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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.