where we have denoted
b c 11 ¼ c 11 þ κ
2 c 44 1 þ k
2
15
À
Á
,
b c 12 ¼ c 12 þ c 66 þ κ
2 c 44 1 þ k
2
15
À
Á :
ð5:101Þ
5.10 Propagation of Thickness-Shear Waves
We consider straight-crested waves propagating in the x 1 direction of a p-doped plate
of crystals of class (6mm) with u 2 ¼ 0 and ∂/∂x 2 ¼ 0 [1]. Then Eqs. (5.100) and
(5.88) 1 reduce to
b c 11 u
1
ð Þ
1,11 À ρω
2
1 u
1
ð Þ
1 À 3h
À2
κe 15 φ
0
ð Þ
,1 À
κe 15 q
ε 11
p
0
ð Þ
,1 ¼ ρ€ u
1
ð Þ
1 ,
Àε 11 φ
0
ð Þ
,11 þ κe 15 u
1
ð Þ
1,1 þ k
2
15
h
2
3
qp
0
ð Þ
,11 ¼ qp
0
ð Þ ,
_
p
0
ð Þ
À p 0 μ
p
11 φ
0
ð Þ
,11 À d
p
11 p
0
ð Þ
,11 ¼ 0:
ð5:102Þ
Let
u
1
ð Þ
1 ¼ A exp i ξ 1 x 1 À ωt
ð
Þ
½
Š ,
φ
0
ð Þ
¼ B exp i ξ 1 x 1 À ωt
ð
Þ
½
Š ,
p
0
ð Þ
¼ C exp i ξ 1 x 1 À ωt
ð
Þ
½
Š ,
ð5:103Þ
where A, B, and C are undetermined constants. The substitution of Eq. (5.103) into
Eq. (5.102) yields the following linear homogeneous equation for A, B, and C:
ρ ω
2
À ω
2
1
À
Á À b c 11 ξ
2
1
À3h
À2
κe 15 iξ 1
À
κe 15 q
ε 11
iξ 1
κe 15 iξ 1
ε 11 ξ
2
1
Àk
2
15
h
2
3
qξ
2
1 À q
0
p 0 μ
p
11 ξ
2
1
Àiω þ d
p
11 ξ
2
1
2
6
6
6
6
4
3
7
7
7
7
5
A
B
C
8
> <
> :
9
> =
> ;
¼ 0: ð5:104Þ
For nontrivial solutions the determinant of the coefficient matrix of Eq. (5.104) has
to vanish, which gives the following equation that determines the dispersion relations of the wave:
ρω
2
¼ ρb ω
2
1 þ b c 11 ξ
2
1 À
p 0 qμ
p
11
ε 11 d
p
11 ξ
2
1 À iω
À
Á
 1 þ
1
3
h
2 k
2
15 ξ
2
1
ρ ω
2
À ω
2
1
À
Á À ξ
2
1 b c 11
Â
à þ κ
2 k
2
15 c 44 ξ
2
1
n
o
,
ð5:105Þ
5.10 Propagation of Thickness-Shear Waves
131