 Pressure-based acoustic analogy with a potential body force

ID: pressure-based-acoustic-analogy-with-a-potential-body-force

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Pressure-based acoustic analogy with a potential body force by

Codex  0 2026-10-06

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.

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