Building Acoustics

180 Building acoustics

the same as the quantity perforation rate ε used in section 5.4.1.5. Here we have to put r
= r'/σ and Z = Z'/σ, hence

00

00

0

'1j

and 1 j ,

r

k

c

c r

Z

ωσ

ρω

ρ σ

σ ρω

=−

=−

(5.49)

where Z will be the characteristic impedance for the equivalent fluid represented by this
bundle of tubes. Z will also be the input impedance for a half-infinite thickness of such a
porous medium, for which we may use Equation (5.4) to calculate the absorption factor
for normal incidence. It should, however, be more realistic to calculate a situation where
the porous medium has a finite thickness, also terminated by a hard reflecting surface.
This will represent a first model simulating a porous sample of e.g. mineral wool placed
against a hard wall in a room. We may modify Equation (5.21) by introducing the
complex wave number k' and also exchanging the characteristic impedance ρ 0 c 0 for air
by the impedance Z. The input impedance Zg will then take the form

Zg=− ⋅jcotg(').Zkd (5.50)

Assuming that the thickness d of the material is much less than the wavelength (k'd <<
1), we may use an approximation for the cotangent function, setting cotg(x) ≈ 1/x - x/3.
Hence
2
00
g 33 j
Z rd o dc
d

ρω ρ

σωσ

⎛⎞

≈+⎜−

⎜

⎝⎠

⎟.

⎟

(5.51)

This expression warrants several comments. First, the real part will only be one-third of
the flow resistance. We will also get some sort of a resonance when the imaginary part is
equal to zero. Normally, however, the stiffness part will dominate, which in practice
gives a relatively high resonance frequency f 0. (Make a calculation of f 0 setting e.g. d
equal 50 mm.)

5.5.2 Simple equivalent fluid models

A model suggested by Delany and Bazley (1970) is, due to its simplicity, widely used for
describing the behaviour of porous materials, being applied to materials ranging from
mineral wool products to porous soil. They developed their model, giving the complex
wave number and the characteristic impedance, in a purely empirical way by
measurements on a broad range of materials having a porosity of approximately one.
This was done by fitting of data to a model having the flow resistivity and the frequency
as parameters. Using the propagation coefficient Γ = j⋅k′ instead of the complex wave
number k′, the expressions are

0.754 0.732

c00

0.700 0.595 0

0

1 0.0571 j 0.087

and j 1 0.0978 j 0.189 where.

Zc E E

f

EEE

cr

ρ

ω ρ

−−

−−

=+⋅−⋅⋅⎡⎤

⎣⎦

Γ= ⎡⎤+ ⋅ − ⋅ ⋅ =

⎣⎦

(5.52)