Environmental Engineering Reference
In-Depth Information
conditions, here we only show a case for the stress-free boundary
condition to reduce the length of the review. In Eq. 4.28,
k
n
is the
corresponding wave vector along the
x
axis, given by the frequency
matching condition
ω
n
)
2
ω
−
2
(
k
n
=
,
(4.31)
v
T
where
ω
n
=
n
π
v
T
/
W
ξ
is the cutoff frequency of the mode
n
in the
region
ξ
. As known, for the “open” channels,
k
n
must be real, i.e.,
those with
ω>ω
n
. Otherwise, wetake
k
n
=
i
|
k
n
|
.
Note that although only the propagating modes will contribute
to the wave transmission, the sums over
n
in Eq. 4.28 include all
the propagating and evanescent modes (imaginary
k
). However, in
real calculations, one generally takes all the propagating modes and
several lowest evanescent modes into account to meet the desired
precision. This is also confirmed by these previous studies [65, 82].
Fortheinterface(
x
=
0)betweenregionsIandII,thedisplacement
ψ
and stress should be continuous in terms of the boundary matching
condition,i.e.,
I
(
x
II
(
x
ψ
=
=
ψ
=
0,
y
)
0,
y
),
(4.32)
II
(
x
,
y
)
/∂
x
]
|
x
=
0
. (4.33)
Now substituting Eq. 4.28 into Eqs. 4.32 and 4.33, we can easily
I
(
x
,
y
)
/∂
x
]
|
x
=
0
=
μ
[
∂ψ
μ
[
∂ψ
get
N
I
N
II
(
A
n
+
B
n
)
I
n
(
y
)
(
A
I
n
+
B
I
n
)
II
n
(
y
),
η
=
η
(4.34)
n
=
0
n
=
0
N
I
N
II
k
n
(
A
n
−
B
n
)
I
n
(
y
)
k
I
n
(
A
I
n
−
B
I
n
)
II
n
(
y
).
η
=
η
(4.35)
n
=
0
n
=
0
To extract the relationships of four constants in these equations,
one multiplies two sides of Eq. 4.34 with
η
I
m
(
y
) and two sides of
Eq. 4.35 with
η
I
m
(
y
) and integrates over the transverse width of the
centralregion.Byusingtheorthogonalityofthewavefunctionsover
the appropriate domain, i.e.,
W
ξ
η
ξ
m
(
y
)
η
n
(
y
)
dy
=
δ
mn
,weget
0
N
II
A
I
m
+
B
I
m
=
D
mn
(
A
I
n
+
B
I
n
),
(4.36)
=
n
0
