Building Acoustics

190 Building acoustics

5.5.4.1 The model of Attenborough

Attenborough (1983,1992), introduces an additional parameter sf, denoted pore shape
factor, as a phenomenological description of the geometrical form of the pores. The
problem with this factor is that it cannot be measured separately but must be estimated by
fitting the model to measured data. As shown in Chapter 3 (section 3.5.3), we may
express the complex characteristic impedance Zc and the propagation coefficient Γ by an
equivalent or effective density ρeff and a corresponding bulk modulus Keff, thus

(^) ceffeff eff
eff
ZK and j.
K
ρ
=Γρ=ω (5.60)
According to the model of Attenbourough we get
eff 0 s
1A
A0A
0
eff
1A
A0 A

1

and 2 J( j) 1 jJ ( j)

.

2 J( Pr j) 1( 1) Pr j J ( Pr j)

k s ss

P K s ss

ρρ

γ

=

⎡⎤−

⎢⎥−

⎢⎥⎣⎦−−

=

⎡⎤⋅−

⎢⎥+−

⎢⎥⎣⎦⋅− ⋅−

(5.61)

The symbol J denotes a Bessel function and the quantity sA is given by

(^) A 0s
f

1 8

.

k s sr

ωρ σ

= (5.62)

The quantity Pr in Equation (5.61) is the so-called Prandtl number given by μ⋅ cp/κ. This
number is a constant for a given fluid that describes the relationship between the
coefficient of viscosity μ, the thermal conductivity κ and the specific heat capacity at
constant pressure cp. For air we get Pr ≈ 0.71.
Assuming sA << 1, which implies low frequency and/or high flow resistivity r,
Attenborough gives the following expressions for characteristic impedance and
propagation coefficient:

1

(^222)
f ssf
c0
00

4 41

jj/ j Prj 33

sr kksr Z c

ωγ ωρ γσ ωρ σ γ σ ωρ

⎡⎤ ⎡⎛⎞−

≈−ΓΓ≈ −−⎢⎥ ⎢⎜⎟

⎢⎥⎣⎦ ⎢⎣⎝⎠ 0

.

⎤

⎥

⎥⎦

(5.63)

As an example on the use of these equations, Figure 5.26 shows measured and predicted
absorption factors for discs of a porous wood (rattan palm) placed at given distances
from a hard wall. In one set of curves the disc thickness d is 5 mm with an air cavity
depth l of 85 mm and in the other set the disc thickness is 10 mm and the cavity depth is
40 mm. Measurements are performed in a standing wave tube, starting with
measurements on a 50 mm thick sample placed directly on the hard backing surface.
From the measured impedance data on this sample, the appropriate material parameters
were extracted by fitting the Attenborough model to the data. We are then in the position