A wire sings with the force on it, not the flow round it

The sound that only leaves found that a flow on its own is about the worst loudspeaker there is. Lighthill’s rearrangement of the equations of motion turns the turbulence into a set of sources in a still medium, and with no walls anywhere those sources can only be quadrupoles: arrangements of four sources whose fields cancel almost exactly at a distance. The residue of that cancellation grows as the eighth power of speed, and only a tiny fraction of a jet’s power ever becomes sound. The sound is what does not cancel made the point sharper: a quadrupole is not a weak source but four strong ones arranged so that almost nothing survives, and every step up the series of multipoles is one more cancellation and one more power of compactness lost.

Both essays left the same door open. Lighthill’s argument assumes the flow is unbounded. Put a solid body in it and the body pushes on the fluid, and a push is not a quadrupole. Curle worked out in 1955 what the body adds: a surface term, a distribution of dipoles whose strength is the force per unit area the surface exerts. For a body small compared with the wavelength, all of it sums to a single dipole, and its far field is the rate of change of the total force, divided by four pi times the sound speed times the distance, along the force’s line.

That is one cancellation fewer than the free flow’s, and the essays above have already said what one cancellation is worth: one power of the compactness in the pressure, two powers of speed in the sound power. A body in a flow radiates as the sixth power of speed, and against the eighth power of the flow’s own noise it wins by one over the Mach number squared. The cleanest place to see it is the oldest aerodynamic sound anybody has written about, the tone of a wire in the wind.

The tone is the force, and the force is the wake’s #

A cylinder across a steady wind does not have a steady wake. When the flow lets go explained why the flow separates from the rear of a bluff body, and the street a steady calculation cannot draw described what it does next: it rolls up into vortices that leave from the top and the bottom alternately, at a frequency that is, over a wide range of conditions, a fifth of the wind speed divided by the diameter. Each vortex that leaves from one side changes the circulation round the cylinder, so the lift swings from one side to the other at the shedding frequency. The drag swings too, but at twice that frequency, because every departure, from either side, tugs the cylinder downstream.

The wire need not move for any of this. The force is the fluid pushing on a rigid surface, and by Newton’s third law the surface pushes back on the fluid with a force that swings across the wind several hundred times a second. That reaction is the dipole. The tone’s pitch is the shedding frequency, a fifth of the wind speed over the diameter: 667 hertz for a three-millimetre wire in a 10-metre-a-second wind, and twice that in twice the wind. The tension of the wire and its length do not appear.

The strength needs three numbers that are measured rather than computed, and the figures borrow them openly: the Strouhal number, 0.2; the root-mean-square lift coefficient, which in the subcritical range of Reynolds numbers from about ten thousand to a hundred thousand scatters between 0.2 and 0.6 in the literature and is taken here as 0.45; and the length along the span over which the shedding stays in step, taken as four diameters, with a measured spread of about two to eight. Everything else follows from Curle’s formula: the force’s mean square, its rate of change, the far-field pressure and the power.

A sixth power, and the wake’s own noise never catches it #

The first figure is a wire three millimetres across and a metre long. The heavy curve is the power of its lift tone against the wind speed; it rises as the sixth power, sixty-four times for every doubling of the wind. The dashed curve is the most the turbulence of its wake can radiate as quadrupoles, bounded deliberately on the generous side: the whole wake, a diameter wide and a metre long, moving at the full wind speed, with the coefficient measured for jets. A real wake’s velocity fluctuations are a fraction of the wind, and the eighth power makes that fraction count heavily, so the true quadrupole noise sits well below the line.

At 10 metres a second the tone radiates 7.7·10⁻⁷ watts, three quarters of a microwatt, and the bound on the wake 7.9·10⁻¹²: a hundred thousand times less. The dial moves the diameter from two millimetres to twelve. A thicker wire sings lower, since the frequency is inversely proportional to the diameter, and louder at the same wind: the force per unit length grows with the diameter, the frequency falls with it, and the correlation length, a fixed number of diameters, grows with it too, so the power grows as the diameter itself.

The second figure turns that comparison into its essential form. The ratio of the tone’s power to the wake’s bound, for any diameter, is a number divided by the square of the wind’s Mach number: about ninety-four thousand at Mach 0.03, a fresh breeze of 10 metres a second, and still a hundred at Mach 0.9, though by then a real cylinder’s shedding has long since changed character. The spread between the curves is the spread of the measured lift coefficient, and it does not matter. At any wind a person stands in, the turbulence is irrelevant to the sound. The body is the source; the flow is only how the force on it is made to fluctuate.

This is the general lesson of Curle’s paper, and it reaches far beyond wires. Every fan, propeller and blower has surfaces in its flow, and every one of them is a dipole source before it is anything else. That is why a desk fan at a few metres a second is audible at all, and why the quieting of fans is almost entirely the business of the forces on their blades — the gusts the blades meet, the wakes they cut — rather than of the turbulence they make.

Where the sound goes #

A dipole has a direction. The lift tone’s dipole lies across the wind, so the tone is loudest heard from the side, across the wind and across the wire, and silent along the wind; the drag tone’s dipole lies along the wind, at twice the frequency, and is loudest upstream and downstream. The third figure plots both around a car aerial six millimetres across at 30 metres a second, a little over a hundred kilometres an hour, two metres away in the plane at right angles to the aerial.

Across the wind the lift tone reaches 0.14 pascals at 1,000 hertz. Along the wind it vanishes, and what is left is the drag tone, 0.03 pascals at 2,000 hertz, about a fifth of the pressure and an octave higher. The ratio is the ratio of the force amplitudes, 0.05 against 0.45, multiplied by two for the doubled frequency, since the far-field pressure is the force’s rate of change. So the character of the whistle changes with the listener’s position, and a microphone placed directly downstream of a wire — a natural place to put one in a wind tunnel — hears the octave and almost none of the fundamental.

There is a subtlety in the word “across”. The lift on a wire lies in the plane perpendicular to the wire, so the dipole’s null is the line of the wind, but along the wire itself every element of the span radiates too, each at its own distance and angle. The figures add the pieces of span as independent sources at their own positions, which for a short aerial seen from two metres is almost the same as a single dipole at the middle.

A long wire adds powers, not forces #

That independence is the third measured number at work. The shedding is in step along the span only over a few diameters; beyond that, one stretch of wire sheds a vortex from the top while another, a few centimetres away, sheds from the bottom. The forces on the two stretches do not add. Their mean squares do. The fourth figure plots the mean square of the total force on a span against the span’s length in correlation lengths, for an exponential fall-off of correlation along it.

A span shorter than a correlation length is coherent and its force grows with its length, so its mean square grows as the square. A long span is a row of independent pieces, and its mean square grows only as the length, times the correlation length. At ten correlation lengths the mean square is less than a tenth of the coherent value. A wire a metre long and three millimetres across is eighty correlation lengths long, so the coherent estimate would overstate its power by a factor of eighty, nineteen decibels.

This is where vibration enters, and why the refuted claim has a grain of truth in it. A wire that vibrates in the wind, as a long taut one can when the shedding frequency comes near one of its natural frequencies, drags the shedding into step with its own motion along its whole vibrating length. The street a steady calculation cannot draw describes that lock-in, and the shake that is not resonance the feedback that makes it self-limiting. For the sound, the effect is to lengthen the correlation length from four diameters to the length of a vibrating loop, and the power grows with it. The Aeolian harp, a box of strings set in a window, works this way: the strings’ resonance picks which tones are reinforced, but every string would sing quietly if held rigid. The helical strakes wound round some car aerials and chimneys do the opposite and shorten the correlation length, so that the forces on neighbouring stretches cancel; the force per unit length is hardly changed, and the sound falls anyway.

Four wires in four winds #

The fifth figure gives the rms pressure of the lift tone across the wind for four everyday cases, each at the distance a person might stand.

A fence wire three millimetres across in a 10-metre-a-second breeze makes 0.008 pascals at a metre, a faint whine at 667 hertz. A bicycle spoke at 8 metres a second makes 0.002 at a metre, which is why a moving bicycle’s spokes are inaudible over its tyres. A car aerial at motorway speed makes 0.14 pascals at two metres, seventeen times the fence wire’s pressure, which is a loud whistle at 1,000 hertz and the reason aerials are wound with strakes or hidden in glass. A guy rope a centimetre thick and ten metres long in a 15-metre-a-second wind makes 0.027 pascals at five metres, at 300 hertz, a low moan that carries.

Two of these sit outside the range the borrowed coefficients describe. The fence wire’s Reynolds number is two thousand and the spoke’s about a thousand, where the measured rms lift coefficient is lower than in the subcritical range and the correlation length longer; the two changes push in opposite directions, and the figure’s numbers for those two should be read as the right order of magnitude rather than as predictions. The car aerial, at a Reynolds number of twelve thousand, and the guy rope, at ten thousand, sit inside it.

The comparison that matters most is the one the figures do not need a measured coefficient for. Double the wind on any of these and the pressure rises eightfold, the power sixty-fourfold. That is why a wire that is silent in a breeze screams in a gale, and why the tone seems to appear suddenly rather than grow.

What was checked, and how #

The formula at the centre of everything is the far field of a compact dipole, and it was checked against the exact field of a harmonic point force on a compressible fluid, near field included. A point force makes a jet only when it is strong found what a steady point force does to a viscous fluid; an oscillating one on a compressible fluid makes sound, and the sixth figure is the exact answer. Close to the force the pressure is far larger than the far-field form: ten times at a tenth of the distance kr = 1 and a hundred times at a hundredth.

None of that excess leaves. The power through a sphere, computed from the exact pressure and velocity, is the same at every radius from a hundredth of kr to a hundred, to eight parts in a million, and equals the far-field formula. The near field’s pressure and velocity are a quarter of a cycle apart and carry no net energy. That is the same lesson the sound is what does not cancel drew for the reason measurements must be made far away, with a practical edge here: a microphone a few centimetres from a singing wire is deep in the near field at a thousand hertz, and its reading says almost nothing about what is radiated.

The span integral was checked in its two limits, the square of the length for a short span and the length times the correlation length for a long one. The line sum, which adds every piece of span at its own distance and angle, was checked to reduce to one compact dipole for a short wire seen from far off and to fall off as one over the square root of the distance beside a long one, as a line source should. The tone’s power was checked to rise by exactly two to the sixth for a doubled wind and the wake’s bound by two to the eighth. The calculation refuses a wind of zero and a negative diameter.

What the calculation leaves out #

The wire is compact. Curle’s single dipole needs the body small compared with the wavelength, and a correlated stretch of it as well. The three-millimetre wire at 10 metres a second has a diameter of four hundredths of a wavelength over two pi, and a correlation length of fifteen hundredths; the car aerial’s correlation length is 0.44, where the compact form begins to lose accuracy. A thick cylinder in a fast wind is not compact, and its tone needs the full surface integral.

The shedding is a pure tone. A real lift signal wanders in frequency and amplitude, so the tone has a width, set by how long the shedding stays in phase. The power is the same; its spread over frequencies is not computed.

The measured coefficients are borrowed. No figure here computes a lift coefficient or a correlation length; the Reynolds number decides both and their measured values scatter by a factor of two or more. The scalings — the sixth power, the square of the Mach number, the length times the correlation length — do not depend on them. The absolute levels do.

Nothing moves the air but the wake. A wire in a gusty wind meets a turbulent inflow, and the fluctuating force it feels from the gusts radiates too, with its own spectrum. That is the dominant source on fans and rotors and is not counted here.

Who worked it out #

The tone of wires in wind was measured by Vincenc Strouhal in 1878, spinning a wire on an arm, who found the frequency proportional to speed over diameter and independent of tension; Lord Rayleigh connected it to the wake, and the dimensionless number now carries Strouhal’s name. Theodore von Kármán’s analysis of the vortex street came in 1911 and 1912. The acoustic analogy is Lighthill’s of 1952; the surface term, and the observation that a body turns a flow’s quadrupoles into dipoles, Curle’s of 1955. O. M. Phillips computed the Aeolian tone’s intensity from the fluctuating lift in 1956, with a correlation length taken from measurement, which is the calculation this essay repeats.

Still open: a surface that moves #

Curle’s surface is held still. A propeller’s or a helicopter’s blades move through the air at speeds that are a sizable fraction of the speed of sound, and their forces are dipoles carried along a circle. Ffowcs Williams and Hawkings generalised Curle’s formula to surfaces in arbitrary motion in 1969, and the Doppler factor of the sound now is the source then moves from a correction to the centre of the calculation. The next question is a rotor’s steady loading — a constant force carried round a circle — which is silent when it stands still and sings at the blade-passing frequency once it turns, and how fast its tone grows with tip Mach number as the Doppler factor approaches its singularity.