Geology Reference
In-Depth Information
Since G(t) (x-x
s
) creates a jump in the first spatial derivative of
the
dependent variable u/ x
, it is responsible for a discontinuity in the required
second derivative
2
u/ x
2
. In wave equation modeling, two questions arise if
delta function superposition methods are used. What
physical quantity
is
discontinuous through the source point? Which
dependent variable
should be
used (remember,
its
first space derivative represents that physical quantity)?
Examples of these subtleties arise in MWD pulser valve modeling.
Positive pressure valves
create
pressure
discontinuities at the source, while
negative pressure valves
create
velocity
discontinuities; velocity jumps do not
exist for the former, while pressure jumps are not found in the latter. These
valves require completely different mathematical formulations; as we will show
later, general MWD valve models lacking pure symmetries or antisymmetries
can be constructed from linear superpositions of positive and negative valve
models. While our discussion has focused on second-order equations in the
context of transverse vibrations, the ideas are universal, applying to general
linear systems. In each case, the exact meaning of impulse is crucial; what
discontinuities they induce must be explored and made relevant to the physics,
and all relevant superposition integrals must be derived.
1.7.2.9 Other delta function properties.
We will list some properties of delta functions for reference, since they are
not usually obvious nor available. Again, we have the best known property
(x-x
s
)
(x) dx = (x
s
)
(1.138)
so that, for example,
(x) (x) dx = (0)
(1.139)
The limits are shown over (- , ) for brevity, but the integration may be carried
out over any domain containing “x
s
.” Also, it can be shown that
(-x) = (x)
(1.140)
(ax) = (x)/|a|
(1.141)
(at-bx) = (t - bx/a)/|a|
(1.142)
x (x) = 0
(1.143)
(x
2
-a
2
) = { (x+a) + (x-a)} /{2|a|}
(1.144)
(x) = -x '(x)
(1.145)
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