Civil Engineering Reference
In-Depth Information
D
i
D
e
AB
e
Figure 9.15
An array of resonators.
As in the case of the material described in Section 9.3.4, the absorption coefficient
of an array of resonators presents a maximum when the imaginary part of the impedance
of the array is equal to 0, at a frequency called the resonance frequency.
The frequency dependence of the impedance and the absorption coefficient of an
array of resonators can be calculated by using the results of the previous sections if
the resonator is a cylinder with a square cross-section. As an example, we consider the
case where there is no porous material in the resonator. The impedance
Z(B
)
can be
calculated by Equation (9.54) where
φ(L)
=
1and
D
i
substituted for
D
in Equations
(9.3) and (9.54). The impedance
Z
0
,
0
(B)
is given by
Z
0
,
0
(B)
=−
jZ
c
cot
ke
(9.57)
where
e
is the length of the cylinder. For frequencies well below
c
0
/D
i
and if the
quantities
B
m,n
can be neglected in Equation (9.40), the terms related to the modes of
higher order can be replaced by an added length effect j
ωρ
0
ε
i
,where
ε
i
can be calculated
by Equation (9.18) if the apertures are circular.
It may be pointed out that the open area ratio
s
i
at
B
is the ratio of the area
S
of the
aperture to the area of the cross-section of the 'internal' elementary cell, i.e.
D
i
,andthe
internal added length
ε
i
is given by
ε
i
=
0
.
48
S
1
/
2
(
1
.
0
−
1
.
14
s
i
)
(9.58)
(S/D
i
)
1
/
2
where
s
i
=
(9.59)
and
Z(B)
is given by
Z(B)
=−
js
i
Z
c
cot
ke
+
jε
i
ρ
0
ω
(9.60)
At
A
, the area of the 'external' elementary cell is
D
e
, and the open area ratio
s
e
and the
added length at
A
are given by
(S/D
e
)
1
/
2
s
e
=
(9.61)
ε
e
=
0
.
48
S
1
/
2
(
1
.
0
−
1
.
14
s
e
)
(9.62)
