Civil Engineering Reference
In-Depth Information
Normal direction
Planar direction
Figure 2.5
A panel of fibrous material. The normal direction is perpendicular to the
surface of the panel, and the planar directions lie in planes parallel to the surface.
to the surface of the panel while in the latter case it flows parallel to the surface of
the layer. The normal flow resistivity
σ
N
is larger than the planar flow resistivity
σ
P
.
The flow resistivity of fibreglass and open-bubble foam generally lies between 1000 and
100 000 N m
−
4
s.
2.5.2 Microscopic and macroscopic description of sound propagation
in porous media
The quantities that are involved in sound propagation can be defined locally, on a micro-
scopic scale, for instance in a porous material with cylindrical pores having a circular
cross-section, as functions of the distance to the axis of the pores. On a microscopic scale,
sound propagation in porous materials is generally difficult to study because of the com-
plicated geometries of the frames. Only the mean values of the quantities involved are of
practical interest. The averaging must be performed on a macroscopic scale, on a homog-
enization volume with dimensions sufficiently large for the average to be significant. At
the same time, these dimensions must be much smaller than the acoustic wavelength.
The description of sound propagation in porous material can be complicated by the fact
that sound also excites and moves the frame of the material. If the frame is motionless,
in a first step, the air inside the porous medium can be replaced on the macroscopic scale
by an equivalent free fluid. This equivalent fluid has a complex effective density
ρ
and
a complex bulk modulus
K
. The wave number
k
and the characteristic impedance
Z
c
of
the equivalent fluid are also complex. In a second step, as shown in Chapter 5, Section
5.7, the porous layer can be replaced by a fluid layer of density
ρ
/
φ
and of bulk modulus
K
/
φ
.
2.5.3 The Laws of Delany and Bazley and flow resistivity
The complex wave number
k
and the characteristic impedance
Z
c
have been measured
by Delany and Bazley (1970) for a large range of frequencies in many fibrous materials
with porosity close to 1. According to these measurements, the quantities
k
and
Z
c
depend mainly on the angular frequency
ω
and on the flow resistivity
σ
of the material.
A good fit of the measured values of
k
and
Z
c
has been obtained with the following
expressions:
Z
c
=
ρ
o
c
o
[1
+
0
.
057
X
−
0
.
754
−
j
0
.
087
X
−
0
.
732
]
(2.28)
ω
c
o
[1
0
.
0978
X
−
0
.
700
j
0
.
189
X
−
0
.
595
]
k
=
+
−
(2.29)
