The sound is what does not cancel

Lighthill’s eighth-power law says that the acoustic power a jet radiates goes as the eighth power of its speed, which is the steepest scaling in engineering and the reason a jet aeroplane got quieter by being made to go slower and fatter rather than by anything done to its noise. The law follows from identifying the sources in a turbulent flow as quadrupoles — and quadrupoles are then usually described as inefficient, weak, or poor radiators.

That description is wrong in a way that matters. The sources are not weak. They cancel.

The sum, done without expanding anything #

A point source of sound radiates $q\,e^{ikr}/r$. Four of them on a square of side $d$ with alternating signs make a lateral quadrupole, and the field at any point is the sum of four such terms — a calculation with no approximation in it at all, and in particular no expansion in $kd$.

Doing that sum on spheres of growing radius gives two quantities. The first is what one source alone would produce at that radius, which is $1/r$ throughout and is what every one of the four is producing. The second is the sum.

Inside the cluster there is no cancellation to speak of. A probe at half the source spacing sees 80 per cent of what a single source would give it, because it is sitting next to one of them and the others are further away. The cancellation is a far-field phenomenon: it requires the four sources to be seen from a direction and a distance where their path differences are small compared with a wavelength.

By one radian of wavelength the surviving fraction is one part in ten thousand, and it stays there — the ratio is constant beyond that, because both quantities are now falling as $1/R$.

Three regions, and the exponents that name them #

The measured decay of the summed field is $R^{-3.00}$ between the source and the wavelength, and $R^{-1.002}$ beyond it. Those two exponents are the whole anatomy of a compact source.

Near field, $R^{-3}$. A quadrupole’s static field falls as the cube of the distance, exactly as an electrostatic quadrupole’s does. It carries energy back and forth every cycle and radiates none of it; the technical word is reactive, and it is what an acoustician means by saying that a compact source’s near field is not sound.

Far field, $R^{-1}$. Beyond a wavelength the residual that failed to cancel behaves like any spherical wave. Its amplitude is small, its decay is the slow one, and it is the only part that reaches a listener.

And the crossover is at $kR \approx 1$, one radian of wavelength — about five centimetres for a kilohertz in air. Everything inside that radius is a pressure fluctuation that goes nowhere.

One power of compactness per cancellation #

The classical ladder is usually written in powers of radiated power: a monopole radiates as $U^4$, a dipole as $U^6$, a quadrupole as $U^8$. Written in amplitude, the same statement is that each order of cancellation costs one factor of $kd$, and the exponent can be measured.

The interpretation is the useful part. A dipole is a monopole that has been cancelled once: two sources of opposite sign, whose fields differ only by the path difference $d\cos\theta$, so what escapes is proportional to $kd$. A quadrupole is a dipole that has been cancelled again, and the factor appears twice.

That is why the ladder is a ladder of constraints rather than of strengths. A monopole requires a net volume change — something must be inflating. A dipole requires a net force on the fluid — something must be pushing. A flow with neither has to be a quadrupole, not because its sources are feeble but because the conservation laws have cancelled the first two orders for it.

Why turbulence has no choice about it #

Lighthill’s rearrangement of the equations of motion turns them into an exact wave equation with a source term that is the divergence of a stress. The mathematical form of that source is a quadrupole, and it is not a modelling choice: it is what is left after the mass and momentum equations have been used.

A turbulent flow adds no mass — mass has nowhere to go — so the monopole term is zero — the fluid is not being created anywhere. A turbulent flow with no solid surfaces applies no net force, so the dipole term is zero too. What is left is the quadrupole, and everything about jet noise follows from the fact that the loudest available source has been cancelled twice by conservation laws.

Put a surface in the flow and the second cancellation is undone: the surface can apply a force, the dipole term returns, and the radiated power goes as $U^6$ instead of $U^8$. At a Mach number of 0.3 that is a factor of ten more sound, which is why an aerofoil in a jet’s shear layer, or a blade passing through a wake, is so much noisier than the same flow left alone — and why installation effects dominate the noise of a modern engine.

What sets the frequency, and where the eighth power comes from #

The cancellation supplies two powers of $kd$ and the rest of the eighth power comes from the eddies themselves, which is worth doing explicitly because the assembly is where the law is usually asserted.

An eddy of size $\ell$ in a jet of speed $U$ turns over in a time $\ell/U$, so it radiates at a frequency $\omega \sim U/\ell$ and therefore at $k \sim U/(c\ell)$ — which makes its compactness $kd \sim k\ell \sim U/c$, the Mach number. The strength of the quadrupole is $\rho U^2\ell^3$ from the Reynolds stress it carries, and putting the pieces together gives a radiated power

$$P \sim \frac{\rho U^3 \ell^2}{1} \times \left(\frac{U}{c}\right)^5,$$

the first factor being the mechanical power passing through the eddy and the second the acoustic efficiency. The fifth power of Mach number is the efficiency and the eighth power of speed is the result, and two of those five powers are the double cancellation measured above.

That decomposition explains why the law is so steep and why it is so robust. The exponents come from counting rather than from any property of turbulence: a source with no volume change and no net force, radiating at a frequency set by its own turnover, has no freedom left. It also explains what would break it — a flow with a surface in it, a flow with combustion adding volume, or a source convecting near the speed of sound, each of which removes one of the constraints.

One rung of the ladder, in a domestic object #

The cleanest demonstration that a cancellation rather than a weakness is at work does not need a jet. It is a loudspeaker driver held in the air with nothing round it.

A cone pushing forward is pulling backward at the same instant, so the two faces radiate with opposite sign from points a cone-diameter apart. That is a dipole, exactly the arrangement one rung below the quadrupole above, and the same arithmetic applies: what escapes is proportional to $kd$, so the output falls by six decibels for every halving of frequency below the point where the path round the cone is a half wavelength. A bare eight-inch driver has almost no bass at all, and it is not because the cone is not moving — it is moving exactly as much as it would in a box.

Put it in a sealed box and the rear radiation is removed altogether. One of the two sources is gone, the cancellation with it, and the driver is a monopole: flat down to its own resonance, radiating what the front face alone produces. The box has not made the cone louder. It has removed the source that was cancelling it, and bought a rung of the multipole ladder for the price of some plywood.

An open-baffle design takes the middle path deliberately, keeping the dipole and simply making $d$ large so that the six-decibel roll-off begins lower. Its characteristic figure-of-eight directivity — full output front and back, nulls at the sides — is the dipole pattern of this essay’s figures, and the nulls are directions in which the two sources are equidistant and the cancellation is exact.

The whole of loudspeaker enclosure design is a decision about which multipole to be, and every one of its trade-offs is a trade-off between how much cancellation is tolerated and what it costs to remove.

And the reason a measurement has to be made a long way off #

The three-region anatomy has a practical edge for anybody who measures sound rather than computing it. A microphone inside $kR \approx 1$ is in the reactive near field, sampling a pressure that oscillates energetically and radiates nothing; the reading is real and has almost no relation to what a distant listener hears.

Escaping it means standing many wavelengths away, and a wavelength at low frequency is large. At 50 hertz it is nearly seven metres, so a genuine far-field measurement of a large source at that frequency means tens of metres of clear space — which is why anechoic chambers are quoted with a lowest usable frequency, why it is set by the room’s size rather than by its lining, and why low-frequency source characterisation is so often done in the near field with an analytical correction rather than directly.

It is the same difficulty aeroacoustics has, at a scale a person can walk across. The quantity wanted is a residue that only exists far away, and getting far enough away is a question about the apparatus rather than about the physics — which is why almost every measurement in this subject is of something else, corrected.

The same shape of argument, elsewhere on this site #

The structure — a large quantity almost entirely cancelling, with the residue being the whole of the effect — appears repeatedly, and it is worth collecting.

A wing’s lift is a small residue of large pressures. The suction on the upper surface and the pressure beneath are each several times the lift per unit area, and the resultant is their difference — which is why a pressure distribution measured to one per cent gives a lift good to rather less than that.

A momentum theorem’s two terms cancel to leave a total that does not move. On every contour round a wing the pressure and momentum-flux contributions run from three per cent to ninety-seven while their sum stays at $\rho U\Gamma$ — the same structure, with the residue being the constant rather than the variable.

A far field keeps three numbers and loses the shape, which is the same operation seen from the other end: everything beyond the third multipole cancels as the distance grows, and the terms that vanish are the ones carrying the detail.

And the dissipation anomaly is a cancellation that does not happen. Two factors moving by a million in opposite directions leave a product that does not move, which is the reverse case: there the survival of the residue is the surprise, and here its smallness is.

What it costs to compute, and why the analogy survives #

There is a practical consequence of the numbers in the first table that decides how aeroacoustics is done, and it is worth stating plainly.

A calculation that solves the compressible equations everywhere and reads the sound off the far field has to carry a near field four orders of magnitude larger than the answer, propagate it across many wavelengths without dispersive error, and extract a residue. Every numerical scheme has a truncation error, and a scheme whose error is one part in a thousand of the near field has an error ten times larger than the sound.

That is why the acoustic analogy is still the standard method seventy years after Lighthill proposed it: the sources are computed in the region where they are large, the propagation is done analytically where it is exact, and the cancellation is performed by an integral rather than by a grid. It is also why the surface formulations matter so much — a control surface drawn round the source region converts a volume integral of quadrupoles into a surface integral, and the arithmetic becomes tractable.

The same difficulty in a different currency appears wherever a small difference of large numbers is wanted. This collection has met it in the momentum deficit behind a bluff body, where the wake survey subtracts two large fluxes and the answer is their difference; and it will meet it again in any calculation whose output is a residual.

What the picture cannot show #

The sources are points and the flow is absent. Nothing here computes a turbulent source distribution: four point sources of alternating sign stand in for a quadrupole, and their strengths are stated. What the calculation establishes is the cancellation, which is a property of the arrangement rather than of any flow.

The medium is at rest and the sources do not move. A real jet convects its sources at a fraction of the speed of sound, which sharpens the radiation forwards and multiplies the power by a Doppler factor raised to a substantial power — an effect this collection computes separately by solving the retarded-time equation rather than expanding it.

The compactness is assumed rather than checked against a flow. The whole ladder holds only while the source is small compared with the wavelength it makes, which for a jet eddy means a low Mach number; at $U/c$ approaching one the eddy is no longer compact, the cancellation is incomplete for a different reason, and the eighth power flattens towards a third power. That transition is measured and is not computed here — and it is why supersonic jet noise is a separate subject with shock-associated sources of its own.

And the coefficient in the eighth-power law is borrowed. The exponents are computed here; the constant $K$ in $P = K\rho U^8D^2/c^5$ is a measurement, it is carried as a borrowed quantity wherever it appears on this site, and no figure quotes a sound power level as though this collection had computed one.

Who found it, and when #

Lighthill’s two papers of 1952 and 1954 contain the acoustic analogy, the quadrupole identification and the eighth-power law; Curle added the surface dipole term in 1955, and Ffowcs Williams and Hawkings extended it to moving surfaces in 1969, which is the form every rotor-noise calculation uses today. The identification of the cancellation as the essential point — rather than the weakness of the sources — is in Lighthill’s own writing and has been steadily lost in retellings since.

The surprising connection is with the phase’s own subject. Every essay here is about an average that throws something away and about what survives it. Radiation is the most brutal average in the collection: the far field is an average over the source region, weighted by phase, and it discards all but one part in ten thousand of what the sources are doing. The part it keeps is not representative of the flow in any sense — it is the flow’s failure to cancel, which is a quantity nothing in the near field particularly notices.

That is also why aeroacoustics is hard to compute. A calculation must resolve a near field that is four orders of magnitude larger than the answer, and then extract the answer as a difference. It is the numerical analyst’s least favourite situation, and it is why the acoustic analogy — computing the sources and then propagating them analytically — remains the standard method rather than solving the compressible equations everywhere.

Where the ladder goes next #

The rung below establishes the ladder and the radiation condition — that a solution which only leaves is a choice imposed at infinity rather than a property of the equations. Above this one lies the surface term: what happens when a body is put in the flow, the dipole returns, and the noise rises by an order of magnitude at subsonic speeds. That is where the engineering of quiet aircraft actually lives, and this collection has not yet built the machinery for it.