The sandbars that reflect the sea

The gap a repeat opens stacked two transparent materials in alternating layers and found a band of frequencies the stack would not carry. No material absorbed them and neither had a resonance there. The repeat itself forbade them, because each interface reflected a little and, at the right wavelength, all the little reflections came back in step. Later arguments put a defect into the repeat, cut it so that its ends held states, and made it reflect from every direction. All of them concerned light, and one essay in the series, the frequency a lattice cannot carry, concerned a chain of masses.

The mathematics does not care what is waving. Any wave whose speed varies periodically along its path will be reflected when its half-wavelength matches the period, and one of the largest periodic structures on Earth is a bed of sandbars. On many gently sloping coasts the seabed carries not one bar but several, parallel to the shore and roughly evenly spaced, tens of metres apart. Waves travelling over them move faster where the water is deeper and slower over each crest. That is a periodic medium for water waves. It reflects the waves it resonates with, and the reflection may be part of how the bars came to be evenly spaced in the first place.

One wavelength sent back #

Water waves in shallow water travel at a speed $\sqrt{gh}$ that depends only on the depth, as the speed that depends on the length showed for waves long compared with the depth. Over a seabed of bars the depth varies up and down, and so does the wave speed, and the equation for the surface elevation becomes a one-dimensional wave equation with a coefficient that repeats. Where the depth changes, a small part of the wave is reflected, just as a small part of light is reflected where the index changes.

The figure computes it. The seabed carries a patch of sinusoidal bars, 50 metres apart and 0.8 metres in amplitude, in water 4 metres deep. A wave arrives from the open sea, is partly reflected and partly transmitted, and the equations are integrated through the patch to find how much comes back. For most wavelengths almost nothing does: the bars’ small reflections arrive back with scattered phases and cancel. At a wavelength near 100 metres, twice the bar spacing, they add. Four bars then reflect 56 per cent of the incoming amplitude, and eight reflect 86 per cent. The eight-bar peak is narrower, because with more bars the phases must match over a longer stretch, and a small change of wavelength spoils the match.

The condition is the one a thousand slits buy for a grating in reflection, and the one a layered mirror satisfies: a wave reflected from one bar and a wave reflected from the next differ in path by twice the bar spacing, which must be one wavelength. So the resonant surface wave is twice as long as the spacing of the bars. For bars 50 metres apart in 4 metres of water, that is a wave with a period of about 16 seconds, long ocean swell.

The waves in this problem are long compared with the depth — $kh$ is about 0.25 at resonance — which is where the shallow-water equations are a fair description. For shorter waves in deeper water the full theory of surface waves is needed, and the reflection is weaker because the wave feels the bottom less. The pattern survives: laboratory experiments in wave tanks with rows of ripples on the floor, by Heathershaw and Davies in the early 1980s, found reflection peaks at twice the ripple spacing with the sizes this kind of theory predicts.

A wave that dies inside the bars #

What the wave does inside the patch is the sharpest way to see that this is a band gap and not a loss.

The figure follows the amplitude of the resonant wave along a line from the open sea through eight bars to the shore side. In front of the patch the incoming and reflected waves overlap and make a partial standing wave: the amplitude swings between 0.15 and 1.85 of the incoming wave’s, with nodes 50.2 metres apart, half the wave’s length and the same as the bar spacing. Through the patch the amplitude decreases steadily from bar to bar. Behind it, 0.52 of the amplitude emerges, carrying 27 per cent of the energy.

Nothing in the equations dissipates energy. The integration checks that the reflected and transmitted energies add to exactly the incident, so the 73 per cent that does not emerge has been sent back. Inside the patch the wave is not a travelling wave at all but a decaying one, the water-wave equivalent of the field inside a mirror’s stack at a colour in its stop band. The mode that lives in the mistake put a defect in such a stack to trap a mode in the middle, and a bar field with one bar out of step would do the same for water waves, holding a patch of higher waves at the defect. Bar fields are not regular enough for that to be seen clearly, but the principle is identical.

The same decay is how a bar field shelters a coast. A beach behind a patch of bars resonant with the prevailing swell receives less of it, not because the waves broke but because they were turned back. Offshore breakwaters built as rows of low submerged ridges have been proposed on exactly this principle, with the ridges spaced for the waves they are meant to stop, and they work for the waves they are designed for and not for others. That narrowness is the next figure’s subject.

More bars, a stronger and narrower mirror #

The strength of the reflection and its bandwidth trade against each other as the number of bars grows.

The figure counts bars. One sinusoidal bar reflects 16 per cent of the amplitude at its best wavelength. Four reflect 56 per cent, ten 92, and twenty essentially all. At first each bar adds its small reflection in step with the others and the reflected amplitude grows in proportion to the count. Then the incoming wave is depleted as it goes, the later bars have less to reflect, and the reflection saturates towards totality. Coupled-wave theory gives that saturation as a hyperbolic tangent of the number of bars, and the computed points follow that shape.

The width of the reflected band, measured in wavelength, falls from 74 metres for a single bar to 13 metres for twenty, roughly as one over the number of bars. A single bar reflects weakly over a wide range, and a long field reflects strongly over a narrow one. The two are the same trade that sets what a thousand slits buy in a spectrometer, where more lines give sharper orders, and it limits what a field of bars can do for a coast. Real swell spans a range of periods, and the directions it arrives from vary, which changes the effective spacing the waves see. A natural bar field is at best a partial mirror for part of the incoming sea.

The spacing a sea asks for #

The resonance condition turns into a prediction about the seabed once real wave periods and depths are put in.

The figure uses the full dispersion relation of surface waves, $\omega^2 = gk\tanh kh$, to find the wavelength of waves of each period at each depth, and plots half of it: the bar spacing that would resonate. In 2 metres of water, waves with an 8-second period call for bars 17 metres apart and 12-second waves for bars 26 metres apart. In 10 metres of water the same waves call for 35 and 57 metres. Swell with periods of eight to fifteen seconds therefore resonates with spacings from about twenty metres to about seventy in the depths where bars form.

Those are the spacings of the multiple parallel bars found in sheltered seas and bays with long, gently sloping floors — the coasts of the Chesapeake Bay, the Gulf of St Lawrence and several enclosed seas have been described this way. The match is suggestive and not conclusive: bar spacings also depend on the tide, on how the slope changes offshore, and on storms that rearrange the whole profile, and many bar systems have spacings that increase offshore in a way the simple condition, applied at each bar’s local depth, partly explains.

A gap in an endless bed #

An infinitely long bed of bars shows the underlying structure most cleanly: a band of frequencies that cannot travel.

The figure computes, from the change a wave undergoes across one bar, the wavenumber a wave can have over an endless repetition of them — its Bloch wavenumber — at each frequency. Away from the Bragg frequency, where a wave’s half-wavelength matches the spacing, the waves travel with wavenumbers close to the flat-bottom ones. Near it, the curve breaks. Between 0.944 and 1.043 of the Bragg frequency, for bars 0.8 metres in amplitude, no real wavenumber exists, and a wave at those frequencies can only grow or decay along the bed. For bars half as high the gap is half as wide, from 0.973 to 1.021.

This is exactly the band structure the gap a repeat opens derived for light in a layered stack, with the depth playing the part of the refractive index. The width of the gap is proportional to the contrast in the medium, the bars’ height relative to the depth here and the index difference there. The strong reflection of a finite patch at these frequencies is the gap made visible: a wave entering a medium where it cannot travel decays, and what does not get through comes back.

Bars that build their own spacing #

The reflection matters for more than the waves, because the partial standing wave in front of a bar field acts on the seabed.

Under a standing wave the water near the bottom oscillates back and forth, most strongly under the antinodes and not at all under the nodes. Oscillating flow over a bed has a boundary layer, and within it the oscillation leaves behind a small steady current, the same effect that the drift a sound leaves behind described for acoustic waves in a tube. Under a standing water wave that steady streaming moves near the bed towards the nodes or towards the antinodes, depending on the height above the bed, and it carries sand with it. Sand gathers into bars with the spacing of the standing wave’s pattern, half the surface wavelength.

So the structure has a feedback loop. Suppose a few bars already exist and reflect part of an incoming swell. The reflected wave and the incoming wave make a partial standing wave in front of the field, with nodes spaced exactly as the bars are, and the streaming beneath it builds sand into the same pattern, extending the field seaward. More bars reflect more, strengthening the standing wave and the streaming. The spacing that emerges is the resonant one, half the dominant wavelength, because only at that spacing is the reflection strong enough to drive the growth. Mei and others worked out this mechanism in the 1980s, and wave-tank experiments with movable sand beds have seen patches of ripples grow in the direction of the incoming waves.

How much of real multiple-bar formation this explains is a separate question. Longshore currents, the undertow that returns water seaward after waves break, and the infragravity waves that set up standing patterns near the shore are all proposed as causes too. The Bragg feedback is attractive because it predicts the spacing without any tuning, and because it explains why bar fields tend to be most regular in sheltered places with steady, narrow-banded swell. On open coasts with storms from many directions, other mechanisms probably dominate.

The same condition, read by a radar #

The Bragg condition works in both directions between the sea and a wave sent to it. Here the sea’s surface waves were reflected by a periodic bed. A radar looking at the sea is a wave reflected by a periodic surface, and the same arithmetic picks out which part of the surface it sees.

The rough surface of the ocean can be decomposed into surface waves of every wavelength travelling in every direction. A radar beam striking it at a low grazing angle is scattered back towards the antenna mostly by the component of the sea whose wavelength is half the radar’s, measured along the line of sight: the reflections from successive crests of those waves arrive back in step, exactly as the reflections from successive bars did. Crombie noticed this in 1955 with a radio transmitter at a wavelength of about 22 metres, and saw that the echo from the sea was concentrated at a single Doppler shift. That shift was the speed of the 11-metre ocean waves moving towards and away from the radar, which is fixed by the dispersion relation of water waves. The echo from waves half the radar’s length, moving at their own speed, is a pair of sharp lines in the Doppler spectrum.

That is now an instrument. Coastal high-frequency radars, with wavelengths of ten to a hundred metres, watch the Bragg lines from ocean waves of five to fifty metres. Any shift of the lines away from the value the dispersion relation predicts is the speed of the current carrying the waves, measured from land over distances of a hundred kilometres or more. Networks of such radars map surface currents along many coastlines every hour, and the measurement rests on the same half-wavelength rule as the sandbars. It also rests on the reflected wave’s frequency being shifted by the motion of what reflects it, which the shift a mirror gives twice describes for any moving reflector.

The two cases are mirror images of one argument. The bars are fixed and select the sea wave twice their spacing; the sea waves move and are selected by a radar wave twice their length. In both, a periodic structure and a wave interact strongly only when one period fits two of the other, and everywhere else they barely notice each other.

What the shallow-water model leaves out #

Three limits of the figures matter.

Depth. The shallow-water equations assume waves much longer than the depth. At resonance here the wave is 25 times the depth, which is in their range; for shorter waves the full theory is needed, and it gives weaker reflection because the wave’s motion decays with depth before reaching the bars.

Direction and spectrum. The figures send a single wave straight at the bars. Real seas have a spread of periods and directions. A wave arriving obliquely sees the bars at a wider effective spacing, which moves its resonance, much as the mirror that works from every direction found tilting light moving a stack’s stop band. A spread of wave periods sees only part of its energy in the reflected band.

Nonlinearity and dissipation. Real waves over bars also steepen, break, lose energy to bottom friction and interact with currents. None of that is in the linear, inviscid equations here. They isolate the reflection, which is the part of the physics that a periodic seabed adds; the other effects add to it rather than replace it.

Still open: whether bar fields tune themselves to the sea #

The feedback between reflection and sand transport predicts that a bar field should grow towards the resonant spacing for the dominant waves, and wave-tank experiments show the mechanism operating. Whether natural bar fields are shaped mainly by it is not settled. Field measurements rarely capture the slow growth of bars over years alongside the wave climate that would test the prediction. Numerical models that couple wave reflection to sediment transport can generate bar fields from a flat bed, but they depend on how sediment transport is parameterised, which is uncertain by factors of two or more. Whether multiple-bar systems are self-organised Bragg reflectors, relics of breaking-wave processes, or both at different times on the same coast is argued with the data available, and the answer probably differs between sheltered seas and open coasts.

The habit worth carrying away is to recognise a periodic medium wherever a wave’s speed repeats along its path. Any wave crossing a repeat reflects the one wavelength whose half fits the repeat, and a finite repeat reflects a band around it with a strength and narrowness that trade against each other as the repeat lengthens. A stack of glass layers, a chain of masses and a seabed of sand are the same arithmetic. In the last of them the reflection can build the very repeat that produces it.