Internal-wave response of a sharp buoyancy interface ($T/W_i=2im/[k(1-S)+im]$)

For a lower uniformly stratified half-space and an upper well-mixed half-space, the vertical velocity amplitude is continuous and its derivative jumps by $[W^{\prime }]=-k^2\Delta bW/{\omega }^2$. A wave incident upwards with vertical phase factor $e^{-imz}$ therefore has upper evanescent amplitude $T/W_i=2im/[k(1-S)+im]$, where $m=k\sqrt{N^2/{\omega }^2-1}$ and $S=\Delta bk/{\omega }^2$. Its magnitude peaks at $S=1$, with amplitude twice the incident amplitude. Lossless reflection carries all incident energy back down; the evanescent upper field carries no mean outgoing vertical energy flux.