Introduction to Acoustics

128 Part A Propagation of Sound
Part A 4.6
Normalized impedance
800
5.7° 400
0.057°
600
400
200
200
0
0
–200
1 10
–200
100 1 10 100
400
200
0
–200
1
Real
–Imaginary
Normalized impedance
0.029° 400
0.018°
200
–200
10 100 1
Frequency (Hz)
0
10 100
Frequency (Hz)
Fig. 4.10 Normalized surface impedance predicted for the
four-layer structure, speed of sound in air = 329 m/s (corresponding
to an air temperature of −4 ◦ C) for grazing angles
between 0.018 ◦ and 5.7 ◦
code known as the fast field program for air–ground
systems (FFLAGS) that models sound propagation in
a system of fluid layers and porous elastic layers [4.79].
This numerical theory (FFLAGS) may be used also
to predict the ground impedance at low frequencies.
In Fig. 4.9, the predictions for the surface impedance
at a grazing angle of 0.018 ◦ are shown as a function of
frequency for the layered porous and elastic system described
by Table 4.1 and compared with those for a rigid
porous ground with the same surface flow resistivity and
porosity.
The influence of ground elasticity is to reduce the
magnitude of the impedance considerably below 50 Hz.
Potentially this is very significant for predictions of lowfrequency
noise, e.g., blast noise, at long range.
Figure 4.10 shows that the surface impedance of this
four-layer poro-elastic system varies between grazing
angles of 5.7 ◦ and 0.57 ◦ but remains more or less constant
for smaller grazing angles. The predictions show
two resonances. The lowest-frequency resonance is the
most angle dependent. The peak in the real part changes
from 2 Hz at 5.7 ◦ to 8 Hz at 0.057 ◦ . On the other hand
the higher-frequency resonance peak near 25 Hz remains
relatively unchanged with range.
The peak at the lower frequency may be associated
with the predicted coincidence between the Rayleigh
wave speed in the ground and the speed of sound in
air (Fig. 4.11). Compared with the near pressure dou-
Propagation velocity (m/s)
360
320
280
240
200
160
120
0.3 0.5 0.7 1 2 4 6 8 10 20
Frequency (Hz)
Fig. 4.11 Rayleigh-wave dispersion curve predicted for the
system described by Table 4.1
Sound level (dB re free field)
8
6
4
2
FFLAGS
prediction
Prediction
using Zeff
0
1 10 100
Frequency (Hz)
Fig. 4.12 Excess attenuation spectra predicted for source
height 2 m, receiver height 0.1 m and horizontal range of
6.3 km by FFLAGS (assumed speed of sound in air of
329 m/s) and by classical theory using impedance calculated
for 0.018 ◦ grazing angle
bling predicted by classical ground impedance models,
the predicted reduction of ground impedance at low frequencies
above layered elastic ground can be interpreted