Physical realizability at higher frequencies

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 $b_{23} \equiv 0$.Then, the coefficients b₁₂ and b₁₃ 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 = v² 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 $k = \omega/v = k_0 + i\alpha$, where $\alpha$ 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.