Geology Reference
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9.3.1 Analytical acoustic model.
In Chapter 3, we introduced the use of harmonic analysis in signal modeling and
considered the problems shown in Figure 9.8. In particular, we studied four
scenarios, namely, Case (a), infinite system, both directions, Case (b), drillbit as
a solid reflector, Case (c), drillbit as open-ended reflector, and Case (d), “finite-
finite” waveguide of length 2L. It is Case (d) that is of interest in intermediate
wind tunnel testing, and in this chapter, we discuss its applications.
Figure 9.8 . Different propagation modes (identical to Figure 3-A-1).
The sketch in Figure 9.8d shows a dipole source centered at x = 0 in a
waveguide of length 2L where x is the propagation coordinate. We will assume
open-ended reflectors satisfying wu/wx = 0 at x = r L where u(x,t) is the fluid
displacement and discuss its physical significance (both ends of the intermediate
wind tunnel are opened to the atmosphere). Again, Z is the rotation frequency, c
is the sound speed and t is time. Since standing waves are found at both sides of
the source, linear combinations of sin Zx/c and cos Zx/c are chosen on each side
to represent u(x,t). Use of Equations 3-A-8 and 3-A-9 at the source x = 0
previously gave us the acoustic pressure solutions
p 1 (x,t) = - {p s /(2 tan ZL/c)} [sin Zx/c + (tan ZL/c) cos Zx/c] e i Zt
on - L < x < 0 (3-A-14)
p 2 (x,t) = - {p s /(2 tan ZL/c)} [sin Zx/c - (tan ZL/c) cos Zx/c] e i Zt
on 0 < x < + L
(3-A-15)
These solutions describe a siren source whose differential pressure 'p, or
signal strength, is p s (t). As the rotor turns, equal and opposite pressure fields are
created which travel in opposite directions, reflect at the open ends, and
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