Civil Engineering Reference
In-Depth Information
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
∞
∫
−
bc t
wW Prt h
=
(,, )e
d.
t
(4.76)
0
s
rc
/
0
An approximate solution to this integral, where e.g. the lowest limit is zero, is given by
3
qW
(
)
w
=
K3
r
qb
,
(4.77)
s
0
2
π
ch
0
where K
0
is the modified Bessel function of zero order. The attenuation coefficient
b
may
be expressed as
(
)
′ ′
bb
=
α
,,
hq
+
α
qm
+
.
(4.78)
s
The quantity , which expresses the attenuation due to the boundary surfaces is, as
indicated, not only a function of the mean
absorption exponent
b
'
(
)
′
=− −
for these
surfaces but is also a function of the ceiling height and the scattering cross section.
α
ln 1
α
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:
WK
α
′
r
⎛
⎞
w
=
⋅
F
⎠
,
(4.79)
⎜
⎟
0
2
π
rc h
h
⎝
0
where
α
′
−
2
α
K
=⋅
with
α
′
=− −
ln(1
α
)
2
α
and
π
⎡
⎤
F() sin()Ci() cos()Si()
x
=
x
⋅
x
−
x
x
−
⎦
.
⎢
⎥
2
⎣
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
