Civil Engineering Reference
In-Depth Information
contact with vacuum. Let
V
T
be the transverse wave speed,
V
L
be the compressional
wave speed,
ξ
be the
x
wave number component of a mode propagating in the
x
direction,
p
2
(ω/V
L
)
2
ξ
2
,and
q
2
(ω/V
T
)
2
ξ
2
. The dispersion equation for the modes can
=
−
=
−
be written
−
q
2
)
−
sin
pl
sin
ql
[
ξ
2
−
4
ξ
2
(ξ
2
pq
(ξ
2
−
q
2
)
2
+
4
ξ
2
pq
(8.52)
+
cos
pl
cos
ql
[4
ξ
4
(ξ
2
q
2
)
2
]
=
0
+
−
where
l
is the thickness of the layer. In Figure 8.6(a) an example of dispersion curves
obtained from a numerical search of the roots of Equation (8.52) is shown. The usual
Lamb dispersion curves for a free plate of thickness 2
l
are shown in Figure 8.6(b). The
main difference with the free plate is the absence of modes without cutoff frequency and
a smaller density of modes. The phase speed is infinite at the cutoff frequencies, and
600
500
400
300
200
100
0
50
100
150
200
250
Frequency*Thickness (Hz*m)
(a)
600
500
400
300
200
100
0
50
100
150
200
250
(b)
Frequency*Thickness (Hz*m)
thickness for: (a) a layer of
thickness
l
of an elastic medium on a rigid substrate, (b) a free layer of thickness 2
l
.
The material density is 14 kg/m
3
,V
L
=
222. m/s,
V
R
=
122. m/s. (Boeckx
et al
. 2005).
Reprinted with permission from Boeckx, L. , Leclaire, P. , Khurana, P., Glorieux, C.,
Lauriks, W. & Allard, J. F. Investigation of the phase velocity of guided acoustic waves
in soft porous layers.
J. Acoust. Soc. Amer.
117
, 545 - 554. Copyright 2005, Acoustical
Society of America.
Figure 8.6
Phase speed as a function of frequency
×
