Acoustic energy conservation

Linear acoustic fields obey

\frac{\partial}{\partial t} (K + W) + \nabla \cdot I = 0,

where

K = \frac{1}{2} ρ_{0} ∣ u ∣^{2}, W = \frac{p ^{'2}}{2 ρ _{0} c _{0}^{2}}, I = p^{'} u .

Table of contents
- Near-field energy imbalance in a spherical acoustic wave Acoustic energy conservation
- Acoustic energy density Acoustic energy conservation

Near-field energy imbalance in a spherical acoustic wave ( $e_{K} - e_{P} = \frac{ρ _{0}}{2} (\frac{F ^{2}}{r ^{4}} - \frac{2 F F ^{'}}{r ^{3}})$ )

For outgoing velocity potential

ϕ = F (r - c t) / r

, the radial velocity is

F^{'} / r - F / r^{2}

and the pressure perturbation is

ρ_{0} c F^{'} / r

. The acoustic energy density separates into kinetic density

ρ_{0} (F^{'} / r - F / r^{2})^{2} /2

and compressional density

ρ_{0} (F^{'} / r)^{2} /2

. They are generally unequal near the source but agree at leading far-field order. The additional

r^{- 2}

velocity term is the spherical near field.

Acoustic energy density ( $E$ )

For a linear acoustic perturbation of a uniform fluid,

E = \frac{1}{2} ρ_{0} ∣ u ∣^{2} + \frac{p ^{'2}}{2 ρ _{0} c _{0}^{2}}

is the sum of kinetic and compressional energy densities. Together with the acoustic energy flux

I = p^{'} u

, it satisfies

E_{t} + \nabla \cdot I = 0

.

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