Civil Engineering Reference
In-Depth Information
u
X
1
F
X
3
Figure 3.10
A panel of anisotropic fibrous material. The vectors
υ
and
F
indicate in
the plane
x
2
=
0 the direction of the flow of air, and the force acting on the air due to
viscosity, respectively.
x
3
=
0. Since the flow resistivity is larger in the normal direction than in the planar
direction, the viscous force
F
acting on air flowing in the direction
υ
is generally not
parallel to
υ
.
The force
F
lies in a direction opposite to that of
v
only if
v
is parallel to a normal or a
planar direction. The porous medium is supposed to be transversely isotropic. Rotations
around the axis x
3
do not modify the acoustical properties of the layer. The case of
transversely isotropic porous media is considered in Chapter 10 in the context of the full
Biot theory. If the frame is motionless, the material can be replaced by a transversely
isotropic fluid. In the transversely isotropic fluid, the bulk modulus is a scalar quantity.
In the system
x
1
,
x
2
,
x
3
the density is a diagonal tensor with two different components
ρ
3
=
ρ
P
.Let
u
be the displacement vector. The Newton equations (1.43)
and the stress-strain equations (1.57) can be written, respectively
ρ
N
,
ρ
1
=
ρ
2
=
ρ
P
∂
2
u
1
ρ
P
∂
2
u
2
ρ
N
∂
2
u
3
∂p
∂x
1
=
∂p
∂x
2
=
∂p
∂x
3
=
−
∂t
2
,
−
∂t
2
,
−
∂t
2
,
(3.48)
K
∂u
1
∂u
2
∂x
2
+
∂u
3
∂x
3
−
p
=
∂x
1
+
(3.49)
The wave equation can be written
∂
2
p
∂x
1
+
∂
2
p
∂t
2
∂
2
p
∂x
2
∂
2
p
∂x
3
K
ρ
P
K
ρ
N
=
+
(3.50)
With the exp
(jωt)
time dependence, this equation becomes
∂
2
p
∂x
1
+
∂
2
p
∂x
2
∂
2
p
∂x
3
+
K
ρ
P
K
ρ
N
ω
2
p
+
=
0
(3.51)
The quantities
ω(ρ
P
/K)
1
/
2
and
ω(ρ
N
/K)
1
/
2
will be denoted by
k
P
and
k
N
k
P
=
ω
ρ
P
K
1
/
2
k
N
=
ω
ρ
N
K
1
/
2
,
(3.52)
A solution of Equation (3.51) is
p(x
1
,
x
2
,x
3
,t)
=
A
exp[
j(
−
k
1
x
1
−
k
2
x
2
−
k
3
x
3
+
ωt)
]
(3.53)
