 SH impedance Riccati equation

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.