Civil Engineering Reference
In-Depth Information
z
2
, not on each height separately. The
right-hand side of this equation is referred to as the Sommerfeld integral. The integral can
be evaluated up to a limit for
ξ
which depends on
z
1
+
The reflected pressure depends on the sum
z
1
+
z
2
because
µ
in the exponential is
imaginary with a positive imaginary part for
ξ
>
k
0
which increases with
µ
. The singu-
larity for
µ
=
0 is removed by using
µ
instead of
ξ
as a variable of integration. A simple
test for the accuracy of the evaluation for a given geometry consists in the comparison
of the evaluation performed for
V
jk
0
R
1
)/R
1
. The Bessel function is
related to the Hankel function of first order
H
0
by
J
0
(u)
=
1andexp
(
−
=
0
.
5
(H
0
(u)
H
0
(
−
−
u))
with
µ(
−
ξ)
=
µ(ξ)
,and
V(
−
ξ/k
0
)
=
V(ξ/k
0
)
. Therefore
p
r
can be rewritten as
∞
j
2
ξ
µ
H
0
(
p
r
=
−
ξr)V(ξ/k
0
)
exp[
jµ(z
1
+
z
2
)
]d
ξ
(7.7)
−∞
From Equation (3.39), the surface impedance at oblique incidence is given by
Z
φ
cos
θ
1
Z
s
(
sin
θ)
=−
j
cotg
kl
cos
θ
1
(7.8)
(ρK)
1
/
2
is the characteristic
impedance in the air saturating the porous medium,
k
is the wave number in the porous
medium,
θ
1
is the refraction angle satisfying
k
sin
θ
1
=
where
l
is the thickness of the layer,
φ
is the porosity,
Z
=
k
0
sin
θ
,andcos
θ
1
is given by
1
n
2
−
1
n
2
cos
2
θ
1
=
1
−
cos
2
θ
(7.9)
where
n
=
k/k
0
. From Equation (3.44) the reflection coefficient
V
is given by
Z
s
(
sin
θ)
−
Z
0
/
cos
θ
V(ξ/k
0
)
=
(7.10)
Z
s
(
sin
θ)
+
Z
0
/
cos
θ
7.3
The complex sin
θ
plane
Equation (7.6) is an integral over the real variable
ξ
. It can be considered as an integral
on sin
θ
sin
θ
plane on the right-hand side of the real axis.
Equation (7.7) is an integral over the whole real sin
θ
axis. It may be advantageous to
use other paths of integration in this plane, to show the contribution of the poles, and/or
to get approximate expressions of
p
r
more tractable for large
r
than Equation (7.7). A
symbolic representation of the path of integration of Equation (7.7) is given in Figure 7.2.
Small displacements of the path and the cuts are performed to show their relative
positions. The reflection coefficient
V
of a layer of finite thickness involves cos
θ
and
cos
θ
1
. For a layer of finite thickness,
Z
s
and
V
are even functions of cos
θ
1
, but
V
depends
on the sign of cos
θ
. At each point in the
s
=
sin
θ
plane the reflection coefficient can
take two values, depending on the choice of the sign of cos
θ
. Following Brekhovskikh
and Godin (1992), we cut the
s
plane with the lines
=
ξ/k
0
in the complex
s
=
s
2
1
−
=
u
1
,
1
real
≥
0
(7.11)
These are shown as dotted lines in Figure 7.2. They lie on the whole imaginary
axis
and
on
the
real
axis
for
0
≤|
s
|≤
1.
On
these
lines,
the
imaginary
part
of
