Acoustic analogy

An acoustic analogy rearranges fluid conservation equations into a chosen linear acoustic propagation operator plus effective forcing. The rearrangement can be exact before source approximations are made. A term classified as forcing can also contain propagation effects omitted from the reference operator; this produces propagation-source ambiguity in an acoustic analogy. The Lighthill acoustic analogy is a familiar homogeneous reference-medium example.

Table of contents
- Propagation-source ambiguity in an acoustic analogy Acoustic analogy
- Pressure-based acoustic analogy with a potential body force Acoustic analogy

Propagation-source ambiguity in an acoustic analogy

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The distinction between generation and propagation depends partly on the reference operator chosen in an acoustic analogy. For example, a homogeneous-Laplacian operator with spatially varying reference sound speed omits the first-gradient terms of the stratified acoustic pressure equation. These terms then reappear as effective forcing, even for a freely propagating small disturbance in a static medium. A right-hand-side term is therefore not automatically an independent physical noise source.

Pressure-based acoustic analogy with a potential body force ( $Q, W_{ij}$ )

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Eliminating momentum between the continuity equation and conservation of momentum gives

ρ_{tt} = \partial_{i} \partial_{j} (ρ u_{i} u_{j} - σ_{ij}) + \nabla^{2} (p + χ)

when the force density is

- \nabla χ

. For

q = p - (P - χ)

and time-independent reference fields,

(c_{0}^{- 2} \partial_{t}^{2} - \nabla^{2}) q = \partial_{i} \partial_{j} W_{ij} + Q_{tt}, W_{ij} = ρ u_{i} u_{j} - σ_{ij}, Q = q / c_{0}^{2} - (ρ - ρ_{0}) .

(1)

No energy equation is needed for this identity. The arbitrary reference fields should match the stationary medium where the outgoing sound is evaluated.