Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 372

10.1.2.4 Amplitude and group velocity dependence.
Synthetic seismograms and field data suggest that some relationship
between wave amplitude and “slowness” exists. For example, Burns et al
(1988) shows that amplitude and “slowness,” taken as the reciprocal of phase
velocity, have very similar shapes when plotted versus depth. A simple proof is
given which, for now only, ignores dissipation. In this limit, our wave energy
conservation law “ E/ t + (c g E)/ x = 0” from Chapter 2 requires that the
energy flux “c g E” must be constant with time for dynamically steady systems.
(This is analogous to the mass conservation law “ / t + (v )/ x = 0,” stating
that density increases in time are balanced by changes in the flux “v ,” where v
is the fluid speed; for constant density flows, the mass flux “v ” must be
constant.) Here, E is the wave energy density, and c g is the required group
velocity; it follows that E is inversely proportional to c g . However, since E is
proportional to the square of the wave amplitude a , it follows that

a 1/ c g (10.10)

Hence, wave amplitude varies as the inverse square root of group velocity;
“slowness,” in general, must be defined with respect to the group velocity, and
not the phase velocity - for the examples under consideration, however, we note
that phase and group velocities happen to be close in magnitude. Moreover,
amplitude and group velocity are not independent; the value of the product c g a 2
is completely determined by the wave source. At the higher frequencies, the
observed correlation with inverse phase velocity is obtained only because the
waves are nondispersive, so that phase and group velocities are approximately
equal, but this equality is true only of Stoneley waves. The extension of
Equation 10.10 to dissipative media is given later.

10.2 Stoneley Wave Kinematics and Dynamics
10.2.1 Energy redistribution within wave packets.
In this section, we begin with the kinematic wave energy equation derived
in Chapter 2, but specialized to homogeneous dissipative media, namely,
E/ t + (c g E)/ x = 2 i E (10.11)
and deduce kinematical and dynamical properties important to identifying
Stoneley waves in borehole seismic data and to signal processing and
permeability prediction. Equation 10.11 embodies two simpler wave limits. For
standing waves, where E/ x = 0, we obtain the familiar “ E/ t = 2 i E” from
mechanics, showing exponential damping with a rate of - 2 i t, whereas for
dynamically steady propagation with E/ t = 0, the wave damps in space
according to (c g E)/ x = 2 i E while it propagates with the group velocity with
an inverse dependence on group velocity seen also from Equation 10.10. These