Civil Engineering Reference
In-Depth Information
0.5
0.45
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
40
60
80
100
120
140
160
180
200
220
Frequency(Hz)
Figure 10.5
Normal surface frame velocity created by an incident unit pressure field.
For a unit incident wave, the acoustic pressure field at the contact surface with air is
multiplied by
(
1
+
V),V
being the reflection coefficient given by
Z
s
cos
θ
Z
0
−
1
Z
s
cos
θ
Z
0
+
1
V
=
(10.72)
and the velocity components given by Equations (10.70) and (10.71) must also be multi-
plied by
(
1
+
V)
. The modulus of the vertical velocity of the frame at the free surface of
a layer of the material of Table 10.1 is given in Figure 10.5 as a function of frequency.
The Lafarge model (Equation 5.35) is used for the bulk modulus and the Johnson
et al
.
model (Equation 5.36) for the effective densities in the evaluation of the coefficients
C
i
.
The thickness of the layer is
l
=
10 cm. The incident field is a unit pressure field at
normal incidence.
There is a peak in the velocity distribution close to 110 Hz. This peak is due to the
compressional quarter wavelength resonance of the frame. For the frame in vacuum, the
speed of the compressional wave which propagates in the direction of the axis of symme-
try is equal to 1
/
Re
(ρ
1
/C)
1
/
2
, close to 120 m/s. The speed of the frame compressional
Biot wave in the
z
direction is very close to this value. The peak is shifted toward high
frequencies because, independently of the resonance, the vertical velocity induced by the
unit field increases strongly with frequency.
For the case of a point source, the Sommerfeld representation can be used as in
Chapter 8 to predict the frame displacement components.
