Civil Engineering Reference
In-Depth Information
in Equation (7.B.19), is used for the case of the porous surface
exp
(
−
jk
0
R
1
)
R
1
p
r
=
$
'
1
−
s
p
√
2
k
0
R
1
exp
(
−
3
πj/
4
)(
1
+
√
πu
exp
(u
2
)
erfc
(
−
u))
×
V
L
(s
0
)
−
%
(
u
exp
−
j
3
π
4
2
k
0
R
1
sin
θ
p
−
θ
0
2
u
=
(7.46)
=
1
+
√
πu
exp
and erfc is the complement of the error function. The expression
W(u)
(u
2
)
erfc
(
−
u)
is obtained for small
|
u
|
with the series development
2
u
2
exp
(u
2
)
1
+
√
πu
exp
(u
2
)
u
2
3
+
u
4
2!5
−
u
6
3!7
+···
W(u)
=
1
+
−
(7.47)
|
u
|
and for large
with the development
sgn
(
Re
(u))
]
√
πu
exp
(u
2
)
1
2
u
2
−
3
(
2
u
2
)
2
+
1
×
1
×
3
×
5
W(u)
=
[1
+
+
+···
(7.48)
(
2
u
2
)
3
The following approximations for
V
and
u
are used
cos
θ
+
cos
θ
p
V(
sin
θ)
=
(7.49)
cos
θ
−
cos
θ
p
=
exp
−
k
0
R
1
/
2
(
cos
θ
0
−
cos
θ
p
)
j
3
π
4
u
(7.50)
the term
1
−
s
p
√
2
k
0
R
1
exp
(
−
3
πj/
4
)/u
in Equation (7.46) can be replaced by 1
−
V(
sin
θ)
=−
2cos
θ
p
/(
cos
θ
0
−
cos
θ
p
)
, and Equation (7.46) can be rewritten
+
√
πu
exp
(u
2
)
erfc
(
exp
(
−
jk
0
R
1
)
R
1
p
r
=
[
V
L
(
sin
θ
0
)
+
(
1
−
V
L
(
sin
θ
0
))(
1
−
u))
]
(7.51)
where
V
L
is the reflection coefficient for an impedance plane
Z
s
Z
0
/
cos
θ
p
. A similar
equation has been given by Chien and Soroka (1975) for the case of a locally reacting
surface. The one difference is that for the porous layer cos
θ
p
is obtained from Equation
(7.25) where
Z
s
depends on the angle of incidence. Equation (7.49) is valid only for
θ
0
close to
π
/2, and if
θ
p
is close to
θ
0
. It is difficult to define the limits of validity
of Equations (7.49) - (7.51), due to the numerous parameters involved. Limits to the
accuracy when
θ
p
or
θ
0
are too far from
π
/2 will appear in the following section.
For a semi-infinite layer, and more generally when a pole is close to the stationary
point without being crossed, there is, at small numerical distances, a contribution to the
=−
reflected pressure in
W
given by
√
πu
exp
(u
2
)
, half the contribution added for large
|
when Re
(u)>
0. A similar contribution, the Zenneck wave, exists in the electric
dipole field reflected by a conducting surface. Descriptions of the Zenneck wave have
u
|
