Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 60

This is consistent with Equation 2.34, but the use of Equation 2.37 importantly
identifies the “c” in Equation 2.34 as the group velocity r k (k).
Now we consider the standing wave limit, where / x = 0. Here, Equation
2.37 becomes a 2 / t = 2

i a 2 , and we obtain a 2

exp (2

i t). Thus,

a(t) exp (

i t)

(2.40)

which is consistent with Equation 2.35. In summary, in using Equation 2.37, we
can replace the complex frequency on the right with i = - r /2Q, noting that the
dimensionless function Q is empirically obtained. The speed “c” used earlier
should be the group velocity C(k) = d

r (k)/dk, where k(x,t) is obtained by

solving Equations 2.24 and 2.25.
Recapitulation. We have introduced the Q-model used in geophysics,
interpreted its meaning in terms of propagating and standing waves, and
embedded these ideas within a group velocity based wave description for
continuum systems which allows both changes in space and time. Simple
recipes follow. When a propagation experiment is conducted with the excitation
frequency fixed, direct measurements can be made for the wavenumber k; the
frequency function r (k) can be obtained from repeated experiments.
Simultaneous attenuation observations yield i (k); then, using Equation 2.36, Q
is in principle known. Sometimes r (k) and Q are available from standing wave
measurements. For such problems, Equation 2.36 can be used to find i (k).
Substitution in Equation 2.37 allows standing wave “Q” measurements to be
used in general propagation simulation.

2.3 KWT in Homogeneous Dissipative Media
Having introduced the essential ideas, we formally develop KWT
assuming uniform media for now, but allowing general frequency-dependent
wave dissipation. By dissipation, we include instabilities as well, since we need
not restrict ourselves to purely decaying or growing waves; in fact, i (k) may
generally alternate in sign. In Chapter 1, we saw that weak dissipation (that is,
| i (k)/ r (k)| << 1) leaves the kinematics unaffected to leading order. The
imaginary frequency i (k) primarily affects amplitude, leaving phase unchanged
except at the second order. In this limit, our axiomatic conservation law for
wave crests need not involve the imaginary frequency, and we again write
k/ t +

/ x = 0 (2.41)

in the form
k/ t +

r (k)/ x = 0

(2.42)

where we have replaced by

r (k). If we carry out the differentiation, we find

that Equation 2.37 is used with

k/ t +

r k (k) k/ x = 0 (2.43)