Civil Engineering Reference
In-Depth Information
The right-hand side of this equation is the right-hand side of Equation (9.77) where the
contribution of the (0, 0) mode has been removed and the impedance
Z
m,n
(B)
is related
to a semi-infinite layer of air instead of a porous layer. Then,
Z
c
is the characteristic
impedance of air, and
k
m,n
for frequencies well below
c
0
/D
is approximately equal to
j
4
π
2
m
2
D
2
k
0
sin
θ
2
1
/
2
2
πn
D
−
k
m,n
=−
+
(9.82)
Equations (9.80) and (9.82) yield the following expression for the added length
2
1
/
2
2
πR
m
2
n
D
−
k
0
sin
θ
2
π
J
1
D
2
+
π
2
m,n
D
v
m
ε
e
=
(9.83)
m
2
2
3
/
2
n
k
0
D
sin
θ
2
π
+
−
It can be shown that when
θ
tends to zero, Equation (9.83) becomes identical to Equation
(9.17) giving
ε
e
at normal incidence. The factor 2 in Equation (9.17) does not exist
in Equation (9.83) because
n
can be positive or negative in this equation while it is
always positive in Equation (9.14). Calculations for
θ
varyingupto80
◦
indicate that the
dependence of
ε
e
on
θ
is weak, and that Equation (9.18) can be used also at oblique
incidence.
9.4.3 Evaluation of the impedance of a faced porous layer at oblique
incidence
As in the case of normal incidence, Equations (9.55) and (9.56) can be used to evaluate
the impedance
Z
in the free air close to the facing
Z(B
)
s
+
j
(ε
e
+
d)ρ
0
ω
Z
=
Z(A)/s
=
(9.84)
s
The impedance
Z(B
)
is now given by Equation (9.77). As an example, the impedance
and the absorption coefficient of the faced material of Section 9.3.4 are represented in
Figures 9.17 and 9.18 for three angles of incidence
θ
=
0,
π/
6,
π/
3. The dependence of
the impedance and the absorption coefficient on the angle of incidence is small, due to
the high flow resistance of the layer M
2
which is much thicker than the layer M
1
.
The model was tested at oblique incidence for different configurations using measure-
ments by Guignouard
et al
. (1991), and as in the case of normal incidence, the trends
predicted by the model are present in the measurements. The centres of the holes are set
on axes parallel to the
Ox
2
direction, defined by the intersection of the incidence plane
and the facing. Measurements at oblique incidence indicate that the surface impedances
do not noticeably vary when the direction of the
Ox
2
axis is modified.
