Environmental Engineering Reference
In-Depth Information
By using the similar scheme, we can also deal with the case at
the interface (
x
=
b
) between regions II and III. Then, similar to the
case for the SH wave elaborated above, we rewrite all the equations
into the form of the matrix. Thus, the total scattering matrix for the
coupled P-SV wave in a given structure shown in Fig. 4.1 can be
defined by
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
A
II
P
A
III
A
I
P
A
I
SV
B
III
S
11
S
12
S
13
S
14
S
21
S
22
S
23
S
24
S
31
S
32
S
33
S
34
S
41
S
42
S
43
S
44
A
I
P
A
I
SV
B
III
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
SV
B
P
B
SV
S
T
=
=
, (4.83)
P
B
III
P
B
III
SV
SV
where
A
I
P
,
B
P
,
A
I
SV
,
B
SV
,
A
II
P
,
B
II
P
,
A
III
SV
,and
B
III
SV
are the column
vectors with the corresponding elements
A
I
Pm
,
B
I
Pm
,
A
I
SVm
,
B
SVm
,
A
II
Pn
,
B
II
Pn
,
A
III
SVn
,and
B
III
SVn
.Becausethesetwowavesgenerallycouple
witheachotherintheinterfacescatteringprocess,thetransmission
coe
cient should be discriminated according to the P, SV incoming
and P, SV outgoing waves, respectively. For the sake of unification,
here we denote them as
τ
α
m
,
β
n
, which represents the transmission
coe
cient from the
m
mode of
α
(
α
=
P
,
SV
)wavetothe
n
mode of
β
(
β
=
P
,
SV
) wave. Thus,
τ
Pm
,
Pn
=
(
|
S
11
|
mn
)
2
k
II
Pn
τ
Pm
,
SVn
=
(
|
S
21
|
mn
)
2
k
III
SVn
k
I
Pm
,
k
I
Pm
,
(4.84)
if the incidentwaveisP wave,and
τ
SVm
,
Pn
=
(
|
S
12
|
mn
)
2
k
II
Pn
τ
SVm
,
SVn
=
(
|
S
22
|
mn
)
2
k
III
SVn
k
SVm
, (4.85)
if the incident wave is SV wave. In these equations above,
k
ξ
Pm
,
k
ξ
Pn
,
k
SVm
,and
k
SVn
are the corresponding wave vector determined by
Eqs.4.66and4.67,respectively.Thus,summingoverthetransmitted
modes coupled to the incoming modes, the transmission coe
cient
of the individualincoming
m
mode for agiven
k
SVm
,
ω
is expressed as
τ
Pm
=
τ
Pm
,
Pn
+
τ
Pm
,
SVn
(4.86)
n
,
ω
Pn
<ω
n
,
ω
SVn
<ω
for the P wave and
τ
SVm
=
τ
SVm
,
Pn
+
τ
SVm
,
SVn
(4.87)
ω
Pn
<ω
ω
SVn
<ω
n
,
n
,
