Civil Engineering Reference
In-Depth Information
This is easy to show at high frequencies, the quantities
G
c
and
G
s
being close to
G
c
(s)
=
s
j/
4
(4.110)
s
j/
3
G
s
(s
)
=
(4.111)
and at low frequencies, where
s
tends to zero and
js
2
/
24
G
c
(s)
=
1
+
(4.112)
js
2
/
15
G
s
(s
)
=
1
+
(4.113)
3
4
s
. It appears from Figure 4.9 that
this property is valid for the entire range of variation of
s
and
s
. Substituting for
s
in
Equation (4.109) the expression of
s
given by Equation (4.92) yields
Figure 4.9 compares
G
c
(s)
and
G
s
(s
)
when
s
=
2
3
8
ωρ
o
σφ
1
/
2
s
=
(4.114)
σ
being the flow resistivity of the porous material having rectangular slits. The quantity
G
s
(s
)
can be replaced by
G
c
(s)
,
s
being given by Equation (4.114).
For a given value of
σ
and
φ
, Equation (4.82), which is valid for circular pores, can
also be used in the case of slits, but Equation (4.80) needs to be modified, since
s
must
be expressed as
c
8
ωρ
o
σφ
1
/
2
s
=
(4.115)
where
(
3
)
1
/
2
c
=
(4.116)
4.7.2 Bulk modulus of the air in slits
Biot did not proceed to obtain a prediction for the frequency dependence of the bulk
modulus. It is now easy to complete his model by using Equations (4.45) and (4.46).
The bulk modulus can be calculated, for the case of slits, in the following way. From
Equation (4.82) the effective density
ρ
is
ρ
=
ρ
o
1
+
jωρ
o
G
c
(s)
σφ
(4.117)
(
3
)
1
/
2
. Using Equation (4.115),
ρ
can be
s
being given by Equation (4.115) with
c
=
rewritten
ρ
o
1
+
js
2
G
c
(s)
8
c
2
ρ
=
(4.118)
