u
0
ð Þ
3 ¼ A 3 exp i ωt þ ξ a x a
ð
Þ
½
Š ,
u
1
ð Þ
b ¼ A b exp i ωt þ ξ a x a
ð
Þ
½
Š ,
p
0
ð Þ
À n
0
ð Þ
¼ B exp i ωt þ ξ a x a
ð
Þ
½
Š ,
ð5:95Þ
where A i and B are constants. The substitution of Eq. (5.95) into Eq. (5.94) results in
κ
2 c 44 þ
e
2
15
ε 11
ÀA 3 ξ a ξ a þ A a iξ a
ð
Þ À
κe 15 q
ε 11
B ¼ Àρω
2 A 3 :
ð5:96Þ
We are interested in long waves with small wave numbers or small ξ a . Then the term
quadratic in ξ a in the above equation can be neglected. For long thickness-shear
waves, the frequency ω is very close to the exact fundamental thickness-shear
frequency of an infinite plate given in Eq. (5.92). Hence we substitute Eq. (5.92)
into Eq. (5.96). This leads to the following approximate version of Eq. (5.96):
A 3 ¼ À
h
2
3
1 þ
e
2
15
c 44 ε 11
iξ a A a þ
4h
2
κe 15 q
π 2 c 44 ε 11
B,
ð5:97Þ
which is equivalent to the following differential relationship:
u
0
ð Þ
3 ¼ À
h
2
3
1 þ k
2
15
À
Á
u
1
ð Þ
a,a þ
4h
2
κe 15 q
π 2 c 44 ε 11
p
0
ð Þ
À n
0
ð Þ
,
ð5:98Þ
where we have denoted the shear electromechanical coupling factor by
k
2
15 ¼
e
2
15
c 44 ε 11
:
ð5:99Þ
Substituting Eq. (5.98) into Eqs. (5.86) and (5.87) 1 , neglecting the third- and higherorder derivatives of u
0
ð Þ
3 under the long-wave approximation, we obtain the following
equations under the thickness-shear approximation:
b c 11 u
1
ð Þ
1,11 þ c 66 u
1
ð Þ
1,22 þ b c 12 u
1
ð Þ
2,12 À ρω
2
1 u
1
ð Þ
1
À3h
À2
κe 15 φ
0
ð Þ
,1 À
κe 15 q
ε 11
p
0
ð Þ
,1 À n
0
ð Þ
,1
¼ ρ€ u
1
ð Þ
1 ,
c 66 u
1
ð Þ
2,11 þ b c 11 u
1
ð Þ
2,22 þ b c 12 u
1
ð Þ
1,12 À ρω
2
1 u
1
ð Þ
2
À3h
À2
κe 15 φ
0
ð Þ
,2 À
κe 15 q
ε 11
p
0
ð Þ
,2 À n
0
ð Þ
,2
¼ ρ€ u
1
ð Þ
2 ,
Àε 11 φ
0
ð Þ
,aa þ κe 15 u
1
ð Þ
a,a þ k
2
15
h
2
3
q p
0
ð Þ
,aa À n
0
ð Þ
,aa
¼ q p
0
ð Þ
À n
0
ð Þ
,
ð5:100Þ
130
5 Extension and Bending of Plates