Geology Reference
In-Depth Information
simplifications have been independently observed from synthetic seismograms
computed from Fourier-based solutions to the complete fluid-elastic
formulation. Equation 10.11 for the wave energy density E generally states that
timewise changes in local energy density are affected by the flux of energy
based on the group velocity c
g
and by nonconservative damping or growth with
an exponential rate based on
i
.
Now , s i nc e c
g
depends on k, and k must vary with x and t as the wave
propagates, the “
/ x” operation requires (c
g
E)/ x = c
g
E/ x + E c
g
/ x, with
the result that
E/ t +
r
/ k E/ x = {2
i
- d
2 r
(k)/dk
2
k/ x} E(x,t) (10.12)
From calculus, the total differential dE is obtained as dE = E/ t dt + E/ x dx.
Next, division by dt yields dE/dt = E/ t + dx/dt E/ x. Comparison with
Equation 10.12 allows us to set dE /dt = {2
i
- d
2
r
(k)/dk
2
k/ x} E, that is,
i
- d
2
r
(k)/dk
2
d log
e
E /dt = {2
k/ x}
(10.13)
provided we simultaneously set the wave speed dx/dt equal to
dx/dt =
r
(k)/ k (10.14)
These two equations describe wave energy variation “following the wave,”
and equivalent numerical results (for any particular set of governing equations)
can be formally obtained using the “method of characteristics.” The derivation
given here is equally rigorous and is clearly conceptually much simpler.
Equation 10.14 shows that the relevant velocity for energy propagation is the
group velocity. Equation 10.13 shows how attenuative loss and dispersion
together affect wave propagation: the anticipated “2
i
” damping term is
augmented by an “apparent dissipation rate” - d
2 r
(k)/dk
2
k/ x due to
frequency dispersion. Now, the “2
i
” term in Equation 10.13 will dampen
all
wave components since it is always negative. However, the “- d
2 r
(k)/dk
2
k/ x” term can be positive, negative or both, depending upon the wave system
under consideration and upon initial conditions.
In order to obtain explicit results specific to low-frequency Stoneley
waves, we differentiate Equation 10.3 twice with respect to k to obtain
r
kk
(k) = + ¼ k
-3/2
> 0 ( 1 0 .1 5 )
which is, importantly,
always
positive. Ignoring “2
i
” for the moment, it is
seen from Equation 10.13 that energy increases or decreases along the path of
propagation accordingly as k/ x is negative or positive. Now, a “thought
experiment” - consider an amplitude waveform as a function of the propagation
coordinate “x” a short time after it has left the transmitter. This waveform is
created by a source that increases in frequency, holds steady at a fixed center
Search WWH ::
Custom Search