Civil Engineering Reference
In-Depth Information
X
1
2R
X
2
D
θ
X
2
X
3
Figure 9.16
The surface of a perforated facing at oblique incidence.
where
m
±
2
...
. As for the case of normal incidence
the modes with opposite
m
can be associated with a cosine factorizing the common space
dependence on
x
2
and
x
3
. The pressure field in the layer
L
at the contact surface with
the facing can be written
=
0
,
±
1
,
±
2
,...
and
n
=
0
,
±
1
,
cos
2
πmx
1
D
exp
(j(
2
πn/D
∞
∞
p(x
1
,x
2
,x
3
)
=
−
k
0
sin
θ)x
2
)
(9.68)
m
=
0
n
=−∞
×{
A
m,n
(L)
exp
(
−
j(k
m,n
(L)x
3
)
+
B
m,n
(L)
exp
(jk
m,n
(L)x
3
)
}
The
x
1
component of velocity at
x
1
D/
2 is equal to zero, due to the symmetry of
the problem. The wave number component
k
m,n
(L)
in the
x
3
direction is given by
=±
k
2
(L)
−
−
(
2
πn/D
−
k
0
sin
θ)
2
1
/
2
2
πm
D
2
k
m,n
(L)
=
(9.69)
As in the case of normal incidence, the (
m
,
n
) mode of propagation has the same wave
vector components in the different media of the stratified material, and Equations (9.41)
and (9.42) relating
B
m,n
(L)
and
A
m,n
(L)
can be obtained in the same way as in the
previous section, the one modification being the substitution of 2
πn/D
−
k
0
sin
θ
for
2
πn/D
.
The pressure field related to the (
m
,
n
) mode of propagation can be written at the
contact surface with the facing
exp
j
2
πn
k
0
sin
θ
x
2
∞
∞
cos
2
πmx
1
D
p(x
1
,x
2
,
0
)
=
D
−
(9.70)
n
=−∞
m
=
0
×
A
m,n
(L)(
1
+
β
m,n
)
