Building Acoustics

Room acoustics 143

1

1

,

4

N

i

i

S

q

V =

= ∑ (4.71)

when a total of N objects with surface areas Si are present in a room of volume V.

0 20 40 60 80 100 120 140 160 180 200

Distance c 0 t (m)

0

0.1

0.2

0.3

0.4

0.5

Probability

q=0.1

1

2

3

4

6

(^810)
Figure 4.20 The probability of a wave (a phonon) hitting a given number of scattering objects, indicated by the
number on the curves, having propagated a path of length c 0 ·t. The scattering cross section q is equal to 0.1m-1.
Figure 4.20 shows the probability density P, according to Equation (4.70), of a
phonon hitting a given number k of objects having propagated a path of length c 0 ·t. The
number k is the parameter indicated on the curves calculated for a scattering cross section
q equal to 0.1 m-1. The Poisson distribution will typically give a high probability for
hitting a single object; however, the corresponding width is small, whereas the
probability for hitting many objects is small but the distribution is broad.
An important quantity relating to these aspects is the mean free path Rof the
sound. This quantity is generally used to characterize the path that the sound is expected
to travel between two reflections. For an empty rectangular room having a volume V and
a total surface area of S, we may show that Ris equal to 4V/S. Introducing scattering
objects into the room (see Figure 4.21) we may, by using the probability function given
by Equation (4.70), calculate the corresponding probability function of the free paths R
and thereby the expected or mean valueR. The outcome is that Ris equal to 1/q.

4.8 Calculation models. Examples

In the literature one will find reported a very large number of different models for
predicting sound propagation in large rooms. A number of these are implemented in
commercial computer software, e.g. CATT™, EASE™, EPIDAURE™ and ODEON™.