Environmental Engineering Reference
In-Depth Information
−
B
II
). (4.45)
Note that Eqs. 4.44 and 4.45 are series of homogeneous linear
equations, and can berewritten in the formof matrix
A
I
D
T
K
I
(
A
I
−
B
I
)
=
K
II
(
A
II
B
I
=
M
(
I
,
II
)
A
II
B
II
,
(4.46)
where
M
11
(
I
,
II
)
M
12
(
I
,
II
)
M
21
(
I
,
II
)
M
22
(
I
,
II
)
M
(
I
,
II
)
=
E
I
D
T
K
I
−
1
DD
K
II
E
I
=
(4.47)
D
T
K
I
−
K
II
−
with
E
I
being the identity matrix. In fact,
M
(
I
,
II
) is the resulting
transfer matrix. However, it is well known that the numerical
scheme with the transfer matrix in general becomes unstable, since
there waves which vary exponentially with the distance [83, 84],
especially at very high frequencies. In order to avoid this numerical
instability,one introduces the scattering matrix
S
11
(
I
,
II
)
S
12
(
I
,
II
)
S
21
(
I
,
II
)
S
22
(
I
,
II
)
S
(
I
,
II
)
=
(4.48)
attheinterface(
x
=
0)betweenregionsIandII,whichisdefinedby
A
II
B
I
S
11
(
I
,
II
)
S
12
(
I
,
II
)
S
21
(
I
,
II
)
S
22
(
I
,
II
)
A
I
B
II
.
=
(4.49)
With the help of the transfer matrix
M
(
I
,
II
), the elements of
S
(
I
,
II
)aregivenby
S
11
(
I
,
II
)
M
12
(
I
,
II
)(
M
22
(
I
,
II
))
−
1
M
21
(
I
,
II
)
S
12
(
I
,
II
)
=
M
12
(
I
,
II
)(
M
22
(
I
,
II
))
−
1
S
21
(
I
,
II
)
=
(
M
22
(
I
,
II
))
−
1
M
21
(
I
,
II
)
S
22
(
I
,
II
)
=
(
M
22
(
I
,
II
))
−
1
. (4.50)
As shown by Li
et al.
[85], the scattering matrix connecting the
left and right of the region II can be written as
=
−
M
11
(
I
,
II
)
0
P
P
0
,
S
(
II
,
II
)
=
(4.51)
where
P
is an
N
-dimensional diagonal matrix with diagonal
elements
P
mn
=
e
ik
I
n
b
δ
mn
,and
b
isthelongitudinallengthofregionII.
