= Zero-frequency scattering through an elastic layer
{title2=$S_0=\frac1{Z_a+Z_b}\begin{pmatrix}Z_a-Z_b&2\sqrt{Z_aZ_b}\\2\sqrt{Z_aZ_b}&Z_b-Z_a\end{pmatrix}$}
At zero <frequency> the graded-layer transfer equation integrates to $\exp[L\begin{pmatrix}0&1\\1&0\end{pmatrix}]$, where $L=\tfrac12\log(Z_b/Z_a)$. 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>.
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