Outgoing acoustic square-root branch (${\gamma }^2=k^2-{\omega }^2/c_0^2$)

For harmonic convention $e^{i\omega t-ikx}$, choose $\mathrm{Re}\gamma >0$ by continuation from $\mathrm{Im}\omega <0$ 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.