Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 3

If so, u(x,t) is the displacement from equilibrium, and c is the disturbance
speed (c 2 = T/ l , where T is tension and l is lineal mass density). * Examples
include waves on violin and guitar strings. The domain of x can be finite,
semi-infinite or infinite, depending on the application. But time is always zero
or positive, and future events must never affect present and past motions. This
“principle of causality” governs equations of evolution, particularly “hyperbolic
equations” such as Equation 1.1. Not all equations are causal. The “elliptic
equations” used in reservoir flow simulation, which may look something like
2 p/ x 2 + 2 p/ y 2 = 0 (with a “+” instead of a “-“ sign) deal with “domains of
influence and dependence” such that every point affects and is influenced by
every other point (Chin, 1993). By contrast, “parabolic equations,” taking forms
similar to T/ t - 2 T/ x 2 = 0 ( > 0 ), deal with diffusion in space and time.
In Equation 1.1, / t and / x represent partial derivatives, or derivatives
of functions of several variables relative to the particular independent variable
shown, with all others held fixed. We may also express partial derivatives using
subscript notation,
u tt - c 2 u xx = 0 (1.2)

Its general solution, obtained two hundred years ago by D' Alembert, is
u(x,t) = f(x+ct) + g(x-ct) (1.3)
and will find significance later, for instance, in MWD signal processing and
echo cancellation. To understand why Equation 1.3 holds, we apply the
derivative rule from calculus, stating that the x-derivative of h{p(x)} is
{h'(p)}{dp(x)/dx} where primes denote differentiation with respect to p. Since
u x = f ' + g ' , u xx = f " + g", u t = c f ' - cg' and u tt = c 2 f " + c 2 g", substitution in
Equation 1.2 proves Equation 1.3. This equation contains much information. If
we consider the u(x,t) = f(x+ct) contribution, we observe that u(x,t) must be
constant if x+ct is constant. But “x+ct = constant” is just the straight line in
Figure 1.1. Since time must increase, the argument x+ct must represent left-
going waves. Similarly, u(x,t) = g(x-ct) is constant along trajectories with x-ct =
constant. Thus, g(x-ct) represents right-going waves. Depending upon the
application, wave solutions may be both up-going and down-going. Equation 1.3
states that solutions of Equation 1.1 can be constructed from general families of
left and right-going waves. Lines for which x+ct and x-ct are constant are
known as “characteristics” or “rays,” and their coordinates represent the natural
or canonical variables describing the wave propagation.

* Two densities, the lineal mass density l and the mass density per unit volume , are used in this
topic. The first is appropriate to vibrating string problems satisfying l 2 u/ t 2 - T 2 u/ x 2 = 0 w he r e
the tension T has units of force. The second applies to drillstring vibrations and borehole acoustics
problems satisfying 2 u/ t 2 - E 2 u/ x 2 = 0 where E, either Young' s modulus or the bulk modulus,
has units of force per unit area.