Civil Engineering Reference
In-Depth Information
260
240
220
200
180
0
0.02
0.04
0.06
0.08
0.1
Thickness (m)
−
20
−
30
−
40
−
50
−
60
−
70
0
0.02
0.04
0.06
0.08
0.1
Thickness (m)
Figure 8.5
Wave number of the modified Rayleigh wave as a function of thickness for
material 1, frequency fixed at 2 kHz.
modified pole in the physical Riemann
ξ
plane is shown in Figure 8.5 as a function of
the thickness for a layer of material 1 (see Table 8.1) at 2 kHz. The wavelength of the
shear Biot wave is equal to 2.73 cm. Noticeable modifications appear when the thickness
has the same order of magnitude as the wavelength.
Sound absorbing porous media are generally available in porous layers of thickness
ranging from 1 to 5 cm. This creates a limitation for the measurement of the wave
speed of the Rayleigh waves at low frequencies. Another limitation is due to the large
structural damping of the porous frames. As a consequence, the Rayleigh wave can
only be detected at small distances from the source. It has been shown by Geebelen
et al
. (2008) that measurements performed at distances from the source as small as three
Rayleigh wavelengths can provide reasonable orders of magnitude for the phase speeds.
8.5
Layer of finite thickness - modes and resonances
8.5.1 Modes and resonances for an elastic solid layer
and a poroelastic layer
In a layer of finite thickness, free on both faces or bonded onto a rigid substrate, there
is an infinite number of frame modes. Due to the partial decoupling, the frame modes in
air or in vacuum must involve, at the same frequencies, similar deformation fields of the
frame.
Modes have been studied at low frequencies by Boeckx
et al
. (2005), for a soft
lossless elastic solid with a face bonded on a rigid substrate, the other face being in
