What is Lighthill's acoustic analogy?

Recasting the flow equations (1952) as a wave equation for still air plus a source term generated by turbulent flow. It splits the work: CFD supplies the sources and the wave equation carries the sound, forming the basis of jet-engine and fan noise prediction. The sources are quadrupoles (the Lighthill stress tensor).

Why does jet noise scale with speed to the eighth power?

Dimensional analysis of quadrupole source strength gives acoustic power W ∝ ρU⁸D²/c⁵. A 0.5 m jet radiates 0.02 W at 100 m/s, 4.9 W at 200 m/s and 124 W at 300 m/s. Doubling speed multiplies power 256-fold (24 dB); a 10% cut lowers it 3.7 dB. That's why jet engines use large fans (high bypass ratio) to lower jet velocity.

How much flow energy becomes sound?

Very little. Acoustic efficiency — acoustic power over jet kinetic power — scales roughly with Mach number to the fifth, about 4×10⁻⁵ (0.004%) even at 300 m/s (Mach 0.87). Yet acoustic power reaches 124 W (140.9 dB), still about 90 dB at 100 m.

The Acoustic Analogy of Lighthill

Category: Coupled Analysis – Acoustics | Revised 2026-10-01

Theory: rewriting flow equations as a sound equation

Overview

🙋

Why are jet aircraft so loud?

🎓

Turbulence as the fast jet mixes with surrounding air produces the sound. In 1952 Lighthill rearranged the flow equations so the left side is the wave equation for sound in still air and the right side a source term generated by turbulent flow. These sources are quadrupoles, and dimensional analysis shows acoustic power scales with velocity to the eighth power: twice the speed means 256 times the power, 24 dB louder. So lowering jet speed slightly cuts noise greatly. Modern jet engines produce thrust by moving lots of air slowly through large fans, lowering jet speed to be quieter.

Lighthill's equation

$$ \frac{\partial^2\rho'}{\partial t^2} - c_0^2\nabla^2\rho' = \frac{\partial^2 T_{ij}}{\partial x_i\,\partial x_j},\qquad T_{ij} = \rho u_i u_j + (p' - c_0^2\rho')\delta_{ij} - \tau_{ij} $$

ρ′ is density fluctuation, c₀ the speed of sound in still air and T_ij the Lighthill stress tensor (Reynolds stresses, entropy fluctuations and viscous stresses). The double divergence on the right represents quadrupole sources.

Eighth-power law and acoustic efficiency

$$ W = K\,\frac{\rho_0 U^8 D^2}{c_0^5},\quad K \approx 3\times10^{-5},\qquad L_W = 10\log_{10}\frac{W}{10^{-12}\,\mathrm{W}},\qquad \eta = \frac{W}{\tfrac12\rho_0 U^3 A} \propto M^5 $$

K is an empirical coefficient, D jet diameter, A jet area and M jet Mach number; an approximation for subsonic cold jets.

Coffee Break Trivia

At the dawn of jet airliners

1952, when Lighthill published the acoustic analogy, was also the year the Comet, the first jet airliner, entered service. Jet noise became a major problem around airports, and the eighth-power law showed that lowering jet velocity was the most effective remedy. This led to high-bypass turbofans moving more air through fans, greatly reducing noise for the same thrust. The acoustic analogy still underpins CFD-based noise prediction.

Worked examples

Example 1: acoustic power of a 0.5 m jet

Jet speed	Mach	Acoustic power	Sound power level	Jet kinetic power	Efficiency	SPL at 100 m
100 m/s	0.29	0.02 W	102.8 dB	0.12 MW	1.6×10⁻⁷	51.8 dB
200 m/s	0.58	4.9 W	126.9 dB	0.94 MW	5.1×10⁻⁶	75.9 dB
300 m/s	0.87	124 W	140.9 dB	3.18 MW	3.9×10⁻⁵	90.0 dB

SPL at 100 m assumes uniform radiation in all directions (real jet noise peaks aft at an angle). A 10% speed cut gives −3.7 dB; doubling gives +24 dB.

Example 2: grid for computing sound (300 m/s)

Strouhal number	Frequency	Wavelength	Grid (10 cells per wavelength)
0.2	120 Hz	2.86 m	286 mm
1.0	600 Hz	0.57 m	57 mm

🙋

Only 0.004% of the flow energy becomes sound, yet it's this loud.

🎓

The jet carries a huge 3 MW, so even 0.004% is 124 W. The ear detects intensities down to 10⁻¹² W/m², so 124 W is very loud. The tiny fraction also makes prediction hard: acoustic fluctuations are far smaller than flow fluctuations, so computing both directly together buries sound in numerical error. A hybrid split is common: compute only the turbulent region finely with LES to obtain sources, then propagate to the far field with integrals such as Lighthill's or FW-H. Even the flow calculation needs grid and time step that resolve sources at the highest frequency of interest.

Noise prediction workflow

Get rough levels and speed sensitivity from simple laws such as the eighth-power law.
Set grid size and time step from the frequencies of interest.
Solve the turbulent region with unsteady methods such as LES to obtain sources.
Compute far-field sound with FW-H or Lighthill integrals.
Compare with measured noise by frequency and direction.

Coffee Break Trivia

“Cutting exhaust speed by 10%”

Noise from airflow leaving a factory exhaust duct slightly exceeded the limit at the site boundary. A silencer was proposed, but an eighth-power estimate first showed that cutting exit velocity by 10% would lower noise about 3.7 dB, meeting the limit. Enlarging the outlet about 10% to slow the flow reduced measured noise almost exactly as calculated — a case where estimating the speed effect paid off before costly measures.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Solving flow and sound directly on one grid	Sound lost in error	Hybrid approach
Sources from RANS	No fluctuations	Use LES or unsteady methods
Coarse grid	High frequencies missing	Size grid from frequency
Ignoring directivity	Wrong observer levels	Evaluate by direction
Eighth-power law at supersonic speed	Overestimation	Check validity range

🙋

I'd like to learn related topics.

🎓

Rate this article

Thank you for your feedback!

Helpful

More details

Report error