Civil Engineering Reference
In-Depth Information
For the locally reacting medium,
a
L
is given by
1
−
s
p
s
p
a
L
=−
2
(7.B.14)
Using
f(s)
=−
j
cos
(θ
−
θ
0
)
gives
f
(s
0
)
=
j/(
1
−
s
0
)
,and
F(s
0
)
s
0
1
−
(
−
1
+
V(s
0
))
2
j(
1
−
s
0
)
−
2
f
(s
0
)
=
(7.B.15)
s
0
For the porous layer, the reflected pressure is given by
k
0
2
πr
1
/
2
exp
−
jπ
4
exp
(
−
jk
0
R
1
)
exp
(
−
jk
0
R
1
)
R
1
p
r
=
+
−
k
0
R
1
q
p
)
erfc
(j
k
0
R
1
q
p
)
×
jπa
exp
(
−
1
/
2
s
0
1
π
k
0
R
1
V(s
0
))
2
j(
1
a
q
p
s
0
)
+
(
−
1
+
−
+
(7.B.16)
s
0
−
After some rearrangement,
p
r
can be rewritten
exp
(
−
jk
0
R
1
)
R
1
p
r
=
$
'
1
s
p
√
2
k
0
R
1
exp
(
+
√
πu
exp
(u
2
)erfc(
−
−
3
πj/
4
)(
1
−
u))
×
V(s
0
)
−
u
s
0
s
p
[1
s
p
)
[2
k
0
l/(
n
2
s
p
sin 2
k
0
l
n
2
%
(
−
(
1
−
−
−
s
p
)
+
1
/(n
2
−
s
p
)
]]
(7.B.17)
where
u
is the numerical distance, defined by
j
k
0
R
1
q
p
u
=−
(7.B.18)
For the locally reacting surface, the expression for
p
r
is very similar
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
√
s
0
s
p
%
(
(7.B.19)
The reflection coefficient
V(s
0
)
in Equation (7.B.17) is given by Equation (7.38).
If
θ
0
is
close
to
π
/2,
at
the
first
order
approximation
in
cos
θ
0
,sin
θ
0
=
1and
