Civil Engineering Reference
In-Depth Information
(see Equation 8.B.12 in Appendix 8.B) the following expression for the radial component
is obtained
∞
J
1
(rξ)
µ
U
s
V(ξ/k
0
)
]
U
x
(ξ)
exp[
jµ(z
1
+
r
(r)
=−
[1
+
z
2
)
]
ξ
d
ξ
(8.55)
0
V)U
z
and
(
1
+
V)U
x
can be written, respectively
In Equations (8.54) and (8.55),
(
1
+
=
2cos
θ
T
11
[
(
1
−
φ)T
32
−
φT
42
]
+
T
12
[
φT
41
−
(
1
−
φ)T
31
]
V(ξ/k
0
)
]
U
z
(ξ)
[1
+
D
1
Z
+
(T
31
T
42
−
T
32
T
41
)
cos
θ
(8.56)
=
2cos
θ
T
21
[
(
1
−
φ)T
32
−
φT
42
]
+
T
22
[
φT
41
−
(
1
−
φ)T
31
]
V(ξ/k
0
)
]
U
x
(ξ)
[1
+
D
1
Z
+
(T
31
T
42
−
T
32
T
41
)
cos
θ
(8.57)
V(ξ/k
0
)
]
U
z
(ξ)
and [1
V(ξ/k
0
)
]
U
x
(ξ)
are the same as
The singularities of [1
+
+
those of
V
given by Equation (8.30).
Measurements and predictions are reported in Allard
et al
. (2007) and Geebelen
et al
.
(2007). Measurements of the normal velocity are performed with the laser beam normal
to the surface using a height of the source around 5 cm, and a radial distance from the
source to the point where the velocity is measured of 1 or 2 cm. Measurements of the
radial velocity are performed with an angle of incidence of the laser beam larger than
60
◦
, and a radial distance larger than 5 cm. Due to the Bessel function J
1
in Equation
(8.45), the radial displacement is negligible at small
r
. A microphone is mounted in the
tube of the source, at the side opposite to the compression driver, and in a first step the
measured velocity is normalized with the pressure signal provided by the microphone. In
a second step, this measured velocity is normalized with respect to the predicted velocity,
to make the comparison easier.
In Figures 8.8 and Figures 8.9, previsions and measurements are presented. Predictions
are performed with the acoustic parameters of material 2 in Table (8.1). The thickness of
the layer is
l
=
2 cm. The acoustic parameters that characterize the rigidity are the Poisson
ratio and the shear modulus. They have been adjusted to fit the measured distribution.
The mode manifestations are the broad peaks in the velocity distributions. For the radial
velocity in Figure 8.9, the maximum is close to the Biot shear wave
λ/
4 resonance. It may
be noticed that the quarter shear wave resonance is generally not observable, contrary
to the Biot frame-borne compressional wave
λ/
4 resonance. For the normal velocity,
distributions for materials with a Poisson coefficient equal to 0.35 or less are peaked
at this resonance (Allard
et al
. 2007, Geebelen
et al
. 2007). For material 2, the Poisson
ratio is large and with the geometry chosen for the measurement set-up, the main peak in
the normal distribution appears at a frequency lower than that of the Biot compressional
wave
λ/
4 resonance. The origin of the main peak has been studied for the simpler case
of a soft elastic medium. In this case, a similar peak corresponds to the contribution of a
