Civil Engineering Reference
In-Depth Information
and the stress components are given by Equations (6.A.2) -(6.A.4). The wave equations
can be written
K
b
+
3
N
1
.
u
s
2
u
s
ω
2
ρ
s
u
s
ω
2
ρ
eq
γ
u
t
∇∇
+
N
∇
=−
−
(6.A.6)
.
u
t
ω
2
ρ
eq
γ
u
s
−
ω
2
ρ
eq
u
t
K
eq
∇∇
=−
(6.A.7)
In the previous equations,
γ
,
ρ
eq
,and
ρ
s
are given respectively by
φ
ρ
12
1
−
φ
γ
=
ρ
22
−
(6.A.8)
φ
ρ
22
φ
2
ρ
eq
=
(6.A.9)
ρ
12
γ
2
ρ
eq
ρ
s
=
ρ
11
−
ρ
22
+
(6.A.10)
Using two scalar potentials
φ
s
and
φ
t
for the compressional waves gives the following
equation of motion
ω
2
[
ρ
]
ϕ
s
ϕ
t
2
ϕ
s
ϕ
t
−
=
[
K
]
∇
(6.A.11)
where [
ρ
]and[
K
] are given, respectively, by
ρ
s
ρ
eq
γ
ρ
eq
γ ρ
eq
,
[
K
]
=
P
0
0
K
eq
[
ρ
]
=
(6.A.12)
where
4
3
N
The matrix [
K
] is diagonal. The wave numbers of the Biot compressional waves and the
ratios
µ
i
,i
=
1, 2 are obtained from Equation (6.A.12). The wave number of the shear
wave and the ratio
µ
3
are obtained using a potential vector. It is shown that with the
Dazel representation, the prediction of the surface impedance of a porous media can be
performed with a mathematical formalism which is simpler than the one associated with
the first formalism.
P
=
K
b
+
The mixed pressure -displacement representation (Atalla
et al.
1998)
In this representation the displacement
u
s
and the pressure
p
are used instead of the couple
u
s
,
u
f
. The developments assume that the porous material properties are homogeneous.
The derivation follows the presentation of Atalla
et al
. (1998). Note that that a more
general time domain formulation valid for anisotropic materials is given by Gorog
et al
.
(1997).
The system (Equations 6.54, 6.55) is first rewritten:
ω
2
ρ
11
u
s
ω
2
ρ
12
u
f
+
div
σ
s
+
=
0
(6.A.13)
ω
2
ρ
22
u
f
+
ω
2
ρ
12
u
s
−
φ
grad
p
=
0
