Sound absorbers 203
(^2) g000
g000
cos
1and
ppcos
Z c
RR
Z c
φ ρ
α
φρ
−
=− =
+
(5.77)
The angle φ 0 denotes the direction of plane wave incidence on the first layer.
Furthermore, we have assumed that the medium on the input side is air having
characteristic impedance ρ 0 c 0.
As mentioned above, our model will turn out much more complicated if we want to
include elastic materials either solid or porous. In the former case, we need at least four
physical variables for description, e.g. the particle velocity and the stress in two
directions. The corresponding matrix to the one given in Equation (5.75) will therefore
be a 4 by 4 matrix instead of a 2 by 2 matrix. It may be shown, however, that if there are
fluid layers on both sides of the elastic layer, this 4 by 4 matrix may be reduced to a
simple 2 by 2 matrix.
Having a porous elastic material, on which we want to use the Biot theory, we end
up with a 6 by 6 matrix. If we want to combine layers described in such a different way
we cannot just multiply the matrices to make a model for the complete system; we shall
have to construct coupling matrices expressing the boundary conditions between the
layers. We have presented results above that have been calculated using such a technique
(see section 5.5.5). Here we shall just illustrate the technique by showing how to find the
components of the 2 by 2 matrix for a porous material described as an equivalent fluid.
For a corresponding description using the Biot theory we shall refer to the literature (see
e.g. Brouard et al. (1995)).
5.7.1.1 Porous materials and panels
We shall characterize a porous material by using the wave number k and the
characteristic impedance Zc, both normally complex quantities. Alternatively, we could
have used the effective density and the bulk modulus. However, the conversions between
these variables are simple if one wishes to use the alternative description. For simplicity,
we shall consider plane wave incidence normal to a layer of thickness d (see Figure
5.37). Our task is, first, to set up the relationships between the sound pressure and the
particle velocity on the two sides of the layer, second, to cast these into the form given in
Equation (5.75).
Figure 5.37 Sound transmission through a porous layer of thickness d.p 1 p 2
v 1 v 2
k, Zc
x=0 x=d
Generally, we may express these variables by the following equations, when assuming
harmonic time dependence