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 Incoming links: Time average of harmonic power

For

r = R r

with the observation distance much larger than the scatterer, expand

∣ r - r^{'} ∣ = R - r \cdot r^{'} + O (R^{- 1})

ψ_{s}^{(1)} (R r) = \frac{e ^{ik R}}{R} f_{\infty}^{(1)} (r, r_{0}) + O (R^{- 2}),

f_{\infty}^{(1)} = \frac{k ^{2}}{4 π} \int_{V} [n (r^{'})^{2} - 1] e^{- i k_{s} \cdot r^{'}} d^{3} r^{'}, k_{s} = k (r - r_{0}) .

This convention includes the factor

1/ (4 π)

f_{\infty}

; moving that factor into the outgoing-wave definition would change the pattern's normalization.

The absolute energy flux depends on what physical amplitude

ψ

represents. For acoustic pressure in a homogeneous fluid of mass density

ρ_{0}

c_{0}

v = \nabla ψ / (i ρ_{0} ω)

⟨ I ⟩ = \frac{1}{2} Re (ψ v^{*}) = \frac{1}{2 ρ _{0} ω} Im (ψ^{*} \nabla ψ) .

The radial derivative of the outgoing factor is

(ik - 1/ R) e^{ik R} / R

. Its imaginary current is therefore

k ∣ f_{\infty} ∣^{2} / R^{2}

to leading order. The incident unit-amplitude plane wave has imaginary current

k r_{0}

. Hence, with

k = ω / c_{0}

E_{s} (R) = \frac{k}{2 ρ _{0} ω R ^{2}} ∣ f_{\infty}^{(1)} ∣^{2} + O (R^{- 3}), E_{i} = \frac{k}{2 ρ _{0} ω} = \frac{1}{2 ρ _{0} c _{0}} .

The scattered energy flux points along

r

at leading order, and the incident energy flux along

r_{0}

. If

ψ

is a velocity-potential amplitude instead, both leading expressions use the common prefactor

ρ_{0} ωk /2

in place of

k / (2 ρ_{0} ω)

. A normalized scalar current uses

k

as the common prefactor. The paper does not specify this amplitude normalization; the ratio in part (c) is independent of that choice.

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}} .