= Complex-frequency SH flux monotonicity
{title2=$S_z'=-\tfrac12\operatorname{Im}\omega[|T|^2/\mu+(\rho+\mu k^2/|\omega|^2)|V|^2]$}
For positive real <mass density> and <shear modulus>, real $k$ and $\operatorname{Im}\omega>0$, the displayed <derivative> of $S_z=-\tfrac12\operatorname{Re}(V^*T)$ follows from the two first-order shear equations. Every nontrivial solution has strictly decreasing $S_z$. Since $S_z=\tfrac12\operatorname{Re}Z|V|^2$, positive-real impedance at a deeper point remains positive at all shallower points; negative-real impedance at a shallower point remains negative deeper down. A zero field is the necessary exception to strictness. At complex <frequency> this is a spectral flux diagnostic, reducing to a physical cycle average on the real axis.
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