Civil Engineering Reference
In-Depth Information
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
Frequency (kHz)
Figure 7.5
Amplitude of the reflected wave calculated with Equation (7.6), semi-infinite
layer of material 1,
z
1
+
z
2
=−
0
.
5m,
r
=
0
.
5m.
0.8
Added contribution
Error
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
2.25 2.5 2.75
3
Frequency (kHz)
Figure 7.6
Modulus of the difference between the exact evaluation and the evaluation of
the reflected pressure with the steepest descent method, and modulus of the term
N/k
0
R
1
in Equation (7.20), semi-infinite layer of material 1,
z
1
+
z
2
=−
0
.
5m,
r
=
0
.
5m.
the same quantities are represented for a semi-infinite layer of material 1, with
r
=
0
.
5m
and
z
1
+
0
.
5 m. The contribution of the lateral wave is not taken into account. The
good agreement between the exact result obtained from Equation (7.6) and the passage
path method shows that this contribution can be neglected. This has been always verified
in our previous studies.
For the chosen examples the reflected field, for frequencies higher than 500 Hz, is
obtained with good precision with the steepest descent method. For frequencies higher
than 1 kHz, the reflected pressure is close to the pressure created by the unit image
source multiplied by the plane wave reflection coefficient. These frequencies decrease
z
2
=−
