How a wave thins out

A shout carries a hundred metres and a whisper carries two. A ripple crosses a pond and is still visible at the far side. A radio signal from a spacecraft four billion kilometres away is received with an antenna and a very cold amplifier. All three are the same question — how does a wave weaken with distance — and the answers differ by exponents that have nothing to do with sound, water or radio.

The argument is a conservation law and a piece of geometry, in that order. If nothing is lost, the power crossing a sphere of radius $r$ is the same for every $r$. The area of that sphere is $4\pi r^2$. So the power per unit area — the intensity — goes as $1/r^2$, and since intensity goes as the square of amplitude, the amplitude goes as $1/r$.

Two quantities, and the factor of two that is always waiting #

The distinction between amplitude and intensity is the source of more arithmetic errors than any other part of the subject, and it is worth fixing before anything else.

The amplitude is the height of the curve and the energy in it is not, which is the distinction the whole essay turns on. The kinetic part of the energy goes as the square of the transverse velocity and the potential part as the square of the slope, and both are proportional to the square of the amplitude. Doubling the height of a drawn wave quadruples what it carries. So a picture of a wave always understates how fast it is dying: amplitude is what a drawing can show and intensity is what a detector reports, and a fall to 55% of the height is a fall to 30% of the intensity — 5.2 dB, or the effect of moving 1.8 times further from a point source.

The power a travelling wave carries along a string is $\tfrac12\mu v\omega^2A^2$, and every wave in physics has an expression of that shape: a property of the medium, times the square of the frequency, times the square of the amplitude. What is conserved as the wave spreads is that power. What the geometry acts on is therefore the intensity, and the amplitude’s behaviour is a consequence.

Why the exponent is a statement about shape #

The same argument in a different number of dimensions gives a different answer, and the pattern is entirely about how the area available grows.

For a ripple on water the energy spreads over a circle of circumference $2\pi r$, so the intensity goes as $1/r$ and the amplitude as $1/\sqrt r$. That is why a ripple stays visible so much further than a shout carries: the amplitude of the ripple has fallen by a factor of ten at a hundred times the distance, where the shout has fallen by a hundred.

Water waves add a complication a sound does not have: they are dispersive, so a disturbance spreads in length as well as in area and its peak amplitude falls faster than the geometric argument alone predicts. Separating dispersion from spreading needs the shape of the pulse, which is why the clean exponents on this page belong to a single frequency and not to a splash.

For a wave in a pipe, a waveguide, an optical fibre or along a string, the area does not grow at all and the exponent is zero. A pressure wave in a straight pipe arrives at the far end with the amplitude it started with, minus whatever the walls took, which is the reason a speaking tube works over a distance at which a shout would be inaudible and the reason a stethoscope works at all. It is also the whole economic argument for an optical fibre: confine the wave and the geometric loss disappears, leaving only the absorption, and a glass pure enough makes that a fifth of a decibel per kilometre.

The part that is a loss #

Geometric spreading is not dissipation — the energy is all still there, spread thinner. Absorption is different in kind: it converts the wave’s energy into heat, and it removes a fixed fraction per unit distance rather than a fixed fraction per doubling.

Because one is a power law and the other an exponential, the exponential wins eventually, always. The only question is where the crossing is, and for sound in air at ordinary frequencies it is kilometres away, which is why absorption plays almost no part in everyday acoustics and a decisive part in long-range propagation. It is also strongly frequency-dependent: absorption in air rises roughly as the square of frequency, so a distant sound is not merely quieter but duller, and the thunder from a nearby strike cracks while the thunder from a distant one rumbles. The crack is still there; its high frequencies were absorbed on the way, and the distance to a storm can be estimated from the timbre of the thunder as well as from the delay after the flash.

The same accounting covers light through the atmosphere, where the removal is by scattering rather than absorption and the frequency dependence is a fourth power. At an airmass of 38 — the sun on the horizon — the blue end is gone entirely and the red is barely touched. Scattering and absorption both remove energy from a beam exponentially, and the only difference between them is whether the energy is thermalised or merely sent somewhere else.

The attenuation length, and the unit that keeps causing trouble #

An exponential loss has a natural length: the distance over which the intensity falls by a factor of $e$, or by half, or by ten, depending on the convention. Any of them turns “how far does this go” into a number.

Where an exponential attenuation comes from is the same argument in every case: each step of the journey carries the same chance of ending it, so the surviving fraction falls by a constant factor per unit distance. That gives the absorption of light in glass, of neutrons in water and of X-rays in bone, and it is why every attenuation law in physics has the same shape whatever is doing the absorbing.

The decibel exists because of exactly this. Two mechanisms that multiply — spreading and absorption — become additive when logarithms are taken, so a propagation budget can be written as a sum: so many dB of spreading, so many of absorption, so many of scattering, so many of the receiver’s inefficiency. It is a bookkeeping convenience and it has one trap, which is that the decibel is defined on power: 20 dB is a factor of ten in amplitude and a hundred in intensity, and 6 dB per doubling of distance is the inverse-square law with the factor of two already applied.

The attenuation length also settles a question that comes up whenever a wave has to get through something. Seawater absorbs radio at metre wavelengths within a few metres, which is why a submarine communicating from depth needs wavelengths of tens of kilometres and a transmitter the size of a county. Tissue absorbs ultrasound at roughly half a decibel per centimetre per megahertz, which is why an abdominal scan runs at 3 MHz and gives coarse pictures and an eye scan runs at 20 MHz and gives fine ones. In each case the frequency is chosen by putting the required depth into an exponential and solving for what survives, and every imaging modality in medicine is that trade made once and built into an instrument.

Two worked cases at opposite ends #

The arithmetic is worth doing twice, because the two cases that matter most sit at opposite extremes of it.

A conversation across a field. Speech radiates about ten microwatts. At one metre that is spread over a hemisphere of four square metres, giving a couple of microwatts per square metre — roughly 60 dB above the threshold of hearing. At a hundred metres the spreading has cost 40 dB, the absorption a fraction of one, and the remaining 20 dB is comfortably above the threshold and hopelessly below the noise of any outdoor environment. The limit on shouting across a field is not the physics of propagation; it is that the wind and the birds got there first.

A spacecraft at the edge of the solar system. Voyager 1 transmits 23 watts into a 3.7-metre dish, which concentrates it into a beam about half a degree wide — a gain of some 48 dB over an isotropic source. At 24 billion kilometres the spreading loss alone is about 300 dB, and the signal arriving at a 70-metre antenna on Earth is around $10^{-19}$ watts. It is still readable, at a few tens of bits per second, because the receiver is cooled to reduce its own noise and the data are coded so that the decision about each bit can be made using thousands of others.

The contrast between the two is entirely in the exponents. Both signals obey $1/r^2$; one is measured in metres and the other in astronomical units, and 300 dB is what twenty orders of magnitude in distance costs.

That the second works at all is a statement about how little power a bit actually needs, not about any loophole in the geometry.

The exponent that comes back doubled #

Every case so far has the source at one end and the listener at the other. A great many instruments put both at the same end, and the arithmetic changes in a way that decides what those instruments can do.

A radar transmits a pulse, which spreads out to a target as $1/r^2$. The target intercepts a small fraction and re-radiates it, more or less in all directions — so the returning wave spreads as $1/r^2$ all over again on the way back. The received power therefore falls as

$$P_r \propto \frac{1}{R^4},$$

and the constant of proportionality contains the transmitter’s power, the antenna’s gain twice over, and the target’s radar cross-section.

The fourth power is brutal and it sets the shape of the whole field. Doubling a radar’s range costs sixteen times the transmitted power, or four times the antenna area, or a sixteenfold improvement in receiver noise. Every specification in the subject is an exercise in finding a factor of sixteen somewhere.

It also explains why stealth is worth so much less than the numbers suggest and so much more than that sounds. Range goes as the fourth root of the cross-section, so reducing an aircraft’s radar cross-section by a factor of ten thousand — an enormous engineering achievement — shortens the range at which it is detected by a factor of only ten. That is nonetheless decisive, because it turns a two-hundred-kilometre detection into a twenty-kilometre one, and twenty kilometres at the speed of an aircraft is not very long.

The same arithmetic gives a strategic asymmetry that has shaped naval warfare. Active sonar pays the fourth power; passive listening pays only the second, because the sound makes the journey once. So a submarine listening quietly detects a surface ship at a range far beyond the range at which that ship’s sonar could find the submarine — and the moment the submarine transmits, it becomes an emitter that can be heard at the same enormous passive range. The physics says: listen, and do not speak. Radar warning receivers on aircraft embody exactly the same inequality.

Every round-trip sensing method shares the exponent. Lidar, ultrasound imaging, seismic reflection surveys and echolocation all send a wave out and wait for a fraction of it to come back, and all of them are fighting a fourth power against a second.

Reading a distance off a brightness #

The spreading law is usually a nuisance and it is also one of the most productive measuring instruments in science, because it can be run backwards.

If the power a source emits is known, then measuring how much arrives per square metre gives the distance directly: $d = \sqrt{L/4\pi F}$. Nothing else is needed — no parallax, no travel time, no signal from the source about where it is.

That is the whole basis of the astronomical distance scale. An object whose intrinsic luminosity can be established some other way is a standard candle, and the entire ladder out to cosmological distances is this one exponent applied repeatedly. Cepheid variables were the first rung: their pulsation period is tightly related to their luminosity, a relation established by Henrietta Leavitt in 1912 from variables in the Magellanic Clouds, all at effectively the same distance. Measure the period, look up the luminosity, measure the flux, invert the inverse-square law. Type Ia supernovae are the next rung and reach far further, and they are calibrated against galaxies whose Cepheids can still be resolved.

The dominant systematic in the whole enterprise is exactly the distinction this essay is about. Dust between here and there removes light exponentially, and light removed by dust is indistinguishable, in a single measurement, from light thinned by distance. An object behind dust looks further away than it is. Correcting for it means measuring the colour as well as the brightness — since extinction is stronger in the blue — and the residual uncertainty in that correction is a substantial fraction of the disagreement about how fast the universe is expanding.

There is one consequence of the same law that is worth stating because it is so often got wrong. An extended object’s surface brightness does not fall with distance. Its total flux falls as $1/d^2$, and the solid angle it covers falls as $1/d^2$ as well, so the two cancel exactly: a wall looks equally bright from one metre and from ten, and a nebula’s brightness per unit area is the same through any telescope. Only its apparent size changes. That is the same conservation of radiance that bounds what a lens can concentrate, and it is why a bigger telescope shows a faint galaxy larger rather than brighter.

Where the model stops #

Near the source, none of it holds. The spreading laws assume the wave has got far enough away to look like it came from a point. Within a wavelength or so of a source there is a near field, in which the fields do not propagate at all but store and return energy, and in which the amplitude falls much faster than any of the exponents above. That region is why an antenna’s impedance depends on what is next to it and why a microphone close to a mouth picks up an enormous bass boost.

A beam is not a sphere. A laser, a searchlight, a dish antenna and a horn all confine the wave into a cone, and inside the cone the intensity falls far more slowly than $1/r^2$ — until the diffraction limit of the aperture takes over and the beam starts spreading anyway. The exponents in the figure are the free-space answers for an isotropic source, and every directional device is a deliberate departure from them.

Reflections change the accounting entirely. In a room, sound spreads spherically for the first few metres and then reaches a regime where the reflected energy dominates and the intensity stops falling with distance at all. The distance at which those are equal has a name, the critical distance, and past it moving away from a talker does not make them quieter — which is the acoustics of every reverberant hall and the reason a recording made across a room sounds distant rather than merely soft. The cue the ear uses for distance indoors is therefore not loudness at all but the ratio of direct to reverberant sound, which is a quantity the spreading law does not contain and which no amount of turning the volume up will change.

The medium can focus. Sound in the ocean travels in a channel where the speed has a minimum, and rays that would have escaped are bent back into it; the intensity there falls as $1/r$ rather than $1/r^2$, and a whale call has been detected across an ocean basin — a distance at which the spherical law would have put it forty decibels below anything measurable. The atmosphere does the same under a temperature inversion, which is why distant sounds carry so much better at night.

The same three exponents, drawn from somewhere else #

Nothing in the spreading argument mentioned waves. What was used was that something is conserved, that it flows outward, and that the surface it flows through grows in a particular way — and any quantity with those three properties obeys the same law.

Static fields do. The electric field of a point charge falls as $1/r^2$ because the flux is conserved and the sphere grows; the field of a charged wire falls as $1/r$ because the surface is a cylinder; the field of a charged plane does not fall at all. Those are the three exponents on this page, arrived at with no wave anywhere and with a conserved flux instead of a conserved power.

So does anything counted. The apparent brightness of a star, the dose from a radioactive source, the number of raindrops per second landing on a target, and the strength of the sun’s light at each planet all follow the same rule for the same reason, and they follow it whether the thing spreading is a wave, a particle or a probability.

The generality is worth one caution. What makes the exponent −2 is not three-dimensional space but a conserved flow in it. Anything that is created or destroyed on the way — a beam being absorbed, a population decaying, a signal being amplified — departs from it immediately, and the departure is the interesting part. The spreading law is the baseline that makes an absorption measurable.

The counting behind the inverse square is worth separating from everything else here, because it is the one part that is pure geometry. The same energy crossing shells at one, two and four times the radius gives intensities in the ratio 1, ¼ and 1/16, and the argument is identical whatever is being counted — which is why the same figure would serve in the electrostatics essays and in a discussion of a star’s apparent brightness. Its content is a statement about the area of a sphere, and it would be false in a space of a different dimension.

What the picture cannot show #

The plots on this page are of a single frequency, and every real wave is a band. Because absorption depends strongly on frequency, a broadband pulse does not merely get quieter with distance — it changes shape, losing its high frequencies first, and its peak amplitude falls faster than any single-frequency curve predicts. A figure drawn at one frequency cannot show a pulse turning into a rumble.

The logarithmic axes also hide how brutal the near part of the curve is. Half of the total spreading loss between a source and a listener ten metres away happens in the first metre. Anyone drawing a straight line on log axes should remember that the left-hand decade is a few centimetres wide in the world.

The domain of validity is: a source small compared with the distance, a uniform medium, no boundaries, no focusing, one frequency, and far enough out for the near field to have gone. That excludes most rooms, all oceans and every antenna measured on a bench — which is the ordinary condition of an idealisation that everybody uses anyway, because the departures are easier to name once the baseline is known.

The ladder from here #

Later rungs on this anchor: the energy and power of a wave derived from the wave equation rather than asserted; the near field and the transition to the far field; directivity and gain, which measure how far a source departs from isotropic; the reverberant field and the critical distance; and the ocean sound channel, where the spreading exponent is changed by refraction rather than by geometry.

The neighbouring ladders are what a travelling wave is, whose energy this essay follows, the inverse-square law for fields, which is the same counting argument about a different conserved quantity, and scattering, which is where the exponential part comes from for light.