Civil Engineering Reference
In-Depth Information
Inhomogeneous
porous material
Solid
Inclusions or
air cavities
Septum
x
1
Plate
Anechoic
termination
x
3
x
2
L
1
−∞
+∞
L
2
p
1
p
2
Γ
Γ
Figure 13.2
A complex porous component in a waveguide (Sgard 2002).
Using interface conditions, Equation (13.16), leads to
∂p
a
∂n
δp
d
S
1
ρ
0
ω
2
I
i
=
δ(pu
n
)
d
S
−
(13.30)
p
i
p
i
The first term amounts to the calculation of a classical coupling matrix. The second
term will be rewritten using the orthogonal modes of the waveguide. The pressure
p
a
on
the emitter side is written as the sum of the blocked pressure
p
b
satisfying
∂p
b
∂n
1
=
0
and the pressure
p
rad
radiated from the surface of the poroelastic medium
p
a
=
p
b
+
p
rad
(13.31)
For a normal mode excitation of amplitude
p
0
the complex amplitude of
p
b
reduces
to 2
p
0
. The radiated pressure may then be expressed in terms of the orthogonal normal
modes
ϕ
mn
in the waveguide
p
rad
(
x
)
=
m,n
B
mn
ϕ
mn
(x
1
,x
2
)e
jk
mn
x
3
(13.32)
where
x
=
(x
1
,x
2
,x
3
)
. For a waveguide with a rectangular cross section
L
1
×
L
2
=
cos
mπx
1
L
1
cos
nπx
2
L
2
ϕ
mn
(x
1
,x
2
)
(13.33)
and
mπ
L
1
2
nπ
L
2
2
k
mn
=
k
2
−
−
(13.34)
