Introduction to Acoustics

124 Part A Propagation of Sound
Part A 4.6
the roughness axes, may be written [4.66]
�
3V 2k3 ��
b
β ≈
1 +
2
δ2
�
+ iVk(δ − 1) , (4.38)
2
where V is the roughness volume per unit area of surface
(equivalent to mean roughness height), b is the
mean center-to-center spacing, δ is an interaction factor
depending on the roughness shape and packing density
and k is the wave number. An interesting consequence
of (4.38) is that a surface that would be acoustically
hard if smooth has, effectively, a finite impedance at
grazing incidence when rough. The real part of the admittance
allows for incoherent scatter from the surface
and varies with the cube of the frequency and the square
of the roughness volume per unit area. The same approach
can be extended to give the effective normalized
surface admittance of a porous surface containing 2-D
roughness [4.65, 69]. For a randomly rough porous surface
near grazing incidence [4.70] it is possible to obtain
the following approximation:
� �� �
〈H〉Rs 2 �
Zr ≈ Zs −
− 1 , Re(Zr) ≥ 0
γρ0c0 ν � ,
(4.39)
Corrected level difference (dB re 19 m)
–5
0
5
10
15
20
25
10
100
1 × 10 4
1 × 10 3
Frequency (Hz)
Fig. 4.6 Parkin and Scholes data for the level difference
between microphones at a height of 1.5 m and at distances
of 19 m and 347 m from a fixed jet engine source (nozzlecenter
height 1.82 m) corrected for wavefront spreading
and air absorption. � and ♦ represent data over airfields
(grass covered) at Radlett and Hatfield respectively with
a positive vector wind between source and receiver of
1.27 m/s (5ft/s); × represent data obtained over approximately
0.15 m thick (6–9 inch) snow at Hatfield and a
positive vector wind of 1.52 m/s(6ft/s). (After [4.67, 68])
where ν = 1 + 2 3 π〈H〉, 〈H〉 is the root mean square
roughness height and Zs is the impedance of the
porous surface if it were smooth. This can be used
with an impedance model or measured smooth surface
impedance to predict the effect of surface roughness for
long wavelengths.
Potentially, cultivation practices have important influences
on ground effect since they change the surface
properties. Aylor [4.71] noted a significant change in
the excess attenuation at a range of 50 m over a soil
after disking without any noticeable change in the meteorological
conditions. Another cultivation practice is
sub-soiling, which is intended to break up soil compaction
300 mm or more beneath the ground surface
caused, for example, by the repeated passage of heavy
vehicles. It is achieved by creating cracks in the compacted
layer by means of a single- or double-bladed
tine with sharpened leading edges. Sub-soiling only has
a small effect on the surface profile of the ground. Plowing
turns the soil surface over to a depth of about 0.15 m.
Measurements taken over cultivated surfaces before and
after sub-soiling and plowing have been shown to be consistent
with the predicted effects of the resulting changes
in surface roughness and flow resistivity [4.65, 72].
4.6.6 Examples of Ground Attenuation
under Weakly Refracting Conditions
Pioneering studies of the combined influences of ground
surface and meteorological conditions [4.67, 68] were
carried out using a fixed Rolls Royce Avon jet engine as
a source at two airfields. The wind speeds and temperatures
were monitored at two heights and therefore it was
possible to deduce something about the wind and temperature
gradients during the measurements. However,
perhaps because the role of turbulence was not appreciated
(Sect. 4.8.3), the magnitude of turbulence was not
monitored. This was the first research to note and quantify
the change in ground effect with type of surface.
Examples of the resulting data, quoted as the difference
between sound pressure levels at 19 m (the reference
location) and more distant locations corrected for the
decrease expected from spherical spreading and air
absorption, are shown in Fig. 4.6. During slightly downwind
conditions with low wind speed (< 2ms −1 )and
small temperature gradients (< 0.01 ◦ /m), the ground
attenuation over grass-covered ground at Hatfield, although
still a major propagation factor of more than
15 dB near 400 Hz, was less than that over the other
grass-covered ground at Radlett and its maximum value
occurred at a higher frequency. Snowfall during the pe-