Àε 11 φ
0
ð Þ
,aa þ κe 15 u
0
ð Þ
3,aa þ u
1
ð Þ
a,a
þ
1
2h
d
0
ð Þ
¼ q p
0
ð Þ
À n
0
ð Þ
,
Àζ 11 φ
1
ð Þ
,aa À 3h
À2 e 31 u
0
ð Þ
a,a À ε 33 φ
1
ð Þ
þ
3
2h
3
d
1
ð Þ
¼ q p
1
ð Þ
À n
1
ð Þ
,
ð5:87Þ
and semiconduction described by p
(0) , p
(1) , n
(0) and n
(1) :
Àp 0 μ
p
11 φ
0
ð Þ
,aa À d
p
11 p
0
ð Þ
,aa þ
1
2hq
j
p 0
ð Þ
¼ À_ p
0
ð Þ ,
Àp 0 μ
p
11 φ
1
ð Þ
,aa À d
p
11 p
1
ð Þ
,aa
À3h
À2
Àp 0 μ
p
33 φ
1
ð Þ
À d
p
33 p
1
ð Þ
þ
3
2h
3 q
j
p 1
ð Þ
¼ À_ p
1
ð Þ ,
ð5:88Þ
Àn 0 μ
n
11 φ
0
ð Þ
,aa þ d
n
11 n
0
ð Þ
,aa þ
1
2hq
j
n 0
ð Þ
¼ _
n
0
ð Þ ,
Àn 0 μ
n
11 φ
1
ð Þ
,aa þ d
n
11 n
1
ð Þ
,aa
À3h
À2
Àn 0 μ
n
33 φ
1
ð Þ
þ d
n
33 n
1
ð Þ
þ
3
2h
3 q
j
n 1
ð Þ
¼ _
n
1
ð Þ ,
ð5:89Þ
with couplings among them.
5.8 Thickness-Shear Vibration
Consider thickness vibrations of the plate in Fig. 5.2. The plate is unelectroded at the
top and bottom. There are no surface loads. The electric field in the surrounding free
space is neglected.
Thickness vibrations are independent of x 1 and x 2 . With the first-order plate
equations we have, we can only study the fundamental thickness-shear vibration
with one nodal plane (the x 1 –x 2 plane) but not the higher-order thickness-shear
vibrations with more nodal planes and thickness-stretch vibrations. We do not
consider lateral field excitation so that the in-plane electric field is zero. For crystals
of class (6mm), the fundamental thickness-shear motion is not coupled to the
thickness electric field. In addition, crystals of class (6mm) are effectively
x 3
x 1
2h
c
Fig. 5.2 An unbounded
plate of crystals of class
(6mm)
128
5 Extension and Bending of Plates
0
ð Þ
,aa þ κe 15 u
0
ð Þ
3,aa þ u
1
ð Þ
a,a
þ
1
2h
d
0
ð Þ
¼ q p
0
ð Þ
À n
0
ð Þ
,
Àζ 11 φ
1
ð Þ
,aa À 3h
À2 e 31 u
0
ð Þ
a,a À ε 33 φ
1
ð Þ
þ
3
2h
3
d
1
ð Þ
¼ q p
1
ð Þ
À n
1
ð Þ
,
ð5:87Þ
and semiconduction described by p
(0) , p
(1) , n
(0) and n
(1) :
Àp 0 μ
p
11 φ
0
ð Þ
,aa À d
p
11 p
0
ð Þ
,aa þ
1
2hq
j
p 0
ð Þ
¼ À_ p
0
ð Þ ,
Àp 0 μ
p
11 φ
1
ð Þ
,aa À d
p
11 p
1
ð Þ
,aa
À3h
À2
Àp 0 μ
p
33 φ
1
ð Þ
À d
p
33 p
1
ð Þ
þ
3
2h
3 q
j
p 1
ð Þ
¼ À_ p
1
ð Þ ,
ð5:88Þ
Àn 0 μ
n
11 φ
0
ð Þ
,aa þ d
n
11 n
0
ð Þ
,aa þ
1
2hq
j
n 0
ð Þ
¼ _
n
0
ð Þ ,
Àn 0 μ
n
11 φ
1
ð Þ
,aa þ d
n
11 n
1
ð Þ
,aa
À3h
À2
Àn 0 μ
n
33 φ
1
ð Þ
þ d
n
33 n
1
ð Þ
þ
3
2h
3 q
j
n 1
ð Þ
¼ _
n
1
ð Þ ,
ð5:89Þ
with couplings among them.
5.8 Thickness-Shear Vibration
Consider thickness vibrations of the plate in Fig. 5.2. The plate is unelectroded at the
top and bottom. There are no surface loads. The electric field in the surrounding free
space is neglected.
Thickness vibrations are independent of x 1 and x 2 . With the first-order plate
equations we have, we can only study the fundamental thickness-shear vibration
with one nodal plane (the x 1 –x 2 plane) but not the higher-order thickness-shear
vibrations with more nodal planes and thickness-stretch vibrations. We do not
consider lateral field excitation so that the in-plane electric field is zero. For crystals
of class (6mm), the fundamental thickness-shear motion is not coupled to the
thickness electric field. In addition, crystals of class (6mm) are effectively
x 3
x 1
2h
c
Fig. 5.2 An unbounded
plate of crystals of class
(6mm)
128
5 Extension and Bending of Plates