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Use the quadratic spectral flux $S_z=-\frac{1}{2}\mathrm{Re}(V^{\ast }T)$. For positive real material mass density and shear modulus, differentiation of the velocity–traction system givesBecause $k$ is real, $\mu p_{\beta }^2=\rho -\mu k^2/{\omega }^2$. If $v=\mathrm{Im}\omega >0$, then $\mathrm{Re}(i{\omega }^{\ast })=v$ and $\mathrm{Re}(-i\omega \mu p_{\beta }^2)=v(\rho +\mu k^2/|\omega {|}^2)$. This proves complex-frequency SH flux monotonicity:Strictness holds for a nontrivial wavefield: if $V,T$ vanished together at one point, uniqueness of the first-order system would make the entire solution zero. The zero solution is the necessary exception to the literal strict inequality. Across a bonded material boundary $V,T$, and hence $S_z$, are continuous; integrating the negative derivative gives strict decrease across any positive-depth interval as well.

Where the SH input impedance is finite, $S_z=\frac{1}{2}\mathrm{Re}Z|V{|}^2$. If $\mathrm{Re}Z(z_1)>0$, then $S_z(z_1)>0$ and all shallower fluxes are larger and positive. They cannot have $V=0$, since that would make $S_z=0$. Thus $\mathrm{Re}Z(z)>0$ for every $z<z_1$. Likewise, negative real SH input impedance at $z_0$ gives negative flux throughout $z>z_0$, excludes velocity zeros there and implies $\mathrm{Re}Z(z)<0$ for every $z>z_0$.

In a uniform region let $q=\mu p_{\beta }$ on the causal vertical wavenumber branch. The constant Riccati solutions are $Z=\pm q$, and integrating the one-way equation givesSince $\mathrm{Re}q>0$, the plus solution carries positive downward flux and the minus solution negative upward flux. Writing $k_{\beta }=a+ib$ with $b>0$, their moduli are proportional to $e^{-bz}$ and $e^{bz}$ respectively: each decays in its direction of energy flow.

For complex frequency the displayed quadratic form is the spectral flux diagnostic specified in the question; a genuinely periodic time average is recovered on the real-frequency axis. The decrease here comes from the upper-half-plane convergence parameter, not from material dissipation. At real frequency in a lossless layer the corresponding flux is constant.

Let the shear-horizontal wave displacement field be $U(z)e^{ikx-i\omega t}$. Its velocity is $V=-i\omega U$, and its traction is $T=\mu U^{\prime }$. The scalar shear-wave equation givesWith $p=k/\omega$ and $p_{\beta }^2={\beta }^{-2}-p^2$, the first-order velocity–traction system isThese equations remain valid for depth-dependent $\mu ,\rho$; differentiating $\mu U^{\prime }$ is essential and one must not incorrectly use $\mu U^{\prime \prime }$ when $\mu$ varies.

Where $V\ne 0$, define the SH input impedance $Z=-T/V$. Differentiating $T=-ZV$ and substituting the two equations gives the SH impedance Riccati equation:The second relation is a one-way scalar equation once the complete SH input impedance is known. It does not discard reflections: $Z$ describes the actual superposition of both wave directions. At a zero of $V$, $Z$ may have a pole while the velocity–traction system stays regular.

The SH input impedance is the negative ratio of vertical traction to particle velocity for a shear-horizontal wave field, with positive depth and tension-positive traction. A superposition of upward and downward waves generally has a depth-dependent complex ratio. A downward progressive wave in a uniform medium has $Z=\mu p_{\beta }$, while the upward wave has $Z=-\mu p_{\beta }$. The input ratio includes reflection from deeper structure and can have poles at velocity nodes.