Part 5

ACOUSTIC WAVE PROPAGATION

IN ANISOTROPIC MEDIA

Review of Fundamentals displacement-strain relation (deformation)

stress-strain relation (constitutive equation)

balance of momentum (Newton's Law) ⇓ equation of motion (Christoffel's equation) Notation: position vector 1 1 2 2 3 3( )x x x= + +x e e e displacement vector u strain matrix ε stress matrix τ stiffness tensor C

Three-Dimensional Problem Displacement-strain relation: vector notation s= ∇ uε indicial notation , ,½ ( )ij i j j iu uε = +

differential notation ½ ( )jiij

j i

uux x

∂∂ε = +

∂ ∂

31 211 22 33

1 2 3, , ,uu u

x x x∂∂ ∂

ε = ε = ε =∂ ∂ ∂

1 212 21

2 1½ ( )u u

x x∂ ∂

ε = ε = +∂ ∂

3223 32

3 2½ ( )uu

x x∂∂

ε = ε = +∂ ∂

3 131 13

1 3½ ( )u u

x x∂ ∂

ε = ε = +∂ ∂

Stress-strain relation: vector notation :=τ εC indicial notation ij ijkl klCτ = ε

11 11 12 12 13 13

21 21 22 22 23 23

31 31 32 32 33 33

ij ij ij ij

ij ij ij

ij ij ij

C C C

C C C

C C C

τ = ε + ε + ε

+ ε + ε + ε

+ ε + ε + ε

Abbreviated Notation

11 12 13

21 22 23

31 32 33

ε ε ε⎡ ⎤⎢ ⎥= ε ε ε⎢ ⎥⎢ ⎥ε ε ε⎣ ⎦

ε 11 12 13

21 22 23

31 32 33

τ τ τ⎡ ⎤⎢ ⎥= τ τ τ⎢ ⎥⎢ ⎥τ τ τ⎣ ⎦

τ

Stiffness matrix

11 12 13 14 15 1611 11

12 22 23 24 25 2622 22

13 23 33 34 35 3633 33

14 24 34 44 45 4623 23

15 25 35 45 55 5631 31

16 26 36 46 56 6612 12

222

C C C C C CC C C C C CC C C C C CC C C C C CC C C C C CC C C C C C

τ ε⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢τ ε

= ⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢

τ ε⎢ ⎥ ⎢⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

⎥⎥⎥⎥⎥⎥⎥⎥

Stress-strain relations for isotropic materials (Hooke's Law) In indicial notation,

2ij kk ij ijτ = λε δ + με

Kronecker delta

1 if and 0 elseij iji jδ = = δ =

11 11

22 22

33 33

23 23

31 31

12 12

2 0 0 02 0 0 0

2 0 0 020 0 0 0 020 0 0 0 020 0 0 0 0

τ ελ μ λ λ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥τ ελ λ μ λ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥τ ελ λ λ μ

=⎢ ⎥ ⎢ ⎥⎢ ⎥τ εμ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥τ εμ⎢ ⎥ ⎢ ⎥⎢ ⎥τ εμ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

++

+

Stress-Displacement Relation

31 2 111

1 2 3 1( ) 2uu u u

x x x x∂∂ ∂ ∂

τ = λ + + + μ∂ ∂ ∂ ∂

31 2 222

1 2 3 2( ) 2uu u u

x x x x∂∂ ∂ ∂

τ = λ + + + μ∂ ∂ ∂ ∂

3 31 233

1 2 3 3( ) 2u uu u

x x x x∂ ∂∂ ∂

τ = λ + + + μ∂ ∂ ∂ ∂

1 212 21

2 1( )u u

x x∂ ∂

τ = τ = μ +∂ ∂

3223 32

3 2( )uu

x x∂∂

τ = τ = μ +∂ ∂

3 131 13

1 3( )u u

x x∂ ∂

τ = τ = μ +∂ ∂

Traction and Body Forces:

σx

x1

x2

x3B3 dx1 dx2 dx3

traction forcesbody forces τ33 dx1 dx2

τ32 dx1 dx2

τ31 dx1 dx2

B1 dx1 dx2 dx3

B2 dx1 dx2 dx3

τ22 dx1 dx3

τ23 dx1 dx3

τ21 dx1 dx3τ12 dx2 dx3

τ13 dx2 dx3

τ11 dx2 dx3

σxσx

x1

x2

x3B3 dx1 dx2 dx3

traction forcesbody forces τ33 dx1 dx2

τ32 dx1 dx2

τ31 dx1 dx2

B1 dx1 dx2 dx3

B2 dx1 dx2 dx3

τ22 dx1 dx3

τ23 dx1 dx3

τ21 dx1 dx3τ12 dx2 dx3

τ13 dx2 dx3

τ11 dx2 dx3

Equilibrium Equations

3111 211

1 2 30B

x x x∂τ∂τ ∂τ

+ + + =∂ ∂ ∂

3212 22

21 2 3

0Bx x x

∂τ∂τ ∂τ+ + + =

∂ ∂ ∂

13 23 33

31 2 3

0Bx x x

∂τ ∂τ ∂τ+ + + =

∂ ∂ ∂

x1-direction:

11 1 1 11 1 2 3 21 2 2 21 2 1 3

31 3 3 31 3 1 2 1 1 2 3

[ ( ) ( )] [ ( ) ( )]

[ ( ) ( )] 0

x dx x dx dx x dx x dx dx

x dx x dx dx B dx dx dx

τ + − τ + τ + − τ

+ τ + − τ + =

31 3 3 31 311 1 1 11 1 21 2 2 21 21

1 2 3

( ) ( )( ) ( ) ( ) ( ) 0x dx xx dx x x dx x Bdx dx dx

τ + − ττ + − τ τ + − τ+ + + =

3111 21

11 2 3

0Bx x x

∂τ∂τ ∂τ+ + + =

∂ ∂ ∂

3212 22

21 2 3

0Bx x x

∂τ∂τ ∂τ+ + + =

∂ ∂ ∂

13 23 33

31 2 3

0Bx x x

∂τ ∂τ ∂τ+ + + =

∂ ∂ ∂

= − ρB u

Balance of Momentum:

∇ ⋅ = ρuτ

Wave Equation

s= ∇ uε

:=τ εC

s:∇ ⋅ ∇ = ρC u u For isotropic materials:

2( )λ+ μ ∇∇ ⋅ + μ∇ = ρu u u indicial notation

, ,( ) j ji i jj iu u uλ+ μ + μ = ρ

detailed differential equation form

22 2 2 2 2 231 2 1 1 1 1

2 2 2 2 21 2 1 31 1 2 3( )( ) ( )uu u u u u u

x x x xx x x x t∂∂ ∂ ∂ ∂ ∂ ∂

λ+μ + + + μ + + = ρ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

22 2 2 2 2 2

31 2 2 2 2 22 2 2 2 21 2 2 32 1 2 3

( )( ) ( )uu u u u u ux x x xx x x x t

∂∂ ∂ ∂ ∂ ∂ ∂λ+μ + + + μ + + = ρ

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

2 2 2 2 22 2

3 3 3 3 31 22 2 2 2 21 3 2 3 3 1 2 3

( )( ) ( )u u u u uu ux x x x x x x x t

∂ ∂ ∂ ∂ ∂∂ ∂λ+μ + + + μ + + = ρ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

Plane Wave Solutions

( t)iA e − ω= k xu p

amplitude A

angular frequency ω

polarization unit vector p wave vector k=k d

propagation unit vector d

wave number 2 2 21 2 3k k k k= + +

sound velocity ckω

=

1 1 2 2 3 3( )1 1

i k d x d x d xi tu A p e e + +− ω=

1 1 2 2 3 3( )2 2

i k d x d x d xi tu A p e e + +− ω=

1 1 2 2 3 3( )3 3

i k d x d x d xi tu A p e e + +− ω=

2

1 1 1 2 1 3 12

1 2 2 2 2 3 22 31 3 2 3 3 3

( ) ( ) ( ) ( ) 0( ) ( ) ( ) ( ) 0

0( ) ( ) ( ) ( )

d d c d d d d pd d d d c d d p

pd d d d d d c

⎡ ⎤λ+μ + μ−ρ λ+μ λ+μ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥λ+μ λ+μ + μ−ρ λ+μ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥λ+μ λ+μ λ+μ + μ−ρ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

Christoffel's Equation

[ ]1

2

3

000

ppp

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥Γ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

0Γ =

Since the material is isotropic, 1 1 2 3( 1, 0)d d d= = = =d e

can be assumed without loss of generality.

21

22

2 3

2 0 0 00 0 0

00 0

c pc p

pc

⎡ ⎤λ+ μ−ρ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥μ−ρ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥μ−ρ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

Longitudinal (or dilatational) wave

d2c λ+ μ

=ρ

and 2 3 0p p= =

Shear (or transverse) wave

sc μ=

ρ and 1 0p =

Symmetry Considerations lack of rotation

ijkl jikl ijlk jilkC C C C= = =

reciprocity

ijkl klijC C=

Independent elastic constants

most general anisotropic 21 orthorhombic 9 cubic symmetry 3 isotropic 2

ABBREVIATED NOTATION

11 12 13

21 22 23

31 32 33

ε ε ε⎡ ⎤⎢ ⎥= ε ε ε⎢ ⎥⎢ ⎥ε ε ε⎣ ⎦

ε 11 12 13

21 22 23

31 32 33

τ τ τ⎡ ⎤⎢ ⎥= τ τ τ⎢ ⎥⎢ ⎥τ τ τ⎣ ⎦

τ

Stiffness matrix:

11 12 13 14 15 1611 11

12 22 23 24 25 2622 22

13 23 33 34 35 3633 33

14 24 34 44 45 4623 23

15 25 35 45 55 5631 31

16 26 36 46 56 6612 12

222

C C C C C CC C C C C CC C C C C CC C C C C CC C C C C CC C C C C C

τ ε⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢τ ε

= ⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢

τ ε⎢ ⎥ ⎢⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

⎥⎥⎥⎥⎥⎥⎥⎥

Simplest Anisotropy, Cubic Symmetry

[100]

[010]

[001]

[110]

[111]

11 11 12 12 11

22 12 11 12 22

33 12 12 11 33

23 44 23

31 44 31

12 44 12

0 0 00 0 00 0 0

0 0 0 0 0 20 0 0 0 0 20 0 0 0 0 2

C C CC C CC C C

CC

C

τ ε⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Isotropic Material

11 11 12 12 11

22 12 11 12 22

33 12 12 11 33

23 44 23

31 44 31

12 44 12

0 0 00 0 00 0 0

0 0 0 0 0 20 0 0 0 0 20 0 0 0 0 2

C C CC C CC C C

CC

C

τ ε⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥τ ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

44 11 122C C C= −

11 12 442 , , ,C C C= λ + μ = λ = μ

λ and μ are Lame's constants

11 11

22 22

33 33

23 23

31 31

12 12

2 0 0 02 0 0 0

2 0 0 020 0 0 0 020 0 0 0 020 0 0 0 0

τ ελ μ λ λ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥τ ελ λ μ λ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥τ ελ λ λ μ

=⎢ ⎥ ⎢ ⎥⎢ ⎥τ εμ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥τ εμ⎢ ⎥ ⎢ ⎥⎢ ⎥τ εμ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

++

+

Transformation of Tensors

Rotation of Rectangular Coordinate Axes

First-Rank Tensor

[ '] = [ ][ ]u a u

[a] denotes the transformation matrix

' ' '

' ' '

' ' '

cos cos cos

[ ] = cos cos cos

cos cos cos

x x x y x z

y x y y y z

z x z y z z

a

⎡ ⎤θ θ θ⎢ ⎥

θ θ θ⎢ ⎥⎢ ⎥

θ θ θ⎢ ⎥⎣ ⎦

z = z'

x

y

x'

y'

θ

θ

u = u'z zu

ux

xu'

uy

yu'

z = z'

x

y

x'

y'

θ

θ

u = u'z zu = u'z zu

uxux

xu'xu'

uyuy

yu'yu'

cos sin 0[ ] = sin cos 0

0 0 1a

θ θ⎡ ⎤⎢ ⎥− θ θ⎢ ⎥⎢ ⎥⎣ ⎦

Rotation of Second-Rank Tensors

Symmetric strain tensor: [ ] = [ ][ ]du dxε

[ '] = [ ][ ]du a du

[ '] = [ ][ ]dx a dx

[ '] = [ ][ ][ ]du a dxε

1[ ] = [ ][ '] = [ ] [ ']Tdx a dx a dx−

[ '] = [ ][ ][ ] [ ']Tdu a a dxε

[ '] = [ ][ ][ ]Ta aε ε

Symmetric stress tensor: [ '] = [ ][ ][ ]Ta aτ τ

Bond Transformation

[ '] = [ ] [ ] [ ]TC M C M

[M] is the so-called Bond transformation matrix

2 2 2

12 13 13 11 11 1211 12 132 2 2

22 23 23 21 21 2221 22 232 2 2

32 33 33 31 31 3231 32 3321 31 22 32 23 33 22 33 23 32 21 33 23 31 22 31 21 32

31 11 32 12 33 13 12 33 13 32 13 31 1

2 2 2

2 2 2

2 2 2[ ]

a a a a a a a a a

a a a a a a a a a

a a a a a a a a aMa a a a a a a a a a a a a a a a a aa a a a a a a a a a a a a

=+ + ++ + 1 33 11 32 12 31

11 21 12 22 13 23 12 23 13 22 13 21 11 23 11 22 12 21

a a a a aa a a a a a a a a a a a a a a a a a

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

+⎢ ⎥⎢ ⎥+ + +⎣ ⎦

The Bond method can be applied directly to elastic constants given in abbreviated notation!

Simple Rotation by angle θ around the z axis

2 2

2 2cos sin 0 0 0 sin 2

sin cos 0 0 0 sin 20 0 1 0 0 0[ ]0 0 0 cos sin 00 0 0 sin cos 0

-½ sin 2 ½ sin 2 0 0 0 cos2

M

⎡ ⎤θ θ θ⎢ ⎥

θ θ − θ⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥θ − θ⎢ ⎥

θ θ⎢ ⎥⎢ ⎥θ θ θ⎣ ⎦

[ '] = [ ] [ ] [ ]TC M C M

11 12' 211 11 44( )sin 2

2C CC C C−

= − − θ

11 12' 212 12 44( )sin 2

2C CC C C−

= + − θ

'13 12C C=

11 12'16 44( )sin 2 cos2

2C CC C−

= − − θ θ

'33 11C C=

'44 44C C=

11 12' 266 44 44( )sin 2

2C CC C C−

= + − θ

the other matrix elements are zero

Coupled Normal Stress and Shear Stain

11 12'16 44( )sin 2 cos2

2C CC C−

= − − θ θ

11 12 13 14 15 1611 11

12 22 23 24 25 2622 22

13 23 33 34 35 3633 33

14 24 34 44 45 4623 23

15 25 35 45 55 5631 31

16 26 36 46 56 6612 12

222

C C C C C CC C C C C CC C C C C CC C C C C CC C C C C CC C C C C C

τ ε⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢τ ε

= ⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢τ ε⎢ ⎥⎢ ⎥ ⎢

τ ε⎢ ⎥ ⎢⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

⎥⎥⎥⎥⎥⎥⎥⎥

11

1612

12

C ∂τ=

∂ε

symmetry direction (or isotropic) off symmetry direction

τ11

ε12 = 0

τ11

ε12 = 0/

Christoffel's Equation for an Anisotropic Solid

2( ) 0ijk j ik kC d d c p− ρδ =

211 12 13 1

212 22 23 2

2 313 23 33

000

c pc p

pc

⎡ ⎤λ −ρ λ λ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥λ λ −ρ λ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥λ λ λ −ρ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

It is customary in the literature to denote the direction cosines 1 2 3, , and by , , andd d d m n .

2 2 2

11 11 66 55 56 15 162 2 2C m C n C mnC n C m Cλ = + + + + +

2 2 222 66 22 44 24 46 262 2 2C m C n C mnC n C m Cλ = + + + + +

2 2 233 55 44 33 34 35 452 2 2C m C n C mnC n C m Cλ = + + + + +

2 2 212 16 26 45 46 25 14 56 12 66( ) ( ) ( )C m C n C mn C C n C C m C Cλ = + + + + + + + +

2 2 213 15 46 35 45 36 13 55 14 56( ) ( ) ( )C m C n C mn C C n C C m C Cλ = + + + + + + + +

2 2 223 56 24 34 44 23 36 45 25 46( ) ( ) ( )C m C n C mn C C n C C m C Cλ = + + + + + + + +

Pure mode longitudinal waves:

p d

i ip d= (i = 1,2,3)

Pure mode shear waves: ⊥p d

0i ip d = ( 1 1 2 2 3 3 0p d p d p d+ + = )

Cubic Crystals

Christoffel's equation:

211 12 13 1

212 22 23 2

2 313 23 33

000

c pc p

pc

⎡ ⎤λ −ρ λ λ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥λ λ −ρ λ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥λ λ λ −ρ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

2 2 211 11 44( )C m n Cλ = + +

2 2 222 11 44( )m C n Cλ = + +

2 2 233 11 44( )n C m Cλ = + +

12 12 44( )m C Cλ = +

13 12 44( )n C Cλ = +

23 12 44( )mn C Cλ = +

three axes of symmetry: [100], [110] and, [111]

Pure Modes Along Symmetry Axes

Sound Wave Propagating along the [100] Direction:

1, 0m n= = =

2

11 12

44 22 344

0 0 00 0 0

00 0

C c pC c p

pC c

⎡ ⎤−ρ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−ρ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−ρ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

Characteristic equation:

2 2 211 44( )( ) 0C c C c−ρ −ρ =

Wave speeds (eigenvalues):

111

Cc =ρ

442 3

Cc c= =ρ

(no birefringence)

Polarizations (eigenvectors): 2

1 11( ) 0p C c−ρ =

22 44( ) 0p C c−ρ =

23 44( ) 0p C c−ρ =

For c1 1 2 31 and 0p p p= = = (pure longitudinal wave)

For c2 or c3 2 21 2 30 and 1p p p= + = (pure transverse waves)

[100]

[010]

[001]

c1c2 = c3

Sound Wave Propagating along the [110] Direction:

1 2 and 0m n= = =/

2

11 44 12 44 12

12 44 11 44 22 344

½ ( ) ½ ( ) 0 0½ ( ) ½ ( ) 0 0

00 0

C C c C C pC C C C c p

pC c

⎡ ⎤+ −ρ + ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + −ρ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−ρ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

Characteristic equation:

2 2 2 2

11 44 12 44 44[( 2 ) ( ) ]( ) 0C C c C C C c+ − ρ − + −ρ =

Wave speeds (eigenvalues):

11 12 441

22

C C Cc + +=

ρ

11 122 2

C Cc −=

ρ

443

Cc =ρ

Polarizations (eigenvectors):

For 1c c= :

12 44 12 44 1

12 44 12 44 2

11 12 3

½ ( ) ½ ( ) 0 0½ ( ) ½ ( ) 0 0

0 0 ½ ( ) 0

C C C C pC C C C p

C C p

− + + ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥+ − + =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− +⎣ ⎦ ⎣ ⎦⎣ ⎦

1 2 3( 1 2) and 0p p p= = =/ (pure longitudinal wave)

For 2c c= :

12 44 12 44 1

12 44 12 44 2

44 11 12 3

½ ( ) ½ ( ) 0 0½ ( ) ½ ( ) 0 0

0 0 ½ ( ) 0

C C C C pC C C C p

C C C p

+ + ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥+ + =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎣ ⎦

44 11 12½ ( ) 0C C C− − ≠

3 0p = and 1 2 1 2( . . , 1 2 and 1 2)p p e g p p= − = = −/ /

pure shear wave polarized in the [1 10 ]

For 3c c= :

11 44 12 44 1

12 44 11 44 2

3

½ ( ) ½ ( ) 0 0½ ( ) ½ ( ) 0 0

0 0 0 0

C C C C pC C C C p

p

− + ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥+ − =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

1 2 30 and 1p p p= = =

pure shear wave polarized in the [001] direction.

[100]

[010]

[001]

[110]

c1c2

c3

Sound Wave Propagating along the [111] Direction:

1 3m n= = = /

2

11 12 12 12

12 11 12 22 312 12 11

000

c pc p

pc

⎡ ⎤λ −ρ λ λ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥λ λ −ρ λ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥λ λ λ −ρ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦

111 11 443

( 2 )C C=λ +

112 12 443

( )C C=λ +

Adding the three rows

2

11 12 1 2 31( 2 )( ) 0c p p pλ + λ − ρ + + =

Characteristic equation: 2

11 12 12 0cλ + λ − ρ =

Wave speed (eigenvalue):

11 12 11 12 441

2 2 43

C C Cc λ + λ + += =

ρ ρ

Polarization (eigenvector):

1 2 3 ( 1 3)p p p= = = /

pure longitudinal mode

Shear modes:

1 2 3 0p p p+ + =

For example, 1 2 3and 0p p p= − = ( 1p is either 1 2/ or 1 2− / ).

Characteristic equation: 2

11 12 0cλ − λ − ρ =

Wave speeds (eigenvalues):

11 12 11 12 442 3 3

C C Cc c λ − λ − += = =

ρ ρ

(no birefringence)

Polarization (eigenvector):

1 2 3and 0p p p= − =

in the (111) plane

[100]

[010]

[001]

[111]

c1c2 = c3

For Nickel, the pure longitudinal wave velocities are:

[100] c1 = 5,299 m/s [110] c1 = 6,027 m/s [111] c1 = 6,251 m/s isotropic cd = 6,032 m/s

Anisotropy Factor for Cubic Crystals

4411 12

2CAC C

=−

1AΔ = − (zero for isotropic materials)

For isotropic materials: 1A =

Ani

sotro

py F

acto

r

0

1

2

3

Sodi

umFl

uorid

e

Yttr

ium

Iron

Gar

net

Fuse

d Si

lica

(Iso

tropi

c)

Tung

sten

Alu

min

um

Dia

mon

d

Silic

on Iron

Nic

kel

Gol

d

Silv

er

Velocity Distributions in the (001) Plane

[100]

[010]longitudinal

shear

Aluminum

[100]

[010]

Nickel

(1 km/s per division)

Anisotropic Phenomena

orientation-dependent acoustic velocity

Transducer

Specimen

dA

d B

Longitudinal

polarization-dependent transverse velocity (birefringence)

Transducer

SpecimenpA

d

pBShear

skewed polarizations (quasi-modes)

deviation between phase and energy directions (beam skewing),

etc.

Birefringence

"Fast" Mode "Slow" Mode

0o

22.5o

90o

45o

67.5o

22 2

2( ) cos ( ) ( )dS t u t

c= θ −

23 3

3( ) cos ( ) ( )dS t u t

c= θ −

3 2cos( ) sin( )θ = θ

2 22 3

( ) cos ( ) ( ) sin ( ) ( )d dS t u t u tc c

= θ − + θ −

Quasi-Modes, Skewed Polarizations

isotropic medium anisotropic medium

(no skewing, no birefringence) (skewing, birefringence)

p L

pS

dx

y

z

pQL

pQS1

dx

y

z

pQS2

Particle orientations are always mutually orthogonal.

Huygens' Principle, Isotropic Case

x

θ

P r,( )θ

y

Time delay to P(r,θ) from source point at x:

( , ; )( , ; ) ( , ; ) ( , ; )( , ; )r xt r x r x s r x

c r xθ

θ = = θ θθ

0( , ; ) ( , ; )t r x r x sθ = θ

0( , ; ) ( sin )t r x r x sθ ≈ − θ

0( , ; ) sin 0t r x s

x∂ θ

≈ − θ =∂

0θ = (no skewing)

Huygens' Principle, Anisotropic Case

cos( , ; ) ( , ; ) ( , ; ) ( sin ) ( )xt r x r x s r x r x srθ

θ = θ θ ≈ − θ θ −

cos( , ; ) ( sin )[ ( ) ]x st r x r x srθ ∂

θ ≈ − θ θ −∂θ

( , ; ) ( ) ( )sin cos st r x s r x s x ∂θ ≈ θ − θ θ − θ

∂θ

( , ; ) ( )sin cos 0t r x ssx

∂ θ ∂≈ − θ θ − θ =

∂ ∂θ

1tan( )

ss

∂θ = −

θ ∂θ

0θ ≠ (skewing)

slowness

curveray (group)direction

d

wave (phase)direction

∂∂θ

s ΔθΔθ s( )θ

Δθ

Slowness Diagrams Velocity:

propagation distancecpropagation time

≡

Slowness: propagation times

propagation distance≡

1s c−≡

Applications:

wave direction is determined by the wave speed

(group velocity, refraction, diffraction, scattering)

Nickel

[100]

[010]

quasi-longitudinaltrue shear

[100]

[010]

quasi-shear

velocity diagram slowness diagram

Wave (Phase) Direction vs Ray (Group) Direction

2D-case

RQLRQS1

RQS2

wavedirection

In general, the three group directions (solid arrows) are different from the wave direction!

Ray Direction Analogy with dispersion:

( )f f t= ω − k x

Phase velocity:

pckω

=

Group velocity:

gck∂ω

=∂

slownesssurface

d

cg

x2

x3

x1

γ

cp

Phase

Direction

RayDirection

Beam ContourPhase Plane

Transmitter