Civil Engineering Reference
In-Depth Information
Equation (9.49) can be rewritten
−
2
d
R
+
4
R
s
1
s
e
Z
=
js
i
Z
c
cotg
ke
+
j(ε
i
+
ε
e
+
d)ωρ
0
+
(9.63)
The resonance frequency
f
0
is implicitly defined by Im Z
=
0:
2
πf
0
e
c
0
1
s
i
c
0
[
ε
i
+
ε
e
+
d
]2
πf
0
cotg
=
(9.64)
If
s
is sufficiently small, this equation is verified for a value of the argument of the
cotangent much smaller than 1, and the resonance frequency
f
0
is given by
1
/
2
c
0
2
π
s
i
e(ε
i
+
f
0
=
(9.65)
ε
e
+
d)
Multiplying the numerator and the denominator in the square root by
D
i
gives
1
/
2
c
0
2
π
S
V(d
+
ε
i
+
ε
e
)
f
0
=
(9.66)
where
V
is the volume of the resonator. Equation (9.66) can be used for resonators of
different shapes. The added lengths are calculated in Ingard (1953) for different positions
and shapes of the aperture and different shapes of the resonator.
9.4
Impedance at oblique incidence of a layered porous
material covered by a facing having circular
perforations
9.4.1 Evaluation of the impedance in a hole at the boundary surface
between the facing and the material
In the previous sections, at normal incidence, the pressure field in the porous medium is
the sum of different waves with wave number components
k
1
=
2
mπ/D
in the direction
x
1
and
k
2
=
2
nπ/D
in the direction
x
2
,
n
and
m
varying from
−∞
to
∞
. These waves
can be considered, by using the Floquet theorem (see for instance Collin (1960) pp 371
and 465), as the different modes transmitted at normal incidence by the doubly periodic
perforated screen with the spatial period
D
. This theorem will be used here for evaluating
the impedance at oblique incidence. The surface of the facing with a plane wave incident
at an angle
θ
=
0 is shown in Figure 9.16. The incidence plane is
x
3
Ox
2
and the circular
perforations are periodically distributed in the two directions
Ox
1
and
Ox
2
with a spatial
period
D
equal to the distance between two holes. The modes transmitted by the doubly
periodic screen in the porous medium have now a space dependence
g(x
1
,x
2
)
given by
exp
j
2
πmx
1
D
2
πn
D
−
k
0
sin
θ
x
2
g(x
1
,x
2
)
=
+
(9.67)
