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da
2
/a
2
= 2
i
dt -
r
kk
(k)/
r
k
(k) dk
(2.66)
log
e
a
2
= 2
i
dt - log
e
r
k
(k)
(2.67)
log
e
a
2 r
k
(k) = 2
i
dt
(2.68)
and finally, the desired results
a
2
r
k
(k)}
-1
exp {2
i
dt}
{
(2.69)
a {
r
k
(k)}
-1/2
exp {
i
dt} (2.70)
These results should be contrasted with the solutions obtained earlier, namely,
A(x) = A
0
exp (-
r
x/2cQ) in Equation 2.34, and A(t) = A
0
exp (
i
t) in Equation
2.35, for our two simpler problems.
2.4 High-Order Kinematic Wave Theory
So far, we have developed KWT assuming that Equation 2.42 (that is,
k/ t +
r
(k)/ x = 0) applies, and assuming that a
2
/ t + (C(k)a
2
)/ x = 2
i
a
2
given by Equation 2.37 holds. Since the former is independent of a(x,t), these
assumptions imply that we can solve for k(x,t) and a(x,t) sequentially. This is
not absolutely true: from Chapter 1, wave phase (and hence, k(x,t))
is
affected
by dissipation through amplitude at the second order. The limitations behind
our ad hoc assertions need to be established, ideally by examining how they
arise as limits of a more complete theory.
In this section, we formally derive both of the above low-order results. In
fact, we give their extensions to two additional orders of magnitude, and
demonstrate the consequences in truncating the new
phase
and
amplitude
expansions. In low-order wave theory, it was sufficient to consider the
kinematics of the motion using the dispersion relation for the uniform plane
wave only, which was completely independent of amplitude. This is not the
case with higher order corrections, even when nonlinearities are completely
neglected. The results obtained here are extremely important to a number of
physical problems, ranging from the study of kinematic shocks, to ray tracing
through dissipative formations in crosswell tomography, to Kirchoff wave
migration in seismics, and to lateral drillstring vibrations deep in the wellbore.
2.4.1 Basic assumptions.
We start with a homogeneous uniform medium, but again allow general
frequency-dependent attenuation or instability. We assume that the complex
dispersion relation corresponding to the uniform sinusoidal wavetrain is
available, obtained, say, using the methods of Chapter 1. Now let us examine a
general linear superposition of monochromatic wave components, each
behaving like exp i(kx-
t) and satisfying the complex dispersion relation
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