Geology Reference
In-Depth Information
which shows, quite naturally, how group velocity emerges as the physically
relevant propagation velocity. Equations 2.94 and 2.95, again, are not
independent. They are mathematically coupled through Equation 2.86, which
we now rewrite in the form
k/ t + / x = 0 (2.96)
Substitution of Equation 2.94 in Equation 2.96, as we had shown earlier, reveals
the group velocity
r k as the relevant transport speed for the wavenumber k(x,t).
2.4.5 The low-order limit.
We are finally in a position to derive, with mathematical rigor, the low-
order limit arising out of Equations 2.94 to 2.95. If in Equation 2.94, we retain
only the term in “
r (k)
” shown, substitution in Equation 2.96 leads to
k/ t +
r (k)/ x = 0
(2.97)
Deletion of the high-order terms in Equation 2.95 leads to
a 2 / t + ( r k a 2 )/ x = 2 i (k)a 2 (2.98)
which are exactly Equations 2.42 and 2.55 assumed earlier. Despite their
apparent generality, our results are limited to linear systems because linear
assumptions were invoked, e.g., Fourier superposition, sinusoidal solutions,
exponential damping, and so on. Also, the derivation disallows explicit
heterogeneities (e.g., a vibrating string with varying density l ( x) or tension
T( t)). But the structure of the high-order terms will apply to weakly
heterogeneous systems, to weakly nonlinear systems, or both, where r and i
are based on the primary wave or harmonic. These high-order terms will figure
importantly in our geophysical applications.
2.5 Effect of Low-Order Nonuniformities
Regardless of discipline, whether we deal with acoustics, electromagnetic
waves, or capillary waves on water, all linear waves are marked by a single
characteristic: they are completely characterized by the dispersion relation
connecting the complex frequency to a real wavenumber k. This relationship,
which may depend explicitly on k, x and t, takes the general form
(k,x,t) = r (k,x,t) + i i (k,x,t) (2.99)
and may be obtained analytically or experimentally. Again, any dependence on
x and t must necessarily be weak, for if not, slowly varying motions would not
result and the consequent motions would invalidate the assumptions behind our
wave model. The presence of heterogeneities, of course, does not affect the
fundamental ideas behind wave crest conservation. When nonconservative
effects are weak, waves are still conserved, and we can still write
r (k,x,t)/ x = 0
k/ t +
(2.100)
Search WWH ::




Custom Search