Civil Engineering Reference
In-Depth Information
D
Figure 9.2
A square grid of holes at normal incidence and the related partition.
X
1
r
M
θ
θ
r
B
X
2
X
3
D
Figure 9.3
The elementary cell of the partition. The circumference at
x
3
=
0isthe
boundary of the aperture in contact with the free air at the left hand side of Figure 9.1.
the polar coordinates
r
,
θ
and the cartesian coordinates
x
1
,
x
2
canbeused.Onthefaces
located at
x
1
D/
2, the velocity components in the directions
x
1
and
x
2
,
respectively, are equal to zero. A simple pressure field in the cell is
=±
D/
2,
x
2
=±
jk
m,n
x
3
)
cos
2
πm
D
x
1
cos
2
πn
p
m,n
(x
1
,x
2
,x
3
)
=
A
m,n
exp
(
−
D
x
2
(9.2)
In this equation,
k
m,n
is given by
k
2
1
/
2
4
π
2
m
2
D
2
4
π
2
n
2
D
2
k
m,n
=
−
−
(9.3)
where
k
is the wave number in the free air. The set of these functions for m and n taking
the values 0, 1, 2,
...
, etc., is an orthogonal basis for the pressure field in the cylinder.
A pressure field in the cylinder can be written
jk
m,n
x
3
)
cos
2
πm
D
x
1
cos
2
πn
D
x
2
p(x
1
,x
2
,x
3
)
=
C
m,n
exp
(
−
(9.4)
m,n
