Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 20

“Kinematic wave theory,” which deals with propagating waves, provides
an important and powerful alternative to physical modeling. The dispersion
functions r (k,x,t) and i (k,x,t) are often available experimentally; for example,
they may be derived from wave speed and attenuation data. “KWT” provides
the formalism through which equation-based modeling can be circumvented,
allowing immediate use of laboratory results in practical problems. This is clear
intuitively: r (k,x,t) and i (k,x,t) already embody the dynamical process, and in
this sense, equation modeling is superfluous.
The word “kinematic” in kinematic wave theory must not be taken lightly:
it indicates that the propagation properties for assumed linear systems do not
depend upon the wave amplitude in the leading description. When nonlinear or
“finite amplitude” effects (as opposed to infinitesimal ) are important, analyses
similar to those given above can be performed, but the dispersion relations
obtained now contain a dependence on wave amplitude. For example, we can
write = r (k,x,t,A 2 ) for nondissipative systems. The formalism extending
KWT to dissipative nonlinear systems, presented in Whitham (1974) for low-
order systems, was further extended by Chin (1976, 1981) to high-order
systems. Some general results are presented in Chapter 2. We now return to our
survey of classical techniques and continue our discussion by introducing
Laplace transforms. This technique is useful in drillstring vibrations modeling,
and in mud pulse telemetry modeling, because it greatly simplifies source
description through “delta function” formulations.

1.5 Laplace Transforms
Laplace transforms provide powerful tools for the solution of boundary
value problems. The foundation underlying this classical technique, developed
by the English engineer Heaviside a century ago, is firmly rooted in the theory
of complex variables, which we will not attempt to reproduce here. Churchill
(1941, 1958) provides readable expositions on the method, treating problems in
a down-to-earth manner. Here, we will give an overview only, preferring to
demonstrate through examples how solutions are obtained; for continuity, we
will continue to focus on solutions to the wave equation.

1.5.1 Wave equation derivation.
Let us derive the equation satisfied by the transverse displacements u(x,t)
of a string acting under its own weight,
2 u/ t 2 - c 2 2 u/ x 2 = -g (1.50)
where g is the acceleration due to gravity. We apply Newton's law F = m a in
the vertical direction to the mass element l dx shown in Figure 1.2, where l is
the lineal mass density and dx is a small length. Along the string segment, the
slope u/ x must equal the vertical-to-horizontal force ratio V/H . For small