Civil Engineering Reference
In-Depth Information
U
s
(see Appendix 8.B) gives the following expression for the radial component
r
(r)
∞
=
−
j
2
π
U
s
ξ
J
1
(rξ)G(ξ) U
x
(ξ)
d
ξ
r
(r)
(8.45)
0
For a line source in the direction
Y
with a spatial dependence h(
x)
, the direct and
inverse spatial Fourier transforms can be used
∞
H(ξ)
=
h(x)
exp
(jξx)
d
x
−∞
(8.46)
∞
1
2
π
h(x)
=
H(ξ)
exp
(
−
jξx)
d
ξ
−∞
The
z
and
x
components of the velocity displacement at the surface are now given by
∞
1
2
π
U
s
H(ξ)U
z
(ξ)
exp
(
z
(x)
=
−
jξx)
d
ξ
(8.47)
−∞
∞
1
2
π
U
s
H(ξ)U
x
(ξ)
exp
(
x
(x)
=
−
jξx)
d
ξ
(8.48)
−∞
8.3
Semi-infinite layer - Rayleigh wave
Rayleigh pole
It was shown by Feng and Johnson (1983) that the Rayleigh wave is the one surface
mode that could be detected experimentally at a semi-infinite porous frame - light fluid
interface. In Feng and Johnson (1983) the different losses predicted by the Biot theory are
not accounted for. A description of the Rayleigh wave for the lossy air-saturated porous
sound absorbing media is performed by Allard
et al
. (2002). Illustrations are given with
the porous medium denoted as material 1 in Table 8.1. Predictions for a semi-infinite
layer are obtained by setting all
r
ij
in Equations (8.7) - (8.9) equal to 0.
Apriori
, due
to the partial decoupling, the Rayleigh wave must be similar to the one that could exist
at the surface of the porous frame in vacuum. For the case of the frame in vacuum, the
velocity of the Rayleigh wave must be slightly smaller than the velocity of the shear
wave. For material 1, at 2 kHz, the wave number
k
3
of the shear Biot wave is equal to
k
3
=
230
.
6-j24.0m
−
1
.
Using for the porous medium the empirical formula in Victorov (1967) that relates,
for an elastic solid, the wave number
k
R
of the Rayleigh wave to the wave number of
the shear wave
1
+
ν
0
.
87
k
R
=
k
3
(8.49)
+
1
.
12
ν
