Civil Engineering Reference
In-Depth Information
where [
ρ
]and[
M
] are respectively
ρ
11
,
PQ
QR
ρ
12
[
ρ
]
=
[
M
]
=
(6.64)
ρ
12
ρ
22
Equation (6.63) can be rewritten
−
ω
2
[
M
]
−
1
[
ρ
][
ϕ
]
=
∇
2
[
ϕ
]
(6.65)
Let
δ
1
and
δ
2
be the eigenvalues, and [
ϕ
1
]and[
ϕ
2
] the eigenvectors, of the left-hand
side of Equation (6.65). These quantities are related by
−
δ
1
[
ϕ
1
]
=
∇
2
[
ϕ
1
]
(6.66)
δ
2
[
ϕ
2
]
=
∇
2
[
ϕ
2
]
−
The eigenvalues
δ
1
and
δ
2
are the squared complex wave numbers of the two com-
pressional waves, and are given by
2
Q ρ
12
−
√
]
ω
2
2
(PR
δ
1
=
Q
2
)
[
P ρ
22
+
R ρ
11
−
(6.67)
−
+
√
]
ω
2
2
(PR
δ
2
=
Q
2
)
[
P ρ
22
+
R ρ
11
−
2
Q ρ
12
(6.68)
−
where
is given by
2
Q ρ
12
]
2
Q
2
)( ρ
11
ρ
22
ρ
12
)
=
[
P ρ
22
+
R ρ
11
−
−
4
(PR
−
−
(6.69)
The two eigenvectors can be written
ϕ
1
ϕ
1
,
ϕ
2
ϕ
1
[
ϕ
1
]
=
[
ϕ
2
]
=
(6.70)
Using Equation (6.60), one obtains
Pδ
i
−
ω
2
ρ
11
ω
2
ρ
12
−
ϕ
i
/ϕ
i
=
µ
i
=
i
=
1
,
2
(6.71)
Qδ
i
or
Qδ
i
−
ω
2
ρ
12
ω
2
ρ
22
−
ϕ
i
/ϕ
i
=
µ
i
=
i
=
1
,
2
(6.72)
Rδ
i
These equations give the ratio of the velocity of the air over the velocity of the frame
for the two compressional waves and indicate in what medium the waves propagate
preferentially. Four characteristic impedances can be defined, because both waves simul-
taneously propagate in the air and the frame of the porous material. In the case of waves
propagating in the
x
3
direction, the characteristic impedance related to the propagation
in the air is
p/(jωu
3
)
Z
f
=
(6.73)
