Civil Engineering Reference
In-Depth Information
2R
B
A
Transmitted
field
Incident field
d
X
3
0
Figure 9.1
A perforated facing in a normal acoustic field.
The facing distorts the flow of air in a region having a thickness which is generally
smaller than the diameter of a perforation. This modification of the flow creates an
increase of kinetic energy. This effect is similar to the effect of tortuosity in porous
media. If a plane wave propagates at the left-hand side of the facing, parallel to the
x
3
axis (see Figure 9.1), the impedance
p/υ
3
in the free air is equal to the characteristic
impedance
Z
c
.At
B
, at the boundary of the facing in front of a perforation, the impedance
would be equal to
sZ
c
if the inertial effects were not present. An evaluation of
Z
B
, carried
out in Section 9.2.2, indicates that
Z
B
is given by
Z
B
=
Z
c
s
+
jωε
e
ρ
0
(9.1)
where
ε
e
ρ
0
is an added mass per unit area of perforation,
ρ
0
is the density of air, and
ε
e
is a length that must be added to the cylindrical perforation at the left of
B
in order to
create the same effect as the distortion of the flow.
9.2.2 Calculation of the added mass and the added length
For the case of normal incidence, if the arrangement of the perforations is the square grid
represented in Figure 9.2 (the perforations are periodically distributed in the two directions
with a spatial period
D
equal to the distance between two holes), it is possible to divide
the space around the facing in separate cylinders having a square shaped cross-section
without modifying the acoustic field.
At the boundary between two cylinders, the velocity components perpendicular to the
boundary are equal to zero, due to the symmetry of the grid, and the different cylinders
can be separated by rigid sheets (it may be noticed that these sheets do not exist, and
the viscous and the thermal interaction with the sheet cannot be taken into account).
An elementary cell for the partition is represented in Figure 9.3. In the plane
x
3
=
0,
