waves are completely coupled and only the principal direction x3 defines three pure modes. Skewing angles of up to 40 ~ for quasi-longitudinal modes and up to 65 ~ for quasi-shear modes are possible for these two cases. The significance and utility of the slowness surfaces can best be demonstrated through their dual relationship with the energy propagation surfaces (also known by the ray surfaces). Derivation of such relationships is the subject of the following section.

3.5

Group velocity

Having stated the phase velocity v - w/~, we now state the group velocity vector gj as

gJ-

Ow Ov = ~cgnj Onj

(3.56)

since ~ is a constant wavenumber. From equation (3.4) with UiUi = 1, we then deduce that

2~vdw = ~2~ijklUiUi(nkdn j + njdnk).

(3.57)

It can be easily shown, by interchanging j with k, i with 1 and using the symmetry relations

"klkji -'- ~lkij -- ,~ijkl

(3.58)

3.5.

55

GROUP VELOCITY

n~/v

n3/v ,-v-

u ........ -.,', "% ",,

/v

n1

",,,,'.... ..... :k .... Z ...... "

(a)

(b)

Figure 3.8: Polar diagrams of (a) slowness and (b) polarization skewing angle for propagation along r = 30 ~ in graphite-epoxy. Scale is (a) 0.15 sec/km-per-division and (b) 10~ that the second term is identical with the first and hence we obtain gj = ()~ijklnkUzUi) /v.

(3.59)

We can also recognize by multiplying both sides of equation (3.59) with nj and comparing with equation (3.45) that gjnj -- v.

(3.60)

From equation (3.60) we have gj dnj + nj dgj = dv

(3.61)

and through comparison with equation (3.56) we conclude that njdgj = O.

(3.62)

This remarkably simple result shows that the propagation direction fi is always normal to the propagating energy surface. Now we prove that the group velocity ~ is normal to the slowness surface. Since the slowness vector Sj is defined by Sj = n j ?2

(3.63)

56

CHAPTER

3.

BULK WAVES

we have v d S j + S j d v = dnj.

(3.64)

Multiplying both sides by gj, noting from equation (3.60) that dv - g j d n j , and from combinations of the equations (3.60) and (3.63) that g j S j - 1 leaves gjdSj = 0

(3.65)

which dictates that t7 be normal to the surface S. We finally show, in the form of an example, that the phase and group velocities coincide for isotropic materials. To this end, we start with equation (3.45) and rewrite it in terms of Cijkl by using the relation (3.5). Subsequently, we specialize it to the constitutive relation (3.46), multiply both sides by gj and use equation (3.60) to arrive at

pvgj -- [~(~ij(~kl + #((~ik(~jl + (~ilbjk)]nkU1Ui.

(3.66)

This equation can be contracted to pvgj = ()~ + # ) U j U k n k + # n j .

(3.67)

Once again, dependence upon the displacement amplitudes disappears in the following two circumstances. In the first, if the polarization is assumed along the normal, namely if Uk = nk and hence Uknk -- 1, then pvgj = (~ + 2~)nj.

Since for this case we showed earlier that pv 2 = (s reduces to gj = v n j

(3.68) then equation (3.68) (3.69)

which implies that ~ is directed along the propagation direction g. Furthermore, by multiplying both sides of equation (3.69) by gj and using the relation (3.60), namely g j n j -- v, we conclude that the energy speed [iT[ equals v. Thus, ~ is identical with vg. E n e r g y (Ray) surface The above described dual relationships between the slowness and the energy surfaces can be used to calculate the energy surfaces. To this end we rewrite equation (3.60) in the alternate manner ItTIcos r = v.

(3.70)

3.5.

GROUP V E L O C I T Y

57

4O

90

.

.

80

i

:

A

ID

3 0

"

-

=

70

-

~" eo

.o 2 0

....... ..~

99 : . . . . . .

/

~ ............

.:

o}

10

.

_o

.

.

.

.

,~ so : ............;,.......

,;,:: . . . .

t.

/

:

.0i.!ii!.i ii.i.ii-,.-!!;! c:: -10

~4o >,

.."

3o g . . . . . . . . . .

.............

~

i

:\

...... : . y ' : "

..... .......

20

.,

-20

0

10

20

30

incident angle (deg)

40

(a)

~

5

10

15

20

25

30

incident angle (deg)

35

40

(b)

Figure 3.9: Energy flow angle vs. incident angles for: (a) InAs cubic (r = 0~ (b)graphite-epoxy (r = 0~ Here ~ is the angle between t7 and g defined as the power flow angle. It is obvious that this angle depends upon the shape of the slowness curve and hence on the propagation direction g. From the duality relation (3.62) and (3.65), we conclude that cosr is equal to the dot product between and the normal to the slowness curve at that location. We can then find the power flow angle from the slowness curve by first finding the normal to it and then dot product this normal with ~ to find cos r By considering all propagation directions we can calculate the corresponding energy flow angles. Depending upon the shapes of the slowness curves, this angle can vary from one propagation to the other and sometimes can be quite large. The two sample calculations shown in figures 3.9a,b corresponding to the slowness curves of figures 3.4a and 3.7a clearly show a wide range of energy deviations from the propagation directions. Having done so, we can then proceed to calculate the energy curves by the following two steps: first we find ]gl from equation (3.60). In the second step we assign this value to the normal to the slowness curve. By doing so for all possible propagation directions we construct the energy flow surfaces. The corresponding energy curves to those of the slowness curves of figures 3.4a and 3.7a are plotted in figures 3.10a,b, respectively. The cusps appearing in these figures correspond to regions of null energy and will be elaborated upon in chapter 13 when dealing with the response of solids to transient sources.

CHAPTER 3. BULK WAVES

58

E3

E3

====================== .... .

/,'//

'~'~I ,l

~', , ,

,"/ I::I (a)

: I ,,

,, 1:1

(b)

Figure 3.10: Energy curves for: (a) InAs cubic (r - 0~ (b) graphite-epoxy (r = 0~ Background circles are in increments of 105 km/sec in (a) and 105 km/sec in (b).

3.6

Energy

flux

Any elastic deformation, and in particular that caused by the propagation of waves, is associated with the transfer of energy. At any instant in time, the total energy contained in the wave front as it propagates in a lossless media is given by the sum of kinetic and strain energies E -- -~

(p~igzi -+ a i j e i j ) d V .

(3.71)

Here, superposed dots denote differentiation with time t and V is the volume of the disturbed region. Let us concentrate a little more on the product aijeij. From the strain-displacement relations (2.33) we write 1 aij eij = -~ aij (ui,j -}- uj,i).

(3.72)

Since aij is symmetric, this reduces to aijeij -- aijui,j.

(3.73)

Furthermore, using the stress-strain relations (2.32) we get O'ijei j

-

-

1 -~Cijkl(~k, l nt- ~tl,k )Ui, j.

(3.74)

3.6. E N E R G Y FLUX

59

Since again, Cijki is symmetric with respect to k and l, it further reduces to

aijeij -- cijklUk,lUi,j

(3.75)

and hence we get

E -- -~

(puiui + CijklUk,lUi,j)dV.

(3.76)

Differentiating equation (3.75) with respect to time t gives

Oaijeij Ot -- Cijkl(Ui'ji~k'l + Uk,lUi,j).

(3.77)

Since Cijkl-~Cklij, this can be written as

Oaijeij Ot -- 2CijklUk,li~i,j = 2aij~ti,j.

(3.78)

By differentiating equation (3.76) with respect to t, we get ot =

(pit~ii~ + ~jit~,j)dV

(3.79)

Ot --

[pi~iui § (aiji~i),j -i~iaij,j]dV

(3.80)

or=

[(piii - aij,j)i~i + (aiji~i),jldV.

(3.81)

The term in the first bracket is the equation of equilibrium and hence vanishes reducing this equation to

OE Ot -- Iv (aiji~i),jdV.

(3.82)

Using Gaussian theorem, we replace the volume integral by the surface integral and get

OE Ot = IS aiji~injdS

(3.83)

which represents the energy crossing a wave surface in unit time. On introducing the energy flux (Poynting) vector Pj, we set

OE = - Is PjnjdS Ot

(3.84)

where

Pj = -aiji~i.

(3.85)

Thus the energy variation within volume V per unit time is represented as the flux of the Poynting vector over the boundary surface.