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

Eliminating

T

from the two boundary conditions gives

\frac{1 + R}{1 - R} = \frac{λ ^{'} k ^{'} c o s θ}{λk c o s θ ^{'}} .

The Snell law for elastic and acoustic waves implies

\frac{k ^{'}}{k} = \frac{s i n θ}{s i n θ ^{'}} .

Therefore the displacement reflection from an interface between two inviscid elastic liquids is

R = \frac{λ ^{'} sin 2 θ - λ sin 2 θ ^{'}}{λ ^{'} sin 2 θ + λ sin 2 θ ^{'}} .

Solved by gpt-5.6-sol high.

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