146 Building acoustics
absorption characterized by the power attenuation coefficient m. All these attenuation
processes may be assembled in a factor exp(–bc 0 t), where b is a total attenuation
coefficient comprising all loss mechanisms.
Now, the idea is to assume that this attenuation takes place gradually along the
whole path covered by a phonon. Thereby, we may assemble all the energy of phonons
arriving by calculating the integral
0 (4.76)
0s
/(,, )ebc td.
rcwW Prth t∞
= −∫
An approximate solution to this integral, where e.g. the lowest limit is zero, is given by
(^) s0( )
0
3
K3
2
qW
wr
πch= qb, (4.77)where K 0 is the modified Bessel function of zero order. The attenuation coefficient b may
be expressed as
bb hq=+′′(αα,,) sqm+. (4.78)
The quantity , which expresses the attenuation due to the boundary surfaces is, as
indicated, not only a function of the mean absorption exponent
b'
α′=− −ln 1( α)for these
surfaces but is also a function of the ceiling height and the scattering cross section.
4.8.1.2 “Direct” sound energy
The expression giving the direct energy density caused by the source and its infinite
number of images (see Equation (4.63)) may approximately be solved by letting this row
of sources be represented by a line source. The following solution is obtained:
(^0)
0
F
2
WK r
w
rc h hα
π,
⎛⎞′
=⋅⎜⎟
⎝⎠
(4.79)
where
2
with ln(1 )
2
K
α α
α α
α′ −
=⋅ ′=− −
and
F() sin()Ci() cos()Si() 0.
2
xxxxx
⎡ π⎤
=⋅−⎢ −⎥
⎣ ⎦The functions Ci and Si are the so-called cosine and sine integral function (see e.g.
Abramowitz and Stegun (1970)). We have thereby arrived at a closed expression for the
energy density in the direct field but without taking the scattered part into account. We
shall have to correct it by the probability exp(–qc 0 t) that a phonon has not been scattered
during the time t. Also taking the excess attenuation due to air absorption into account,
we finally may express the direct (or the non-scattered) energy density by