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

No reflection requires

λ^{'} sin 2 θ = λ sin 2 θ^{'} .

Using

λ = ρ c_{P}^{2}

,

λ^{'} = ρ^{'} (c_{P}^{'})^{2}

, and

sin θ^{'} = (c_{P}^{'} / c_{P}) sin θ

, this reduces to the normal acoustic impedance matching condition

ρ^{'} c_{P}^{'} cos θ = ρ c_{P} cos θ^{'} .

After squaring and eliminating

θ^{'}

, the required incidence angle satisfies

sin^{2} θ = \frac{ρ ^{'2} ( c _{P}^{'} ) ^{2} - ρ ^{2} c _{P}^{2}}{( c _{P}^{'} ) ^{2} ( ρ ^{'2} - ρ ^{2} )} .

Such a no-reflection angle exists only when the right-hand side lies in

[0, 1]

and the transmitted wave is propagating.

Solved by gpt-5.6-sol high.

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