Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 4

Figure 1.1. The characteristic plane.
In linear theory, characteristics within each wave family are parallel. The
“method of characteristics,” used analytically and numerically, solves problems
by tracing values of the dependent-variable along rays. We will use this method
in Chapter 2 to obtain closed form solutions for use in kinematic wave theory.
From Figure 1.1, the slope c in the characteristic x-t plane is the wave or “sound
speed,” e.g., the line x-ct = constant has the slope c, which is obviously the
propagation velocity with which disturbances travel. In general, a disturbance
propagates with speed c to both the left and the right. Equation 1.1 states that c
is the only available wave speed. In many problems, other speeds are possible.
In three-dimensional waveguides, additional speeds are possible. Waves with
different lengths (or, equivalently, frequencies) may travel at different speeds:
an initially confined wave group may disperse and lose its identity.
“Wave dispersion” is also possible in one-dimensional problems, e.g.,
bending waves on beams. We deal with the subject of dispersion later, but it is
important to recognize that Equation 1.1 does not describe dispersion at all.
And since, as discussed, disturbances satisfying Equation 1.1 propagate with
shape and amplitude both intact, the classical equation does not describe
dissipative, attenuative or nonconservative effects, terms often used
interchangeably. It applies to “conservative” or undamped wave motions only.
Also, Equation 1.1 does not contain variable coefficients: the motion occurs in
uniform or homogeneous media, in contrast to nonuniform or heterogeneous
media. Heterogeneous media may contain spatial and temporal
inhomogeneities, that is, variable coefficients in x and t. Such general waves are
easily modeled using kinematic wave theory. In Equation 1.3, “+” appears in
one function, while “-“ appears in the other. Different conventions are used in
different applications; u(x,t) = f(t+x/c) + g(t-x/c) may appear in one context, but
u(x,t) = g(-t - x/c) - f(-t + x/c) may be more convenient in others. Sometimes f