Civil Engineering Reference
In-Depth Information
The coefficient
λ
1
is obtained from Equation (8.23) where
σ
zz
=−
φ
. The velocity
components, the surface impedance and the reflection coefficient are given by
T
12
s
1
)
−
φT
42
+
(
1
−
φ)T
32
U
z
=
(T
11
+
(8.26)
T
31
T
42
−
T
32
T
41
T
22
s
1
)
−
φT
42
+
φ)T
32
T
31
T
42
−
T
32
T
41
(
1
−
U
x
=
(T
21
+
(8.27)
D
1
T
31
T
42
−
v
z
=
(8.28)
T
32
T
41
Z
s
(ξ/k
0
)
=
(T
31
T
42
−
T
32
T
41
)/D
1
(8.29)
=
−
D
1
Z
0
+
(T
31
T
42
−
T
32
T
41
)
cos
θ
V(ξ/k
0
)
(8.30)
D
1
Z
0
+
(T
31
T
42
−
T
32
T
41
)
cos
θ
where
D
1
is given by
D
1
=
[
−
φT
42
+
(
1
−
φ)T
32
][
(
1
−
φ)T
11
+
φT
21
]
(8.31)
+
[
φT
22
+
(
1
−
φ)T
12
][
φT
41
−
(
1
−
φ)T
31
]
When the external source is the unit stress
τ
zz
=
jξx)
, a plane wave similar
to the reflected wave in the previous case exists, with a
z
dependence exp
(jzk
0
cos
θ)
.
At the surface of the layer in the free air, the pressure
p
e
and the normal velocity
v
z
are
related by
exp
(
−
p
e
v
z
Z
0
/
cos
θ
=−
(8.32)
Using Equation (8.10), Equation (8.32) can be rewritten
p
e
=
s
2
λ
1
+
s
3
λ
2
(8.33)
with
s
2
and
s
3
given by
Z
0
cos
θ
[
(
1
−
φ)T
11
+
φT
21
]
s
2
=−
(8.34)
Z
0
cos
θ
[
(
1
s
3
=−
−
φ)T
12
+
φT
22
]
(8.35)
Equation (8.23) becomes
(T
31
+
φs
2
)λ
1
+
(T
32
+
φs
3
)λ
2
=
0
(8.36)
Equation (8.24) with
τ
s
=
1 becomes
[
T
41
+
(
1
−
φ)s
2
]
λ
1
+
[
T
42
+
(
1
−
φ)s
3
]
λ
2
=
1
(8.37)
The parameters
λ
1
and
λ
2
can be obtained from this set of equations. For an excitation
of unit amplitude, the frame velocity components at the surface of the layer are given by
U
z
=
[
T
11
(T
32
+
φs
3
)
−
T
12
(T
31
+
φs
2
)
]
/D
2
(8.38)
U
x
=
[
T
51
(T
32
+
φs
3
)
−
T
52
(T
31
+
φs
2
)
]
/D
2
(8.39)
