B þ
e 15
ε 11
A ¼ C,
Àc
0
44 ξ
2
1 A À
1
2h
Ac 44 ξ 2 þ e 15 Bξ 1
ð
Þ ¼ À ρ
0
ω
2 A,
ε
0
11 ξ
2
1 C þ
1
2h
ε 11 Bξ 1 þ ε 0 ξ 1 C
ð
Þ ¼ qP,
ÀqiωP þ iξ 1 Àqp 0 μ
p
11 iξ 1 C À qD
p
11 iξ 1 P
À
Á ¼ 0,
ð6:30Þ
which is a system of linear homogeneous equations for A, B, C, and P. For nontrivial
solutions the determinant of the coefficient matrix has to vanish:
e 15
ε 11
1
À1
0
ρ
0
ω
2
À c
0
44 ξ
2
1 À
c 44 ξ 2
2h
À
e 15 ξ 1
2h
0
0
0
ε 11 ξ 1
2h
ε 0 ξ 1
2h
þ ε
0
11 ξ
2
1
Àq
0
0
qp 0 μ
p
11 ξ
2
1
Àqiω þ qD
p
11 ξ
2
1
¼ 0, ð6:31Þ
which determines the dispersion relations between ω and ξ 1 of the surface waves. In
terms of the surface wave speed v ¼ ω/ξ 1 , Eq. (6.31) can be written in the following
form:
v
2
v 0 2
T
À 1
c
0
44
c 44
2hξ 1 À
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
v
2
v 2
T
s
þ k
2
15
¼
k
2
15
1 þ
ε 0
ε 11
þ
ε
0
11
ε 11
ξ 1 2h þ
qp 0 μ
p
11 2h
ε 11 D
p
11 ξ 1 À iv
À
Á
,
ð6:32Þ
where Eq. (6.13) has been used, and
v
0 2
T ¼
c
0
44
ρ 0 , k
2
15 ¼
e
2
15
ε 11 c 44
:
ð6:33Þ
We make the following observations from Eq. (6.32):
(i) When h ¼ 0, i.e., the semiconductor plate does not exist, Eq. (6.32) reduces to
v
2
¼ v
2
T 1 À
k
4
15
1 þ ε 11 =ε 0
ð
Þ
2
"
#
¼ v
2
BG ,
ð6:34Þ
which is the speed of the well-known Bleustein-Gulyaev wave [2], an anti-plane or
shear-horizontal (SH) surface wave that relies on piezoelectric coupling and does not
have an elastic counterpart.
146
6 Composite Structures
e 15
ε 11
A ¼ C,
Àc
0
44 ξ
2
1 A À
1
2h
Ac 44 ξ 2 þ e 15 Bξ 1
ð
Þ ¼ À ρ
0
ω
2 A,
ε
0
11 ξ
2
1 C þ
1
2h
ε 11 Bξ 1 þ ε 0 ξ 1 C
ð
Þ ¼ qP,
ÀqiωP þ iξ 1 Àqp 0 μ
p
11 iξ 1 C À qD
p
11 iξ 1 P
À
Á ¼ 0,
ð6:30Þ
which is a system of linear homogeneous equations for A, B, C, and P. For nontrivial
solutions the determinant of the coefficient matrix has to vanish:
e 15
ε 11
1
À1
0
ρ
0
ω
2
À c
0
44 ξ
2
1 À
c 44 ξ 2
2h
À
e 15 ξ 1
2h
0
0
0
ε 11 ξ 1
2h
ε 0 ξ 1
2h
þ ε
0
11 ξ
2
1
Àq
0
0
qp 0 μ
p
11 ξ
2
1
Àqiω þ qD
p
11 ξ
2
1
¼ 0, ð6:31Þ
which determines the dispersion relations between ω and ξ 1 of the surface waves. In
terms of the surface wave speed v ¼ ω/ξ 1 , Eq. (6.31) can be written in the following
form:
v
2
v 0 2
T
À 1
c
0
44
c 44
2hξ 1 À
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
v
2
v 2
T
s
þ k
2
15
¼
k
2
15
1 þ
ε 0
ε 11
þ
ε
0
11
ε 11
ξ 1 2h þ
qp 0 μ
p
11 2h
ε 11 D
p
11 ξ 1 À iv
À
Á
,
ð6:32Þ
where Eq. (6.13) has been used, and
v
0 2
T ¼
c
0
44
ρ 0 , k
2
15 ¼
e
2
15
ε 11 c 44
:
ð6:33Þ
We make the following observations from Eq. (6.32):
(i) When h ¼ 0, i.e., the semiconductor plate does not exist, Eq. (6.32) reduces to
v
2
¼ v
2
T 1 À
k
4
15
1 þ ε 11 =ε 0
ð
Þ
2
"
#
¼ v
2
BG ,
ð6:34Þ
which is the speed of the well-known Bleustein-Gulyaev wave [2], an anti-plane or
shear-horizontal (SH) surface wave that relies on piezoelectric coupling and does not
have an elastic counterpart.
146
6 Composite Structures