Civil Engineering Reference
In-Depth Information
It was pointed out by Ingard (1953) that
R
B
given by Equation (9.19) is too small, and
that a better prediction of the measured values is obtained with
R
B
=
2
R
s
.At
A
,the
two added masses at the two sides of the facing, the two added viscous corrections, and
the mass and the flow resistivity in the aperture must be taken into account, and
Z
A
is
given by
2
d
R
+
4
R
s
+
Z
A
=
(
2
ε
e
+
d)jωρ
0
+
Z
c
s
(9.21)
In this equation,
(
2
d/R)R
s
is the flow resistance of the circular hole of length
d
.This
term can be obtained using Equation (5.26) with
α
∞
=
1and
=
R
. The effective
density can be written
ρ
0
δ
jρ
0
δ
ρ
0
α(ω)
=
ρ
0
+
−
(9.22)
The term
ρ
0
δ/
can be neglected, because it is very small compared with
ρ
0
at acoustical
frequencies for usual perforations. The density
ρ
0
corresponds to the term
jωdρ
0
in
Equation (9.21). In the same way,
jρ
0
δ/
will correspond to
ρ
0
δωd/R
in Equation
(9.21). By substituting
(
2
η/ρ
0
ω)
1
/
2
for
δ
in this term, one obtains 2
dR
s
/R
.Itmay
be noticed that when the material is in contact with a porous layer, the viscous term
(
2
d/R
−
+
4
)R
s
is generally negligible, compared with
Z
c
s
.
9.2.4 Apertures having a square cross-section
An elementary cell having a square aperture is represented in Figure 9.4. As in the case
of a circular aperture, the amplitude
U
of the velocity in the hole is considered uniform.
Equation (9.6) can be rewritten
v
m,n
a/
2
−
a/
2
D
2
4
U
cos
2
πmx
1
D
cos
2
πnx
2
D
k
m,n
C
m,n
ωρ
0
d
x
1
d
x
2
=
(9.23)
a/
2
−
a/
2
and
C
m,n
is given by
π
2
mn
sin
πma
D
sin
πna
D
m
=
4
v
m,n
U
ωρ
0
k
m,n
1
=
0
,n
=
0
C
m,n
(9.24)
πn
sin
πna
D
m
4
v
m,n
Uωρ
0
Dk
m,n
a
C
m,n
=
=
0
,n
=
0
(9.25)
Uωρ
0
a
2
D
2
k
C
m,n
=
m
=
n
=
0
(9.26)
The average pressure over the aperture is
a
2
a/
2
a/
2
C
m,n
cos
2
πmx
1
D
cos
2
πnx
2
D
1
p
=
dx
1
dx
2
(9.27)
−
a/
2
−
a/
2
m,n
