Civil Engineering Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
3
3.2
3.4
3.6
3.8
4
4.2
Frequency (kHz)
Figure 7.17
Normalized difference between the exact total pressure
p
t
and the pressure
p
t
predicted with the passage path method. Material 2,
l
=
2cm,
z
1
=
z
2
=−
2
.
5cm,
r
=
1m.
the passage path method though the contributions of the numerous poles crossing the
path of integration are neglected.
This good agreement also shows that the pole contributions are negligible for the
chosen geometry at high frequencies. Using Equation (7.A.3) for
θ
0
=
π/
2 gives at
zeroth order in cos
θ
0
=−
2
Z
s
(
sin
θ
0
)
Z
0
2
N
(7.67)
and from Equation (7.20)
p
r
is given by
V(
sin
θ
0
)
−
−
2
j(Z
s
(s
0
)/Z
0
)
2
k
0
R
1
exp
(
−
jk
0
R
1
)
R
1
p
r
=
(7.68)
The same expression can be obtained with the modified formulation when only the
leading term 1/(2
u
2
) in Equation (7.48) is retained. Pressures predicted with the passage
path method and the modified formulation close to grazing incidence for large numerical
distances are similar. This explains why predictions obtained under these conditions with
the modified formulation are valid.
In Figure 7.16 the total pressure for material 2 is predicted with a large error in the
medium-frequency range. This is an illustration of a general trend. The error increases
when the flow resistivity decreases. The error is negligible for materials with a large flow
resistivity because the material becomes locally reacting and the modified formulation
becomes identical to the original formulation of Chien and Soroka which can be used,
as indicated in Appendix 7.C, over the whole frequency range. The influence of the
thickness on the error is shown in Figure 7.18.
The error decreases when the thickness increases and the peak in the medium-
frequency range disappears for semi-infinite layers. There is only one pole for a
semi-infinite layer and the surface impedance is less dependent on the angle of incidence
