Civil Engineering Reference
In-Depth Information
X
X
k
I
ϕ
k
R
k
R
Z
Z
(a)
(b)
Figure 7.11
The real and imaginary wave number vector of the plane wave associated
with the pole for a semi-infinite layer: (a) perfect fluid, (b) air.
is represented in Figure 7.11(a) for a semi-infinite layer saturated by a perfect fluid, and
in Figure 7.11(b) for an air saturated semi-infinite layer. The modulus of the imaginary
wave number vector is equal to 0 in Figure 7.11(a),
k
R
=
k
0
,
k
x
=
k
0
cos
ϕ
,
k
z
=
k
0
sin
ϕ
.
Formally, changing the sign of cos
θ
changes
V
into 1/
V
(see Equation 7.10), and the
reflected wave becomes the incident wave. The reflected wave related to the pole for the
semi-infinite layer can be considered as a plane wave at an angle of incidence where the
reflection coefficient is equal to 0. This angle is real in the absence of damping and equal
to
π/
2
− ϕ
. This angle is the Brewster angle
θ
B
of total refraction. A similar Brewster
angle of total refraction exists for T. M. electromagnetic waves (Collin 1960). The angle
is complex for air saturated porous media. From Equation (7.10)
θ
B
and
θ
p
are related by
cos
θ
B
=−
cos
θ
p
(7.37)
The Brewster angle is not real for air saturated porous media, but cos
θ
B
can lie close
to the real cos
θ
plane in the complex cos
θ
plane. Then the modulus of the reflection
coefficient can present a minimum around
θ
close to
θ
B
. The predicted modulus of the
reflection coefficient is shown in Figure 7.12 at 4 kHz as a function of cos
θ
. The minimum
appears close to cos
θ
B
=
0
.
65.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
cos
q
Figure 7.12
The modulus of the reflection coefficient of a semi-infinite layer of material
1 at 4 kHz. The predicted Re cos
θ
B
=−
Re cos
θ
p
=
0
.
65.
