Building Acoustics

186 Building acoustics

absorption. Taking porous materials as an example, we find that an angle of incidence
50–60° will give a maximum absorption factor. The mean value obtained when
averaging over all angles of incidence, i.e. the statistical absorption factor αstat, is of even
more practical interest. We may determine this factor by using our models to calculate
the mean value for incidence angles in the range 0 to 90°. As shown earlier on, assuming
local reaction such that the input impedance Zg is independent of the angle of incidence
φ, the statistical absorption factor is expressed as

(^222)
stat
00
2()sincosd21Rp sincosd
ππ
α ==αφ φ φ φ ⎡⎤⎢⎥−φ φ φ,

∫∫⎣⎦

(5.55)

where Rp is the pressure reflection factor given by

g0n

g0n

cos cos 1

.

p cos cos 1

ZZZ

R

ZZZ

φ φ

φφ

− −

==

+ +

(5.56)

100 1000

200 400 600 800 2000 4000

Frequency (Hz)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Abso

rption facto

r

60

70

80

89

0

30

Figure 5.23 Absorption factor for a 50 mm porous sample with hard backing. The parameter on the curves is
the angle of incidence in degrees. The dashed curve shows the statistical absorption factor. Delany-Bazley
model with r = 10kPa⋅s/m^2.

In the last expression we have normalized the input impedance by the characteristic
impedance Z 0 for air. Inserting this last expression into Equation (5.55) we get

(^) {}
(^22)
stat n 2
0 n
sin cos
8Re d ,
cos 1

Z

Z

π

φφ

αφ

φ

=

+

∫ (5.57)