Civil Engineering Reference
In-Depth Information
the system
X,Y,Z
. The coefficients
N
j
must be chosen for the boundary conditions to
be satisfied. In a first step, the displacement components
a
i
and
b
i
are predicted in the
system
X
,Y
,Z
for each wave. Let
ψ
be the angle between the axes
X
and
X
(see
Figure 10.7). The angle
ψ
is defined by
cos
ψ
=
q
x
/q
P
sin
ψ
(10.103)
=
q
y
/q
P
The displacement components in the system
X
,Y
,Z
and the system
X
,Y
,Z
are
related by
a
1
=
a
1
cos
ψ
−
a
2
sin
ψ
a
2
=
a
1
sin
ψ
a
2
cos
ψ
a
3
=
+
(10.104)
a
3
The same relations are satisfied by the components
b
i
and
b
i
. The stress components
can be evaluated in the system
X
,Y
,Z
from the displacement components
a
i
,b
i
and
the slowness vector components with Equations (10.21) - (10.27). In a last step, the con-
tribution
a
i
and
b
i
of each wave to the total velocity are evaluated with the relations
a
1
cos
ϕ
a
3
sin
ϕ
a
1
=
+
a
2
a
3
=−
a
2
=
(10.105)
a
1
sin
ϕ
a
3
cos
ϕ
+
The same relations hold for the
b
i
and the
b
i
. For each wave the total stress compo-
nents in the system
X,Y,Z
and in the system
X
,Y
,Z
are related by
σ
zz
=
σ
x
x
sin
2
ϕ
−
2
σ
x
z
sin
ϕ
cos
ϕ
σ
z
z
cos
2
ϕ,
+
(10.106)
σ
zx
=−
σ
x
x
sin
ϕ
cos
ϕ
σ
z
x
(
cos
2
ϕ
−
sin
2
ϕ)
+
(10.107)
σ
z
z
sin
ϕ
cos
ϕ,
+
σ
zy
=−
σ
x
z
sin
ϕ
+
σ
x
y
cos
ϕ
(10.108)
q
y
y))
applied on the frame at the free
surface of the layer creates a plane wave in air with a spatial dependence exp[
The unit stress field
τ
zz
=
exp
(
−
jω(q
x
x
+
−
j(ωq
x
x
+
ωq
y
y
−
k
0
z
cos
θ)
]wherecos
θ
is given by
(k
0
/ω)
2
−
q
x
−
q
y
k
0
/ω
cos
θ
=±
(10.109)
The choice in the physical Riemann sheet corresponds to Im cos
θ
≤
0. The pressure
p
e
and the velocity component
v
z
are related by
p
e
v
z
Z
0
/
cos
θ
=−
(10.110)
where
Z
0
is the characteristic impedance in the free air. If the predictions are restricted
to the case of a semi-infinite layer, there are no upward waves and only the contribu-
tions in the porous medium of the four other waves must be taken into account. Let
