Civil Engineering Reference
In-Depth Information
The impedance
Z(M
2
)
at the surface of the material is given by
p(M
1
)
cos
ϕ
υ(M
1
)nπR
2
p(M
1
)
φυ(M
1
)
Z(M
2
)
=
=
(4.150)
p(M
1
)
and
υ(M
1
)
being the pressure and the velocity of air in a pore at the surface
of the material. The effective density of the air in the material must be calculated in
the macroscopic direction of propagation, which is perpendicular to the surface of the
layer. In the case of circular cross-sectional shape, the Newton equation (4.19) in the
microscopic directions of propagation
x
1
and
x
2
can be rewritten, with Equation (4.128)
∂p(x)
∂x
i
=
jωρ
o
υ
i
(x)
+
8
ρ
0
ω
s
2
G
c
(s)υ
i
(x)
−
i
=
1
,
2
(4.151)
where
G
c
is given by Equation (4.128) and
s
by Equation (4.145).
In the
x
direction, by the use of Equations (4.141), (4.145) and
∂p(x)
∂x
=−
1
cos
ϕ
∂p(x)
∂x
i
−
(4.152)
Equation (4.151) can be rewritten in the macroscopic direction of propagation
∂p(x)
∂x
=
−
jωρ
o
α
∞
υ(x)
+
σφG
c
(s)υ(x)
(4.153)
where
υ(x)
is the velocity component in a pore in the macroscopic direction of propa-
gation. The effective density can be written
α
∞
ρ
0
1
+
σφG
c
(s)
jωα
∞
ρ
0
ρ
=
(4.154)
The high-frequency limit of
ρ
is now
ρ
=
α
∞
ρ
0
(
1
+
2
/jδ/R)
(4.155)
It may be shown that Eq. (4.155) is valid for all cross-sectional shapes if
R
is replaced
by the hydaulic radius
α
∞
ρ
0
(
1
+
2
/jδ/
¯
r)
ρ
=
(4.156)
For the general case of arbitrary cross-sectional shapes, Equation (4.151) can be used
with
s
given by
c
8
ωρ
o
α
∞
σφ
1
/
2
s
=
(4.157)
8
ωρ
0
α
∞
σφ
1
/
2
1
¯
r
c
=
(4.158)
The bulk modulus is not modified when the pores are oblique and is given by Equation
(4.126) with
=
1
1
+
G
c
(Bs)
σφ
jB
2
ωρ
o
α
∞
F(B
2
ω)
(4.159)
