3.13 Nonlinear Waves
Consider the rod shown in Fig. 3.24. It is doped into an n-type semiconductor with
p ffi 0. We study the propagation of extensional waves with consideration of
electrical nonlinearity [10]. The analysis is somewhat crude with several approximations. It is hoped that the solution can still capture some of the physics involved.
The governing equations are as follows:
T 3,3 ¼ ρ € u 3 ,
D 3,3 ¼ q Àn þ N
þ
D
À
Á
,
J
n
3,3 ¼ q _
n,
ð3:175Þ
T 3 ¼ c 33 S 3 À e 33 E 3 ,
D 3 ¼ e 33 S 3 þ ε 33 E 3 ,
J
n
3 ¼ qnμ 33 E 3 þ qD 33 n ,3 ,
ð3:176Þ
S 3 ¼ u 3,3 ,
E 3 ¼ Àφ ,3 ,
ð3:177Þ
Fig. 3.23 Distribution of Δn along the rod at different time instants. m ¼ 0–15, r ¼ 0–7. (a)
t ¼ 4 Â 10
À11
s. (b) t ¼ 10.7 Â 10
À11
s. (c) t ¼ 18 Â 10
À11
s. (d) t ¼ 22 Â 10
À11
s
72
3 Extension of Rods
Consider the rod shown in Fig. 3.24. It is doped into an n-type semiconductor with
p ffi 0. We study the propagation of extensional waves with consideration of
electrical nonlinearity [10]. The analysis is somewhat crude with several approximations. It is hoped that the solution can still capture some of the physics involved.
The governing equations are as follows:
T 3,3 ¼ ρ € u 3 ,
D 3,3 ¼ q Àn þ N
þ
D
À
Á
,
J
n
3,3 ¼ q _
n,
ð3:175Þ
T 3 ¼ c 33 S 3 À e 33 E 3 ,
D 3 ¼ e 33 S 3 þ ε 33 E 3 ,
J
n
3 ¼ qnμ 33 E 3 þ qD 33 n ,3 ,
ð3:176Þ
S 3 ¼ u 3,3 ,
E 3 ¼ Àφ ,3 ,
ð3:177Þ
Fig. 3.23 Distribution of Δn along the rod at different time instants. m ¼ 0–15, r ¼ 0–7. (a)
t ¼ 4 Â 10
À11
s. (b) t ¼ 10.7 Â 10
À11
s. (c) t ¼ 18 Â 10
À11
s. (d) t ¼ 22 Â 10
À11
s
72
3 Extension of Rods