Environmental Engineering Reference
In-Depth Information
N
I
D
nm
k
I
m
(
A
I
m
−
B
I
m
)
=
k
I
n
(
A
I
n
−
B
I
n
).
(4.37)
n
=
0
This is a set of linear equations that determine the relationships
between these expansion coe
cients. In these above equations,
D
mn
represents the coupling strength of different modes, which is
expressed by the overlap of the wave functions for the regions I
and II
W
I
I
m
(
y
)
η
I
n
(
y
)
dy
.
D
mn
=
η
(4.38)
0
Substituting Eq. 4.29 into the above equation, we can derive the
explicitform of
D
mn
, i.e.,
W
I
W
II
, if
m
=
0,
n
=
0;
D
mn
=
(4.39)
2
W
II
W
I
sin
n
π
W
I
, if
m
=
0,
n
=
0;
1
n
π
D
mn
=
(4.40)
W
II
D
mn
=
0, if
m
=
0,
n
=
0;
(4.41)
sin
m
π
+
1
√
W
I
W
II
1
n
π
W
I
W
II
D
mn
=
W
I
+
W
II
m
n
,if
m
sin
m
π
1
n
W
I
W
II
0,
m
n
W
II
;
(4.42)
+
π
−
=
=
W
I
=
0,
n
W
I
−
W
II
m
n
W
I
W
II
, if
m
=
0,
n
=
0,
m
n
W
II
.
D
mn
=
W
I
=
(4.43)
For the sake of making the matrix calculation, one defines
matrices
D
,
K
I
,
K
II
whose elements are
D
mn
,
k
I
m
δ
mn
,
k
I
n
δ
mn
,
respectively. Similarly, four column vectors
A
I
,
B
I
,
A
II
,and
B
II
are
alsoconstructedbythecorrespondingelements
A
I
m
,
B
I
m
,
A
I
n
,and
B
I
n
.
In thisnotation,wehave
A
I
B
I
D
(
A
II
B
II
),
+
=
+
(4.44)
