Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 66

to both sides, and noting that

i / t = ( -

r 0 ) G dk

(2.78)

(-i) n

n

/ x n = (k - k 0 ) n G dk

(2.79)

leads to an important identity that serves as the exact partial differential equation
satisfied by

(x,t),

i / t = i

i 0

+ {(-i) n /n!}

n

/ x n (2.80)

nk,0

Equation 2.80 will serve as the focal point of our subsequent discussion, one
which determines its asymptotic wave properties.

2.4.3 Method of multiple scales.
In many physical problems, multiple time scales exist. For example, the
solution to the damped wave equation given by Equation 1.44 is characterized
by a period defined by the undamped system, a second time scale associated
with amplitude decay, and a third which introduces dissipation-dependent
changes to wave phase. Or consider the oscillations of a mass-spring-damper
system. The obvious time scale is the period, which depends for the most part
on mass and spring stiffness. The damping term affects amplitude over larger
time scales, and again, after longer times, induces changes to the period itself.
In the method of multiple scales, this a priori knowledge of the physical
system is explicitly exploited (Cole, 1968; Nayfeh, 1973). We specifically seek
separable, asymptotic, wave-like (a.k.a., “WKB”) solutions of the form

(x,t) = a(X,T) e i (x,t) (2.81)

where (x,t), a rapidly varying disturbance phase function, is motivated by the
discussion leading to Equations 2.8 and 2.9. Its space and time derivatives are
related to wavenumber and frequency as before, but following Equation 2.6, we
insist that the latter vary slowly in space and time.
To develop the theory formally, we introduce new uppercase wavenumber
and frequency functions K and . We require that they vary slowly with respect
to x and t, and accomplish this by having them depend explicitly on the large-
scale variables X and T where
X = x (2.82)
T = t (2.83)
Again, is the small parameter inferred from Equation 2.72, which states that i
(k)/ r (k) O( ). That is, K and may vary significantly, but only over large
space and time scales for which X and T are O(1), or equivalently, for which x
and t are O(1/ ). Thus, we formally assume that