Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 40C / c / i / Solution

Tangential phase matching givesA transmitted propagating wave exists when its normal wavenumber is real, equivalentlyFor incidence angles between zero and $\pi /2$, this is automatic if $c_P^{\prime }\le c_P$ and imposes the corresponding critical-angle bound if $c_P^{\prime }>c_P$.