Civil Engineering Reference
In-Depth Information
ϕ
i
can be written
ϕ
i
a
i
exp
(
=
∓
jα
i
z
−
jξx)
(k
i
−
ξ
2
)
1
/
2
(8.1)
α
i
=
i
=
1
,
2
,
3
Let
σ
αβ
and
σ
αβ
be the components of the Biot stress tensors for the frame, and the
air, respectively. They are related to forces acting on the frame and on the air, for a unit
surface of porous medium. The stress components in the Biot theory are given by
σ
ij
=
[
(P
−
2
N)θ
s
Qθ
f
]
δ
ij
+
2
Ne
s
ij
+
(8.2)
σ
ij
=
(Qθ
s
+
Rθ
f
)δ
ij
(8.3)
where
θ
f
and
θ
s
are the dilatation of the air, and the dilatation of the frame, respectively,
and the terms
e
s
ij
are the strain components of the frame.
In order to simplify the calculations for layers of finite thickness, three independent
displacement fields which satisfy the boundary conditions at the lower face of the layer
are defined. They are obtained by associating to each potential function exp
(
−
jα
i
z)
,
i
1,2,3. The boundary conditions at
the lower face when the layer is glued to a rigid impervious backing are
=
1,2,3, three potential functions
r
ik
exp
(jα
k
z)
,
k
=
u
s
z
=
0
(8.4)
u
z
=
0
(8.5)
u
s
x
=
0
(8.6)
The coefficients
r
ji
are given in Appendix 8.A. Let
b
1
,
b
2
,and
b
3
be the frame
displacement fields defined by (the factor exp
(
−
jξx)
being removed)
j
ω
[
∇
b
1
=−
(
exp
(
−
jα
1
z)
+
r
11
exp
(jα
1
z)
(8.7)
+
r
12
exp
(jα
2
z))
+
∇
∧
n
r
13
exp
(jα
3
z)
]
b
2
=
−
j
ω
[
∇
(r
21
exp
(jα
1
z)
+
exp
(
−
jα
2
z)
(8.8)
+
r
22
exp
(jα
2
z))
+
∇
∧
n
r
23
exp
(jα
3
z)
]
b
3
=
−
j
ω
[
∇
(r
31
exp
(jα
1
z)
+
r
32
exp
(jα
2
z))
+
∇
∧
n
(
exp
(
(8.9)
−
jα
3
z)
+
r
33
exp
(jα
3
z))
]
λ
3
b
3
satisfies the boundary conditions at the
lower face. Let
p
e
be the pressure and
v
z
be the normal component of the velocity of air in
the free air close to the upper face of the layer. At the upper face, the boundary conditions
for the case of an incident plane wave can be written (see Equations 6.98 - 6.100)
φ
u
z
+
Any linear combination
λ
1
b
1
+
λ
2
b
2
+
φ)
u
s
z
=
v
z
(
1
−
(8.10)
