Civil Engineering Reference
In-Depth Information
5
Sound propagation in porous
materials having a rigid frame
5.1
Introduction
In Chapter 4, porous materials with cylindrical pores were considered. In the case of com-
mon porous materials, a similar analytical description of sound propagation that takes
into account the complete geometry of the microstructure is not possible. This explains
why the models of sound propagation in these materials are mostly phenomenological
and provide a description only on a large scale. A review of the models worked out
before 1980 can be found in the work by Attenborough (1982). Many models have been
presented since 1980. Moreover direct time-domain analysis has brought new tools for
the modelling and the measurements of these materials (Carcione and Quiroga-Goode
1996, Fellah
et al
. 2003). In order to give a physical basis to the description of sound
propagation in porous media, we have selected in the frequency domain a series of
semi-phenomenological models involving several physical parameters. A brief descrip-
tion of methods used to measure these parameters is given to clarify their physical nature.
As for the case of cylindrical pores, the air in the porous frame is replaced by an equiv-
alent fluid that presents the same bulk modulus
K
as the saturating air and a complex
density
ρ
that takes into account the viscous and the inertial interaction with the frame.
As in Chapter 4, the wave number
k
=
ω(ρ/K)
1
/
2
and the characteristic impedance
Z
c
=
(ρK)
1
/
2
can describe the acoustical properties of the medium. A detailed descrip-
tion of the conditions where an equivalent fluid can be used has been given by Lafarge
(2006). The main condition is the long-wavelength condition. The wavelength is much
larger than the characteristic dimensions of the pores, and the saturating fluid can behave
as an incompressible fluid at the microscopic scale. At the end of the chapter, the homog-
enization method for periodic structures introduced by Sanchez-Palencia (1974, 1980),
Keller (1977), and Bensoussan
et al
. (1978), is presented with the dimensionless analysis
developed by Auriault (1991). Real porous media are generally not periodic. However,
