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The lower-layer dispersion relation givesThe incident phase is $e^{-i{m}_{0}z}$; its vertical group velocity is positive, even though its vertical phase velocity is negative. Write the lower field as $W=w_0{e}^{-i{m}_{0}z}+r{e}^{i{m}_{0}z}$ and the upper bounded field as $W=T{e}^{-kz}$. Interface continuity gives $T=w_0+r$ andThus the internal-wave response of a sharp buoyancy interface isAt fixed incident $k,\omega ,N_0$ this is largest at $S=1$, where $|T|=2|w_0|$. Consequently ${\omega }^2=\Delta bk=2{\omega }_{\mathrm{free}}^2$, or $\omega =\sqrt{2}{\omega }_{\mathrm{free}}$. The maximum does not coincide with the free frequency for two homogeneous layers: the stratified lower radiating layer has a different dynamic pressure response. The upper field is evanescent, so in this ideal lossless problem it carries no mean outgoing vertical energy flux; the reflected wave has $|r|=|w_0|$.