SH impedance Riccati equation ($Z^{\prime }=-i\omega (Z^2/\mu -\mu p_{\beta }^2)$)

With $e^{ikx-i\omega t}$ phasors, positive depth $z$, shear modulus $\mu =\rho {\beta }^2$, vertical traction $T$ and particle velocity $V$, the shear-wave equations give $V^{\prime }=-i\omega T/\mu$ and $T^{\prime }=-i\omega \mu p_{\beta }^{2}V$, where $p_{\beta }^2={\beta }^{-2}-k^2/{\omega }^2$. Define $Z=-T/V$ wherever $V\ne 0$. Differentiation gives the displayed Riccati equation and $V^{\prime }=i\omega ZV/\mu$. A pole of $Z$ can be an ordinary zero of $V$, not a singular physical wavefield.

In a uniform layer, put $q=\mu p_{\beta }$, $\kappa =\omega p_{\beta }$ and $d=z-z_r$. Initial impedance $Z_r$ gives the displayed fractional-linear solution and $V(z)=V_r[\cos \kappa d+i(Z_r/q)\sin \kappa d]$. Expanding into exponentials yields forward and backward shear-horizontal waves. Negative $d$ propagates a bottom load upward. If the denominator vanishes, the nonsingular pair $(V,T)$ must be propagated through the impedance pole.

Set the lower half-space load to $q_b={\mu }_b{p}_{\beta ,b}$ using the causal vertical wavenumber branch, which excludes a wave arriving from depth infinity. Apply the uniform-layer SH impedance update upwards through every layer, keeping particle velocity and traction continuous. For upper half-space impedance $q_a$, incident velocity $V_0$ and reflected velocity $R{V}_0$, the top values are $V=V_0(1+R)$ and $T=-q_a{V}_0(1-R)$. Their ratio gives the displayed reflection coefficient, incorporating every underlying interface and multiple reflection.

For positive real mass density and shear modulus, real $k$ and $\mathrm{Im}\omega >0$, the displayed derivative of $S_z=-\frac{1}{2}\mathrm{Re}(V^{\ast }T)$ follows from the two first-order shear equations. Every nontrivial solution has strictly decreasing $S_z$. Since $S_z=\frac{1}{2}\mathrm{Re}Z|V{|}^2$, 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.