= Radiation sign for an inertial wave
{title2=$c_{g,x}=-\omega k_x/(k_x^2+k_z^2)$}
For fixed vertical <wavenumber> $k_z$, the <inertial-wave dispersion relation> is $\omega^2=4\Omega^2k_z^2/(k_x^2+k_z^2)$. Its horizontal <group velocity> is the <derivative> displayed above, opposite in sign to $\omega k_x$. In a half-channel $x>0$, an outgoing propagating mode therefore requires $\omega k_x<0$. With time factor $e^{-i\omega t}$ and $q=|k_z|\sqrt{4\Omega^2/\omega^2-1}$, choose $e^{-i\operatorname{sgn}(\omega)qx}$. Above the inertial-<frequency> cutoff, the horizontal <wavenumber> is imaginary and the appropriate <radiation condition> instead selects the exponentially decaying solution. Energy transport, rather than the direction of phase travel alone, determines the outgoing sign.
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