 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/40c/c/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

x < 0

, write

p = e^{- ik x} + R e^{ik x} .

In the cavity, the rigid condition

u (L) = 0

gives

p = C cos k (x - L)

. Continuity of fluid velocity at the membrane and the membrane equation

- ω^{2} m ξ = p (0^{-}) - p (0^{+}), u (0) = iω ξ,

give, after eliminating

C

and

ξ

R = \frac{cos β + ( α - i ) sin β}{cos β + ( α + i ) sin β}, α = \frac{ωm}{ρ _{0} c _{0}}, β = \frac{ω L}{c _{0}} .

∣ R ∣ = 1

. No mean acoustic energy can pass through the rigid end, so all incident energy is ultimately reflected.

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