where μ 33 ¼ μ
n
33 and D 33 ¼ D
n
33 . With the use of Eq. (3.177), we write Eq. (3.176) as
T 3 ¼ c 33 u 3,3 þ e 33 φ ,3 ,
D 3 ¼ e 33 u 3,3 À ε 33 φ ,3 ,
J 3 ¼ Àqnμ 33 φ ,3 þ qD 33 n ,3 :
ð3:178Þ
The substitution of Eq. (3.178) into Eq. (3.175) yields three equations for u 3 , φ, and
n:
c 33 u 3,33 þ e 33 φ ,33 ¼ ρ € u 3 ,
e 33 u 3,33 À ε 33 φ ,33 ¼ q N
þ
À n
ð
Þ ,
Àμ 33 φ ,33 n À μ 33 φ ,3 n ,3 þ D 33 n ,33 ¼ _
n:
ð3:179Þ
The first two terms in Eq. (3.179) 3 are nonlinear.
Assuming low doping and low carrier concentration, we make an approximation
by dropping the charges q(N
+
À n) on the right-hand side of Eq. (3.179) 2 . This
reduces Eq. (3.179) into two one-way coupled systems of
c 33 u 3,33 þ e 33 φ ,33 ¼ ρ€ u 3 ,
e 33 u 3,33 À ε 33 φ ,33 ¼ 0,
ð3:180Þ
and
Àμ 33 φ ,33 n À μ 33 φ ,3 n ,3 þ D 33 n ,33 ¼ _
n:
ð3:181Þ
Equation (3.180) is the classical theory of piezoelectricity within which harmonic
extensional waves in the rod are described by
u 3 ¼ A cos ψ, φ ¼ B cos ψ,
ð3:182Þ
where
x3
x2
x1
c
Fig. 3.24 A piezoelectric
semiconductor rod of
crystals of class (6mm)
3.13 Nonlinear Waves
73