Building Acoustics

118 Building acoustics

where V is the room volume and w is the energy density (J/m^3 ) in the room. We shall, for
simplicity, initially assume that the room boundaries are the only absorbing surfaces,
thereby relating the first term to the absorption factors αj of these surface areas Sj. Hence

( )

()

abs ( abs)

ib

j j j,

j j

WW

WIS

α ==

⋅

(4.24)

where Wi is the power incident on all boundaries (walls, floor and ceiling) and Ib is the
corresponding sound intensity.

L

M

W

Sj

Figure 4.7 Room with a sound source, emitting a power W.

Having assumed that the energy density is everywhere the same implies that the
latter quantities are independent of the position on the boundary. Equation (4.23) may
therefore be written as

(^) bb
d
,
j jj dd
w
WI SV IAV
tt

= ⋅∑α +⋅ = ⋅+⋅

dw

(4.25)

where A (m^2 ) is the total absorbing area of the room. It remains to find the relationship
between the energy density w and the intensity Ib. It should be noted that the total sound
intensity at any position in the room is ideally equal to zero because the energy transport
is the same in all directions but certainly, we may associate an effective intensity with the
energy transport in a given direction. The idea is then to calculate the part of the energy
contained in a small element of volume that per unit time impinges on a small boundary
surface element, thereafter integrating the contributions from the whole volume. We shall
skip the details in this calculation, which results in

2

2 0 2

000

b

00

.

444

p

c

wc c p

I

c

ρ

ρ

⋅

⋅

== =





(4.26)

Additionally, we have introduced the relationship between the sound energy density and
the sound pressure in a plane progressive wave, this due to our assumption that the sound