Civil Engineering Reference
In-Depth Information
0
0.4
Exact
Simulated measurement
Exact
Simulated measurement
−
0.25
0.2
0.5
−
0
−
0.75
−
0.2
1
−
−
0.4
−
1.25
−
0.6
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
Frequency(kHz)
Frequency(kHz)
Figure 7.14
Comparison
between
the
exact
cos
θ
p
and
a
simulated
measurement
obtained with Equation (7.53). Material 1,
l
=
3cm,
r
=
1m,
z
1
=
z
2
=−
5cm.
0
Exact
Simulated measurement
−
0.1
Exact
Simulated measurement
−
0.1
−
0.15
−
0.2
−
0.2
−
0.3
−
0.25
−
0.4
−
0.3
−
0.5
−
0.35
−
0.6
−
0.4
−
0.7
−
0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Frequency (kHz)
Frequency (kHz)
Figure 7.15
Comparison between the exact cos
θ
p
and a simulated measurement
obtained with Equation (7.56) from the variation of the total pressure on an axis perpendic-
ular to the surface of the layer. Material 1,
l
z
2
=−
0
.
5cm,
z
2
=−
2
.
5cm,
=
3cm,
z
1
=
r
=
1m.
are presented in Allard
et al
. (2003a, b) for thin layers of porous foam. In the domain
of validity of Equation (7.53),
θ
p
is close to
π
/2, and from Equation (7.25)
Z
s
(
sin
θ
p
)
=
−
Z
0
/
cos
θ
p
, the surface impedance at an angle of incidence
θ
p
close to
π
/2, can be
evaluated from cos
θ
p
. This surface impedance can be an important parameter in room
acoustics. The reflection coefficient at an angle of incidence equal to
π
/2 is - 1, and there
is no absorption. However, a major part of sound absorption can occur at large angles of
incidence where the impedance remains close to the impedance at an angle of incidence
equal to
π
/2.
Measurements on layers of glass beads and sand having a large flow resistivity and
a small porosity are presented in Hickey
et al
. (2005). The thickness of these layers
was sufficiently large for their reflection coefficient to be very similar to the reflection
coefficient of semi-infinite layers. The modulus of the surface impedance close to grazing
incidence is much larger than the characteristic impedance of air and the pole at Re
θ>
0
is located close to
θ
=
π/
2. As indicated in Section (7.5.2), the Brewster angle
θ
B
of
total refraction is related to
θ
p
by cos
θ
B
=−
cos
θ
p
. The Brewster angle can be evaluated
from the measured cos
θ
p
.
