Civil Engineering Reference
In-Depth Information
The macroscopic displacements of the frame and the air are parallel to the
x
3
direc-
tion, and by the use of Equation (6.3), Equation (6.73) can be rewritten for the two
compressional waves
Q/µ
1
)
δ
1
φω
Z
1
=
(R
+
(6.74)
Q/µ
2
)
δ
2
φω
Z
2
=
(R
+
(6.75)
The characteristic impedance related to the propagation in the frame is
Z
s
=−
σ
33
/(jωu
s
3
)
(6.76)
By the use of Equation (6.2), Equation (6.76) can be rewritten for the two compres-
sional waves
Qµ
i
)
δ
i
ω
Z
i
=
(P
+
(6.77)
i
=
1
,
2
6.5.2 The shear wave
As in the case for an elastic solid, the wave equation for the rotational wave can be
obtained by using vector potentials. Two vector potentials,
ψ
s
f
, for the frame and
and
ψ
for the air, are defined as follows:
u
s
s
=
∇
∧
ψ
(6.78)
u
f
f
=
∇
∧
ψ
(6.79)
Substitution of the displacement representation, Equations (6.78) and (6.79), into
Equations (6.54) and (6.55) yields
ω
2
ρ
11
ψ
s
ω
2
ρ
12
ψ
f
2
s
−
−
=
N
∇
ψ
(6.80)
ω
2
ρ
12
ψ
s
ω
2
ρ
22
ψ
f
−
−
=
0
(6.81)
The wave equation for the shear wave propagating in the frame is
ρ
11
ρ
22
ω
2
N
ρ
12
−
2
s
s
∇
ψ
+
ψ
=
0
(6.82)
ρ
22
The squared wave number for the shear wave is given by
ρ
11
ρ
22
−
ω
2
N
ρ
12
δ
3
=
(6.83)
ρ
22
and the ratio
µ
3
of the amplitudes of displacement of the air and of the frame is given
by Equation (6.81)
µ
3
=−
ρ
12
/ ρ
22
(6.84)
