Building Acoustics

192 Building acoustics

of material parameters). Concerning the size of these parameters, one will find values in
the range of some tenths of a micrometer to some hundred micrometers for typical
porous foam materials. The ratio between the two parameters will tell us something of
what the pores look like. We have tried to illustrate this in Figure 5.27. Certainly, the two
parameters will be equal in a material where the pores have simple tube like shape,
whereas Λ < Λ' when the connections between the pores are small and narrow. This is
due to the fact that the viscous length Λ will mainly be determined by the contributions
from areas having large velocity amplitudes, i.e. where the passages are narrow.

Λ = Λ’ Λ < Λ’

Figure 5.27 Sketches of the form of the pores in a porous material.

The expressions for effective density and bulk modulus according to this model is

eff 0 s J

0s

0

eff 1

2 J

0

1G()and

j

.

8

(1)1 G(Pr)

jPr

r

k

k

P

K

σ

ρρ ω

ωρ

γ

μ

γγ ω

ωρ

−

⎡⎤

=+⎢⎥

⎣⎦

=

⎡ ⎤

−− +⎢ ′ ⋅⎥

⎢⎣ ⋅Λ′ ⎥⎦

(5.65)

The functions GJ and GJ ́ are given by

1

(^22)
s0
J 222
1
(^22)
0
J

4

G( ) 1 j

and

Pr

G(Pr ) 1 j.

16

k

r

μρ ω

ω

σ

ρω

ω

μ

⎡⎤

=+⎢⎥

⎢⎥⎣⎦Λ

⎡ Λ′ ⎤

′ ⋅=+⎢ ⎥

⎢⎣ ⎥⎦

(5.66)

Figure 5.28 may serve as an example showing the importance of these characteristic
lengths. Calculations are performed for a 50 mm thick sample directly on to a hard
backing, keeping the characteristic thermal length constant while varying the
corresponding viscous length between 10 and 100 μm. The other parameters used in this
example are given in the figure caption.