Environmental Engineering Reference
In-Depth Information
themode
n
intheoutgoingleadthroughallthescattering interfaces
isdescribe as
2
k
III
n
k
I
m
τ
mn
=|
t
mn
|
(4.58)
where
|
t
mn
|=|
S
11
|
mn
(4.59)
and
k
I
m
,
k
III
n
is given by Eq. 4.31. Now summing over all the
propagatingmodes
n
intheoutgoinglead,onegetsthetransmission
coe
cient
τ
m
forthe individualincomingmode
m
at frequency
τ
m
(
ω
=
τ
mn
)
(4.60)
n
,
ω
n
<ω
with
ω
n
=
n
π/
W
III
denoting the cutoff frequency of mode
n
in
the outgoing lead. Thus, the total transmission coe
cient is easily
obtained by the linear summation of the transmission coe
cient of
each mode.
τ
(
ω
)
=
τ
m
(
ω
).
(4.61)
ω
m
<ω
m
,
However, for the coupled P-SV waves in the
x
-
y
plane, their
displacement determined by Eqs. 4.23 and 4.24 can be obtained
by solving Eqs. 4.8 and 4.11 under the appropriate boundary
conditions.AsshownbyLi
et al.
[58],themixedboundaryconditions
can be expressed by
u
y
=
=
=
0 at the edges with
T
xy
and
T
zy
denoting the stress tensors, and are considered here. Under
these conditions, the displacements in each region
ξ
(
ξ
=
I, II and
III)can beexpressed as
T
xy
T
zy
N
ξ
A
ξ
Pn
e
ik
ξ
Pn
(
x
−
x
ξ
)
−
B
ξ
Pn
e
−
ik
ξ
Pn
(
x
−
x
ξ
)
(
k
ξ
Pn
)
η
n
(
y
)
u
x
=
=
n
0
+
B
SVn
e
−
ik
SVn
(
x
−
x
ξ
)
n
W
ξ
η
n
(
y
)
(4.62)
A
ξ
SVn
e
ik
SVn
(
x
−
x
ξ
)
+
A
ξ
Pn
e
ik
ξ
Pn
(
x
−
x
ξ
)
+
B
ξ
Pn
e
−
ik
ξ
Pn
(
x
−
x
ξ
)
N
ξ
n
W
ξ
u
y
=
χ
n
(
y
)
−
=
n
0
A
ξ
SVn
e
ik
SVn
(
x
−
x
ξ
)
−
B
SVn
e
−
ik
SVn
(
x
−
x
ξ
)
(
k
SVn
)
χ
n
(
y
)
+
(4.63)
