The sound that only leaves

A jet engine at take-off is one of the loudest things ever built. It is also, as a loudspeaker, about as bad as a machine can be: a hundredth of a per cent of a per cent of its power leaves as sound. Both of those statements are consequences of the same argument, and the argument is about multipole order — about what a flow with no moving surfaces is allowed to be, acoustically.

There is a second condition in the problem that is easy to miss because it is never stated. The wave equation is perfectly happy with sound converging on a jet from the whole of space and being absorbed by it. That solution has the same energy, the same frequency and the same equation as the one everybody uses, and it is ruled out by a condition imposed at infinity.

What a flow is not allowed to be #

Sound is radiated by three kinds of source, in decreasing order of effectiveness.

A monopole is a volume that changes: a loudspeaker cone, a bursting balloon, a bubble. A flow with constant density has none, because there is nothing in it whose volume can change.

A dipole is a force applied to the fluid from outside: a vibrating surface, a fan blade, a body in the stream. A flow with no surfaces in it has none, because there is nothing to push against.

A quadrupole is a stress in the fluid acting on itself — the momentum flux $\rho u_i u_j$ of the turbulence, which is not a force on the fluid from anywhere and does not integrate to one. That is what is left, and it is the weakest radiator of the three by two powers of the source’s compactness each time.

The whole of Lighthill’s argument is that list. Rearranging the exact equations of motion into a wave operator on the left and everything else on the right shows the everything-else to be the divergence of a divergence of a stress — which is to say, a quadrupole distribution — and the scalings follow.

From $(kd)^{2n}$ to $U^{2n+4}$ #

The ladder measured above is in $kd$ — the size of the source measured in radians of wavelength. The step to a power of speed needs one physical statement about turbulence, and it is the one every scaling argument in this collection uses: an eddy turns over in about the time it takes to cross itself.

So an eddy of size $\ell$ carrying velocity $u$ has frequency $\omega \sim u/\ell$, and

$$kd \sim \frac{\omega \ell}{c} \sim \frac{u}{c} = M.$$

The compactness of a turbulent eddy is its Mach number. With the source strength scaling as the stress $\rho u^2 \ell^2$, the radiated power comes out as

$$P \sim \frac{\rho\, U^8 D^2}{c^5},$$

which is Lighthill’s eighth-power law. Each rung of the multipole ladder is two powers of $U$: a monopole would give $U^4$, a dipole $U^6$, the quadrupole $U^8$.

Two hundred and fifty-six #

The eighth power is the most consequential exponent in the subject, and its consequence is a design rule rather than an observation. Doubling the jet speed multiplies the noise by 256. Halving it divides the noise by 256 — and, at fixed thrust, halving the jet speed means doubling the mass flow, which means a bigger fan.

That is exactly what happened. Airliner engines went from turbojets throwing a thin fast jet to turbofans throwing a wide slow one, and the noise fell by twenty decibels or more. It was not achieved by making anything quieter. It was achieved by moving along an exponent, and the same change improves the propulsive efficiency for the same reason — a lot of air moved a little is better than a little air moved a lot, in the noise budget as in the energy one.

The efficiency numbers are worth quoting because they are so small. A one-metre jet at 300 m/s radiates about 1.7 kilowatts of sound out of 12.7 megawatts of mechanical power: an acoustic efficiency of $1.4\times10^{-4}$. A jet engine is a hundred-thousandth of a loudspeaker. It is deafening only because the power it is a tiny fraction of is measured in megawatts.

Why cancellation is so expensive #

The ladder deserves a physical reading, because why each cancellation costs two powers is not obvious from the fitted slopes.

A monopole radiates because it displaces fluid. Put two of opposite sign a distance $d$ apart and, seen from far away, the displacements very nearly cancel — what survives is the difference between two nearly equal signals, which is smaller by the fraction of a wavelength between them, $kd$. Power goes as amplitude squared, so the power is down by $(kd)^2$. Add a second cancellation and the argument repeats.

At $kd = 0.05$ — an eddy at Mach 0.05, which is a gentle breeze — a quadrupole radiates $4\times10^{-7}$ of what a monopole of the same strength would. Turbulence is quiet because it cancels itself, and it stops being quiet only when the cancellation becomes imperfect, which happens when the source is no longer small compared with a wavelength. That is the same statement as the Mach number rising.

The Doppler factor, solved rather than expanded #

Turbulent eddies do not sit still: they are convected downstream at a fair fraction of the jet speed, and a moving source is louder in front of itself than behind.

The factor is exact and it is a Jacobian. An observer at time $t$ hears what was emitted at the time $t_e$ satisfying $c(t-t_e) = |\mathbf{x} - \mathbf{y}(t_e)|$ — one nonlinear equation — and differentiating its solution gives $\partial t_e/\partial t = 1/(1 - M\cos\theta)$.

One detail in that figure is worth its own sentence, because it looks like a solver error and is physics. The angle in the formula is the angle the sound was emitted at, not the angle the observer sees it from. They differ by how far the source moved while the sound was in transit — three degrees at Mach 0.3 and sixty degrees of observer angle, which is a quarter of a per cent in the answer. The first version of this computation compared the solved Jacobian against the formula evaluated at the observer’s angle, disagreed by that quarter of a per cent, and was right about everything except which angle it meant.

For a compact quadrupole the intensity carries five of those factors, so the convective amplification ahead of a jet is $(1 - M_c\cos\theta)^{-5}$ and is worth many decibels at the angles where jet noise is loudest. That exponent is quoted here rather than derived: it counts Doppler factors from the source strength, the retarded time and the solid angle, and each of them is a separate argument.

The condition that makes sound leave #

Now the part that is a boundary condition rather than a scaling.

The wave equation admits $p = A\cos(kr - \omega t)/r$ and $p = A\cos(kr + \omega t)/r$. Same wavelength, same $1/r$ envelope, same energy density to twelve figures, and both satisfy the equation to $10^{-11}$. The first carries energy away from the origin for ever; the second brings it in from infinity and delivers it to whatever is there.

The Sommerfeld radiation condition — that far away the waves go outwards — is not derivable from the equations of motion. It is an extra statement, imposed at a boundary nobody can reach, and it is the acoustic member of the family this collection’s essay on counting conditions sets out: the equation is the same equation, and what was told at the edge decides the answer.

Its physical justification is causality rather than fluid mechanics. The incoming solution requires the whole of space to have been arranged, in the infinite past, so that a spherical wave would arrive at this jet at this moment — a conspiracy rather than a state. Ruling it out is the same move as choosing retarded potentials in electromagnetism, and it has the same status: a statement about the direction of time, imported into a time-symmetric equation from outside.

What the eighth power does not say, which is where the noise goes #

The law is about total radiated power, and nobody has ever been annoyed by a total radiated power. Two further scalings sit between the exponent and what a person under a flight path experiences, and both of them move in the same direction as the exponent does — which is a piece of luck the industry has spent sixty years collecting.

The spectrum scales too. The eddies that radiate have a size set by the jet’s diameter and a velocity set by its speed, so their frequencies scale as $U/D$: jet noise peaks at a Strouhal number $fD/U$ of about a fifth, and that is where the spectrum’s broad hump sits for every subsonic jet ever measured. Halving the jet speed therefore does two things at once. It divides the power by 256, and it halves every frequency in the spectrum; doubling the diameter to keep the thrust halves them again.

That second effect is worth as much as part of the first, because neither the atmosphere nor the ear is flat. Air absorbs high frequencies far more strongly than low ones, so a kilometre of propagation removes a great deal more of a turbojet’s spectrum than of a turbofan’s — and the ear’s own weighting peaks in the few-kilohertz region, which is where a small fast jet puts its energy and a large slow one does not. A quieter engine is quieter three times over: less power, more of it absorbed on the way, and less of what arrives in the band people hear best. Certification measures perceived noise rather than acoustic power for exactly that reason, and the twenty decibels quoted above is the smaller of the two numbers.

It cuts the other way for anything small. A drone’s rotor, a computer fan, a hair dryer: each is a low-power source whose dimensions are small, so its Strouhal peak lands high, in the band the ear weights most and the air absorbs least over the few metres involved. Their acoustic powers are laughable beside a jet’s and their nuisance is not, and the whole of that discrepancy is spectrum rather than power.

And the radiation is not spread evenly. Even setting aside the refraction the previous section mentions, the convective factor concentrates the sound into a cone at a shallow angle to the jet axis — the peak is around thirty degrees downstream of the exit, not to the side — so the power that does escape leaves along the ground track behind the aeroplane rather than out to the sides. Every noise contour drawn around an airport is that directivity plotted against a map, and it is why the regulatory measuring points sit where they do: one under the departure path, one to the side, one on approach, each sampling a different part of a pattern rather than three estimates of one number.

None of this contradicts the exponent; all of it is what the exponent has to be combined with before it says anything about a person. The scaling argument establishes that the total power goes as $U^8$, and it is silent on how that power is distributed in frequency, in angle and in what survives the journey. A law that predicts a total to within a coefficient still needs three more distributions before it predicts a measurement, and each of them has its own scaling with the same two variables — which is why the design change that satisfied all of them at once was a single one, and why nothing since has been anything like as effective.

What the picture cannot show #

Nothing here computes a source. The multipole clusters are model sources with prescribed strengths; the turbulence that actually radiates is not in this calculation, and its statistics are what an aeroacoustic prediction actually needs. What this essay establishes is the scaling that any source of the right type must obey.

The constant is borrowed. $K \approx 10^{-4}$ in $P = K\rho U^8 D^2/c^5$ is a measurement. No figure in this collection quotes a sound power level, and none should: the exponent is computed here and the coefficient is not.

The analogy assumes the sound does not affect the flow, and at high jet speeds it does — refraction by the jet’s own shear and temperature gradients bends the radiation into a cone, which is why real jet noise has a directivity that this argument does not predict.

And it is all subsonic. A supersonic jet radiates by Mach-wave emission, which is a completely different and far more efficient mechanism: the eddies convect faster than sound, so the warning cannot arrive, the $(1-M_c\cos\theta)$ factor passes through zero, and the compactness argument that gives the eighth power is void. Screech and broadband shock noise belong to that regime and are named here rather than computed.

What the analogy actually rearranges #

It is worth seeing how little is done and how much follows, because the acoustic analogy has a reputation for depth that the algebra does not support.

Start with the exact equations of motion for a compressible fluid — continuity and momentum, nothing linearised, nothing dropped. Differentiate the first with respect to time, take the divergence of the second, subtract, and add and subtract $c^2\nabla^2\rho$. What is left has a wave operator acting on the density on the left, and on the right the double divergence of a tensor, $T_{ij} = \rho u_i u_j + (p - c^2\rho)\delta_{ij} - \tau_{ij}$.

That is the whole manoeuvre, and it is exact: nothing has been assumed about the flow. What has been gained is a form. The left-hand side is the operator whose solutions are sound in a uniform medium at rest, and the right-hand side is everything else, now wearing the clothes of a source distribution.

The step that costs something comes next, and it is an approximation about scale rather than about physics: the source term is evaluated as though the flow it describes were unaffected by the sound it makes. For a low-Mach jet that is excellent, since the acoustic efficiency computed above is $10^{-4}$ and the back-reaction is smaller still. For a resonating cavity, a screeching supersonic jet or a flame it is not, because the sound field feeds back into the source that made it — and the whole class of problems where that happens is the class the analogy cannot touch.

Who found it, and when #

James Lighthill published the acoustic analogy in 1952 and 1954, at Manchester, in response to a practical crisis: the first jet airliners were about to enter service and nobody could say how loud they would be or what to do about it. The rearrangement at the heart of it is algebraically trivial — move terms from one side of the exact equations to the other — and its consequence was a design programme that has occupied the industry ever since.

The surprising connection is with the essay on asking for a pressure distribution, which is also Lighthill’s, from seven years earlier. The two look nothing alike and share a habit: take a problem nobody can solve, rewrite it as a problem somebody has already solved with an awkward right-hand side, and then find out what the right-hand side is allowed to be. In the inverse problem it is three integrals. Here it is a multipole order — and the answer, that a flow can only be a quadrupole, is what makes flight bearable to live near.

Where the ladder goes next #

The rung above is the surface term Lighthill’s analogy acquires when there are bodies in the flow: Curle’s 1955 extension, in which a surface supplies a dipole and therefore $U^6$ rather than $U^8$. That is why a small propeller can be louder than a large jet, and why every fan in every appliance is a more efficient radiator than a jet engine — it has a surface, and a surface can push.

The one beside it is the same argument applied to a rotor, where the sources are periodic rather than random and the sound is tonal. Ffowcs Williams and Hawkings’ 1969 formulation is what a helicopter is designed against, and it is the acoustic analogy with the surface moving — which brings the retarded-time factor above into the middle of the calculation rather than leaving it as a correction.