= Outgoing acoustic square-root branch
{title2=$\gamma^2=k^2-\omega^2/c_0^2$}
For harmonic convention $e^{i\omega t-ikx}$, choose $\operatorname{Re}\gamma>0$ by continuation from $\operatorname{Im}\omega<0$ and impose vertical factors $e^{-\gamma y}$ above the interface and $e^{\gamma y}$ below it. At positive real frequency, $\gamma=i\sqrt{k_0^2-k^2}$ for propagating Fourier components and is positive real for evanescent components. This analytic prescription fixes the radiation branch and how the Fourier contour passes poles.
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