where ∇ ¼ i 1 ∂ 1 + i 2 ∂ 2 is the two-dimensional gradient operator and ∇
2
¼ ∂
2
1 þ ∂
2
2
is the two-dimensional Laplacian. Introduce [2]
ψ ¼ φ À
e 15
ε 11
u 3 ,
ð6:5Þ
then
T 23 ¼ c 44 u 3,2 þ e 15 ψ ,2 ,
T 31 ¼ c 44 u 3,1 þ e 15 ψ ,1 ,
ð6:6Þ
D 1 ¼ Àε 11 ψ ,1 ,
D 2 ¼ Àε 11 ψ ,2 ,
ð6:7Þ
c 44 ∇
2 u 3 ¼ ρ€ u 3 ,
∇
2 ψ ¼ 0,
ð6:8Þ
where
c 44 ¼ c 44 þ
e
2
15
ε 11
¼ c 44 1 þ k
2
15
À
Á
, k
2
15 ¼
e
2
15
ε 11 c 44
:
ð6:9Þ
For a surface wave solution, we require that
u 3 , φ ! 0, x 2 ! þ1:
ð6:10Þ
Consider possible solutions in the following form:
u 3 ¼ A exp Àξ 2 x 2 À h
ð
Þ
½
Š exp i ξ 1 x 1 À ωt
ð
Þ
½
Š ,
ψ ¼ B exp Àξ 1 x 2 À h
ð
Þ
½
Š exp i ξ 1 x 1 À ωt
ð
Þ
½
Š ,
ð6:11Þ
where A and B are undetermined constants and ξ 2 should be positive for decaying
behavior away from the surface. Equation (6.11) 2 already satisfies Eq. (6.8) 2 . For
Eq. (6.11) 1 to satisfy Eq. (6.8) 1 , we must have
c 44 ξ
2
1 À ξ
2
2
À
Á ¼ ρω
2 ,
ð6:12Þ
Propagation
direction
x2
x1
2h
Semiconductor plate
Free space
Piezoelectric
half space
Fig. 6.1 A piezoelectric
dielectric half space with a
nonpiezoelectric
semiconductor plate
142
6 Composite Structures