Solution
= Solution
Substitution of the monochromatic ansatz gives
$$
\widehat w''+k^2(N^2/ω^2-1)\widehat w=0.
$$
Thus the outer solutions oscillate with $m=k\sqrt{N_0^2/ω^2-1}$ and the middle solutions are combinations of $\cosh kz$ and $\sinh kz$. The incident wave has phase $kx-mz-ωt$; its group velocity is upward, whereas the reflected sign has downward group velocity. The transmitted wave must again use the upward-energy sign. This gives exactly the three forms stated.