Civil Engineering Reference
In-Depth Information
The displacements induced by the rotational waves are parallel to the
x
1
0
x
3
plane, and
only the
x
2
component of the vector potential is different from zero. This component is
ψ
2
=
A
3
exp
(j(ωt
A
3
exp
(j(ωt
−
k
33
x
3
−
k
t
x
1
))
+
+
k
33
x
3
−
k
t
x
1
))
(11.23)
The air displacement potentials are related to the frame displacement potentials by
ϕ
i
=
µ
i
ϕ
i
i
=
1
,
2
(11.24)
and
ψ
2
=
µ
3
ψ
2
(11.25)
The ratio
µ
i
of the velocity of the air over the velocity of the frame for the two
compressional waves and
µ
3
for the shear wave are given by Equations (6.71) and (6.84),
respectively. The displacements field for the frame and the air are known completely if
A
1
,A
2
,A
3
,A
1
,A
2
and
A
3
are known, while the stresses can be calculated by Equation
(6.2) and (6.3).
Matrix representation
The acoustic field in the porous layer consists of six waves and can be described by
Equations (11.22) and (11.23). The acoustic field in the porous layer can be predicted
everywhere if the six amplitudes
A
1
,A
2
,A
3
,A
1
,A
2
and
A
3
are known. However, instead
of these parameters, one may choose six independent acoustic quantities. The six acoustic
quantities that have been chosen are three velocity components and three elements of the
stress tensors; the two velocity components
v
1
and
v
3
of the frame, the velocity component
v
3
of the fluid, the two components
σ
33
and
σ
13
of the stress tensor of the frame, and
σ
33
in the fluid. If these six quantities are known at a point
M
in the layer, the acoustic
field can be predicted everywhere in the layer. Moreover, the values of these quantities
anywhere in the layer depend linearly on the values of these quantities at
M
.Let
V
p
(M)
be the vector
V
p
(M)
=
[
v
1
(M)
v
3
(M)
v
3
(M)
σ
33
(M) σ
13
(M)
σ
33
(M)
]
T
(11.26)
The superscript T indicates a transposition,
V
(M)
being a column vector. These six
quantities are written as:
#
$
v
1
=
jω
∂ϕ
1
∂ϕ
2
∂ψ
2
∂x
3
∂x
1
+
∂x
1
−
jω
∂ϕ
1
∂ϕ
2
∂ψ
2
∂x
1
v
3
=
∂x
3
+
∂x
3
+
(11.27)
%
jω
∂ϕ
1
∂ϕ
2
∂x
3
+
∂ψ
2
∂x
1
v
3
=
∂x
3
+
