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For the purposes of this paper, we will take (b12nocc) and
(b13nocc) to be the low frequency limits of the drag coefficients
and also assume that .Then, the coefficients *b*_{12} and *b*_{13}
must be modified at higher frequencies
in order to assure that the theory as a whole preserves obvious physical requirements such as
nonnegative dissipation for all modes at all times.
(If the theory always predicts nonnegative dissipation, then we will
say it is ``realizable.'' If the theory predicts negative
dissipation for any of the modes of propagation, then the theory
is not realizable, and further effort will be required to make the theory
fully realizable.)
This issue arises naturally when we have obtained the solution
*x* = *v*^{2} to (Newton) for any one of the three compressional
modes.
Then, taking the complex square root, we get two roots that differ only
by + and - signs. We want the solution
that has both positive real velocity and a negative imaginary part.
This is so because , where should be a
physical attenuation coefficient such that

i(kz-t) = (-z)i k_0(z-v_0 t)
leads to a decrease in the overall amplitude of the compressional
mode.
If, for any of the three compressional modes,
no root exists with both positive real velocity and negative
imaginary part, then the dispersion relation is unphysical and
the approximations we have made in deriving it are suspect.

In our examples to follow, we will tentatively take the results
from Appendix B as the proper way to modify the drag coefficients
at higher frequencies, but must always be careful to check that
this choice does not lead to unphysical behavior.

** Next:** EXAMPLE
** Up:** THE DISPERSION RELATION AND
** Previous:** Solution of the dispersion
Stanford Exploration Project

4/20/1999