The arrival that keeps arriving

A wave is a shape that travels, and the equation it obeys is the same equation in any number of dimensions. This essay is about a property of its solutions that the equation does not advertise and that turns out to decide whether sound can carry a message.

The same equation, three answers #

The wave equation is $\nabla^2 u = c^{-2}\partial_t^2 u$ in every dimension, and the response to a point source firing once is called its Green’s function. With the speed set to one:

$$G_3 = \frac{\delta(t-r)}{4\pi r},\qquad G_2 = \frac{\theta(t-r)}{2\pi\sqrt{t^2-r^2}},\qquad G_1 = \frac{\theta(t-|x|)}{2}$$

The delta in the first is a spike of zero duration: everything arrives at $t=r$ and nothing arrives at any other time. The second is a function, not a spike, and it is nonzero for every time after the arrival. The third is a step and stays up for ever.

Nothing was assumed to get those. They are the solutions, and the difference between them is not a modelling choice.

It is worth reading the second one slowly, because its shape is the whole subject. It is zero until $t=r$, so nothing arrives early and the speed is the same $c$ in every dimension. At $t=r$ it is infinite, which is the front — an integrable infinity that a source of finite width turns into a peak of finite height. And after that it decreases, as $1/t$ for large $t$, without ever reaching zero. Three separate statements, and only the first two have counterparts in three dimensions.

The one-dimensional case has a feature of its own worth noticing: the response is a step, so the medium does not return to where it started at all. Strike a long rod on the end and the far end moves and stays moved. That is not a tail so much as a permanent displacement, and it is the same fact in its most extreme form.

The one-dimensional travelling wave is where everybody’s intuition is formed, and it is the one case where the shape is preserved exactly. That is worth stating carefully, because the statement that survives generalisation is narrower than it looks: a shape travelling unchanged is what a wave equation does in one dimension for a periodic disturbance. What it does to a single pulse, even in one dimension, is a different statement — and in two dimensions it is not true at all.

Spreading is the smaller difference #

There is a familiar difference between dimensions and it is the wrong one to reach for here. Energy from a point source in three dimensions is spread over a sphere, so the intensity falls as $1/r^2$ and the amplitude as $1/r$; in two it spreads over a circle, so the amplitude falls as $1/\sqrt{r}$.

Spreading is the smaller difference, and it is the one everybody already knows about. Amplitude falls with distance at a rate set by the number of dimensions, and every such curve is a statement about the size of the arriving signal. Nothing in it says anything whatever about the signal’s shape — which is the difference this essay is about, and the reason spreading is not an answer to it.

How a wave thins out is that argument, and it is about how much arrives. The tail is about when, and the two are independent: normalising each curve to its own peak, as the first figure does, removes the spreading entirely and leaves the difference in shape standing.

That is the point worth being careful about. A two-dimensional wave is not merely quieter or louder than a three-dimensional one. It arrives at the same time, at the same speed, and then does something the three-dimensional one does not do at all.

Where the tail comes from #

The reason is geometric and can be seen without any analysis, by asking what a two-dimensional point source is.

A disturbance in a plane, considered as a three-dimensional problem, is a disturbance uniform along the third axis: an infinite line of sources, all firing together. The listener hears the nearest point of the line at $t=r$, and thereafter hears the pair of points at slant distance $t$, which run off toward infinity as time goes on. The tail is the far end of that line arriving late.

The arithmetic confirms it exactly. Adding up the whole line numerically, with every source a sharp three-dimensional arrival and nothing about two dimensions assumed anywhere, reproduces the two-dimensional Green’s function to half a per cent at the times checked. This construction is Hadamard’s method of descent, and it is how the even-dimensional solutions are obtained in the first place: not solved for, but projected down from the odd-dimensional ones above them.

What Huygens’ principle actually claims #

Every point on a wavefront is a source of wavelets, and the next front is their envelope. The construction predicts refraction, diffraction and the law of reflection, and it is one of the most useful pictures in the subject.

Huygens’ construction draws circles from points on one front and takes the next front as their envelope, and what it asserts silently is the part that matters: that there is nothing behind the front. The medium the wave has already passed through is taken to have returned to rest. In three dimensions that is true and the construction is exact; in two it is false, and the construction is a good approximation whose error is precisely the tail.

The silent assertion is the one that fails. In two dimensions the medium behind the front is not at rest and never becomes so; the tail is precisely the disturbance the construction says is not there. The strict statement — that a sharp disturbance stays sharp, so the solution at a point depends on the source only on the light cone and not inside it — is called Huygens’ principle in the strong sense, and it is a theorem about dimension: it holds in three, five, seven and every odd number above one, and fails in every even one and in one dimension.

Huygens could not have known. The construction was published in 1690 and the theorem is nineteenth- and twentieth-century work; and in three dimensions, which is where he was working, it happens to be true.

Why odd dimensions and not even ones #

The pattern — sharp in three, five and seven, diffuse in two, four and six, and diffuse in one — is not a coincidence of three separate calculations. The solutions in $d$ and $d+2$ dimensions are related by one differential operator: apply $(1/r)\,\partial_r$ to the $d$-dimensional Green’s function and the $(d+2)$-dimensional one comes out, up to a constant.

So the whole family is generated from two starting points. One dimension gives three, five, seven; two dimensions gives four, six, eight. A delta function stays a delta function under that operator, so everything descended from the one-dimensional case is a clean front — and the one-dimensional case itself is the awkward member, a step rather than a spike, because it is the bottom of the ladder rather than a rung on it. Everything descended from the two-dimensional case keeps a tail, because a tail differentiated is still a tail.

That is also why the question is usually invisible. Almost every wave anybody meets is either a periodic disturbance, where the tail has nothing to be distinguished from, or a three-dimensional one, where there is no tail. The two-dimensional impulse response is the case in which the difference has somewhere to show.

How long the tail lasts #

A tail that decays is not necessarily a tail that stops mattering, and the rate decides which.

A power law and an exponential are different kinds of ending. An exponential decay has a time after which it is gone; a power law has only a time after which it is small, and ten times the wait buys a tenth of the amplitude and no more. Twelve pulse-lengths after the front has passed, the two-dimensional signal is still a tenth of its peak.

That is what decides whether a second event can be told from the remains of the first.

A message is a sequence of events, and in two dimensions each event is heard on top of the ones before it. Speech in a two-dimensional world would not work — not because the medium is lossy or the room is bad, but because the wave equation in an even number of dimensions does not deliver a sequence as a sequence. The sharpness that makes an echo a separate thing from the sound that caused it, and that makes a syllable a separate thing from the one before it, is a consequence of a three that could have been a two.

Why a picture is possible #

The essay’s remark about speech has a visual counterpart that is worth drawing out, because it is the reason anything can be seen at all.

The field of a moving charge at a given place and instant depends on where that charge was at one earlier moment — the retarded time — and on nothing else about its history. That is a direct consequence of the three-dimensional Green’s function being a spike: the source contributes only on the light cone, so the field carries a single snapshot of the source rather than a weighted memory of it.

Which is what makes an image. Light arriving at an eye from a scene carries the state of each point of that scene at one instant of that point’s own past; the points are at different distances, so those instants differ, but each contributes one. Nothing arrives from a point’s earlier or later history, so the received field is a picture rather than a smear.

Run the same argument in two dimensions and it collapses. The field there depends on an integral over the source’s entire past, weighted by the tail — so every point of a scene would contribute not its state at one moment but a fading average of everything it has ever done. There would be no instant to see, and no amount of optics could recover one, because the information was combined before the light left.

That is a strong statement about a very quiet property of the wave equation, and it generalises the speech argument rather than repeating it. The sharpness of the odd-dimensional Green’s function is what makes it possible for a signal to carry a state rather than a history — for sound to carry syllables, for light to carry images, and for a radar to carry a range.

The tail that curvature puts back #

The theorem above is a statement about a wave equation in flat space, and it does not survive being taken seriously in the presence of gravity.

In a curved spacetime the wave equation acquires terms involving the curvature, and its Green’s function acquires support inside the light cone as well as on it. Part of the disturbance travels along the cone and arrives sharply, as before; part of it is scattered by the curvature on the way, arrives late, and keeps arriving. Huygens’ principle in the strong sense fails, in four dimensions, in any spacetime that is not flat.

The size of the effect is set by how much curvature the wave crossed. For a signal travelling through the solar system the tail is minute; for a disturbance of a black hole it is not. Ring a black hole and the signal has three parts in sequence: a sharp burst, then a decaying oscillation at the hole’s own frequencies, and then — after the oscillation has died — a power-law tail that goes on for ever. The tail is backscatter off the curvature outside the hole, and its exponent depends on the multipole of the disturbance rather than on anything about the source.

That is the same phenomenon as this essay’s two-dimensional tail arriving by a completely different route. In the flat two-dimensional case the disturbance behind the front comes from parts of an extended source arriving late; in the curved four-dimensional case it comes from the wave scattering off the geometry it is travelling through. Both leave a signal that never quite ends, and in both the ending is a power law rather than an exponential.

It also means the sharpness the previous section leaned on is a property of flat space rather than of three dimensions alone. An image is possible because spacetime is very nearly flat where the light crossed, not because the dimension is right — and near something strongly curved, it is not.

The tail as an instrument #

A tail is usually a nuisance and here it carries information the front does not.

The two-dimensional response falls as $1/\sqrt{t^2-r^2}$, and the distance appears in that expression twice: once as the moment the front arrives, and once as the scale over which the tail decays. A receiver that hears both therefore has two independent estimates of the range from a single impulse and a single sensor — which a three-dimensional receiver does not have, because its impulse response contains only the arrival time.

Everything here is linear, which is what makes the tail an instrument rather than a nuisance. A message is a sum of impulses and the received message is the sum of their responses, so a known tail can be subtracted — and filling in the gap left by a missing measurement is the same operation run backwards. Seismology and medical ultrasound both depend on it, and both would be impossible if the tail were not reproducible.

This is used. Guided seismic waves in the crust are effectively two-dimensional at long range, and the coda that follows a first arrival is read for distance as well as for structure; a plate inspected with an ultrasonic pulse gives back a tail whose shape locates a flaw along the plate. In both, the practical work is the same: model the tail the geometry demands, and treat what is left over as the thing being measured.

Where two-dimensional waves actually are #

The case is not hypothetical. Waves on a membrane, on a plate, on the surface of shallow water, and along an interface are two-dimensional, and they behave this way.

A drum is the everyday one. The long ring after a strike is partly its own modes — whose frequencies are not in a harmonic series — and partly this effect, and the two contributions are hard to separate by ear. That is one reason the tail went unremarked for so long in a phenomenon everybody has heard: it arrives mixed with something else that also rings.

A struck plate rings on far longer than its damping alone accounts for. Ripples on a pond keep arriving at a floating leaf long after the first ring has passed it, and the ones arriving late were not made late — they left with the rest. A seismic wave guided in the crust arrives smeared for the same reason, and the smearing has to be undone before the arrival time is usable.

Circular fronts expanding from a stationary source are the picture almost everybody has of a wave in a plane, and it is the picture that hides the point. What it cannot draw is that the water between the rings is not flat. In two dimensions the medium behind the front is still moving, and that residual motion is the tail — invisible in a diagram of crests, and the entire subject of this essay.

Why one dimension is diffuse too #

The one-dimensional case looks like an exception and is not. A step that stays up for ever is the extreme version of a tail that never ends, and the reason is the same: a point in one dimension is a plane in three, and a plane of sources keeps delivering sound from further and further out along it, for ever, with no decay at all.

One dimension is diffuse too, and it is worth ending on because it corrects the intuition the essay started with. A pulse arriving at a change of medium reflects and transmits with its shape intact, which is why one-dimensional intuition feels so clean — but that is intuition about a pulse train in a bounded system rather than about a free pulse in an unbounded one. Widen the setting and the cleanliness goes.

That the sharp case is the odd-dimensional one, and that it improves as the dimension rises through the odd numbers, is worth stating plainly: three dimensions is the smallest number in which a wave can carry a message cleanly, and the world sound is listened to in is the first one that works.

And why a two-dimensional simulation misleads #

There is a practical consequence for anybody who computes wave problems, and it catches people out regularly.

A three-dimensional wave calculation is expensive, and the standard economy is to run it in two dimensions — take a cross-section, assume nothing varies along the third axis, and solve there. For questions about geometry that is often defensible: reflection angles, interference patterns, the shape of a diffracted front.

For anything about a transient it is not, and the reason is this essay. The two-dimensional calculation has a tail the three-dimensional problem does not have, and the tail is not a small correction — it is a signal at a tenth of the peak, twelve pulse-lengths after the arrival, decaying as a power law. A reverberation time estimated from a two-dimensional model, an impulse response computed from one, or a signal-to-noise figure for a pulsed system will all be wrong, and wrong in the direction of predicting a worse result than reality.

The confusion is compounded by the fact that the error looks like physics. A computed tail is smooth, plausible and reproducible, and it is easy to attribute to scattering, to the boundaries or to numerical dispersion — all of which produce something similar and none of which is responsible. The way to tell is to remove them: run the calculation in an unbounded, uniform, non-dispersive medium, and if a tail remains it is the dimension.

The general form of the caution is worth stating beyond simulation. A model reduced by a symmetry has not merely lost the detail in the suppressed direction; it may have changed the kind of solution the equation has. Two dimensions is not three with something ignored.

What the pictures cannot show #

Every calculation here is lossless, uniform and unbounded. A real medium absorbs, which cuts the tail off; a real room has walls, which add a much larger reverberation of a different origin. The tail is what remains when both are removed, and it is why a measurement in an anechoic chamber of a plate’s response still shows one.

The source is a Gaussian pulse of finite width. A true impulse has no width, and in three dimensions its response is a delta function that no figure can draw. The width is a stand-in for a real source and the tail’s shape does not depend on it, which was checked by varying it.

Nothing here is dispersive. Water waves, plate waves and most real two-dimensional systems also have a wave speed that depends on wavelength, which smears a pulse in its own way — a packet that will not keep its shape — and in a real measurement the two effects arrive together. The tail computed here is present in a medium with no dispersion at all.

And the three-dimensional case is sharp only for a wave in free space. A wave in a duct or an optical fibre is guided, its speed depends on frequency, and its impulse response has a tail from the guiding rather than from the dimension. The channel with no walls is the same equation with a boundary in it.

The ladder from here #

Later rungs on this anchor: the method of descent stated properly, and the Hadamard theorem about which dimensions are sharp; the wave equation with a source distribution rather than a point, and the retarded integral that solves it; the same question for the Klein–Gordon equation, which is diffuse in every dimension because it has a mass term; and the electromagnetic case, where the sharp three-dimensional Green’s function is what makes a field point where the charge is now.

The neighbouring ladders are how a wave thins out, which is the amplitude question this essay sets aside, and Huygens’ construction, whose hidden assumption this essay is about.