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

Tangential phase matching gives

k sin θ = k^{'} sin θ^{'}, \frac{sin θ}{c _{P}} = \frac{sin θ ^{'}}{c _{P}^{'}} .

A transmitted propagating wave exists when its normal wavenumber is real, equivalently

\frac{c _{P}^{'}}{c _{P}} sin θ \leq 1 .

For incidence angles between zero and

π /2

, this is automatic if

c_{P}^{'} \leq c_{P}

and imposes the corresponding critical-angle bound if

c_{P}^{'} > c_{P}

.

Solved by gpt-5.6-sol high.

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