Civil Engineering Reference
In-Depth Information
Ta b l e 8 . 1
Parameters for different materials.
Material
Frame
Poisson
Shear
Tortuosity Porosity
Flow
Characteristic
modulus
α
∞
φ
density
ratio
resistivity
dimensions
ρ
1
(kg/m
3
) νN
(kPa)
σ
(N m
−
4
s)
,
(
µ
m)
1
25.
0.3
75
+
j15
1.4
0.98
50 000
50, 150
2
24.5
0.44
80
+
j12
2.2
0.97
22 100
39, 275
one obtains
k
R
=
248 - j25.9 m
−
1
. There are no explicit expressions for the location of
the singularities of
U
z
and
U
x
given by Equations (8.38) - (8.39), but the
x
wave number
components close to the predicted
k
R
can be obtained by recursive methods. Two poles
are obtained at
ξ
R
=
247
.
7
+
j25.7 m
−
1
for a set of three
Biot waves having a decreasing amplitude with increasing z in the porous medium. The
wave in air above the porous medium has a dependence on
z
given by exp
(jkz
cos
θ)
,
with cos
θ
=
249
.
1-j26.5m
−
1
and
ξ
R
j
6
.
69for the second
pole. The first pole is on the physical Riemann sheet. Both cos
θ
are almost opposite,
both sin
θ
are almost equal.
=−
0
.
73
−
j
6
.
73 for the first pole, and cos
θ
=
0
.
71
+
Rayleigh wave contribution for a circular source
The simplest source is a normal periodic point force
F
exp(
jωt
)
δ(x)δ(y)
.Itcanbe
replaced by a superposition of axisymmetric fields
∞
F
2
π
Fδ(x)δ(y)
=
J
0
(ξr)ξ
d
ξ
(8.50)
0
Using Equation (8.44), the vertical velocity at a radial distance
r
from the source can
be written
∞
F
2
π
U
s
J
0
(ξr) U
z
(ξ)ξ
d
ξ
z
(r)
=
(8.51)
0
U
z
is given by Equation (8.38) and presents a singularity at
ξ
=
ξ
R
.Using
where
=
0
.
5
(H
0
(u)
H
0
(
the relation J
0
(u)
u))
the integral on half the real
ξ
axis can be
replaced, as in Chapter 7, by an integral on the whole
ξ
axis. This path of integration can
be deformed in different ways to cut the Rayleigh poles. An example of integration on
different paths is given in Allard
et al
. (2004) for water - poroelastic interfaces. Predictions
obtained for a radial distance
r
−
−
=
25 cm with Equation (8.51) and a fast Fourier transform
are shown in Figure 8.2 for a semi-infinite layer of material 1 in air. The vertical force
is a burst centred at 2 kHz.
The part of the signal denoted as the 'Rayleigh wave' is almost identical to the
contribution of the residues related to the Rayleigh pole in the physical Riemann sheet. It
is shown in Appendix 8.C how to evaluate this contribution. Simulated measurements for
different radial distances give a velocity of the Rayleigh wave close to
ω/
Re
ξ
R
≈
52 m/s
at 2 kHz.
