Introduction to Acoustics

sure and the continuity of the normal component of air
particle velocity.
4.6.2 Attenuation of Spherical Acoustic
Waves over the Ground
The idealized case of a point (omnidirectional) source of
sound at height zs and a receiver at height z, separated by
a horizontal distance r above a finite-impedance plane
(admittance β) is shown in Fig. 4.5.
Between source and receiver, a direct sound path
of length R1 and a ground-reflected path of length R2
are identified. With the restrictions of long range
(r ≈ R2), high frequency (kr ≫ 1, k � �
z + zs ≫ 1, where
k = ω/c and ω = 2π f , f being frequency) and with
both the source and receiver located close (r ≫ z + zs)
to a relatively hard ground surface (|β| big ≪ 1), the total
sound field at (x, y, z) can be determined from
p(x, y, z) = e−ikR1
4π R1
+ e−ikR2
4π R2
+ Φp + φs , (4.24)
where
Φp ≈ 2i √ � �1/2 12
π kR2 β e −w2
erfc(iw) e−ikR2
.
4π R2
(4.25)
and w, sometimes called the numerical distance, is given
by
w ≈ 1 2 (1 − i)� kR2(cos θ + β) . (4.26)
φs represents a surface wave and is small compared with
Φp under most circumstances. It is included in careful
computations of the complementary error function
erfc(x) [4.43]. In all of the above a time dependence of
e iωt is understood.
After rearrangement, the sound field due to a point
monopole source above a locally reacting ground be-
Source
θ
R1
R 2
Receiver
Fig. 4.5 Sound propagation from a point source to a receiver
above a ground surface
comes
Sound Propagation in the Atmosphere 4.6 Ground Effects 121
p(x, y, z) = e−ikR1
4π R1
× e−ikR2
4π R2
+ � Rp + � � �
1 − Rp F(w)
, (4.27)
where F(w), sometimes called the boundary loss factor,
is given by
F(w) = 1 − i √ πwexp(−w 2 )erfc(iw) (4.28)
and describes the interaction of a spherical wavefront
with a ground of finite impedance [4.44]. The term in
the square bracket of (4.27) may be interpreted as the
spherical wave reflection coefficient
Q = Rp + � �
1 − Rp F(w) , (4.29)
which can be seen to involve the plane wave reflection
coefficient Rp and a correction. The second term of Q allows
for the fact that the wavefronts are spherical rather
than plane. Its contribution to the total sound field has
been called the ground wave, in analogy with the corresponding
term in the theory of amplitude-modulated
(AM) radio reception [4.45]. It represents a contribution
from the vicinity of the image of the source in
the ground plane. If the wavefront is plane � R2 →∞ �
then |w|→∞ and F → 0. If the surface is acoustically
hard, then |β|→0, which implies |w|→0and
F → 1. If β = 0, corresponding to a perfect reflector, the
sound field consists of two terms: a direct-wave contribution
and a wave from the image source corresponding
to specular reflection and the total sound field may be
written
p(x, y, z) = e−ikR1
4π R1
+ e−ikR2
4π R2
This has a first minimum corresponding to destructive interference
between the direct and ground-reflected components
when k � � � �
R2 − R1 = π, orf = c/2 R2 − R1 .
Normally, for source and receiver close to the ground,
this destructive interference is at too high a frequency
to be of importance in outdoor sound prediction. The
higher the frequency of the first minimum in the ground
effect, the more likely that it will be destroyed by
turbulence (Sect. 4.8).
For |β|≪1 but at grazing incidence (θ = π/2), so
that Rp =−1and
p(x, y, z) = 2F(w)e −ikr /r , (4.30)
the numerical distance w is given by
w = 1 2 (1 − i)β√kr . (4.31)
.
Part A 4.6