Building Acoustics

142 Building acoustics

• The sound scattering objects are assumed to be point like and the energy of the
  incident wave is scattered evenly in all directions.


• The scattering phenomenon follows a Poisson process. The energy emitted by the
  source is sent out in discrete quantities as “phonons” or sound packages having
  energy W⋅Δt, where W is the sound power of the source.

0123

Angle (radian)

90

100

110

120

Sound p

ressu

re level (dB)

SURFACE

Flat - Anal.

Flat - FEM

Diffusor - FEM

Figure 4.19 Total sound pressure level as a function of angle calculated on the circle with radius r equal one
meter; situation as depicted in Figure 4.18. The source (S) power is equal to 1.0 W. Thick solid line – diffuser
surface (FEM). Thin solid line – flat surface (FEM). Dashed line – flat, infinitely large surface (analytical).

The validity of the first hypothesis will depend on the ratio of the dimensions of the
scattering object and the actual wavelength. Initially assuming that an object scatters
sound, not only reflects sound in a specular way, we shall put up a limit on the
relationship between a typical dimension D and the wavelength λ, demanding that D/λ >
1/2π.
From the second hypothesis follows that the probability density Pk of a phonon
hitting a number k scattering objects within a time interval tk is given by

(^0) () 0
0
e
()
!
qc tk k k
kk
qc t
Pct
k

− ⋅

= , (4.70)

where c 0 is the usual wave speed and q is the average scattering cross section per unit
room volume, a quantity also denoted the scattering frequency.
The determination of q is difficult for scattering objects having a complicated
shape. A common practice is equalizing the scattering effect (at high frequencies) of an
object having a total surface area S by the one offered by a sphere of equal surface area.
The average scattering cross section may then be expressed as