Environmental Engineering Reference
In-Depth Information
where
2
/
π
/
=
W
ξ
cos(
n
y
W
ξ
), if
n
0
η
n
(
y
)
=
1
(4.64)
/
=
W
ξ
,
if
n
0
2
/
W
ξ
sin(
n
π
y
/
W
ξ
), if
n
=
0
0,
χ
n
(
y
)
=
(4.65)
if
n
=
0
−
(
ω
ξ
Pn
)
2
v
L
ω
2
k
ξ
Pn
=
(4.66)
−
(
ω
SVn
)
2
v
T
ω
2
k
SVn
=
(4.67)
ω
ξ
Pn
=
ω
SVn
=
π
v
L
/
π
v
T
/
with
W
ξ
.FromEqs.4.62and
4.63, it is seen that there exist four expansion coe
cients
(
A
ξ
P
,
B
P
,
A
ξ
SV
,
B
SV
)ineachregioncorrespondingtothetransmitted
and reflected components of the P and SV waves, respectively. Thus,
when the P wave solely goes into the waveguide, the reflection at
theinterfacesmayleadtothemodeconversion,namelyitsreflected
wave and the transmitted wave may simultaneously contain both P
and SV waves. For the SV wave incidence, the situation is similar.
To determine the relationships of these expansion coe
cients, we
imposethecontinuityconditionofthedisplacementandstressatall
the interfaces (
x
=
0,
b
) betweenadjacent regions, i.e.,
u
I
(
II
)
n
W
ξ
and
n
u
II
(
III
)
=
(4.68)
x
x
u
I
(
II
)
=
u
II
(
III
)
(4.69)
y
y
+
λ
∂
u
I
(
II
)
+
λ
∂
u
II
(
III
)
(
λ
+
2
μ
)
∂
u
I
(
II
)
∂
y
=
(
λ
+
2
μ
)
∂
u
II
(
III
)
x
∂
x
y
x
∂
x
y
∂
y
(4.70)
+
μ
∂
u
I
(
II
)
+
μ
∂
u
II
(
III
)
μ
∂
u
I
(
II
)
=
μ
∂
u
II
(
III
)
y
∂
x
y
∂
x
x
∂
y
x
∂
y
.
(4.71)
0) between regions I and
II. By substituting Eqs 4.62 and 4.63 into Eqs. 4.68-4.71, we can
Firstly, we consider the interface (
x
=
