Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 32

1.7.2 Properties of point loads.
The point load F(t) is associated with more properties, which are easily
illustrated by the action of a bow on a violin string (by simply writing F( t) , w e
imply that its magnitude can be controlled, and that adequate feedback and
control makes this mechanically possible). First, while F is responsible for a
local discontinuity or jump in u/ x, it does not directly dictate the exact value
of u/ x or u, as a prescribed boundary condition would: F is associated with
differences in u/ x only. Second, waves heading towards x = x s from left or
right will pass through x s without change. Waves on a violin string pass through
the bow contact point: they do not terminate there and reflect backwards. In
other words, point forces are transparent to propagating waves. This property,
for example, is essential to modeling mud pulsers in MWD telemetry - pulsers
create signals, but their reflections pass through them without interaction.
1.7.2.1 Example 1-13. Boundary conditions versus forces.
Consider a finite string of length L, rigidly attached to a wall at x = 0 so
that u(0,t) = 0. We will discuss modeling options for a sinusoidal excitation at
the end x = L. The “boundary condition” u(L,t) = A 2 sin t might be a
reasonable model, if everything that can conceivably happen can be described
by a prescribed displacement. But it would be restrictive if the actual excitation
were due to an external point load entity close to x = L, and another constraint
describing another feature characterizing x = L were needed: we would have
“used up” the boundary condition at x = L, thus limiting ourselves in modeling
options. This statement will be clear in modeling rate-of-penetration in
drillstring vibrations: we cannot enforce “u(L,t) = A 2 sin t” at the bit since this
would preclude any drilling ahead, but this sinusoidal prescription is used
elsewhere in an extended formulation, e.g., refer to Chapter 4 for details.
By using a forcing function model instead, e.g., l 2 u/ t 2 + u/ t - T
2 u/ x 2 + l g = A 3 sin t (x-L+ ), where > 0, we open up more classes of
modeling options because we have not yet “used up” the boundary condition at
x = L. These others might be u(L,t)/ x = A 1 sin t, u(L,t) = A 2 sin t, or a
radiation condition u(x,t) f (x-ct) to model semi-infinite effects at a point.
Also, boundary conditions are restrictive and not transparent: they isolate “left”
and “right” events in this sense. Suppose that the force in Figure 1.4 were
replaced by a rigid wall, and that a boundary condition excitation u(x s ,t) = A 2
sin t was mechanically enforced at the wall x s . Then the strings to the left and
right of the wall will act independently; they will not “see” each other. This
wall separates events in x < x s from those in x > x s . If any of these reflect back
from finite boundaries, their motions terminate here and re-reflect at the wall,
because the exact level of u has been prescribed.