Civil Engineering Reference
In-Depth Information
Substituting the right-hand side of Equation (4.93) for
ρ
o
ω/s
in Equation (4.28)
yields
∂p
∂x
3
=
φ υσG
s
(s
)
−
jωρ
o
υ
+
(4.94)
where
j
1
/
2
s
tanh
(s
j
1
/
2
)
3
1
G
s
(s
)
=
(4.95)
tanh
(s
j
1
/
2
)
s
j
1
/
2
−
The limiting value of
G
s
(s
)
at
ω
=
0 is 1 as in the previous case, and Equation
(4.94) can be rewritten
∂p
∂x
=
σφυ
−
(4.96)
At the first-order approximation in 1/
ω
, Equations (4.74) and (4.75) yield
6
5
ρ
0
+
σφ
jω
ρ
=
(4.97)
γP
0
γ
−
(γ
−
1
)/
6
K
=
(4.98)
σφ
jB
2
ωρ
0
5
+
In the high-frequency range, Equations (4.71) and (4.72) yield
ρ
=
ρ
0
1
+
2
/j
δ
2
a
(4.99)
γP
0
−
1
)/
1
+
√
2
/j
K
=
(4.100)
δ
2
Ba
γ
−
(γ
4.6.3 High- and low-frequency limits of the effective density and the
bulk modulus for pores of arbitrary cross-sectional shape
At low frequencies the limit of the effective density is at the 0 order approximation in
ω
, from the definition of flow resistivity, Equation (4.84)
φσ
jω
ρ
=
(4.101)
From Equation (4.46), the limit for the bulk modulus at the first-order approximation
in
ω
is
γP
0
γ
−
j(γ
−
1
)ωB
2
ρ
0
/(φσ)
K
=
(4.102)
