Geology Reference
In-Depth Information
compressibility of the medium, whereas by “hydraulics,” we refer to the
incompressible constant density flow associated with material transport “from
here to there.” The pressures associated with these flows are additive: just as
children can shout in a noisy wind, acoustic pulses can travel in flowing mud.
Interactions with mud and MWD pulser create sound, but once sound waves
leave the source, they are “on their own.” These subjects and their subtleties
will be discussed later.
1.2 Functional Representation
In mathematics, functions can be represented by various series, each useful
for particular applications. Why might one wish to represent any function in
anything more than classical Taylor series? The answer is simple: the required
functional representation is determined by the boundary value problem one
wishes to solve and the geometry of the problem domain. Since the
possibilities for problem formulations are numerous, there is no shortage of
representations, e.g., Legendre, Bessel and Hankel series. Refer to Hildebrand
(1948) for a complete discussion. Fortunately, for the one-dimensional analyses
considered in this topic, no more than classical Taylor and Fourier series are
required. We now discuss these conventional representations.
1.2.1 Taylor series.
Suppose we wish to represent y = f(x) using a power series in x. We recall
from calculus that we may expand f(x) about any value of x, say x
0
, where the
value of the function and its derivatives are known. This assumes that the latter
exist, which may not be the case; y = x
-1/2
, for example, is infinite and has
singular (infinite) derivatives at x = 0, and therefore no Taylor series
representation there. But away from x = 0, this representation applies. If f(x)
and its derivatives f
n
(x) exist at x = x
0
, then its Taylor series expansion is
f(x)
= f(x
0
) + f '(x
0
)(x-x
0
) + 1/2 f "(x
0
)(x-x
0
)
2
+ . . .
f
n
(x
0
)(x-x
0
)
n
/n!
(1.7)
where superscripts indicate the order of the derivative.
1.2.2 Fourier series.
Now let us represent y = f(x), defined on 0 < x < L, using Fourier series.
Because its values on -L < x < 0 are not significant, we can assume f(x) to be
even with respect to x = 0. Since f(x) = f(-x), we can write
f
e
(x) = A
0
+ A
n
cos n x/L
(1.8)
provided
A
0
= { f
e
(x) dx}/L
(1.9)
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