Harmonic Analysis: Elementary Pipe and Collar Models - Measurement While Drilling Signal Analysis, Optimization and Design - page 93

w 2 u/wt 2 - c 2 w 2 u/wx 2 = 0 (3.B.1)
c 2 = B/U (3.B.2)
p = - B wu/wx (3.B.3)
u(x,t) = X(x) e i Zt (3.B.4)
d 2 X(x)/dx 2 + Z 2 /c 2 X = 0 (3.B.5)
X 1 (x) = A sin Zx/c + B cos Zx/c (3.B.6)
X 2 (x) = C sin Zx/c + D cos Zx/c (3.B.7)
X 3 (x) = E exp (-iZx/c) (3.B.8)
Equations 3.B.6 and 3.B.7 allow standing waves to form in Sections 1 and
2, while Equation 3.B.8 for Section 3 states that u 3 = E e i Z(t - x/c) which implies
that the wave is traveling to the right without reflection. This wave will
attenuate, but only at a distance far from the dimensions shown in Figure 3.B.1.
At x = L c , continuity of volume velocity (A c u 2 = A p u 3 ) and of acoustic pressure
(wu 2 /wx = wu 3 /wx) require that
C sin ZL c /c + D cos ZL c /c = (A p /A c ) E exp (-iZL c /c) (3.B.9)

C cos ZL c /c - D sin ZL c /c = - i E exp (-iZL c /c) (3.B.10)

At x = L m , continuity of displacement yields
A sin ZL m /c + B cos ZL m /c = C sin ZL m /c + D cos ZL m /c

(3.B.11)

The pulser develops a pressure discontinuity at x = L m of the form

p 2 - p 1 = 'p = P s exp (iZt) (3.B.12)

Since p = - B wu/wx, we have

A cos ZL m /c - B sin ZL m /c - C cos ZL m /c + D sin ZL m /c = cP s /(ZB)
(3.B.13)
At the drillbit x = 0, the assumption of an open reflector, i.e., Case (e) ,
requires wu 1 /wx = 0 or A = 0, while the assumption of a solid reflector, that is,
Case (f) , requires u 1 = 0 or B = 0, In either event, we have five equations for the
five complex unknowns A , B , C , D and E which can be solved exactly in closed
analytical form. Some algebra shows that

E solid = {(cP s )/(ZB)} sin ZL m /c exp (+iZL c /c) / {(A p /A c ) cos ZL c /c + i sin ZL c /c}
(3.B.14)

E open = - {(cP s )/(ZB)} cos ZL m /c exp (+iZL c /c)/ {(A p /A c ) sin ZL c /c - i cos ZL c /c}
(3.B.15)

Since p 3 (x,t) = - B wu 3 /wx = i(BZ/c) E e i Z(t - x/c) , then