Kinematic Wave Theory - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 54

(x,t) = kx - t (2.9)

the interchangeability of the differentiation

xt (x,t) = tx (x,t) leads to a statement

describing “wave crest conservation.” That is,

x / t = t / x implies k/ t = -

/ x, or

k/ t + / x = 0 (2.10)

In fluid mechanics, the mass conservation law / t + ( q)/ x = 0 states
that timewise changes in density must be balanced by differences in the mass
flux q, where q is the transport velocity. Similarly, Equation 2.10 states that
timewise changes in the number of waves must be balanced by the flux of
frequency through the control length. Equation 2.10, which follows from an
asymptotic solution to a Fourier integral applicable to all physical systems, is
taken as a fundamental premise in kinematic wave theory.
In three-dimensional problems, the dispersion relation may depend on three
wavenumbers, with = (k 1 ,k 2 ,k 3 ,x,t) = ( k ,x,t). For such problems, the
vector description for wave crest conservation is k / t + = 0, and a special
irrotationality condition for k applies; we consider three-dimensional problems
later, dealing with acoustic waveguide and cross-well geophysical applications.
It turns out that Equation 2.10 appears as an asymptotic result in many linear
and nonlinear problems in inhomogeneous and uniform media as well. In KWT,
we take this ki n e ma ti c consequence as axiomatic . We will demonstrate that this
is consistent with both conventional analysis and physical reality.

2.1.5 Simple consequences of KWT.
In order to illustrate the power behind KWT, we now draw upon a low-
order corollary of a general amplitude theorem we derive later (e.g., see
Equation 2.95). As in Equations 2.4 and 2.5, we later show without invoking
specific engineering equations that
a 2 / t + (C(k)a 2 )/ x = 0 (2.11)

that is,
a t + C(k) a x + 1/2 a C'(k) k x = 0 (2.12)

where
C(k) = d (k)/dk (2.13)
is the “group velocity” introduced by nineteenth-century physicists and well
known among electrical and acoustics engineers. What are the consequences of
Equation 2.11? To understand the implications, we apply the chain rule of
calculus to Equation 2.10, and rewrite it as k/ t + d (k)/dk k/ x = 0, or
k t + C( k) k x = 0 (2.14)