T ab,a þ
1
2h
T 2b x 2 ¼ h
ð
ÞÀT 2b x 2 ¼ Àh
ð
Þ
½
¼ ρ€ u b ,
D a,a þ
1
2h
D 2 x 2 ¼ h
ð
ÞÀD 2 x 2 ¼ Àh
ð
Þ
½
¼ q Δp
ð Þ,
q
∂ Δp
ð Þ
∂t
þ J a,a þ
1
2h
J 2 x 2 ¼ h
ð
ÞÀJ 2 x 2 ¼ Àh
ð
Þ
½
¼ 0,
ð6:26Þ
where u a , T ab , D a , J a , and Δp are independent of x 2 and are understood to be the
averages of the corresponding fields along the plate thickness. The specific motion of
the plate relevant to the present problem is described by u 3 (x 1 , t) and is called the
face-shear (FS) wave of a plate. Consider the following wave solution:
u 3 ¼ A exp i ξ 1 x 1 À ωt
ð
Þ
½
, φ ¼ C exp i ξ 1 x 1 À ωt
ð
Þ
½
,
Δp ¼ P exp i ξ 1 x 1 À ωt
ð
Þ
½
,
ð6:27Þ
where P is an undetermined constant. Equation (6.27) already satisfies the continuity
of displacement between the plate and the ceramic half space and the continuity of
electric potential between the plate and the free space. We consider a plate of silicon
which is a cubic crystal of class (m3m). The elastic and dielectric constants are given
by
c
0
11
c
0
12
c
0
12
0
0
0
c
0
12
c
0
11
c 12 0
0
0
c
0
12
c
0
12
c
0
11
0
0
0
0
0
0 c
0
44
0
0
0
0
0
0 c
0
44
0
0
0
0
0
0 c
0
44
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
,
ε
0
11
0
0
0 ε
0
11
0
0
0 ε
0
11
0
B
@
1
C
A:
ð6:28Þ
Then the constitutive relations take the following specific form:
T 13 ¼ c
0
55 S 13 ¼ c
0
44 u 3,1 ¼ c
0
44 iξ 1 A exp i ξ 1 x 1 À ωt
ð
Þ
½
,
D 1 ¼ ε
0
11 E 1 ¼ Àε
0
11 φ ,1 ¼ Àε
0
11 iξ 1 C exp i ξ 1 x 1 À ωt
ð
Þ
½
,
J
p
1 ¼ Àqp 0 μ
p
11 φ ,1 À qD
p
11 Δp
ð Þ ,1
¼ Àqp 0 μ
p
11 iξ 1 C À qD
p
11 iξ 1 P
À
Á
exp i ξ 1 x 1 À ωt
ð
Þ
½
:
ð6:29Þ
The substitution of Eqs. (6.11), (6.15), (6.17), (6.18), (6.27), and (6.29) into the
continuity condition of the electric potential between the ceramic half space and the
plate, Eq. (6.26) 1 for b ¼ 3, and Eq. (6.26) 2,3 yields
6.1 Surface Waves
145
1
2h
T 2b x 2 ¼ h
ð
ÞÀT 2b x 2 ¼ Àh
ð
Þ
½
¼ ρ€ u b ,
D a,a þ
1
2h
D 2 x 2 ¼ h
ð
ÞÀD 2 x 2 ¼ Àh
ð
Þ
½
¼ q Δp
ð Þ,
q
∂ Δp
ð Þ
∂t
þ J a,a þ
1
2h
J 2 x 2 ¼ h
ð
ÞÀJ 2 x 2 ¼ Àh
ð
Þ
½
¼ 0,
ð6:26Þ
where u a , T ab , D a , J a , and Δp are independent of x 2 and are understood to be the
averages of the corresponding fields along the plate thickness. The specific motion of
the plate relevant to the present problem is described by u 3 (x 1 , t) and is called the
face-shear (FS) wave of a plate. Consider the following wave solution:
u 3 ¼ A exp i ξ 1 x 1 À ωt
ð
Þ
½
, φ ¼ C exp i ξ 1 x 1 À ωt
ð
Þ
½
,
Δp ¼ P exp i ξ 1 x 1 À ωt
ð
Þ
½
,
ð6:27Þ
where P is an undetermined constant. Equation (6.27) already satisfies the continuity
of displacement between the plate and the ceramic half space and the continuity of
electric potential between the plate and the free space. We consider a plate of silicon
which is a cubic crystal of class (m3m). The elastic and dielectric constants are given
by
c
0
11
c
0
12
c
0
12
0
0
0
c
0
12
c
0
11
c 12 0
0
0
c
0
12
c
0
12
c
0
11
0
0
0
0
0
0 c
0
44
0
0
0
0
0
0 c
0
44
0
0
0
0
0
0 c
0
44
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
,
ε
0
11
0
0
0 ε
0
11
0
0
0 ε
0
11
0
B
@
1
C
A:
ð6:28Þ
Then the constitutive relations take the following specific form:
T 13 ¼ c
0
55 S 13 ¼ c
0
44 u 3,1 ¼ c
0
44 iξ 1 A exp i ξ 1 x 1 À ωt
ð
Þ
½
,
D 1 ¼ ε
0
11 E 1 ¼ Àε
0
11 φ ,1 ¼ Àε
0
11 iξ 1 C exp i ξ 1 x 1 À ωt
ð
Þ
½
,
J
p
1 ¼ Àqp 0 μ
p
11 φ ,1 À qD
p
11 Δp
ð Þ ,1
¼ Àqp 0 μ
p
11 iξ 1 C À qD
p
11 iξ 1 P
À
Á
exp i ξ 1 x 1 À ωt
ð
Þ
½
:
ð6:29Þ
The substitution of Eqs. (6.11), (6.15), (6.17), (6.18), (6.27), and (6.29) into the
continuity condition of the electric potential between the ceramic half space and the
plate, Eq. (6.26) 1 for b ¼ 3, and Eq. (6.26) 2,3 yields
6.1 Surface Waves
145