Waves

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

15 min read 6 figures Made into an instrument The same equation again

Assumes: The distance that takes the treble out · What adding does to the energy

The first rung of this ladder treats absorption as a number that removes energy from a wave, and gets a long way with it: the loss per cycle is roughly fixed, the number of cycles per metre goes as the frequency, so absorption climbs as $f^2$ and a distant thunderclap rumbles where a near one cracks.

It ends by noting what it has not asked. The energy removed has to go somewhere, and the answer turns out to have two halves that are usually taught as separate subjects.

A wave carries energy. A wave also carries momentum — its momentum flux is the intensity over the wave speed — and absorbing the energy means absorbing the momentum too. So an absorber in a beam feels a force, and the medium in which the absorption happens feels a body force spread through it.

A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one. — Fig. 1 The force a fully absorbed acoustic beam exerts, against its power. It is the power divided by the speed of sound: a watt in water gives 675 micronewtons, the weight of sixty-eight milligrams. The consequence is that acoustic power is measured by weighing — an absorbing target on a laboratory balance is the primary standard for ultrasonic output, and every medical transducer is calibrated against one.

Why the acoustic version is the large one

Light has a pressure and it is famously difficult to detect. A kilowatt of sunlight per square metre pushes with about three micropascals, which is not enough to blow a grain of dust away unless the grain is small; a solar sail needs to be square kilometres in size to be worth having; Nichols and Hull needed a torsion balance in a vacuum to see it at all.

The acoustic version is the same expression with a different denominator.

F = \frac{P}{c}.

For light, $c$ is $3 \times 10^8$ metres per second. For sound in water it is 1,482. The ratio is two hundred thousand, so an acoustic watt pushes two hundred thousand times harder than an optical one — which turns a delicate measurement into a routine one.

That single factor is why the two subjects developed differently. Optical radiation pressure is a phenomenon to be demonstrated; acoustic radiation force is a piece of laboratory equipment. The National Physical Laboratory and its counterparts calibrate ultrasonic transducers by suspending an absorbing target above one on a balance and reading the weight, and a well-built balance measures a milliwatt.

The same force is what an acoustic levitator uses, what separates cells in a microfluidic channel, and what is measured to check that a physiotherapy machine is delivering what its dial says — an effect made into an instrument rather than merely observed.

Both halves of the same coefficient

The heating is the half everybody expects, and putting the two on one page shows they are not two facts.

The same absorption, counted as a temperature. The rate at which an absorbed beam raises the temperature of the medium, against intensity. It is 2αI over the volumetric heat capacity, and it is the same α that produced the force in the previous figures — the ratio of the two is ρc_p/c, a property of the medium with no intensity in it, checked here at two intensities four decades apart. Diagnostic ultrasound at a few hundred milliwatts per square centimetre raises tissue by hundredths of a degree; a focused therapeutic beam at ten kilowatts per square centimetre raises it at tens of degrees a second and cooks a millimetre-sized volume in seconds. Both are the same coefficient at different amplitudes, and the safety limits in the first case are computed from the second. — Fig. 2 The rate at which an absorbed beam raises the temperature of the medium, against intensity. It is 2αI over the volumetric heat capacity — the same α that produced the force. Their ratio is ρc_p/c, a property of the medium with no intensity in it at all, checked here at two intensities four decades apart.

Divide one by the other and the absorption cancels:

\frac{\text{body force}}{\text{heating rate}} = \frac{\rho c_p}{c}.

Nothing about the beam survives. That is the sense in which the force and the heating are one measurement: knowing either, and the medium, gives the other, and any experiment that measured both would be measuring the same thing twice.

The practical consequence appears in every ultrasound safety standard. A diagnostic scanner’s output is characterised by a mechanical index and a thermal index, and the two are computed from the same acoustic measurement — the pressure field — because the force and the heating are not independent quantities to be measured separately.

The numbers span an enormous range. Diagnostic ultrasound at a few hundred milliwatts per square centimetre raises tissue by hundredths of a degree over a scan. A focused therapeutic beam at ten kilowatts per square centimetre raises it at tens of degrees a second, and destroys a millimetre-sized volume of tissue in a few seconds without touching anything in between, which is the geometry a focused beam buys applied to sound. Both are the same coefficient at different amplitudes.

Where the momentum comes from, exactly

The derivation above is a sentence long and hides a genuine subtlety, and it is worth opening because the subject spent decades arguing about it.

A sound wave in a fluid does not carry momentum in the way a stream of particles does. To first order in the amplitude the fluid’s velocity averages to zero over a cycle, so the average momentum is zero. The momentum flux that matters appears at second order — from the correlation between the density fluctuation and the velocity fluctuation, which do not average to zero when multiplied together.

That is why every result in this rung is quadratic in the amplitude while the wave itself is linear, and it is why the effect exists at all in a medium with no net flow. What is being transported is not the fluid but a correlation.

The consequence is that the “momentum of a sound wave” is not a quantity a fluid element possesses; it is a flux across a surface. Asking where the momentum is inside the beam has no clean answer, and asking how much crosses a plane has an exact one. That distinction is the whole of why radiation force is defined on a target rather than in the medium, and it is the same distinction the energy of a field runs into: a flux is unambiguous and a density is a bookkeeping choice.

The optical version of the same argument is the long-running dispute about whether a photon in a medium carries $n$ times or $1/n$ times its vacuum momentum. Both answers are right about different quantities — one is the momentum transferred to a mirror and the other the momentum imparted to the medium — and the acoustic case has the same two-sided structure with the same resolution.

The flow it drives

The body force does not merely heat the fluid. It pushes it, and since the push is steady rather than oscillating, the fluid flows.

The steady flow a sound beam pushes. The speed of the steady flow driven by an absorbed beam of radius 5 millimetres, against the beam's intensity, for 3 media. The absorbed momentum is a body force in the fluid, and balancing it against the viscous stress over the beam's own width gives a speed proportional to the intensity — not to its square root, which is how the oscillation itself scales. So doubling the drive doubles the streaming and only increases the oscillation by forty per cent, and above some intensity the steady flow is the conspicuous thing. Millimetres per second in water at ordinary diagnostic intensities is enough to see, and it is what an ultrasonic cleaning bath is really doing. — Fig. 3 The steady flow driven by an absorbed beam, against the beam’s intensity. The absorbed momentum is a body force, and balancing it against the viscous stress across the beam’s own width gives a speed proportional to the intensity — not to its square root, which is how the oscillation itself scales. Millimetres per second in water at diagnostic intensities is enough to see.

The scaling is the surprise, and it is what makes streaming conspicuous rather than negligible.

An acoustic wave’s particle velocity goes as the square root of the intensity, because the intensity is quadratic in the amplitude. The streaming velocity goes as the first power of the intensity, because it comes from a term that is already second order. So the ratio of the steady flow to the oscillation grows as the amplitude does, and past some intensity the flow is the dominant motion.

That is why an ultrasonic cleaning bath works, and it is a different mechanism from the one usually cited. Cavitation gets the credit and does much of the scrubbing; the streaming is what carries the loosened material away and what refreshes the liquid at the surface being cleaned, and in a degassed bath where cavitation is suppressed the streaming is the whole effect — a diffusion of momentum sideways driven by a beam rather than by a wall.

It is also a nuisance. A thermocouple placed in an ultrasound beam to measure the heating reads the wrong answer, because the streaming cools it convectively. The measurement is made instead in a gel whose viscosity suppresses the flow, and the difference between the two readings is a standard laboratory demonstration of exactly this rung.

The steady flow a sound beam pushes. The speed of the steady flow driven by an absorbed beam of radius 2 millimetres, against the beam's intensity, for 3 media. The absorbed momentum is a body force in the fluid, and balancing it against the viscous stress over the beam's own width gives a speed proportional to the intensity — not to its square root, which is how the oscillation itself scales. So doubling the drive doubles the streaming and only increases the oscillation by forty per cent, and above some intensity the steady flow is the conspicuous thing. Millimetres per second in water at ordinary diagnostic intensities is enough to see, and it is what an ultrasonic cleaning bath is really doing. — Fig. 4 Water at one megahertz against water at ten, in a narrower beam. Raising the frequency raises the absorption by a hundredfold and the streaming with it, while narrowing the beam reduces it as the square of the radius, because the viscous stress the force has to overcome grows as the beam gets thin. Both are the same balance with different numbers, and both are how a device is designed to stream or not to.

The dependence on the beam radius is worth noticing: it goes as the square, so a beam a tenth the width streams a hundredth as fast. That is the same competition between a driving force and a viscous stress across a length that sets every boundary-layer thickness, arriving here as a design rule for whether a device streams.

The measurement, and why it is the primary one

A radiation-force balance is worth describing in a little detail, because it is one of the few places where a physical quantity is measured against a mass rather than against another instance of itself.

The apparatus is an absorbing target — usually a wedge or a cone of a rubber loaded to match the acoustic impedance of water — suspended in a tank from an analytical balance, with the transducer beneath it pointing up. Switching the transducer on changes the reading. The change, times $g$ , is the force; the force times the speed of sound is the power.

What makes it a primary standard is that nothing in that chain requires a calibration against another acoustic instrument. A balance is traceable to a kilogram, the speed of sound in water is known to a part in $10^5$ , and the geometry is measurable. There is no acoustic reference in the loop at all, which is exactly what a primary standard means.

The competing method is calorimetry: absorb the beam in a known mass of liquid and measure the temperature rise. It measures the same absorption through the other half of this rung’s identity, it is slower and less accurate, and the two agree — which is the experimental confirmation that the force and the heating really are one coefficient.

The uncertainties are dominated by things the ideal picture leaves out. The target reflects a per cent or two rather than nothing; the beam is not plane, so the momentum flux across the target is not quite the power over $c$ ; convection from the streaming pushes on the balance; and the water heats over the course of a measurement, changing its sound speed. A good balance reaches two or three per cent, and improving on that is a research problem rather than an engineering one.

The interlaboratory comparisons are the check that matters, and they are humbling. National standards laboratories circulate the same transducer and measure its output independently; the spread between them has historically been several per cent, larger than any one laboratory’s stated uncertainty. That is the usual outcome of such an exercise and it is the reason they are done: an uncertainty budget assembled from known effects is a lower bound, and the comparison is what finds the effects nobody listed.

The trap a standing wave makes

Everything above is a travelling wave, where the force points where the wave goes. A standing wave has no net momentum flux and still exerts a force, and the force has a structure.

The mechanism is the same momentum bookkeeping applied to a spatially varying field. A particle in a gradient of acoustic energy density scatters more from one side than the other, and the imbalance is a force pushing it toward the pressure node — for a particle denser and stiffer than the fluid, which covers most solids and liquids in air.

Acoustic levitation is a laboratory technique rather than a curiosity because of what it makes possible. A droplet held in mid-air touches no container, so it can be supercooled far below its freezing point with no wall to nucleate on, and it can be studied while it evaporates without any contamination. Protein crystallisation, containerless melting of refractory materials and the measurement of surface tension in undercooled liquids all use it.

The same force in a microfluidic channel sorts cells. A standing wave across the channel pushes cells toward the node, and because the force depends on size and density, different cell types collect at different rates and can be separated continuously — which is how circulating tumour cells are extracted from blood without labelling anything.

The trap a standing wave makes. The time-averaged force on a 5-micrometre particle of density 1050 kilograms per cubic metre in a standing wave in water at 2000 kilohertz, against position through one wavelength, in units of the particle's own weight. The force vanishes at the pressure nodes and points toward them from either side, so the nodes are traps — checked here rather than asserted. At an acoustic pressure amplitude of 200 kilopascals the peak force is 0.2 times the weight, which is why a droplet can be held in mid-air on nothing. The mechanism is the same absorbed-momentum bookkeeping as the travelling case; what a standing wave adds is a gradient, and a gradient is a trap. — Fig. 6 The same construction for a cell in water at two megahertz — the microfluidic case rather than the levitation one. The wavelength is under a millimetre, so the trap spacing is a few hundred micrometres and fits across a channel, and the force on a ten-micrometre cell is a small fraction of its weight and entirely sufficient because the cell is neutrally buoyant. Changing one frequency and one medium moves the technique from holding a droplet in air to sorting blood.

What the coefficient does not cover

There is a limit to the whole picture that is worth being explicit about, because it is where the rung’s central identity fails.

The force and the heating are the same coefficient only if the absorption is dissipative — if the energy taken out of the wave becomes heat in the medium at the point where it was removed. That covers viscous and thermal absorption, and it covers molecular relaxation, which is where air’s and seawater’s absorption comes from.

It does not cover scattering. A beam attenuated by scattering off inhomogeneities loses intensity from the forward direction without any energy becoming heat, so the heating is small and the force is not — the momentum is redirected rather than absorbed, and a scatterer feels a force that depends on the scattering pattern rather than on any coefficient.

Distinguishing the two in a real medium is a genuine measurement problem. Biological tissue attenuates by both, in a ratio that varies between tissues and with frequency, and the safety indices that assume all of the attenuation becomes heat are therefore conservative by an unknown factor. The measurement that separates them — comparing the attenuation of a collimated beam with the total energy deposited in a calorimeter — is difficult enough that the tabulated ratios carry uncertainties of tens of per cent.

So the identity at the centre of this rung is exact for a dissipative medium and an approximation for a real one, and the direction of the error is known: assuming all attenuation is absorption overestimates the heating and gets the force right.

Where this stops being right

The force expressions assume a plane wave and total absorption. A real target reflects some of what arrives, which doubles the force in the limit of perfect reflection, and a real beam is not plane. Radiation-force balances correct for both, and the corrections are the largest term in their uncertainty budget.

The streaming estimate is a scaling argument. Balancing the body force against a viscous stress across the beam radius gives the right dependence and a number good to a factor of a few; the exact answer requires solving for the flow field with the beam’s actual profile, and depends on the geometry of the container as well.

Acoustic radiation force is not the same as acoustic radiation pressure, and the literature is a minefield about it. There are at least two definitions — Langevin’s and Rayleigh’s — differing in whether the fluid is held at constant volume or constant pressure, and they give different answers for the same experiment. The force on an absorbing target in an open bath is unambiguous, which is why the standard is defined that way.

And nonlinearity has been ignored. At the intensities where streaming is conspicuous, the wave is also distorting — its peaks travel faster than its troughs, it steepens, and it generates harmonics that are absorbed far more strongly than the fundamental. That enhanced absorption is what a therapeutic beam relies on, and it is not in any expression here.

What the pictures cannot show

The force figures draw a force against a power, and what a balance actually reads is a mass. Converting between them requires knowing the local gravity and correcting for buoyancy on the target, and the standard’s uncertainty is dominated by neither of those but by the acoustic field’s not being the plane wave the conversion assumes.

The streaming figure draws a single speed, and a streaming flow is a circulation — fluid pushed forward along the beam has to return around it, so there is a return flow the drawing has no room for and whose speed depends on the container rather than on the beam.

And the standing-wave figure draws a force on a particle that is not there. What is computed is the force a particle would feel, from a field the particle would itself disturb; for a particle small compared with the wavelength that disturbance is negligible, and the figure’s own guard refuses a particle large enough for it not to be.

Where this ladder goes next

Three rungs stand on attenuation. The first found that absorption removes the treble as well as the volume. The second found that absorption and refraction are one function tied by causality. This one asks where the removed energy goes and finds it appears twice, from one coefficient.

The habit worth carrying away is about conserved quantities travelling together. A wave that carries energy carries momentum in a fixed ratio, so any process that removes one removes the other, and a measurement of either is a measurement of both. That is why an absorption coefficient has a mechanical consequence at all, and the same reasoning applies to a wave carrying angular momentum — an absorbed vortex beam exerts a torque, which is measured, and is the acoustic and optical version of the same sentence.

What is left on this ladder is what happens when the wavelength gets short enough that a medium stops being a medium. Everything here treats absorption as a smooth coefficient with a smooth frequency dependence. Take the photon energy up to where it can eject an electron from an atom, and the coefficient acquires steps — a material that becomes suddenly more opaque to a harder photon, which nothing smooth does.

Part 3 of 6

This essay is one argument about Attenuation. The others:

$Absorption and refraction, drawn as one function. The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else.$ The answer that cannot come first Part 2 — A medium absorbs at some frequencies and bends light at all of them, and those look like two separate facts to be measured separately. They are one fact. The requirement that a material respond after it is asked rather than before ties the absorption at every frequency to the refraction at every other, by an integral — and a material given the two halves independently answers before the question arrives.