Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 39A / a / Solution

Substitution of the monochromatic ansatz givesThus the outer solutions oscillate with $m=k\sqrt{N_0^2/{\omega }^2-1}$ and the middle solutions are combinations of $\cosh kz$ and $\sinh kz$. The incident wave has phase $kx-mz-\omega t$; 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.