Geology Reference
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We discuss several solutions to the acoustic wave equation applicable to
wave propagation where the signals are created by dipole sources within the
acoustic channel. These solutions correspond to the waveguides shown in
Figure 3.A.1. In order to illustrate key ideas, uniform cross-sectional areas are
assumed for simplicity (for more detailed analysis, use the six-segment
waveguide model of Chapter 2). Again we take the Lagrangian displacement
u(x,t) as the dependent variable, where t is time and x is the propagation
coordinate. Then, away from the source, in the absence of attenuation, we have
w
2
u/wt
2
- c
2
w
2
u/wx
2
= 0
(3.A.1)
c
2
= B/U
(3.A.2)
p = - B wu/wx
(3.A.3)
where c is the sound speed, B is the bulk modulus, U is fluid density and p is the
acoustic pressure.
Figure 3.A.1
. Different propagation modes.
3.1.1 Case (a), infinite system, both directions.
The “infinite-infinite” system in Figure 3.A.1a is considered. Both ends
are far away and reflections do not return to the dipole source located at x = 0.
As our system is linear, we consider a Fourier component of the excitation
pressure having frequency Z. We seek separable solutions of the form
u(x,t) = X(x) e
i
Zt
(3.A.4)
in which case
d
2
X(x)/dx
2
+ Z
2
/c
2
X = 0 (3.A.5)
This has linearly independent solutions of the form sin Zx/c, cos Zx/c, e
i
Zx/c
and
e
-
i
Zx/c
, different combinations of which may be used to satisfy boundary
conditions. For this example, we include all the mathematical details in order to
illustrate the method. For the example in Figure 3.A.1a, complex exponentials
are appropriate and we assume
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