The even wave train that cannot stay even

The pulse two failures keep alive found a balance between two effects that each destroy a wave on its own: dispersion, which spreads a packet because its components travel at different speeds, and nonlinearity, which makes the wave’s speed depend on its height. At exactly one height for each width they cancel, and the result is a soliton that travels unchanged. The solitons a hump already contains found that a hump of any other shape resolves into a set of solitons, fixed in number and height before anything moves.

This essay starts from a wave that looks like the least interesting possible case: an endless, perfectly even train of identical waves, every crest the same height. It is a solution of the equations, and it seems as if the balance between dispersion and nonlinearity should leave it alone. In water that is deep compared with the wavelength, and in light travelling through an optical fibre with the right kind of dispersion, it does not. The even train is unstable, it comes apart into groups, and what it does next is one of the stranger stories in wave physics.

Ripples that grow on an even train #

The envelope of a train of deep-water waves, or of an intense light wave in a fibre, obeys the nonlinear Schrödinger equation: a dispersive term that spreads variations in the envelope, and a nonlinear term that shifts the local frequency in proportion to the local intensity. On deep water that shift raises the frequency where the waves are higher; in a fibre, the refractive index rises with intensity. Combined with dispersion of the appropriate sign, both have the same effect. A part of the train that is slightly higher than its neighbours bends the wave’s phase so that energy flows towards it, which makes it higher still.

The drawing gives the growth rate of a small modulation of each wavelength, computed from the equation linearised about the even train. Every modulation longer than a certain length grows, with a rate that rises from zero, peaks and falls back to zero at a wavenumber of twice the train’s amplitude. The fastest grows at a rate equal to the amplitude squared, so a train twice as high is unstable to modulations twice as short and destroys itself four times as fast. There is no amplitude small enough to be stable: every even train of finite height on deep water is an unstable equilibrium, like a pencil balanced on its point.

The instability was found in water independently by Thomas Brooke Benjamin and Jim Feir, whose experiments at the National Physical Laboratory in the mid-1960s could not make a regular train survive the length of their tank, and by several theorists at about the same time, and in optics it appears as the spontaneous break-up of a continuous laser beam in a fibre into a train of pulses. It is the reason the regular swell of textbook drawings does not exist on the open ocean over long distances.

Two sidebands fed by the carrier #

The same instability looks different, and perhaps clearer, in the language of frequencies. A modulation of an even train at wavenumber $K$ is the same thing as two weak extra waves added to the strong one, at wavenumbers a little above and a little below the carrier’s — sidebands, the pair that a heterodyne or any beating produces. The nonlinearity couples the three. Two quanta of the strong carrier can be converted into one quantum of each sideband, conserving energy because the sidebands’ frequencies sum to twice the carrier’s, and conserving momentum when the dispersion lets the mismatch be made up by the nonlinear shift. When that bookkeeping balances, the carrier pumps both sidebands at once and they grow together.

That is a parametric amplifier, the mechanism by which a swing is pumped rather than pushed: something periodic about the system itself, here the carrier’s intensity, modulates a property that the sidebands feel, and energy flows into them at a rate proportional to how much they already have. In optics the process is called four-wave mixing, and the modulational instability of a laser beam in a fibre is the spontaneous version of it, seeded by noise. The recurrence in the next drawing is the carrier running low and the sidebands giving their energy back, as the pump in a parametric amplifier is eventually depleted and reversed.

Why deep water and not shallow #

The instability needs the nonlinearity and the dispersion to have opposite effects on a group, so that a slightly higher part of the train focuses energy into itself rather than spreading it. On deep water they do. In shallow water they do not: when the water’s depth falls below about 1.36 divided by the wavenumber, the sign of the nonlinear term reverses, a higher part of the train spreads rather than gathers, and even trains become stable. That is the regime of shallow-water solitons, where a single hump can travel unchanged and a train stays a train. The dispersion that separates the two regimes is the one that makes wave speed depend on wavelength differently in deep and shallow water.

The same division appears in optical fibres. A fibre’s dispersion changes sign at a wavelength that depends on its design — about 1.3 micrometres for standard fibre — and a continuous laser beam on the long-wavelength side, where the dispersion is called anomalous, breaks up into pulses, while one on the other side does not. Telecommunications systems operate on the anomalous side, which is why the instability had to be understood and suppressed before long-distance fibre links could carry intense signals, and why it can be used deliberately to generate trains of short pulses from a steady beam.

The same shape of instability as a collapsing cloud #

The growth-rate curve has a familiar shape for anyone who has met a disturbance that grows instead of travelling. In a self-gravitating gas, pressure resists compression at short wavelengths and gravity wins at long ones, so disturbances larger than the Jeans length grow while smaller ones oscillate. In an even wave train, dispersion resists the gathering of energy at short modulation lengths and the nonlinear focusing wins at long ones, so modulations longer than a critical length grow. In both, a restoring effect that is strong at short scales competes with an attractive one that does not weaken at long scales, and the boundary between stable and unstable is a length set by their ratio.

The analogy runs further than the shape of a curve. A beam of intense light in a transparent medium whose refractive index rises with intensity suffers the same instability across its width rather than along its length: small ripples in the beam’s cross-section grow, and the beam breaks into filaments, each a narrow thread of light that focuses itself until something else — ionisation of the medium, in the case of ultrashort laser pulses in air — stops it. The filaments that intense lasers draw through the atmosphere, kilometres long, begin as a transverse modulational instability of an initially smooth beam.

A crest that grows and gives the energy back #

The linear growth rate says only how the instability starts. To see where it goes, the drawing integrates the full nonlinear equation from an even train with a ripple of one per cent at the fastest-growing wavenumber, by the split-step Fourier method, and follows the height of the tallest crest. For a while nothing visible happens; the ripple grows exponentially from a per cent, and on a linear scale it takes several time units to be noticed. Then it rises sharply. Near $t = 5.6$ all the energy of one modulation period has gathered into a single crest, 2.41 times the original height.

Then the unexpected happens. The crest does not break or keep growing; it spreads out again, and the train returns almost exactly to the even train it started as, its highest point within a per cent of the original height. The energy that was gathered into the crest is handed back to the whole train. And the cycle repeats: near $t = 17$ a crest rises again to the same height. The wave’s total energy — its norm, conserved by the equation and checked by the drawing to a part in a hundred million — never changes. Only its arrangement does.

The height of the crest, $1 + \sqrt2$, is not an accident of the numerics. The equation has an exact solution, found by Nail Akhmediev and colleagues in the 1980s, that describes precisely this: an even train, a modulation that grows, a crest, and a return. For the fastest-growing modulation its crest is exactly $1 + \sqrt2$ times the background, which the drawing checks against the integration.

A train that breathes #

Snapshot by snapshot the train breathes. At first it is even. A crest appears at the centre of each modulation period, with a deep trough on either side where the energy has been drawn from — the troughs falling almost to zero, to seven hundredths of the original height. Then the crest subsides, the troughs fill, and the train is even again, before the cycle repeats.

The return is the same phenomenon that Enrico Fermi, John Pasta and Stanislaw Ulam found in 1955 when they put energy into one mode of a chain of weakly nonlinear oscillators and expected it to spread among all the modes, as equipartition demanded: instead it spread into a few modes and then came back, energy that refuses to be shared. The nonlinear Schrödinger equation is integrable, like the equation that a hump’s solitons come from, and integrable equations do not thermalise: their motion is confined by infinitely many conserved quantities, and a modulation that grows must eventually undo itself. The recurrence has been seen in water tanks and, very clearly, in optical fibres, where the whole cycle can be followed over kilometres of fibre.

A wave out of nowhere #

Take the Akhmediev solution to the limit where the modulation’s wavelength becomes infinitely long, and something remarkable is left. Howell Peregrine found it in 1983: a solution that is an even train everywhere and at all times except in one small region of space and one short interval of time, where a single crest rises out of the train to exactly three times the background height, with two points of perfect calm on either side of it where the envelope drops to zero, and then sinks back and leaves the train exactly as it was.

It is localised in time as well as space. Long before its moment and long after it, the train is uniform; there is no warning and no aftermath. That property is what has made the Peregrine solution the leading mathematical model of rogue waves — waves more than twice the height of those around them, long dismissed as sailors’ exaggeration and now recorded by instruments, most famously the twenty-six-metre wave that struck the Draupner platform in the North Sea on New Year’s Day 1995 in a sea whose significant height was about twelve. The Peregrine breather was observed in an optical fibre in 2010 and in a water tank in 2011, rising to very nearly three times the background as predicted.

Whether real rogue waves on the ocean are Peregrine breathers is another matter. The ocean is not a single even train but a spread of wavelengths and directions, which weakens the modulational instability, and rogue waves can also arise from the chance superposition of many ordinary waves that happen to arrive in phase at one point — linear focusing, with no instability in it. Which mechanism dominates depends on how narrow the sea’s spectrum is, and the answer seems to be both, in different seas.

How long an even sea lasts #

For real water waves the equation’s units translate into the waves’ steepness — their amplitude times their wavenumber, the ratio that decides how nonlinear they are. The fastest modulation grows at a rate of half the wave frequency times the steepness squared, so its growth takes a number of wave periods that falls as the inverse square of the steepness. Gentle swell, with a steepness of a few hundredths, needs well over a hundred periods to grow by a factor of $e$ — many kilometres of travel — and by then it has been disturbed by wind, currents and crossing seas far more than by its own instability. A steep storm sea, at a steepness of a tenth or more, destroys its own regularity within a few tens of periods.

That is why the effect matters most in exactly the conditions where waves are steep and the sea’s spectrum is narrow: a long-crested swell running into an opposing current that steepens it, or a storm sea whose waves all come from one direction. These are the conditions in which unusually high waves are reported most often, and forecasting services now compute a measure of how prone the sea is to modulational instability — the Benjamin–Feir index, the ratio of the waves’ steepness to the width of their spectrum — alongside the wave height.

Where the equation stops #

Narrow band. The nonlinear Schrödinger equation describes an envelope varying slowly compared with the waves inside it. When the instability focuses energy into a crest a few waves long, the envelope is no longer slow, and higher-order versions of the equation, or the full equations of water waves, are needed; they predict crests that are asymmetric and somewhat different in height, and eventually breaking, which the equation cannot describe at all.

One dimension. The drawings follow a train along a line. Real seas spread in two dimensions, and a modulation can grow obliquely; the instability survives in two dimensions but is weaker when the sea’s energy is spread over many directions, and the recurrence largely disappears.

No dissipation, no wind. The equation conserves energy exactly. Real waves lose energy to breaking and viscosity and gain it from wind, and those change the recurrence into a more irregular sequence of focusings. Even so, the early growth is predicted accurately, which is what matters for how quickly an even sea stops being even.

What the envelopes do not show #

The envelopes drawn are the heights of the wave groups, not the waves themselves. Inside each envelope, individual crests travel at twice the speed of the group on deep water, the packet moving at another speed than its own crests, so a rogue crest in the ocean is not one wave that rises and falls but a succession of crests passing through the focus of the group, each of them briefly the tallest. And they do not show the spectrum: as a crest forms, the train’s spectrum broadens into sidebands on either side of the carrier, and in a fibre that broadening is a source of new colours, one of the mechanisms behind the white-light supercontinuum sources used in microscopy.

Still open: predicting a rogue wave #

Rogue waves are no longer disputed; how to predict them is. Statistics can say how often a sea of given steepness and spectral width produces a wave twice its significant height, and the Benjamin–Feir index changes that probability measurably. Predicting a particular rogue wave a minute ahead, at a particular place, would need a measurement of the surrounding sea detailed enough to integrate the equations forward, and radar systems on ships and platforms that measure the sea surface over kilometres are being tested for exactly that. Whether the ocean’s nonlinear dynamics are predictable over the minute or so of warning that would be useful, given how sensitively a focusing depends on the phases of the waves involved, is an open question.

The habit worth carrying away is to ask whether a uniform state is stable before treating it as the natural one. An even train of waves on deep water is a solution of the equations and an unstable one, and its instability does not dissolve it into randomness but lends its energy to a crest and takes it back — so a sea can produce a wave three times the height of its neighbours without anything arriving from outside, and give no sign of it before or after.