Borehole Acoustics –New Approaches to Old Problems - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 371

Phase
velocity

Decrement

Frequency Frequency

Figure 10.2. Phase velocity and attenuation decrement.
That “mirror images” must be the case is seen by combining Equations
10.3 and 10.4 to form

r (k) = Vk + i (k) = Vk - | i |. Then, division by “k”

shows that
c p (k) + | i | /k = V (10.8)
c p (k) being the phase velocity and | i | /k the so-called “attenuation decrement”
or damping rate per wavelength. Thus, if one quantity increases, the other
decreases, and vice-versa, as observed numerically, the two quantities
representing exact “mirror images.” While phase velocity is ordinarily
unimportant, it does provide a measure of attenuation rate in the case of
Stoneley waves. Again, the two quantities are not independent; when one is
known, the other can be calculated, assuming that V is available. Importantly,
their sum is identically equal to the parameter V, a constraint which may prove
useful in interpretation. Equation 10.8 also provides a check point for the
slowness and attenuation calculations, which are presently performed using
various signal processing methods without knowledge of their inter-relationship.
10.1.2.3 Relative magnitudes, phase and group velocities.
Numerical solutions also show that group velocities always exceed phase
velocities for the same frequency, e.g., see Chang and Toksoz (1981). That the
property “c g > c p ” is generally valid for our frequencies of interest is easily
deduced from Equations 10.5 and 10.6. Since the latter always subtracts more
of the positive quantity k -1/2 from the same V, it immediately follows that

c g > c p (10.9)

The two velocities are almost equal in the high frequency limit, but the
inequality applies at lower frequencies. This is important in the numerical
propagation of waveforms from a receiver station to successive stations, since
the two are in general not equal. Later, we will show that energy propagation, in
fact, must follow rays defined by the group velocity and not the phase velocity.