Civil Engineering Reference
In-Depth Information
k
R
X
k
I
X
k
I
ϕ
Z
k
R
Z
(a)
(b)
Figure 7.8
The wave number vector is
k
=
k
R
+
j
k
I
. The wave is evanescent in the
direction opposite to
k
I
and propagates in the direction
k
R
: (a) thin layer saturated by a
perfect fluid, (b) air-saturated thin layer.
wave number components
k
x
and
k
z
are given by
k
x
=
k
0
sin
θ
p
=
k
R
(7.32)
k
z
=−
k
0
cos
θ
p
=
jk
I
(7.33)
The wave number vector of a thin layer saturated by air is represented in Figure 7.8(b).
For a layer of material 1 of thickness
l
=
4 cm, the pole with sin
θ
p
closest to 1 has been
localized with a simple recursive algorithm; sin
θ
p
is represented in the complex s plane
in Figure 7.9(a) and cos
θ
p
is represented in Figure 7.9(b) in the complex cos
θ
p
plane in
the 50 - 1000 Hz frequency range.
For small thicknesses, the imaginary and the real part of cos
θ
p
are negative and the
imaginary part of sin
θ
p
is negative. As shown in Figure 7.8(b), the real part of the wave
number vector makes an angle
ϕ
with the axis
x
. The wave number components are now
given by
k
R
cos
ϕ
jk
I
sin
ϕ
k
x
=
k
0
sin
θ
p
=
−
(7.34)
k
z
=−
k
0
cos
θ
p
=
k
R
sin
ϕ
+
jk
I
cos
ϕ
(7.35)
0
0
−
0.05
50 Hz
−
0.05
50 Hz
−
0.1
575 Hz
−
0.15
−
0.1
−
0.2
500 Hz
−
0.25
−
0.15
−
0.3
−
0.2
1000 Hz
1000 Hz
−
0.35
−
0.4
−
0.25
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
−
0.5
−
0.4
−
0.3
−
0.2
−
0.1
0
Re sin
q
p
Re cos
q
p
(a)
(b)
Figure 7.9
Trajectory of sin
θ
p
and of cos
θ
p
.Material1,
l
=
4cm.
