Introduction to Acoustics

haps the most sophisticated scheme for this purpose
is NORD2000 [4.17]. A European project HAR-
MONOISE [4.18] is developing a comprehensive
source-independent scheme for outdoor sound pre-
4.3 Spreading Losses
Distance alone will result in wavefront spreading. In the
simplest case of a sound source radiating equally in all
directions, the intensity I [W−2 ] at a distance rm from
the source of power P [W], is given by
I = P
, (4.1)
4πr2 This represents the power per unit area on a spherical
wavefront of radius r. In logarithmic form the relationship
between sound pressure level Lp and sound power
LW, may be written
Lp = LW − 20 log r − 11 dB . (4.2)
From a point sound source, this means a reduction of
20 log 2 dB, i. e., 6 dB, per distance doubling in all directions
(a point source is omnidirectional). Most sources
appear to be point sources when the receiver is at a sufficient
distance from them. If the source is directional
then (4.2) is modified by inclusion of the directivity
index DI.
Lp = LW + DI − 20 log r − 11dB . (4.3)
The directivity index is 10 log DFdB where DF is
the directivity factor, given by the ratio of the actual
Sound pressure level (dB re 1m)
0
–10
–20
–30
Spherical spreading
–40
0.1 1
Cylindrical spreading
Finite line
10
Distance/Line length
Fig. 4.1 Comparison of attenuation due to geometrical
spreading from point, infinite line and finite line sources
Sound Propagation in the Atmosphere 4.3 Spreading Losses 115
diction. As in NORD2000 various relatively simple
formulae, predicting the effect of topography for example,
are being derived and tested against numerical
predictions.
intensity in a given direction to the intensity of an omnidirectional
source of the same power output. Such
directivity is either inherent or location-induced. A simple
case of location-induced directivity arises if the point
source, which would usually create spherical wavefronts
of sound, is placed on a perfectly reflecting flat plane.
Radiation from the source is thereby restricted to a hemisphere.
The directivity factor for a point source on
a perfectly reflecting plane is 2 and the directivity index
is 3 dB. For a point source at the junction of a vertical
perfectly reflecting wall with a horizontal perfectly
reflecting plane, the directivity factor is 4 and the directivity
index is 6 dB. It should be noted that these
adjustments ignore phase effects and assume incoherent
reflection [4.19].
From an infinite line source, the wavefronts are cylindrical,
so wavefront spreading means a reduction of
3 dB per distance doubling. Highway traffic may be approximated
by a line of incoherent point sources on an
acoustically hard surface. If a line source of length l consists
of contiguous omnidirectional incoherent elements
of length dx and source strength P dx, the intensity at
a location halfway along the line and at a perpendicular
distance d from it, so that dx = rdθ/cos θ, where
r is the distance from any element at angle θ from the
perpendicular, is given by
I =
�l/2
−l/2
This results in
�
P P
dx = 2tan
2πr2 2πd
−1
� ��
l
,
2d
Lp = LW − 10 log d − 8
+ 10 log
�
2tan −1
� l
2d
��
dB . (4.4)
Figure 4.1 shows that the attenuation due to wavefront
spreading from the finite line source behaves as that from
an infinite line at distances much less than the length of
the source and as that from a point source at distances
greater than the length of the source.
Part A 4.3