Lighthill's acoustic analogy

Builds on Acoustic wave equation Navier–Stokes equations Compressible flow Speed of sound

Statement

Let a compressible fluid have density $\rho$ , velocity $u = (u_1, u_2, u_3)$ , pressure $p$ and viscous stress tensor $\tau_{ij}$ . Assume that it satisfies the mass and momentum equations of the compressible Navier–Stokes equations with no external force: $\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u_j)}{\partial x_j} = 0, \qquad \frac{\partial (\rho u_i)}{\partial t} + \frac{\partial (\rho u_i u_j + p \delta_{ij} - \tau_{ij})}{\partial x_j} = 0,$ with summation over repeated indices. Let $\rho_0$ , $p_0$ be constants, let $c_0 > 0$ be a constant, and set $\rho' = \rho - \rho_0$ and $p' = p - p_0$ . Define the Lighthill stress tensor $T_{ij} = \rho u_i u_j + (p' - c_0^2 \rho') \delta_{ij} - \tau_{ij}.$ Then $\frac{\partial^2 \rho'}{\partial t^2} - c_0^2 \, \Delta \rho' = \frac{\partial^2 T_{ij}}{\partial x_i \, \partial x_j}.$

Remarks

The proof takes the time derivative of the mass equation and the divergence of the momentum equation, and subtracts the second from the first; this eliminates $\rho u_i$ . Then $c_0^2 \Delta \rho'$ is subtracted from both sides. No approximation is made, so the equation is exact; it is a rearrangement of the flow equations.

The analogy reads the left side as the acoustic wave equation of a fluid at rest with speed of sound $c_0$ . It reads the right side as a given quadrupole source. In a jet, $T_{ij}$ is small outside the turbulent region, so far from the jet $\rho'$ is the sound radiated by the turbulence. A scaling argument on this source gives Lighthill's eighth power law: for a low Mach number jet of speed $U$ and diameter $D$ , the radiated acoustic power scales like $\rho_0 U^8 D^2 / c_0^5$ .

Going deeper

For a student

The analogy turns a nonlinear flow problem into a linear wave problem with a known right side, which can be solved with the free-space Green's function of the wave equation. The source $\partial_i \partial_j T_{ij}$ is a double divergence, so it radiates like a distribution of quadrupoles, which are inefficient radiators at low Mach number. This explains the strong dependence $U^8$ of jet noise on the jet speed. The Ffowcs Williams–Hawkings equation extends the analogy to flows with moving solid surfaces, where monopole and dipole sources appear too.

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Ffowcs Williams–Hawkings equation

Tag	`J31W7DJ90H`
URI	`m2://J31W7DJ90H`
Kind	statement, proposition
Writing	W0 stub
MSC	76Q05
Status	This copy of the site has no build data. Run `m2 status J31W7DJ90H`.
About	Acoustic wave equation
Uses	Acoustic wave equation, Navier–Stokes equations, Compressible flow, Speed of sound
Changed	3 November 2026

Lighthill's acoustic analogy

Statement

Remarks

Going deeper

For a student

Reading path

Where this leads

Further reading

What this page needs for W1

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