Sound absorbers - Building Acoustics - page 186

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 α stat , 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 Z g is independent of the angle of incidence
ϕ, the statistical absorption factor is expressed as

π π

2 2

2

⎡ ⎤

∫ ∫

α = 2( αϕ ϕ ϕϕ ) sin s d21 = − R sin s ϕ ϕϕ d ,

(5.55)

stat

⎢

p

⎥

⎣ ⎦

0 0

where R p is the pressure reflection factor given by

Z cos ϕ − Z

Z cos ϕ − 1 .

g 0

n

R = =

(5.56)

p

Z cos ϕ + Z Z cos ϕ + 1

g 0 n

1.0

0.9

60
70

0.8

0.7

0.6

80

0.5

0.4

30

0.3

0

0.2

89

0.1

0.0

200 400 600 800 2000 4000

100 1000

Frequency (Hz)

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

π

2

2

sin ϕϕ cos

{}

∫

α = 8Re Z d , ϕ (5.57)

stat n

2

Z cos ϕ + 1

n

0