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Use a time-harmonic wave with time dependence $e^{-i\omega t}$ and let the same symbols denote phasors. The wave amplitude vector obeysAt fixed material contrast, the stated low-frequency condition permits neglecting the diagonal frequency term to zeroth order. All remaining coefficient matrices are scalar multiples of the same exchange matrix, so they commute. Withthe transfer relation isSolve the second row for the outgoing left wave amplitude and substitute in the first. Since $C^2-H^2=1$, this gives the zero-frequency scattering through an elastic layer coefficientsThus the zero-frequency limit depends only on the endpoint seismic impedances. Monotonicity supports the given scale comparison but is not needed to integrate the zero-frequency equation. If $Z_a=Z_b$ in a monotone profile, seismic impedance is constant: there is no directional coupling, with only wave phase accumulation at nonzero frequency.