Sound absorbers - Building Acoustics - page 178

frame is completely stiff resulting only in a movement of the air contained in the pores,
this being subjected to viscous and thermal conditions. It should then not come as a
surprise that most models describing sound propagation in porous materials may be
characterized as being phenomenological. One will find models based on one or more
macroscopic material properties, flow resistivity, porosity etc. We shall present some of
these models, which in the literature are termed equivalent fluid models. The material
behaves like a fluid in a macroscopic perspective, and the sound propagates in the form
of a simple compressional wave.
When we cannot assume that the frame is completely stiff the modelling gets more
difficult. The movement of the fame will be coupled to the movement of the air in the
pores resulting in a significant influence on the properties in certain frequency ranges.
There will be several types of wave propagating and, furthermore, it is generally not so
that one type of wave propagates through the frame and another in the air particles in the
pores. The models commonly used in this case are based on Biot theory (see e.g. Allard
(1993)), which, in fact, was developed for quite another purpose, modelling sound
propagation in porous, fluid-filled rock formations. We shall not go into details on this
theory but give a short overview illustrated by examples.
5.5.1 The Rayleigh model
As a very simple model for a porous material we may envisage a bundle or matrix of
very thin tubes. We assume that the tubes have a circular cross section and being
sufficiently thin so as to make the air movements in them governed by viscous forces.
We may then apply the calculations performed connected to the mode of operation of the
Helmholtz resonator. A sketch of the cross section of the material is shown in Figure
5.17.
Figure 5.17 Simple model of a porous material, a bundle of thin tubes imbedded in a solid matrix.
Initially, we look at one of the tubes assuming that the particle velocity is
represented by a mean value < v x > taken over the cross section. We shall also use the
approximation leading to Equation (5.32), assuming that the quantity s from Equation
(5.28) is less than two. This implies that the diameter (2 a ) of the tube should be less than
0.5 mm if the approximation is to be valid for frequencies up to 1000 Hz.