Civil Engineering Reference
In-Depth Information
The average pressure over the aperture is
πR
2
R
2
π
1
C
m,n
cos
2
πmx
1
D
cos
2
πnx
2
D
p
=
r
d
r
d
θ
(9.13)
0
0
m,n
and the impedance
p/U
is given by
v
m,n
J
1
2
π
D
(m
2
n
2
)
1
/
2
+
4
m,n
p
U
=
sωρ
0
k
+
ωρ
0
(9.14)
π(m
2
n
2
)k
m,n
+
The prime over the symbol
means that the term
(m
=
0
,n
=
0
)
is excluded from the
summation.
For frequencies well below
c
0
/D,k
m,n
given by Equation (9.3) is nearly equal to
k
m,n
=−
j
2
π
D
(m
2
n
2
)
1
/
2
+
(9.15)
and Equation (9.14) becomes
jωρ
0
m,n
v
m,n
2
D
J
1
[2
π(R/D)(m
2
n
2
)
1
/
2
]
+
p
U
=
Z
c
s
+
(9.16)
π
2
(m
2
+
n
2
)
3
/
2
A comparison of Equations (9.1) and (9.16) yields the following expression for the added
length:
v
m,n
2
D
J
1
[2
π(R/D)(m
2
n
2
)
1
/
2
]
+
ε
e
=
(9.17)
π
2
(m
2
+
n
2
)
3
/
2
m,n
The added length
ε
e
can be approximated for
s
1
/
2
<
0
.
4by
ε
e
=
0
.
48
S
1
/
2
(
1
−
1
.
14
s
1
/
2
)
(9.18)
where
S
is the area of the aperture. The limit 0
.
48
S
1
/
2
for
s
=
0 can be obtained by
calculating the radiation impedance of a single circular aperture in a plane.
9.2.3 Flow resistance
Due to the viscous dissipation in the aperture and the surface of the plate, which was
neglected in Section 9.2.2, the impedance
Z
B
presents also a resistive component
R
B
that must be added to
Z
c
s
in Equation (9.1). A calculation by Nielsen (1949) gives the
following expression for
R
B
:
R
B
=
R
s
(9.19)
where
R
s
is a surface resistance defined in Lord Rayleigh (1940) Vol II, p. 318:
1
2
(
2
ηρ
0
ω)
1
/
2
R
s
=
(9.20)
