Sound absorbers - Building Acoustics - page 181

It is assumed that the quantity E lies inside the range 0.01-1.0. As mentioned above, the
model has been and still is widely used. One should, however, bear in mind that the
materials used in the development were highly porous mineral wool products. A slightly
different model is given by Mechel (1976) (see also Mechel (1988)), where the equations
are partly theoretically based, i.e. when describing the behaviour at low frequencies,
partly by a curve-fitting procedure on measured data at the higher frequency range. The
model also includes the porosity as a parameter but, unfortunately, this parameter only
affects the low frequency part. In effect, this is again a one-parameter model and to fit
the various expressions together it is advisable to set the porosity parameter equal to
0.95.

When E < E x (see Table 5.1 for the transition between equations), we get

ω γ

π

Γ=Γ + Γ j = j 1 − j 2

real imag

c E

0

(5.53)

2
00

ρ
ωγσ

c

and ZZ = + j Z = − j Γ ,

c real imag

where γ is adiabatic constant for air (≈ 1.4). When E > E x , Mechel gives the following
expressions

ω

( )

⎡

− 0.6193 − 0.6731

⎤

Γ= 0.2082 E + ⋅ + j 1 0.1087 E

⎣ ⎦

c

0

(5.54)

( )

⎡ ⎤

− 0.717 − 0.6601

and Z = ρ c 1 + 0.06082 E − ⋅ j 0.1323 E .

c 0 0

⎣ ⎦

The limiting value for E , denoted E x , that determines whether one shall use Equations
(5.53) or (5.54) for the real and imaginary components of Γ and Z c , respectively, is
shown in Table 5.1.

Table 5.1 Limiting values for the components in the model by Mechel (1988).

Component Limiting value E x

Γ real 0.04

Γ imag 0.008

Z real 0.006

Z imag 0.02

Calculations on a 50 mm thick porous material having a flow resistivity of 10
kPa⋅s/m 2 and backed by a hard wall is shown in Figures 5.18 and 5.19, using the models
of Delany-Bazley and Mechel. The porosity is put equal to 0.95 in Mechel's model. The
first figure gives the real and imaginary part of the input impedance, the second one the
corresponding absorption factor. It is observed that the differences are quite small in this
example. However, Mechel's model does “repair” the anomaly at the lower frequencies.
Wilson (1997) has also developed a simple phenomenological model for sound
propagation in porous materials, based on viewing the thermal and viscous diffusion in
porous media as relaxational processes. Of particular interest compared to the models