The packet that moves at another speed than its own crests

A group of waves crossing deep water does something that takes a minute of watching to believe, and the group does not keep its shape while doing it. The group as a whole advances. Individual crests advance faster, running forward through the group from its back edge, growing as they reach the middle, shrinking as they approach the front, and disappearing. Crests are continuously created at the rear and destroyed at the front, and the group outlives all of them.

Two speeds, in one wave, both genuine. The crests move at the phase velocity; the group moves at the group velocity; and on deep water the second is half the first.

Why there can be two #

A single infinite sine wave has one speed and no group. It stretches from one end of the universe to the other, every crest identical, and there is nothing about it that could arrive anywhere — which is the point at which the idealisation stops being useful, because every real wave is a finite disturbance.

A single sine wave is the object with only one speed. It has no beginning and no end, every crest is identical to every other, and nothing about it could be said to arrive — so the question of how fast it carries anything does not arise. Two speeds require a shape, a shape requires more than one frequency, and everything in this essay follows from that.

A finite disturbance is a sum of sine waves with a range of wavelengths, added by superposition. The figures here build one explicitly: sixty-one components with a Gaussian spread of wavenumbers around a central value, added point by point.

If every component travelled at the same speed, the sum would travel at that speed too, unchanged, and there would be nothing to discuss. The two speeds appear only when the medium gives different wavelengths different speeds — when it is dispersive. Then the components drift relative to one another, and the shape their sum makes moves at a speed that is not any of theirs.

That control case is worth its space. The two-speed behaviour is often described as though a wave group were an inherently strange object. It is not: in air, in vacuum, on a taut string, groups travel at exactly the speed of their own crests and nothing surprising happens. Water is strange, and so is glass, and so is a plasma.

The count that has to balance #

There is a bookkeeping consequence of the two speeds that is easy to miss and hard to unsee afterwards.

If crests move forward through a group, appearing at the back and vanishing at the front, then the number of crests inside the group is not conserved. Count them at one instant and again a few seconds later and the total is the same only because the group’s length is unchanged — but no individual crest has survived the interval. The group is a persistent object made of a continuously replaced population, in the way a wave on a rope is a persistent shape made of continuously replaced material.

That is the same distinction the travelling wave rung draws between the medium and the shape, applied one level up. There, the water does not travel and the shape does. Here, the crests do not travel with the group and the group does. Each level has something that moves and something that only appears to, and identifying which is which is most of the difficulty of the subject.

The slope and the chord #

Both speeds are read off one curve — the dispersion relation $\omega(k)$, which is the medium’s whole contribution to the subject.

$$v_p = \frac{\omega}{k}, \qquad v_g = \frac{\mathrm{d}\omega}{\mathrm{d}k}.$$

The picture makes the relationship between them geometric rather than algebraic. A straight relation through the origin has its chord and its tangent equal everywhere, which is the non-dispersive case. A relation that bends downward has a tangent shallower than its chord, so the group lags the crests. A relation that bends upward — capillary ripples, where surface tension rather than gravity restores the surface — has the group outrunning them, and in a patch of ripples the crests appear at the front and run backwards through the group.

For deep water, $\omega = \sqrt{gk}$, so $v_p = \sqrt{g/k}$ and $v_g = \tfrac12\sqrt{g/k}$ exactly. The factor of a half is what the hero figure measures as 0.51, having located an envelope peak and a crest at two times and divided by the interval, without either formula being used.

Beats are the same thing with two components #

The simplest packet has two components rather than sixty-one, and it is a figure this site already carries.

Two frequencies close together give an envelope pulsing at their difference — a slow modulation that neither component contains. That is the simplest packet there is, and it already has the two speeds: the crests inside the envelope move at one rate and the envelope at another, and adding more components narrows the envelope without changing the arithmetic.

Two waves of nearby frequency give a carrier at the mean frequency inside an envelope at half the difference frequency. The carrier’s crests move at the mean phase speed. The envelope moves at $\Delta\omega/\Delta k$ — a difference quotient, which is the group velocity before it is refined into a derivative.

That makes beats the two-component special case of everything on this page, and it is the cheapest way to get the idea into the ear rather than the eye: two tuning forks a few hertz apart produce an audible throbbing that neither fork emits.

Where two components are in step the sum is large and where they are out of step it cancels — and since they move at different speeds in a dispersive medium, the places where they agree move at a speed of their own. That is the group velocity, and it is worth seeing that it is a property of the pattern of agreement rather than of anything travelling.

Which speed is the real one #

Both are, and they answer different questions, but only one of them can carry information.

A crest is not a thing. It is a place where two or more components happen to be in step, and that place can move at any speed whatever — including faster than light, in media where the phase velocity exceeds $c$, which happens routinely. An X-ray beam in glass and a radio wave in the ionosphere both have phase velocities above the speed of light, and neither carries a signal that fast, because no information is attached to a crest. A crest that arrives is indistinguishable from the crest before it.

Information requires a change — a beginning, an end, a modulation — and a change is a feature of the envelope. The envelope moves at the group velocity, which in every ordinary medium is below $c$, and that is the resolution of an apparent paradox that would otherwise have relativity in trouble.

The honest version has one further qualification. In strongly absorbing media the group velocity itself can exceed $c$ or turn negative, and experiments have shown pulses whose peak emerges from a cell before the incoming peak entered it. Nothing is transmitted faster than light in those experiments: the medium reshapes the pulse, amplifying its leading edge and absorbing its tail, so the peak that leaves is built from the part of the input that had already arrived. The front of a disturbance — the first non-zero disturbance of any size — never travels faster than $c$ in any medium, and that front, not the peak, is what carries a signal. How long a train of waves stays in step with itself is a separate question again, and the one that decides whether two beams can interfere at all.

Where the packet goes as it travels #

A packet in a dispersive medium does not merely move; it spreads. Components with different wavelengths travel at different speeds, so a group that starts compact is longer later, and the spreading rate is set by the curvature of the dispersion relation rather than by its slope.

The two figures above are drawn at 1.6 s apart, over which the spreading is small enough to be hard to see. Over longer times it dominates. It is why the swell arriving on a beach from a distant storm is sorted by wavelength, the long waves arriving first, sometimes days ahead of the short ones — an ocean acting as a spectrometer over thousands of kilometres. Reading the arrival times backwards gives the distance to the storm, which is how the first estimates of Southern Ocean storm positions were made from Californian beaches.

It is also the limit on an optical fibre. A pulse of light is a packet, glass is dispersive, and a pulse that starts one nanosecond long arrives longer. Pack the pulses too close and they merge, so the dispersion of the glass sets the maximum bit rate over a given length — which is why fibre systems are specified in bandwidth times distance rather than in bandwidth, and why the wavelength around 1.3 micrometres, where silica’s dispersion passes through zero, was worth an industry rearranging itself around.

The observation that came before the theory #

The half was noticed on water long before it was derived, and by people who were not looking for it.

Scott Russell, riding along a canal in 1834, followed a solitary wave for two miles and started the study of solitons. The related observation about groups is Stokes’s and Reynolds’s: that a group of waves advances at half the speed of the waves composing it, so that crests seem to march through it. Reynolds gave the energy argument in 1877 — a group carries energy, the energy in a deep-water wave travels at half the phase speed, so the group must too — and Rayleigh gave the general derivation, showing that $\mathrm{d}\omega/\mathrm{d}k$ is the quantity in question whatever the medium.

What makes it a good historical example is that the phenomenon is freely available to anyone standing on a bridge and is almost never seen, because seeing it requires knowing that there are two things to watch. A reader who has followed the argument on this page and then looks at a canal will see crests running forward through a group within a few seconds. The same reader, a week earlier, would have seen waves.

That gap between looking and seeing is the usual state of affairs with wave phenomena, and it is the reason this site draws two frames rather than one: the second frame is what makes the two speeds a measurement rather than a claim.

How the packet in these figures is built #

The construction is worth stating because it is the assertion the figures make, and because the choice of components is not arbitrary.

Sixty-one sine waves are added, with wavenumbers spread symmetrically about a central $k_0$ and amplitudes following a Gaussian of width $0.14\,k_0$. Each component is given the angular frequency the stated dispersion relation assigns it, and the sum is evaluated at every plotted point at each of the two times.

The envelope is not fitted. It is the quadrature sum: the same components added again with sines in place of cosines gives a second curve, and the square root of the sum of the squares of the two is the analytic envelope. That is exact rather than approximate, which matters because the envelope’s peak is one of the two things being measured.

Both speeds are then measured, not calculated. The envelope’s maximum is located by a search refined to six figures at each time, and the difference divided by the interval is the group speed. The crest nearest the peak is located the same way, and its counterpart at the later time is found near where the phase speed would put it. The numbers printed on the figure — 1.26 m/s and 2.47 m/s, ratio 0.51 — come out of that procedure. The closed forms $\sqrt{g/k_0}$ and half of it appear nowhere in the drawing, and the agreement between the drawn measurement and the theoretical half is the check the figure exists to make.

The residual is the honest part. The measured ratio is 0.51 rather than 0.500 because a packet of finite width samples a range of $k$ over which $\mathrm{d}\omega/\mathrm{d}k$ varies, so the envelope’s speed is an average over the spectrum rather than the derivative at the centre. Narrowing the spread drives the ratio toward 0.500 and lengthens the packet until it no longer looks like a group — which is the same trade the uncertainty relation describes, appearing here as a property of a drawing.

What the figures cannot show #

The envelope drawn around the wiggles is computed exactly, by taking the quadrature sum of the same components, which is the analytic envelope rather than a curve fitted by eye. Three things are still outside the drawing.

The spreading, at this timescale. Two frames 1.6 s apart show the two speeds cleanly and the broadening hardly at all. That is a choice: a time long enough to show the packet spreading is long enough for the tracked crest to be ambiguous, and the figure prefers the measurement it can make honestly.

Which crest is which. The measurement follows the crest nearest the envelope peak, then looks for it again at the place the phase speed predicts. That is a sound procedure and not an infallible one — in a packet only a few wavelengths wide, crests are born and die within the group, and following “the same crest” through the whole journey is not always meaningful.

The third speed. Energy in a wave travels at its own velocity, which for the media here coincides with the group velocity and does not in general. Where they differ — in absorbing or anisotropic media — the phrase “the speed of the wave” has three answers and each of them is the right one for a different question.

The wake behind every boat #

The most public consequence of the half is the shape of a ship’s wake, and it is a shape that does not depend on the ship.

Every vessel on deep water — a duck, a rowing boat, a supertanker — trails a V whose half-angle is 19.47°, independent of its speed and its size. The number is $\arcsin(1/3)$, and the $1/3$ comes from the two speeds on this page: the pattern is built from waves whose crests travel at the phase speed while the energy that sustains them travels at half of it, and the interference of the components a moving disturbance generates is stationary only inside that wedge.

The independence of speed is the striking part. Nearly every other wake phenomenon — a bow wave, a shock cone, the Mach cone of a supersonic aircraft — has an angle that narrows as the source goes faster, because the source is outrunning a fixed wave speed. Deep-water waves have no fixed speed: long waves go faster, so a faster ship simply makes longer waves, and the geometry comes out the same. The angle is a property of the dispersion relation rather than of the ship.

It fails where the relation changes. In shallow water, where the speed becomes $\sqrt{gd}$ for every wavelength and dispersion disappears, the wedge widens as the vessel approaches that speed and the pattern collapses into a bow shock — which is why a boat in a canal drags a wall of water and why river craft are speed-limited by the depth rather than by their engines.

Seventeen metres per second #

The section above dealt with the group velocity made larger than $c$ and found nothing was being signalled. The opposite extreme turned out to be far more useful, and it is the sharpest demonstration available that the two speeds are genuinely independent quantities.

The group velocity is $\mathrm{d}\omega/\mathrm{d}k$, so it is small wherever the dispersion curve is nearly flat — where a large change in wavenumber buys almost no change in frequency. Ordinary transparent media never manage that, but a medium can be made to, by using one laser to open a narrow transparency window in an otherwise opaque atomic transition. The window is narrow in frequency and the index swings steeply across it, which is exactly a flat stretch of the curve.

Hau’s group did this in a sodium condensate in 1999 and measured a light pulse crossing the cloud at 17 metres per second — slower than a bicycle, in a medium whose phase velocity was still essentially $c$. The group index was of order $10^7$.

Turning the coupling laser off while the pulse is inside stops it altogether: the pulse’s information is held in the atoms’ internal states and is released, as light, when the laser is turned back on. Nothing about that is a violation of anything on this page. The envelope’s speed and the crests’ speed were never the same quantity, and this is the case that separates them by seven orders of magnitude.

Where the ladder goes next #

The rungs from here: the derivation of $v_g = \mathrm{d}\omega/\mathrm{d}k$ from two components taken to a limit; capillary waves, where the group outruns the crests and a ship’s wake gets its shape from both regimes at once; wave dispersion as a spectrometer, in the ocean and in a fibre; the uncertainty relation between a packet’s length and its spread of wavenumbers, which is a theorem about Fourier transforms rather than about physics and which becomes Heisenberg’s when the packet is a particle; solitons, where a nonlinearity exactly cancels the spreading and a packet keeps its shape indefinitely; and the quantum case, where a particle’s phase velocity exceeds $c$, its group velocity is the particle’s own speed, and the two-speed structure on this page is the reason the wave picture survives.

The idea to carry forward is the reading of the curve. One graph of $\omega$ against $k$ contains both speeds — a chord and a tangent at the same point — and the entire question of whether a medium does anything interesting to a wave is the question of whether that graph is a straight line through the origin.