Civil Engineering Reference
In-Depth Information
The dimensionless acoustical pressure
p
∗
, velocity
v
∗
, the acoustic air density
ξ
∗
,and
the other space-dependent quantities involved are expressed in the form of asymptotic
expansions in powers of
ε
p
∗
(
x
,
y
)
p
(
0
)
(
x
,
y
)
εp
(
1
)
(
x
,
y
)
ε
2
p
(
2
)
(
x
,
y
)
=
+
+
+
...
(5.83)
υ
∗
(
x
,
y
)
υ
(
0
)
(
x
,
y
)
ευ
(
1
)
(
x
,
y
)
ε
2
υ
(
2
)
(
x
,
y
)
=
+
+
+
...
(5.84)
ξ
∗
(
x
,
y
)
ξ
(
0
)
(
x
,
y
)
εξ
(
1
)
(
x
,
y
)
ε
2
ξ
(
2
)
(
x
,
y
)
=
+
+
+
...
(5.85)
The superscript
i
in
p
i
,
υ
i
,and
ξ
i
denote the different terms in the developments, not
powers. The different
p
i
,
υ
i
,and
ξ
i
are periodic with respect to
y
with the same periodic-
ity as the microstructure. The gradient operator and the divergence are also dimensionless
and are given by
ε
−
1
∇
=
∇
+
∇
(5.86)
x
y
2
ε
−
1
ε
−
2
y
=
x
+
∇
∇
+
(5.87)
x
y
Equation (5.73) gives, at the order O(
ε
−
1
)
y
p
(
0
)
∇
=
0
,
(5.88)
And, at the order O(
ε
0
)
η
∗
y
υ
(
0
)
+
(λ
∗
+
η
∗
)
∇
y
(
∇
y
·
υ
(
0
)
)
−∇
y
p
(
1
)
−∇
x
p
(
0
)
=
jω
∗
ρ
0
υ
(
0
)
(5.89)
Equation (5.74) gives, at the order O(
ε
−
1
)
(
0
)
y
.
υ
=
0
(5.90)
0
)
And, at the order O(
ε
jω
∗
ξ
(
0
)
ρ
0
∇
x
·
υ
(
0
)
ρ
0
∇
y
·
υ
(
1
)
+
+
=
0
(5.91)
Equation (5.75) gives the relations
(
0
)
/
s
=
υ
(
1
)
/
s
=···=
0
υ
(5.92)
0
)
, Equation (5.80) gives
At the order O(
ε
κ
∗
y
τ
(
0
)
−
jω
∗
ρ
0
c
p
τ
(
0
)
=−
jω
∗
p
(
0
)
(5.93)
0
)
, Equation (5.81) gives
At the order O(
ε
p
(
0
)
P
0
ξ
(
0
)
ρ
0
+
τ
(
0
)
T
0
=
(5.94)
At zeroth order the pressure does not depend on the microscopic space variable and
the saturating air is not compressible at the scale of the pore. The method confirms
the validity of the description of sound propagation in cylindrical pores in Chapter 4,
because the period is arbitrary in the direction X
3
of the pores and at the microscopic
