Civil Engineering Reference
In-Depth Information
x
l
porous
layer
air
Figure 6.8
A layer of porous material bonded on to an impervious rigid wall, in a
normal acoustic field.
compressional waves. The stresses in the material are given by
σ
xx
(x)
Z
1
[
V
i
exp
(
V
r
exp
(jδ
1
x)
]
=−
−
jδ
1
x)
−
(6.93)
Z
2
[
V
i
exp
(
V
r
exp
(jδ
2
x)
]
−
−
jδ
2
x)
−
φZ
1
µ
1
[
V
i
exp
(
σ
xx
(x)
V
r
exp
(jδ
1
x)
]
=−
−
jδ
1
x)
−
(6.94)
φZ
2
µ
2
[
V
i
exp
(
V
r
exp
(jδ
2
x)
]
−
−
jδ
2
x)
−
At
x
=
0, where the wall and the material are in contact, the velocities are equal to
zero
u
s
(
0
)
u
f
(
0
)
=
=
0
(6.95)
l
, the porous material is in contact with the free air. Let us consider a thin
layer of air and porous material, including this boundary. This layer is represented in
Figure 6.9.
Let us denote by
p(
At
x
=−
−
l
−
ε)
the pressure in the air on the left-hand side of the thin
ε)
and
σ
xx
(
layer, while
σ
xx
(
ε)
are the stresses acting on the air and on the
frame on the right-hand side. The resulting force
F
acting on the thin layer is
−
l
+
−
l
+
σ
xx
(
σ
xx
(
F
=
p(
−
l
−
ε)
+
−
l
+
ε)
+
−
l
+
ε)
(6.96)
s
s
xx
s
f
xx
-/
0
x
porous
layer
air
Figure 6.9
A thin layer of air and porous material including the boundary.
