Civil Engineering Reference
In-Depth Information
The continuity of the normal displacement at the radiating surface,
∂p
a
∂n
1
ρ
0
ω
2
φ(u
n
−
u
s
n
)
u
s
n
=
+
leads to
ρ
0
ω
2
δ
pu
s
n
d
S
∂p
a
∂n
δp
d
S
1
I
p
=
−
(13.60)
In the semi-infinite domain, the acoustic pressure
p
a
is the sum of the blocked pres-
sure
p
b
and the radiated pressure
p
r
. So at the surface
∂p
a
/∂n
∂p
r
/∂n
, and in the
numerical implementation context, the discrete form associated to the second term of
Equation (13.60) reads
=
[
C
]
∂p
r
∂n
ρ
0
ω
2
∂p
a
∂n
δp
d
S
−
1
−
1
ρ
0
ω
2
=
δp
(13.61)
where [C] is the classical coupling matrix given by the assembling of the elemental
matrices
(
e
)
:
=∪
[
C
]
=
e
e
N
e
{
N
e
}
d
S
e
(13.62)
{
N
e
N
e
with
}
denoting the vector of the used surface element shape functions,
its
transpose and
e
the assembling process.
The planar porous material being inserted into a rigid baffle, the radiated acoustic
pressure is related to the normal velocity via Rayleigh's integral
∂p
r
(x
,y
,
0
)
∂n
G(x,y,z,x
,y
,
0
)
d
S
p
r
(x,y,z)
=−
(13.63)
where
G(x,y,z,x
,y
,
0
)
e
−
jkR
/
2
πR
is the baffled Green's function,
k
ω/c
0
,isthe
acoustic wave number in the fluid,
c
0
, the associated speed of sound and
R
is the distance
=
=
between point
(x,y,z)
and
(x
,y
,
0
)
:
R
=
(x
−
x
)
2
+
(y
−
y
)
2
+
z
2
.
The integral form associated with Equation (13.63) is given by
∂p
r
(x
,y
,
0
)
∂n
G(x,y,
0
,x
,y
,
0
)δp
d
S
d
S
p
r
(x,y,
0
)δp
d
S
=−
(13.64)
Using Equation (13.62), the associated discrete form is
δp
[
C
]
{
p
r
}=−
δp
[
Z
]
∂p
r
∂n
(13.65)
with,
G(x
e
,y
e
,
0
,x
e
,y
e
,
0
)
N
e
}
d
S
e
d
S
e
N
e
[
Z
]
=
e
{
(13.66)
e
e
e
