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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 40D / b / i / Solution

Use complex fields with time factor

e^{iω t}

and put

k_{\pm} = ω / c_{\pm}

. The outgoing potential in

x > 0

is

Φ_{+} (x) = T e^{- i k_{+} x} .

Continuity of pressure and velocity at

x = 0

gives

Φ_{-} (0) = \frac{ρ _{+}}{ρ _{-}} T, Φ_{-}^{'} (0) = - i k_{+} T .

Hence

Φ_{-} (x) = T [\frac{ρ _{+}}{ρ _{-}} cos (k_{-} x) - i \frac{c _{-}}{c _{+}} sin (k_{-} x)] .

The piston condition

Φ_{-}^{'} (- L) = iω ϵ

, with

λ = k_{-} L

, gives

T = ϵ c_{-} \frac{i ( ρ _{+} / ρ _{-} ) sin λ - ( c _{-} / c _{+} ) cos λ}{( ρ _{+} / ρ _{-} ) ^{2} sin ^{2} λ + ( c _{-} / c _{+} ) ^{2} cos ^{2} λ} .

This is the transmitted velocity potential from a piston through an acoustic interface.

Solved by gpt-5.6-sol high.

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