Radiation condition

A radiation condition selects the wave solution whose energy propagates away from its source. For an internal gravity wave, the vertical phase and group velocities have opposite signs, so upward radiation fixes the sign of the vertical wavenumber.

For a slowly switched-on plane internal gravity wave in the half-space $z>0$, the method of multiple scales gives transport of its amplitude at the vertical group velocity. For ${\omega }^2=N^2{k}^2/(k^2+m^2)$, $c_{gz}=-\omega m/(k^2+m^2)$. Initially undisturbed fluid and forcing at the bottom require $c_{gz}>0$. The outgoing radiation condition therefore selects $m<0$ when $\omega >0$, although the vertical phase velocity is downward.

Select outgoing waves as a limit of a problem with weak absorption. With time convention $e^{-i\omega t}$ use $\mathrm{Im}\omega >0$; with $e^{i\omega t}$ use $\mathrm{Im}\omega <0$. The resulting Green function and transform-contour prescriptions determine which poles represent incoming or outgoing waves. This is especially useful when real-axis singularities obscure a Wiener-Hopf factorization.