 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/40c/b/iv/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / iv / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

If

c_{0} ≪ ωa

, then

x ≫ 1

. The sphere is acoustically large, and

⟨ K ⟩ \sim ⟨ W ⟩, ∣ ⟨ I_{r} ⟩ ∣ \sim c_{0} ⟨ K + W ⟩ .

Pressure and velocity are nearly in phase, so the near-surface field behaves locally like a progressive radiating wave. The power approaches

2 π a^{2} ρ_{0} ω^{2} ϵ^{2} c_{0} .

If

c_{0} ≫ ωa

, then

x ≪ 1

. The sphere is acoustically compact, and

\frac{⟨ W ⟩}{⟨ K ⟩} \sim x^{2} ≪ 1, \frac{c _{0} ⟨ K + W ⟩}{∣ ⟨ I _{r} ⟩ ∣} \sim \frac{1}{2 x ^{2}} ≫ 1.

The surface field is mainly reactive kinetic energy: pressure and velocity are nearly in quadrature, energy is stored and returned during each cycle, and only a small mean acoustic energy flux is radiated. The mean power scales as

⟨ P ⟩ \sim \frac{2 π ρ _{0} ω ^{4} ϵ ^{2} a ^{4}}{c _{0}} .

 Read the full article

 New to topics? Read the docs here!