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 Incoming links: Mean acoustic power of a pulsating sphere

For a plane acoustic wave propagating along

k

, the linear momentum equation and dispersion

ω = c_{0} k

give

p = ρ_{0} c_{0} u_{∥}

and

u = u_{∥} k

. Therefore the instantaneous acoustic intensity is

p u = ρ_{0} c_{0} ∣ u ∣^{2} k .

⟨ I ⟩ = \frac{1}{2} ρ_{0} c_{0} ∣ u ∣^{2} k

v = r ϕ

. The spherical wave equation reduces to

v_{tt} = c_{0}^{2} v_{rr}

, so

ϕ (r, t) = \frac{F ( t - r / c _{0} ) + G ( t + r / c _{0} )}{r} .

G = 0

. Linearizing the moving-surface condition at

r = a

gives

ϕ_{r} (a, t) = iωa ϵ e^{iω t}

, with real parts understood. Write the outgoing solution as

ϕ = A e^{iω (t - (r - a) / c_{0})} / r

. Its radial derivative at

a

- A (1 + iωa / c_{0}) e^{iω t} / a^{2}

ϕ (r, t) = Re {- \frac{iω a ^{3} ϵ}{1 + iωa / c _{0}} \frac{e ^{iω (t - (r - a) / c_{0})}}{r}} .

In the far field the velocity amplitude has magnitude

(ω / c_{0}) ∣ A ∣/ r

⟨ P ⟩ = \frac{2 π ρ _{0} ω ^{2}}{c _{0}} ∣ A ∣^{2} = \frac{2 π ρ _{0} ϵ ^{2} a ^{6} ω ^{4}}{c _{0} [ 1 + ( ωa / c _{0} ) ^{2} ]} .

This is positive outward radiated power, evaluated to leading quadratic order in the small real displacement amplitude.

f (t) = Re (f e^{iω t}), g (t) = Re (g e^{iω t}),

define the period average by

⟨ f g ⟩ = \frac{ω}{2 π} \int_{0}^{2 π / ω} f (t) g (t) d t = \frac{1}{2} Re (f g^{*}) .

r = a

, put

x = ωa / c_{0}

u_{r} (a) = iω ϵ, p (a) = - i ρ_{0} ω ϕ (a) = - \frac{ρ _{0} ω ^{2} ϵ a}{1 + i x} .

Hence the mean outward work rate per unit area is

⟨ p^{'} u_{r} ⟩ = \frac{1}{2} Re (p u_{r}^{*}) = \frac{ρ _{0} ω ^{4} ϵ ^{2} a ^{2}}{2 c _{0} ( 1 + x ^{2} )} .

4 π a^{2}

yields

⟨ P ⟩ = \frac{2 π ρ _{0} ω ^{4} ϵ ^{2} a ^{4} / c _{0}}{1 + ω ^{2} a ^{2} / c _{0}^{2}} = 2 π a^{2} ρ_{0} ω^{2} ϵ^{2} c_{0} \frac{ω ^{2} a ^{2}}{c _{0}^{2} + ω ^{2} a ^{2}} .