Sound absorbers - Building Acoustics - page 190

5.5.4.1 The model of Attenborough
Attenborough (1983,1992), introduces an additional parameter s f , 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 Z c and the propagation coefficient Γ by an
equivalent or effective density ρ eff and a corresponding bulk modulus K eff , thus

ρ

eff

Z = K ρ and Γ = ω j .

(5.60)

c eff eff

K

eff

According to the model of Attenbourough we get

1

ρρ = k and

eff 0 s

⎡ ⎤

2 J( s − j)

1A

⎢ 1 − ⎥

s − jJ ( s

P

− j)

⎢ ⎥

⎣ ⎦

A 0 A

(5.61)

γ

0

K = ⎡ .

eff

⎤

2 J( Pr ⋅ s − j)

1 A

1( ) Pr +− γ

⎢ ⎥

⋅ s − j J ( Pr ⋅ s − j)

⎢ ⎥

⎣ ⎦

A 0 A

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

1 8 ωρ

σ

k

0s

(5.62)

s = .

A

s r

f

The quantity Pr in Equation (5.61) is the so-called Prandtl number given by μ⋅ c p / 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 c p . For air we get Pr ≈ 0.71.
Assuming s A << 1, which implies low frequency and/or high flow resistivity r ,
Attenborough gives the following expressions for characteristic impedance and
propagation coefficient:

1

⎡

2

⎤ ⎡ ⎛

2

⎤
⎥
⎦

⎞

sr 4 k ω 4 γ − 1 k sr

2

f s s f

Z ≈ j ωρ − j / Γ Γ ≈ j γσ − Pr − j . (5.63)

⎢ ⎥ ⎢⎜ ⎟

c 0

ωρ 3 σ c 3 γ σ ωρ

⎢ ⎥ ⎢ ⎝

⎠

⎣ ⎦ ⎣

0 0 0

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