Civil Engineering Reference
In-Depth Information
X
1
a
X
2
X
3
Figure 9.4
The elementary cell of the partition. The square at
x
3
=
0 is the boundary
of the aperture in contact with the free air.
This equation can be rewritten
U
ωρ
0
s
sin
2
πna
D
∞
v
0
,n
ωρ
0
k
0
,n
1
π
2
n
2
p
=
k
+
8
n
=
1
D
sin
2
πma
D
sin
2
πna
∞
∞
4
D
2
a
2
v
m,n
ωρ
0
1
π
4
m
2
n
2
+
(9.28)
k
m,n
m
=
n
=
1
1
where
k
m,n
at sufficiently low frequencies is given by
j
2
π
D
(m
2
n
2
)
1
/
2
k
m,n
=−
+
(9.29)
A comparison of Equations (9.1) and (9.28) yields
sin
2
πna
D
∞
4
D
π
3
ν
0
,n
n
3
ε
e
=
n
=
1
sin
2
πma
D
sin
2
πna
D
∞
∞
2
D
3
a
2
π
5
ν
m,n
m
2
n
2
(m
2
+
(9.30)
+
n
2
)
1
/
2
m
=
1
n
=
1
Ingard (1953) indicated that
ε
e
can be approximated for
s
1
/
2
<
0
.
4by
ε
e
=
0
.
48
(S)
1
/
2
(
1
−
1
.
25
s
1
/
2
)
(9.31)
