Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 30

Farfield conditions such as u x + 1 / c u t = 0 or u x - 1/c u t = 0, known as
“radiation conditions” in classical physics, were routinely used at box edges to
filter disturbances that might propagate to undesired directions. They allow
unidirectional wave simulation on small computationally inexpensive boxes,
which otherwise create unacceptable reflections. Such directional filters are
used in geophysics too. “Imaging algorithms” are tested on “synthetic data”
generated by the three-dimensional wave equation for finite reservoir volumes,
where “outgoing wave conditions” are used at computational walls to eliminate
spurious reflections (e.g., Bleistein (1984), Claerbout (1985a,b)).
1.7 External Forces Versus Boundary Conditions
The distinction between applied external forces and prescribed boundary
conditions is sometimes unclear since both are used to excite physical systems.
Their differences will be clarified here, since both are needed to pursue MWD
and drilling vibration applications.
1.7.1 Single point forces.
To visualize the problem, we return to the “transverse displacements”
u(x,t) of a stretched string, with lineal mass density l and tension T. Everyday
guitar strings execute such oscillations, and such motions are ideal for
illustrating key mathematical ideas. In the equation
l 2 u/ t 2 + u/ t - T 2 u/ x 2 + l g = 0 (1.98)
we have introduced a term l g proportional to “body force,” g being the
acceleration due to gravity, and also, an “internal viscous dissipation” u/ t.
These do not affect any “jump properties” to be discussed, but they are retained
in our discussion to show why they won't. When an external point force of
strength F(t) is applied to x = x s , as shown in Figure 1.4, the “Dirac delta
function” representation
l 2 u/ t 2 + u/ t - T 2 u/ x 2 + l g = F(t) (x-x s ) (1.99)
is used, where (x-x s ) is the “delta function” at x = x s . We follow the usual
convention, prefixing F with a “+” sign when the force points upward.
x = x s
Figure 1.4. Single point force acting on a string.