Civil Engineering Reference
In-Depth Information
p
e
=
p
(10.63)
p
e
σ
zz
−
=
(10.64)
where
v
z
and
p
e
are the
z
velocity component in the free air and the pressure in the free
air at the contact surface with the porous layer. The coefficients
N
i
which satisfy the last
three boundary conditions for unit pressure are solutions of the following set of equations
$
'
(
=
q(k)a
1
(k)
c
a
3
(k)
r
ik
1
c
a
3
(i)
1
q(i)a
1
(i)
N
i
+
−
−
0
,
(10.65)
%
i
=
1
,
2
,
3
k
=
1
,
2
,
3
N
i
1
−
B
8
1
c
b
1
(i)
+
q(i)b
3
(i)
c
B
6
a
1
(i)
+
B
7
q(i)a
3
i
=
1
,
2
,
3
r
ik
1
c
B
6
a
1
(k)
B
7
q(k)a
3
(k)
+
−
(10.66)
k
=
1
,
2
,
3
−
B
8
1
c
b
1
(k)
−
q(k)b
3
(k)
=−
1
/jω,
N
i
1
B
7
1
q(i)b
3
(i)
c
B
3
a
1
(i)
B
4
q(i)a
3
c
b
1
(i)
+
−
+
i
=
1
,
2
,
3
r
ik
1
c
B
3
a
1
(k)
−
B
4
q(k)a
3
(k)
+
(10.67)
k
=
1
,
2
,
3
B
7
1
q(k)b
3
(k)
c
b
1
(k)
−
−
=
1
/jω.
The coefficients
N
1
,N
2
,N
3
are obtained from Equations (10.65) - (10.67). Using
the relation
u
s
φ)
u
s
φ
u
f
,the
z
component of velocity in the free air
+
w
=
(
1
−
+
is given by
N
i
(a
3
(i)
b
3
(k))r
ik
jω
i
b
3
(i))
(a
3
(k)
v
z
=
+
+
+
(10.68)
k
and the surface impedance is given by
v
z
=
1
jω
i
N
i
(a
3
(i)
+
b
3
(i))
+
k
(a
3
(k)
+
b
3
(k))r
ik
p
e
Z
s
=
(10.69)
For unit pressure, the
z
component
v
z
of the frame velocity at the surface of the layer
and the
x
component
v
x
are given by
$
'
jω
i
v
z
=
a
3
(i)
a
3
(k)r
ik
N
i
+
(10.70)
%
(
=
1
,
2
,
3
k
=
1
,
2
,
3
$
'
jω
i
v
x
=
a
1
(i)
a
1
(k)r
ik
N
i
+
(10.71)
%
(
=
1
,
2
,
3
k
=
1
,
2
,
3
