Civil Engineering Reference
In-Depth Information
It may be noticed that Equations (9.48) and (9.14) can be related in a simple way.
Equation (9.14) can be rewritten
J
1
2
π
R
n
2
)
1
/
2
D
(m
2
+
4
m,n
p
U
=
kZ
c
k
m,n
ν
m,n
sZ
c
+
(9.52)
π(m
2
+
n
2
)
and Equation (9.48) can be obtained from Equation (9.52) by substituting
Z
m,n
(B)/φ(L)
for
Z
c
k/k
m,n
(the ratios
k/k
0
,
0
and
k(L)/k
0
,
0
(L)
do not appear explicitly in the contri-
butions of the (0, 0) mode because they are equal to 1). Both quantities
Z
m,n
(B)/φ(L)
and
Z
c
k/k
m,n
have the same physical meaning since
Z
m,n
(B)/φ(L)
for the case of a
semi-infinite layer of air is equal to
Z
c
k/k
m,n
.
9.3.2 Evaluation at normal incidence of the impedance for the case
of square holes
From Equation (9.28), the impedance
Z(B)
for the case of square holes and a semi-infinite
layer of air can be written
sin
2
πna
D
π
2
n
2
∞
kZ
c
k
0
,n
Z(B)
=
sZ
c
+
8
ν
0
,n
n
=
1
(9.53)
sin
2
πma
D
sin
2
πna
D
∞
∞
4
D
2
a
2
kZ
c
k
m,n
+
ν
m,n
π
4
m
2
n
2
m
=
1
n
=
1
The previously defined substitution can be used to calculate
Z(B)
when a porous layered
material is substituted for the semi-infinite layer of air. Equation (9.53) becomes
sin
2
πna
D
π
2
n
2
∞
s
φ(L)
Z
0
,
0
(B)
8
φ(L)
Z(B
)
=
+
ν
0
,n
Z
0
,n
(B)
n
=
1
(9.54)
sin
2
πma
D
sin
2
πna
D
∞
∞
D
2
a
2
4
φ(L)
+
ν
m,n
Z
m,n
(B)
π
4
m
2
n
2
m
=
1
n
=
1
As in the case of circular apertures, the inertial term j
(ε
e
+
d)ρ
0
ω
,where
ε
e
is now given
by Equation (9.30), must be taken into account, but the effect of the viscous forces in the
aperture and on the facing around the aperture can be neglected when
Z(A)
is evaluated
Z(B
)
Z(A)
=
+
j(ε
e
+
d)ρ
0
ω
(9.55)
The impedance
Z
in the free air close to the facing is
Z
=
Z(A)/s
(9.56)
