Civil Engineering Reference
In-Depth Information
X
. The plane harmonic waves with a slowness 1/
c
in the direction
X
and a polarization in
the meridian plane are described with the notations of Vashishth and Khurana (2004). The
frame displacement components and the discharge displacement components are written
a
1
exp
jω
t
qz
x
c
−
u
s
x
=
−
(10.35)
a
2
exp
jω
t
c
−
qz
x
u
s
y
=
−
(10.36)
a
3
exp
jω
t
c
−
qz
x
u
s
z
=
−
(10.37)
b
1
exp
jω
t
c
−
qz
x
w
x
=
−
(10.38)
w
y
=
b
2
exp
jω
t
−
c
−
qz
x
(10.39)
b
3
exp
jω
t
qz
x
c
−
w
z
=
−
(10.40)
d
1
exp
jω
t
c
−
qz
x
u
x
=
−
(10.41)
d
2
exp
jω
t
c
−
qz
x
u
y
=
−
(10.42)
d
3
exp
jω
t
c
−
qz
x
u
z
=
−
(10.43)
where
d
k
=
1
,
2
,
3. The symbol
q
is used to denote the
z
slowness vector
component. The displacement components
u
y
and
w
y
, and the wave number component
in the direction
y
, are equal to 0. The displacement components are solutions of a system
of six equations which can be written
=
a
k
+
b
k
/φ,k
2
B
1
+
B
2
c
2
B
5
)
q
c
B
6
c
2
B
6
q
c
B
5
q
2
ρ
−
−
−
(B
3
+
+
ρ
0
B
5
)
q
c
B
5
c
2
B
7
q
c
a
1
a
3
b
1
b
3
B
4
q
2
B
7
q
2
−
(B
3
+
ρ
−
−
ρ
0
+
B
6
c
2
B
7
q
c
B
8
c
2
B
8
q
c
+
ρ
0
−
+
C
1
−
B
6
q
c
B
8
q
c
B
7
q
2
B
8
q
2
ρ
0
+
−
−
+
C
3
0
0
0
0
=
(10.44)
B
1
c
2
+
q
2
B
5
−
ρ
a
2
b
2
−
ρ
0
=
0
(10.45)
ρ
0
C
1
The nondependence of
a
2
and
b
2
on
a
1
,a
3
,b
1
and
b
3
shows the decoupling of pseudo
SH
waves
polarized
in
the
direction
y
and
the
waves
polarized
in
the
meridian
plane.
