Civil Engineering Reference
In-Depth Information
the term tanh
(s
j
1
/
2
)
is equal to 1. The Newton equation (4.28) becomes
υ
3
η
a
2
ρ
o
ω
1
/
2
∂p
∂x
3
=
(
1
j)
√
2
+
−
jωρ
o
υ
3
+
(4.71)
and the bulk modulus
K
is
K
=
γP
o
1
+
2
(
−
1
+
j)(γ
−
1
)/(Bs
)
√
2
(4.72)
At low frequencies, the following approximation can be used:
tanh
(s
j
1
/
2
)
1
−
tanh
(s
j
1
/
2
)
1
s
j
1
/
2
1
s
j
1
/
2
3
(s
j
1
/
2
)
2
+
1
5
=
(4.73)
The Newton equation (4.28) becomes
∂p
∂x
3
=
6
5
jωρ
o
υ
3
+
3
η
a
2
−
υ
3
(4.74)
and the bulk modulus is
P
o
1
(Bs
)
2
1
3
j
γ
−
1
K
=
+
(4.75)
γ
4.6
Evaluation of the effective density and the bulk
modulus of the air in layers of porous materials
with identical pores perpendicular to the surface
First, the cylindrical pore having a circular cross-section and the slit are considered.
Next, general models with adjustable parameters, used for the evaluation of the effective
density and the bulk modulus of the air in pores with other cross-sectional shapes, are
presented. The flow resistivity is used as an acoustical parameter.
4.6.1 Effective density and bulk modulus in cylindrical pores having
a circular cross-section
The sample of porous material represented in Figure 4.8 has
n
pores of radius
R
per unit
area of cross-section. The flow resistivity
σ
given by Equation (2.27) is equal to
p
1
υnπR
2
e
p
2
−
σ
=
(4.76)
The air in the pores is submitted to two opposite forces due to the viscosity and the
pressure gradient. Using Equation (4.67) at
ω
=
0gives
R
2
8
η
∂p
∂x
υ
=
−
(4.77)
