Civil Engineering Reference
In-Depth Information
The chosen material provides a good illustration of partial decoupling, because the
frame is much heavier and stiffer than the air. Some porous materials used for sound
absorption have a frame whose bulk modulus
K
s
has the same order of magnitude as
the bulk modulus
K
f
of the air, and a density
ρ
1
which is about 10 times larger than
the density
ρ
o
of the air. For these materials, the partial decoupling does not exist at low
frequencies, up to an upper bound depending on the flow resistivity and the density of the
material. A simple expression is given by Zwikker and Kosten (1949) for this frequency
φ
2
σ
ρ
1
1
2
π
f
o
=
(6.90)
It may be pointed out that at frequencies higher than
f
o
, the frame-borne wave can be
noticeably different from the compressional wave propagating in the frame in vacuum,
and the airborne wave can be noticeably different from the compressional wave in the
same material with a rigid frame.
6.6
Prediction of surface impedance at normal incidence
for a layer of porous material backed by an impervious
rigid wall
6.6.1 Introduction
A layer of porous material in a normal plane acoustic field is represented in Figure 6.8.
In order to obtain simple boundary conditions at the wall - material interface, the material
is glued to the wall. In a normal acoustic field, the shear wave is not excited and only the
compression waves propagate in the material. The description of the acoustic field, and
the measurements, are easier for this case than for oblique incidence. The Biot theory is
used in this section to predict the behaviour of the porous material in a normal acoustic
field. The parameter that is used to represent the behaviour of the material is the surface
impedance.
6.6.2 Prediction of the surface impedance at normal incidence
Two incident and two reflected compressional waves propagate in directions parallel to
the
x
axis. The velocity of the frame and the air in the material are respectively
u
s
(x)
V
r
exp
(jδ
1
x)
+
V
i
exp
(
−
jδ
2
x)
+
V
r
exp
(jδ
2
x)
V
i
exp
(
=
−
jδ
1
x)
+
(6.91)
u
f
(x)
µ
1
[
V
i
exp
(
V
r
exp
(jδ
1
x)
]
=
−
jδ
1
x)
+
(6.92)
µ
2
[
V
i
exp
(
V
r
exp
(jδ
2
x)
]
+
−
jδ
2
x)
+
In these equations, the time dependence exp
(
j
ωt)
has been removed,
δ
1
and
δ
2
are
given by Equations (6.67) and (6.68), and
µ
1
and
µ
2
by Equation (6.71). The quantities
V
i
,
V
r
,
V
i
and
V
r
are the velocities of the frame at
x
0 associated with the incident
(subscript
i
) and the reflected (subscript
r
) first (index 1) and second (index 2) Biot
=
