Geology Reference
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under consideration. Let us consider, for example, the acoustic field in the
drillpipe. Recall that we had used the separation of variables u(x,t) = U(x)e iZt .
The assumption taken in Equation 2.8a, namely, U p (x) = C 1 exp (-iZx/c mud ),
implies that the time-dependent solution is u(x,t) = C 1 exp {iZ(t-x/c mud )} which
is the representation for a propagating wave. Similar comments apply to
Equation 2.8f for the annulus. In contrast, the “sine” and “cosine”
representations naturally describe standing waves inside the waveguide.
2.4.2.4 Matching conditions at impedance junctions.
Continuity of volume velocity requires that the product between
waveguide cross-sectional area “A” and longitudinal velocity wu(x,t)/wt remain
constant through an impedance junction , that is, any point through which an
acoustic impedance mismatch exists. Since we have chosen the Lagrangian
displacement as the dependent variable, the area-velocity product takes the form
Awu(x,t)/wt. Because u(x,t) = U(x)e i Zt , this quantity is Aw{U(x)e i Zt }/wt or
i AUZe i Zt . And furthermore, since the coefficient i Ze i Zt is the same on both sides
of an impedance change despite changes in the modal function U, it follows that
the continuity of volume velocity, at least for time harmonic disturbances,
requires only that we enforce the continuity of product AU (hence, we have
“A 1 U 1 = A 2 U 2 ”). Continuity of the acoustic pressure p = - B wu(x,t)/wx, on the
other hand, requires that the derivative quantity BU'(x) remain invariant
(consequently, we obtain “B 1 U' 1 = B 2 U' 2 ”), where B represents the bulk
modulus. Although these impedance matching conditions superficially involve
real quantities only, the equations for the coefficients C which arise from
substitution of Equations 2.8a - 2.8f are complex, because combinations of
sinusoidal and complex exponential solutions have be used to fulfill boundary
conditions and radiation conditions. Hence, the coefficients C will, in general,
be complex in nature. For a more detailed discussion on boundary and matching
conditions, the reader is referred to the classical acoustics topics of Morse and
Ingard (1968) and Kinsler et al (2000).
Let us now give the algebraic equations that result from the assumed
sinusoidal or exponential forms for U and the required matching conditions
(refer to our nomenclature list for all symbol definitions). After some algebra, a
detailed set of coupled linear complex equations is obtained, which is solved
analytically and exactly in closed form and then evaluated numerically (phase
errors of the type found in finite difference and finite element methods are not
obtained in the present approach). In the following summary, the particular
impedance junction considered is listed and underlined, and followed,
respectively, by matching conditions obtained for volume velocity and pressure.
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