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strength 'p(Z) that may depend on frequency, but the exact dependence will
vary with valve design, flow rate, and so on. Of course, this separation of
variables is not as limiting as it initially appears; any transient signal can be
reduced to superpositions of harmonic components using Fourier integral
methods. Together with the assumption
ā'pā = 'p e
iZt
(2.4)
we consistently assume, still restricting our discussion to the MWD drill collar, a
Lagrangian displacement of the form
u
c
(x,t) = U
c
(x)e
iZt
(2.5)
Substitution of Equations 2.4 and 2.5 in Equation 2.1 leads to a simple ordinary
differential equation, better known as the one-dimensional Helmholtz equation
governing the modal function U
c
(x),
U
c
xx
(x) + (Z
2
/c
mud
2
) U
c
= - ('p/B
mud
) G(x-x
s
) (2.6)
which is satisfied by all fluid elements residing in the drill collar (here, we have
introduced the sound speed defined by c
mud
= Ā(B
mud
/U
mud
)). This real
inhomogeneous differential equation can be solved using standard Green's
function or Laplace transform techniques (had we allowed attenuation, a more
complicated complex equation would have been obtained). Its general solution,
to be given later, can be represented in the usual manner as the superposition of
a homogeneous solution (with two arbitrary integration constants) satisfying
U
c
xx
(x) + (Z
2
/c
mud
2
) U
c
= 0, and a particular solution (without free constants)
satisfying Equation 2.6 exactly. Now that the fundamental mathematical ideas
have been discussed in the context of drill collar analysis, let us consider the
remaining elements of the waveguide.
2.4.2.3 Governing partial differential equations.
We have discussed in detail the properties of the differential equation
governing the acoustics within the drill collar. The equations governing other
sections of the one-dimensional waveguide are similar, although simpler,
because they are not associated with MWD sources. Again, using separation of
variables like u(x,t) = U(x)e
iZt
, we obtain a sequence of Helmholtz equations for
our waveguide sections. For clarity and completeness, going from the right to
the left of Figure 2.5, these are
U
p
xx
(x) +
(Z
2
/c
mud
2
)U
p
=
0 (2.7a)
U
c
xx
(x) +
(Z
2
/c
mud
2
)U
c
= - ('p/B
mud
) G(x-x
s
) (2.7b)
U
mm
xx
(x) +
(Z
2
/c
mm
2
)
U
mm
=
0 (2.7c)
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