Civil Engineering Reference
In-Depth Information
7.7.2 Localization from the vertical dependence of the total pressure
If the pressure field above the porous layer is the same as that above a locally react-
ing medium of surface impedance
Z
s
(
sin
θ
p
)
, the surface impedance of the layer is
Z
s
(
sin
θ
p
)
. This is verified in what follows when
θ
p
≈
z
2
|
are much smaller than
r
. The total pressure
p
t
is the sum of
p
r
and of the direct field
exp
(
π/
2and
θ
0
≈
π/
2, if
|
z
1
|
and
|
−
jk
0
R
2
)/R
2
,
R
2
being the distance from the source to the receiver
exp
(
−
jk
0
R
2
)
R
2
p
t
=
cos
θ
p
2
k
0
R
1
exp
−
√
π
exp
(u
2
)
erfc
(
exp
(
−
jk
0
R
1
)
R
1
3
jπ
4
+
{
1
−
−
u)
}
(7.54)
where
u
is given by Equation
(7.50). At
θ
0
=
π/
2,
∂R
2
/∂z
2
=
∂R
1
/∂z
2
=
0, and
∂θ
0
/∂z
2
=
1
/r
(the
z
axis is directed toward the porous layer). The derivative of
2
/
√
π
(Abramovitz and Stegun 1972,
exp
(u
2
)
erfc
(
u)
is
w
(u)
w(u)
=
−
=
2
uw(u)
+
j
3
π/
4
)
√
k
0
r/
2cos
θ
p
.
Chapter 7), where
u
can be replaced by
−
exp
(
−
The derivative
∂u/∂z
2
at
θ
0
=
π/
2isgivenby
∂z
2
=−
2
k
0
r
exp
1
2
r
sin
θ
0
∂u
j
3
π
4
−
(7.55)
where sin
θ
0
is close to 1, and from Equation (7.55)
∂p
t
/∂z
2
=
p
t
jk
0
cos
θ
p
(7.56)
Z
0
/
cos
θ
p
,
whichisequaltoZ
s
(sin
θ
p
). This impedance can be evaluated in a free field from
pressure measurements close to the surface on a normal to the surface at
z
2
and
z
2
.The
velocity
v
z
is given by
(j/ωρ
0
)∂p
t
/∂z
2
where the pressure derivative
∂p
t
/∂z
2
can be
approximated by [
p(z
2
)
−
p(z
2
)
]
/(z
2
−
z
2
)
. The measured surface impedance
Z
s
can be
obtained from
The
surface
impedance
is
the
ratio
p
t
/v
z
=−
p
t
jωρ
0
/(∂p
t
/∂z
2
)
=−
j(z
2
−
z
2
)
p(z
2
)
(p(z
2
)
−
p(z
2
))
Z
s
=
(7.57)
ωρ
0
or by the equivalent expression equivalent for small
z
1
−
z
2
j(z
2
−
z
2
)
ωρ
0
1
ln
(p(z
2
)/p(z
2
))
Z
s
=
(7.58)
In Figure 7.15, the evaluated quantity is not
Z
s
, but cos
θ
p
=−
Z
0
/Z
s
. Simulated
measurements are shown for a layer of thickness
l
3 cm of material 1 the exact total
pressure being calculated with Equation (7.6). Equation (7.58) is used to evaluate
Z
s
.
The systematic error increases faster with frequency than with the previous method.
Measurements are presented in Allard
et al
. (2004).
The
z
2
dependence of a pole contribution
p
pole
is exp
(jk
0
cos
θ
p
z
2
)
.Thisgivesfor
the pole contribution derivative the following expression
=
∂p
pole
/∂z
2
=
p
pole
jk
0
cos
θ
p
(7.59)
