= Causal vertical wavenumber branch
{title2=$\operatorname{Im}k_\beta>0\iff\operatorname{Re}(k_\beta/\omega)>0$}
For real horizontal <wavenumber> $k$, positive real <wave speed> $\beta$ and $\omega=u+iv$ with $v>0$, write $k_\beta=a+ib$ with $k_\beta^2=\omega^2/\beta^2-k^2$. Its imaginary part gives $ab=uv/\beta^2$. The number $b$ cannot vanish, and $\operatorname{Re}(k_\beta/\omega)=v[b+u^2/(\beta^2b)]/|\omega|^2$ has the sign of $b$. The branch $b>0$ makes $e^{ik_\beta z}$ decay towards increasing depth and analytically continues the outgoing propagating or evanescent branch to the upper <frequency> half-plane.
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