Civil Engineering Reference
In-Depth Information
−
0.06
−
0.04
−
0.08
0.05
−
200 Hz
0.1
4000 Hz
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−
0.06
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0.12
4000 Hz
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0.07
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0.14
−
0.08
200 Hz
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0.16
−
0.09
−
0.18
1150 Hz
950 Hz
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0.1
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0.2
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0.22
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0.12
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0.24
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0.75
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0.85
0.9
0.95
1
−
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−
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−
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−
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−
0.3
−
0.2
Re sin
q
p
Re cos
q
p
(a) (b)
Figure 7.7
Trajectories of the pole of the reflection coefficient for a semi-infinite layer
of material 1.
This pole is called the main pole. Simple iterative methods can be used to predict
cos
θ
p
without approximations. For thin layers, it was shown in Allard and Lauriks (1997)
that the other poles are far from the real sin
θ
axis and cannot contribute significantly to
the reflected field. Equation (7.28) can be rewritten
cos
θ
p
=−
jφk
0
l
γP
0
ρ
0
ρ
K
−
(7.29)
For a perfect fluid without viscosity and thermal conduction, cos
θ
p
is given by
α
−
1
cos
θ
p
=−
jφk
0
l(
1
−
)
(7.30)
∞
and is imaginary with a negative imaginary part.
7.5.2 Planes waves associated with the poles
Thin layers
For an angle of incidence
θ
θ
p
the plane reflected wave satisfies the boundary condi-
tions with no incident wave. The space dependence of this wave is exp[
=
−
z
cos
θ
p
)
]. For a thin layer without damping, the reflected wave is propagative in the
x
direction and evanescent in the direction opposite to
z
.
As indicated at the end of Section 3.2 the axes of evanescence and of propagation
are perpendicular in a medium with a real wave number. In the present case the wave is
evanescent in the direction perpendicular to the surface of the porous layer and opposite
to the
z
axis. Using a complex wave number vector
k
=
k
R
−
jk
0
(x
sin
θ
p
+
j
k
I
for the reflected wave,
with
(k
R
)
2
(k
I
)
2
k
0
, the spatial dependence of the plane wave is given by
−
=
j(
k
R
j
k
I
)
OM
]
=
exp [
−
j(k
x
x
k
z
z)
(k
x
x
k
z
z)
]
exp[
−
+
+
+
+
(7.31)
The direction of
k
I
is opposite to the direction of evanescence. The wave number is
represented symbolically in Figure 7.8(a). For a thin layer saturated by a perfect fluid the
