Zero-frequency scattering through an elastic layer ($S_0=\frac{1}{Z_a+Z_b}\left(\begin{array}{cc}{Z}_a-Z_b& 2\sqrt{Z_a{Z}_b}\\ 2\sqrt{Z_a{Z}_b}& Z_b-Z_a\end{array}\right)$)

At zero frequency the graded-layer transfer equation integrates to $\exp [L\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)]$, where $L=\frac{1}{2}\log (Z_b/Z_a)$. Eliminating the outgoing wave amplitude at the left endpoint gives the displayed incoming-to-outgoing scattering matrix. It is real orthogonal, so it preserves total incoming and outgoing flux. It coincides with continuity of velocity and traction at a sharp seismic impedance interface; the profile between the endpoints enters at nonzero frequency.