Civil Engineering Reference
In-Depth Information
when the distance
R
1
from the image of the source to the receiver increases. Similar
results are obtained for semi-infinite layers. However predictions performed with the
steepest descent method in the same range of frequencies and similar
R
1
for
θ
0
close
to
π
/2 can give wrong results. A new formulation of the problem must be performed,
which takes into account the location of the singularities of the reflection coefficient.
7.5
Poles of the reflection coefficient
7.5.1 Definitions
The singularities of the plane wave reflection coefficient are located at sin
θ
p
=
s
p
satis-
fying
Z
0
Z
s
(s
p
)
cos
θ
p
=−
(7.25)
leading to a denominator of
V
in Equation (7.10) equal to 0.
For a locally reacting medium, Z
s
does not depend on the angle of incidence and a
pole can only exists at
s
p
satisfying cos
θ
p
=−
Z
0
/
Z
s
. There are two poles in the complex
s
plane for both determinations of
1
−
cos
2
θ
p
.
For a semi-infinite layer, it may be shown that only one cos
θ
p
is related to a zero of
the denominator of
V
and cos
θ
p
is given by
n
2
1
/
2
−
1
ρ
2
/(ρ
0
φ)
2
cos
θ
p
=−
(7.26)
−
1
There is a minus sign before the square root because Re cos
θ
p
is negative. Both
related
s
p
are given by
ρ
2
/(ρ
0
φ)
2
1
/
2
n
2
−
sin
θ
p
=±
(7.27)
ρ
2
/(ρ
0
φ)
2
−
1
The pole trajectory as a function of frequency is represented in the complex sin
θ
plane in Figure 7.7(a) and in the complex cos
θ
plane in Figure 7.7(b).
For a layer of finite thickness, there is at any frequency an infinite number of poles.
It will be shown in Section 7.6 that if one pole is located at
θ
p
close to
π
/2, the one
parameter which characterizes the porous layer for the prediction of the reflected field at
an angle of specular reflection
θ
0
close to
π
/2 is cos
θ
p
.When
|
Z
s
(s
p
)
|
Z
0
in Equation
(7.25), the pole related to
s
p
is located at
θ
p
close to
π
/2. This happens for instance for
media with a large flow resistivity, and for porous layers having a small thickness. A
thin layer is a layer where
|
k
1
l
|
1. Any layer having a finite thickness is a thin layer at
a sufficiently low frequency. The first-order development of 1
/Z
s
(s
p
)
is
jφkl
cos
2
θ
1
/Z
,
and cos
2
θ
1
, neglecting the second-order term cos
2
θ/n
2
in Equation (7.9), can be replaced
by 1-1
/n
2
. The related solution of Equation (7.25) is (Allard and Lauriks 1997; Lauriks
et al
. 1998)
−
1
)
φZ
0
nZ
jk
0
l(n
2
cos
θ
p
=−
(7.28)
