(ii) When k
2
15 ¼ 0 , i.e., the half space is non-piezoelectric, electromechanical
coupling disappears, and the wave is purely elastic. In this case Eq. (6.32)
reduces to
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
¼ 0,
ð6:35Þ
which is the equation that determines the speed of Love wave, an anti-plane surface
wave in an elastic half space carrying an elastic plate, in the limit when the plate is
very thin compared to the wavelength (ξ 1 h < <1). Love waves are known to exist
when the relevant elastic stiffness of the plate is smaller than that of the half space.
(iii) When semiconduction is small, Eq. (6.32) can be solved by an iteration or
perturbation procedure. As the lowest (zero-) order of approximation, we
neglect the small semiconduction and denote the zero-order solution by v (0) .
Then, from Eq. (6.32),
v
2
0
ð Þ
v 0 2
T
À 1
!
c
0
44
c 44
2hξ 1 À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
v 2
0
ð Þ
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
,
ð6:36Þ
which determines a dispersive wave. For the next order of approximation we
substitute v (0) into the right-hand side of Eq. (6.32) and obtain the following equation
for v (1) :
v
2
1
ð Þ
v 0 2
T
À 1
!
c
0
44
c 44
2hξ 1 À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
v
2
1
ð Þ
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 0
ð Þ
À
Á
,
ð6:37Þ
which is complex and suggests a wave that is both dispersive and dissipative.
6.2 Extension of Rods
Consider the composite rod shown in Fig. 6.2. (x,y,z) correspond to (x 1 ,x 2 ,x 3 ). It
consists of two piezoelectric dielectric layers such as polarized ceramics (PZT) and a
nonpiezoelectric semiconductor layer such as silicon (Si). The composite rod is
within |z| < L. It is under the action of a pair of equal and opposite axial forces F [3].
6.2 Extension of Rods
147
2
15 ¼ 0 , i.e., the half space is non-piezoelectric, electromechanical
coupling disappears, and the wave is purely elastic. In this case Eq. (6.32)
reduces to
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
¼ 0,
ð6:35Þ
which is the equation that determines the speed of Love wave, an anti-plane surface
wave in an elastic half space carrying an elastic plate, in the limit when the plate is
very thin compared to the wavelength (ξ 1 h < <1). Love waves are known to exist
when the relevant elastic stiffness of the plate is smaller than that of the half space.
(iii) When semiconduction is small, Eq. (6.32) can be solved by an iteration or
perturbation procedure. As the lowest (zero-) order of approximation, we
neglect the small semiconduction and denote the zero-order solution by v (0) .
Then, from Eq. (6.32),
v
2
0
ð Þ
v 0 2
T
À 1
!
c
0
44
c 44
2hξ 1 À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
v 2
0
ð Þ
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
,
ð6:36Þ
which determines a dispersive wave. For the next order of approximation we
substitute v (0) into the right-hand side of Eq. (6.32) and obtain the following equation
for v (1) :
v
2
1
ð Þ
v 0 2
T
À 1
!
c
0
44
c 44
2hξ 1 À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À
v
2
1
ð Þ
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 0
ð Þ
À
Á
,
ð6:37Þ
which is complex and suggests a wave that is both dispersive and dissipative.
6.2 Extension of Rods
Consider the composite rod shown in Fig. 6.2. (x,y,z) correspond to (x 1 ,x 2 ,x 3 ). It
consists of two piezoelectric dielectric layers such as polarized ceramics (PZT) and a
nonpiezoelectric semiconductor layer such as silicon (Si). The composite rod is
within |z| < L. It is under the action of a pair of equal and opposite axial forces F [3].
6.2 Extension of Rods
147