Geoscience Reference
In-Depth Information
curve linking peak A on each record defines the phase velocity for peak A. The
phase velocity for the frequency of peak A at each distance is the inverse slope of
the dashed curve at that distance. The dashed lines linking the subsequent peaks
B, C and D also determine the phase velocity as a function of frequency. The
slopes of all these dashed lines indicate that, in this example, the phase velocity
decreases as the frequency of the surface waves increases (i.e., the phase velocity
increases with the period).
The group velocity of surface waves is the velocity at which surface-wave
energy of a given frequency travels. It is a constant for a given frequency. There-
fore on Fig. 4.5(a) straight lines passing through the origin mark the surface-wave
signal of any particular frequency on each successive record. Such lines for the
three frequencies
f
1
,
f
2
and
f
3
,where
f
1
<
f
3
, are shown in Fig. 4.5(a). The
group velocities for these frequencies are U
1
,U
2
and U
3
,where U
1
>
f
2
<
U
3
.
In this example, the group velocity decreases as the frequency of these surface
waves increases (i.e., the group velocity increases with the period).
A plot of velocity against period, called a
dispersion curve
,isthe usual way of
presenting this velocity-frequency information. Figure 4.5(b) illustrates disper-
sion curves appropriate for the records of Fig. 4.5(a). Notice that, in this example,
the group velocity is less than the phase velocity. To repeat, energy travels with
the group velocity, not the phase velocity.
Theoretically, the group velocity
U
(
f
) and phase velocity
V
(
f
) are linked
U
2
>
by
d
V
d
f
U
=
V
+
f
(4.7a)
where
f
is frequency. When expressed in terms of wavenumber, the group
velocity
U
(
k
) and phase velocity
V
(
k
) are linked by
d
V
d
k
U
=
V
+
k
(4.7b)
In the example shown in Fig. 4.5,d
V
/
d
f
is negative (phase velocity decreases as
frequency increases), so
U
V
.
First, consider Rayleigh waves. In the ideal theoretical situation, in which
the elastic properties of the Earth are constant with depth, Rayleigh waves are
not dispersive and travel at a velocity of approximately 0.92
<
(i.e., slower than
S-waves). However, the real Earth is layered, and, when the equations of motion
for Rayleigh waves in a layered Earth are solved (which is far beyond the scope
of this text), then Rayleigh waves are found to be dispersive (see Fig. 4.18(b)).
Next, consider Love waves. Love waves can exist only when the shear-wave
velocity increases with depth or, in the layered case, when the shear-wave velocity
of the overlying layer
β
β
1
<β
2
). Love
waves are
always
dispersive. Their phase velocity is always between
β
1
is less than that of the substratum
β
2
(i.e.,
β
1
and
β
2
in magnitude. The low-frequency (long-period) limit of the phase velocity is
β
2
,
β
1
. Generally, Love-wave group
and the high-frequency (short-period) limit is
