Civil Engineering Reference
In-Depth Information
At these frequencies, for samples of thickness
l
around 3 cm or less, cot
kl
can be
developed to first order in
kl
, and the impedance
Z(M
2
)
is given by
jK
φωl
Z(M
2
)
=−
(5.11)
The reflection coefficient at M
2
is given by
Re
Z(M
2
)
j
Im
Z(M
2
)
Re
Z(M
2
)
+
Z
c
+
j
Im
Z(M
2
)
−
Z
c
+
R(M
2
)
=
(5.12)
where
Z
C
is the characteristic impedance of the free air. This equation can be rewritten,
under the conditions
|
Im
Z(M
2
)
|
Z
c
and
|
Im
Z(M
2
)
|
Re
Z(M
2
)
, which are fulfilled
at sufficiently low frequency,
1-2
Z
c
Re
Z(M
2
)
Im
2
Z(M
2
)
exp
(
R(M
2
)
=
−
jϕ)
(5.13)
where
ϕ
is a small real angle,
ϕ
=−
2
Z
c
/
Im
Z(M
2
)
. The function in the square brackets
in Equation (5.13) is a development at the first-order approximation of the modulus of
R
. If the losses out of the porous sample are neglected, an incident pressure amplitude
p
at M
1
corresponds to a total pressure
p
T
given by
p
T
=
p
[1
+
R(M
2
)
exp
(
−
2
jωd/c)
]
(5.14)
where
c
is the speed of sound. This equation can be rewritten
p
1
1-2
Z
c
Re
Z(M
2
)
Im
2
Z(M
2
)
cos
ϕ
j
sin
ϕ
2
ω
d
c
2
ω
d
c
p
T
=
+
+
−
+
(5.15)
The minimum value for
|
p
T
|
is given by
p
2
Z
c
Re
Z(M
2
)
Im
2
Z(M
2
)
min
|
p
T
|=
(5.16)
and is obtained for
ω
satisfying the relation
ϕ
+
2
ωd/c
=
π
. For small variation of
ω
around this value, cos
(ϕ
+
2
ωd/c)
is stationary. The variation
ω
related to an increase
|→
√
2min
of the amplitude min
|
p
T
|
p
T
|
is given by
ω
2
d
2
Z
c
Re
Z(M
2
)
Im
2
Z(M
2
)
c
=
(5.17)
At sufficiently low frequency, Re
K
can be replaced by
P
0
in Equation (5.11), and
Im
K
is given by
ω
ω
dP
0
2
φlγ
Im
K
=
(5.18)
An example is presented in Figure 5.2 for layers of steel beads of thickness
l
ranging
from 2 to 19 cm. The mean diameter of the beads is 1.5 mm, and the viscous static perme-
