Solution

c_{0} ≪ ωa

, then

x ≫ 1

. The sphere is acoustically large, and

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

(498)

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} .

(499)

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.

(500)

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}} .

(501)

Solved by gpt-5.6-sol high.

 Ancestors (12)

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