Geology Reference
In-Depth Information
and g are reversed for convenience. These ideas, for instance, are relevant to
multiple-transducer signal processing in MWD for noise cancellation.
To solve the general initial value problem on an infinite unbounded
domain, subject to initial conditions prescribing u(x,0) and the velocity u t (x,0) at
t = 0, we have D' Alembert' s solution,
x+ct
u(x,t) = {u(x-ct,0) + u(x+ct,0)}/2 + {1/(2c)} u t (x',0) dx'
(1.4)
x-ct
Consider an initial prescribed displacement with vanishing velocity. The
integral term disappears, leaving u(x,t) = {u(x-ct,0) + u(x+ct,0)}/2, showing
through the “+” and “-“ sign differences that half of the initial wave will
propagate to the left, while the remaining half propagates to the right.
1.1.2 Reflection properties.
Equation 1.3 is significant to wave propagation in many facets of
petroleum engineering, e.g., borehole acoustics, MWD telemetry, and others.
Unfortunately, many math courses do not address engineering application, and
the consequences of Equation 1.3 often remain buried in the obscurity of
subscripts and theorems.
We have already alluded to Equation 1.1 as a model for transverse string
vibrations. It is under this interpretation that much of partial differential
equations is taught at the undergraduate and graduate level. As a prelude to
drillstring vibrations and seismics, we also envision u(x,t) as the longitudinal
elastic displacement of a one-dimensional bar, in which case c 2 = E/ where E
is Young' s modulus and is the material density or mass density per unit
volume. The normal stress satisfies Hooke' s law = E u/ x, where u/ x
represents the strain. Let us start with u(x,t) = f(x+ct) + g(x-ct), and calculate,
using the chain rule, the space derivative u x = f '(x+ct) + g'(x-ct) and the
timewise derivative u t = cf '(x+ct) - c g'(x-ct). Now assume that x = 0 represents
the termination end of a finite or semi-infinite one-dimensional system. How
are f and g related, and what are the physical implications?
1.1.2.1 Example 1-1. Rigid end termination.
Suppose that our termination end can be modeled by a rigid wall, so that
the displacement satisfies u(0,t) = 0. Then, it is clear that the choice
u(x,t) = f(x+ct) - f(-x+ct) (1.5)
solves the problem, since u(0,t) = f(ct) - f(ct) yields u(0,t) = 0 for any choice of
the function f. By differentiating, the end velocity u t (0,t) is also zero. At x = 0,
the derivative u x (x,t) = f '(x+ct) + f '(-x+ct) equals 2f '(ct), so that stress doubles
at a rigid end relative to the incident f '(ct) value (as we will show, this is
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