Lighthill's eighth power law¶

An aeroacoustic scaling law stating that far-field sound power from compact low-Mach turbulent flow scales with the eighth power of characteristic velocity.

Version: v2 · 2026-09-19 · History
Domain-specific #: 5326
Domain group: Applied Sciences & Engineering
Origin domain: Aviation & Aeronautics
Subdomain: Aeroacoustics → Aviation & Aeronautics

Core Idea¶

Lighthill's law estimates jet-noise acoustic power as proportional to density, squared source length, and U⁸ divided by the fifth power of sound speed, under compact-source and low-Mach assumptions. The acoustic analogy treats turbulence as an effective quadrupole source; dimensional scaling and retarded propagation make radiated efficiency rise approximately as Mach number to the eighth power. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

Faster Jets, Way Louder

When a jet engine blows out air really fast, the swirly air makes a roar. Lighthill's eighth power law says the roar grows super fast as the air gets faster. If the air goes twice as fast, the sound power gets hundreds of times bigger, not just twice as big.

The Speed-to-the-Eighth Noise Rule

Lighthill's eighth power law is a rule for how much sound power the swirling air from a jet produces. The power goes up with the jet speed multiplied by itself eight times. So doubling the speed makes 2×2×2×2×2×2×2×2 = 256 times more sound power. The rule also depends on how thick the air is, how big the noisy region is, and the speed of sound. It only works when the jet is not too close to the speed of sound and the noisy region is small compared to the sound waves it makes.

Jet Noise U⁸ Scaling Law

Lighthill's eighth power law estimates the total acoustic power radiated by a turbulent jet. The power is proportional to air density times the square of a source length times jet velocity to the eighth power, divided by the speed of sound to the fifth power. It comes from Lighthill's acoustic analogy, which treats the turbulence as an equivalent sound source of a particular type, a quadrupole. The steep U^8 growth is why faster jets are disproportionately noisy. The law depends on assumptions: the source must be compact, the flow low-Mach, the surrounding medium uniform, and the measurement must be total power in the far field. Outside that regime the eighth-power scaling should not be trusted.

Lighthill's eighth power law gives the scaling of far-field acoustic power from turbulent jet noise: P ∝ ρ L² U⁸ / c⁵, where ρ is density, L a characteristic source length, U a characteristic velocity and c the sound speed. It comes from Lighthill's acoustic analogy, which rewrites the flow equations as a wave equation driven by an effective quadrupole source representing the turbulence. Dimensional analysis plus retarded-time propagation from a compact source shows that radiation efficiency grows roughly as Mach number to the eighth power. Validity requires that the source be compact and low-Mach, the medium uniform, the characteristic velocity and length well defined, and the quantity be total far-field acoustic power. The practical lesson is the extreme sensitivity of jet noise to exhaust velocity. Applying U⁸ outside its assumptions, such as high-Mach or non-compact sources, is a misuse of the law.

Scope of Application¶

Lighthill's eighth power law belongs to aeroacoustics and is useful where the analyst can specify the typed aeroacoustics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the turbulent source is compact and low-Mach in a uniform medium, characteristic velocity and length are defined, far-field total acoustic power is measured, and the U-to-the-eighth scaling regime is respected. The scope is broad within that domain but bounded by the need for the turbulent source is compact and low-Mach in a uniform medium, characteristic velocity and length are defined, far-field total acoustic power is measured, and the U-to-the-eighth scaling regime is respected. Conceptual acoustic scaling identity only; no aircraft, propulsion, industrial-noise, or hazardous-flow operating procedure is provided.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the turbulent source is compact and low-Mach in a uniform medium, characteristic velocity and length are defined, far-field total acoustic power is measured, and the U-to-the-eighth scaling regime is respected the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Lighthill's eighth power law. Lighthill's eighth power law compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed aeroacoustics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the turbulent source is compact and low-Mach in a uniform medium, characteristic velocity and length are defined, far-field total acoustic power is measured, and the U-to-the-eighth scaling regime is respected independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of aeroacoustics because they reuse the typed aeroacoustics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, The acoustic analogy treats turbulence as an effective quadrupole source; dimensional scaling and retarded propagation make radiated efficiency rise approximately as Mach number to the eighth power., and type the carrier, state every parameter and convention in the definition, test that the turbulent source is compact and low-Mach in a uniform medium, characteristic velocity and length are defined, far-field total acoustic power is measured, and the U-to-the-eighth scaling regime is respected, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Lighthill's eighth power law Domain-specific

Parents (1) — outgoing typed relations

Lighthill's eighth power law is a kind of Allometry and Scaling Law Prime

What makes it its own entry: the domain-specific identity determined by the turbulent source is compact and low-Mach in a uniform medium, characteristic velocity and length are defined, far-field total acoustic power is measured, and the U-to-the-eighth scaling regime is respected.

Hierarchy path (1) — routes to 1 parentless root

Lighthill's eighth power law → is a kind of Allometry and Scaling Law → is a kind of Scaling and Scale Dependence → is a decomposition of Scale

Neighborhood in Abstraction Space¶

Lighthill's eighth power law sits in a moderately populated region (52^nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Fluid Flow & Aeroacoustic Phenomena (8 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25