Environmental Engineering Reference
In-Depth Information
independently to avoid dealing with the vector wave equations.
According to the normalization condition of the vector potential
H
=
0,
H
x
,
H
y
can be easily eliminated and then Eq. 4.25 can be
further simplified as
v
T
∇
2
u
z
+
ω
2
u
z
=
0.
(4.26)
Interestingly, Eq. 4.26, which describes the vibrational displace-
ment of the out-of-mode, has formally the same structure as Eqs.
4.9-4.11. As expounded in the elastic textbook by Graff [81], the
in-plane modes are generally split into the longitudinal polarized
P wave, the vertically polarized SV wave, and the out-of-plane
mode is renamed as the horizontally polarized shear SH wave.
Their polarization directions are along the
x
,
y
,and
z
directions,
respectively. Note that because both the P wave and the SV wave in
x
-
y
plane couple with each other, the detailed discussion of their
ballistic transmission properties in quantum systems becomes very
complicated, whichisalso different from the case for the SH wave.
4.2.3
Scattering-Matrix Method
For the convenience of elaboration in the following text, we present
asimple2Dmodelstructure,whichhasbeendoneinmanyprevious
studies, as shown in Fig. 4.1. In such a structure, we usually divide
all the system into three regions: the left semi-infinite region I, the
central region II, and the right semi-infinite region III, of which
the corresponding transverse width is denoted by
W
I
,
W
III
and
W
II
,
respectively. It is assumed that the contacts between the two leads
and reservoirs are perfect without any reflection and the elastic
scattering only occurs at attached interfaces between two blocks
withdifferent width.
Firstly, we consider the simplest case for the SH wave incidence,
whichis described by
v
T
∇
2
2
2
ψ
+
ω
ψ
=
0
(4.27)
2
2
/∂
x
2
2
/∂
y
2
. The solution of the above
with
ψ
=
u
z
,
∇
=
∂
+
∂
equation defined in each region
I, II, and III) are given in
terms of linear combination of the transmitted and reflected waves
ξ
(
ξ
=
