Civil Engineering Reference
In-Depth Information
The
x
3
component of the velocity field is given by
exp
j
2
πn
D
−
k
0
sin
θ
x
2
∞
∞
cos
2
πmx
1
D
v
3
(x
1
,x
2
,
0
)
=
(9.71)
n
=−∞
m
=
0
β
m,n
)
k
m,n
(L)
Z
c
(L)k(L)
×
A
m,n
(L)(
1
−
As in the case of normal incidence, the velocity amplitude is assumed to be uniform in
each aperture. Let
U
0
be the velocity on the aperture C
0
. In the porous material of porosity
φ(L)
in contact with the facing, this velocity must be multiplied by 1
/φ(L)
. Multiplying
the left-hand side of Equation (9.71) by cos
(
2
πmx
1
/D)
exp
(
−
j
[
(
2
πn/D)
−
k
0
sin
θ
]
x
2
)
and integrating over the aperture C
0
one obtains
exp
j
2
πn
k
0
sin
θ
x
2
cos
2
πmx
1
D
r
d
r
d
θ
R
2
π
U
0
φ(L)
I
=
−
D
−
(9.72)
0
0
In the integral the exponential function can be replaced by a cosine with the same
argument, because the sine in the decomposition of the exponential is an odd function of
x
2
and does not contribute to the integral. The use of Equations (9.7) - (9.9) yields
exp
j
2
πn
k
0
sin
θ
x
2
cos
2
πmx
1
D
r
d
r
d
θ
R
2
π
−
D
−
0
0
2
1
/
2
2
πR
m
2
(9.73)
n
D
−
k
0
sin
θ
2
π
R
=
J
1
D
2
+
m
2
D
2
+
2
n
D
−
k
0
sin
θ
2
π
The quantity
I
can also be evaluated by multiplying the right-hand side of Equation
(9.71) by cos
(
2
πmx
1
/D)
exp
(
k
0
sin
θ)x
2
)
and integrating over the aper-
ture. Equating these two evaluations of
I
yields
−
j(
2
πn/D
−
2
U
0
φ(L)D
Z
c
(L)k(L)
k
m,n
(L)
1
1
−
β
m,n
(L)
ν
m
A
m,n
(L)
=
2
1
/
2
2
πR
m
2
n
D
−
R
k
0
sin
θ
2
π
(9.74)
×
2
1
/
2
J
1
D
2
+
m
2
D
2
n
D
−
k
0
sin
θ
2
π
+
where
ν
m
=
0and
ν
m
=
1
/
2for
m
=
1for
m
=
0. As in the case of normal incidence,
the impedance
Z(B
)
is estimated by
R
2
π
p(x
1
,x
2
,
0
)r
d
r
d
θ
0
0
Z(B
)
=
(9.75)
πR
2
U
0
