The surface that cancels whatever skims it

The bend at the boundary derived Snell’s law from the requirement that a wave on both sides of a surface agree about its phase along the surface. The law that only asks about one component made that the whole of the argument: a boundary cannot change the part of the wave that runs along it. Later essays followed the same constraint into absorbing media, negative refraction, time-varying boundaries and, in the index that falls below one, X-rays, for which every material reflects only at grazing angles.

How much a surface reflects, as opposed to which way the reflected and transmitted waves go, is the business of Fresnel’s equations, which the direction of the shaking used to explain why reflected glare is polarised. This essay takes one limit of those equations, the wave that arrives almost parallel to the surface, and finds that it erases every difference between materials. At grazing incidence glass, water, soil and polished metal all reflect alike — completely, and with the wave turned upside down. A wave travelling along any surface meets its own inverted reflection there and cancels. The consequences reach from a nineteenth-century demonstration with a single mirror to the reason mobile phone masts are tall.

Every surface at the edge #

For a wave going from a medium of index $n_1$ into one of index $n_2$, the amplitude reflected with its electric field along the surface is

$$r_s = \frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t},$$

and the amplitude reflected with its field in the plane of incidence has the cosines and indices paired the other way. Straight on, the two are equal in size and depend on the materials: a fifth of the amplitude for glass, four per cent of the power, and four-fifths for water at radio frequencies, where its index is about nine. Going round towards grazing, $r_p$ falls to zero at Brewster’s angle, where the reflected light is purely s-polarised, and changes sign.

Then, at grazing incidence, $\cos\theta_i$ goes to zero. In both formulas the terms containing it vanish, the numerator and denominator are left equal and opposite, and both coefficients go to exactly −1, whatever $n_2$ is — provided only that it is not equal to $n_1$. At 89 degrees glass already reflects 97 per cent of the amplitude with its sign reversed, and water 99.6 per cent.

The reason is the same constraint as the earlier essays, read at its limit. The incident wave’s wavenumber along the surface is $k\sin\theta_i$, which at grazing is its whole wavenumber; its wavenumber across the surface, $k\cos\theta_i$, goes to zero. The transmitted wave must match the first, and the second medium, any second medium, then responds with a wave whose variation across the surface is not small at all. Seen from the incident side, a wave that barely varies across the boundary meets one that varies rapidly, and the mismatch of impedance across the boundary is effectively infinite, as at the end of a string fixed to a wall that what happens where the medium changes found reflects everything inverted. A finite difference between two media becomes, for a grazing wave, the largest difference there is.

A dark fringe where the paths are equal #

Humphrey Lloyd, in Dublin in 1834, put a slit of light just above a flat mirror and looked at a screen beyond it. Light reached the screen directly and by reflection from the mirror at a grazing angle, as though from the slit and from its image below the mirror, and the two made interference fringes, a simpler arrangement than Fresnel’s prisms or Young’s two slits for the same effect. The decisive observation was at the edge of the pattern. Where the screen meets the plane of the mirror, the direct and reflected paths are exactly equal in length, so if reflection did nothing to the wave the fringe there would be bright. It is dark.

The reflection had turned the wave over. Lloyd’s mirror was the first direct demonstration of the half-cycle phase change at reflection, and the fact that it was seen at grazing incidence on any mirror at all — not only on a glass surface from the air side, as the usual rule about denser media says — is what the Fresnel coefficients predict: at grazing incidence the sign reversal does not depend on which medium is denser. The two beams in Lloyd’s arrangement are coherent because they come from one source, which is what two lamps can never be; and the reflected beam, inverted, makes a node exactly on the mirror.

A signal that falls with the fourth power #

The same geometry, scaled up by a factor of a million, governs radio over flat ground. A transmitter on a mast and a receiver near the ground are linked by a direct path and by a path reflected from the ground between them, and at any distance much larger than the mast’s height the reflection is at grazing incidence. The ground’s coefficient is then −1, whether the ground is wet soil, dry sand or sea water.

The two paths differ in length by about $2h_t h_r/d$, where $h_t$ and $h_r$ are the antennas’ heights, and that difference shrinks with distance. Close to the mast it is many wavelengths, and as the receiver moves the two waves pass in and out of step, so the signal swings between nearly double and nearly nothing. Beyond the break distance, $4h_t h_r/\lambda$ — 540 metres for a 30 metre mast and a receiver at head height at 900 megahertz — the difference is less than half a wavelength and still shrinking. From there on the reversed reflection cancels the direct wave more completely with every metre, and since the residue is proportional to the shrinking path difference, which goes as 1/d, the field falls as $1/d^2$ and the power as $1/d^4$, forty decibels per decade instead of free space’s twenty.

The ground, which might have been expected to add a second path, subtracts the first. It is the main reason a radio signal across open country fades so much faster than the inverse-square law, and engineers planning a mobile network use the two-ray model as a first estimate of coverage. Its remedy is in the break distance: raise either antenna and the path difference grows, the break moves outward, and the fourth-power fall starts later. Masts are tall, and a phone held up gets a better signal than one held at the waist, for this reason as much as for the obvious one of clearing obstacles.

A radar that cannot see the horizon #

Turned upward, the same interference makes a radar’s coverage over the sea into a fan of lobes. The direct beam and its reflection from the water combine with a phase that depends on the elevation angle, giving maxima of twice the free-space field and nulls between them, spaced in angle by $\lambda/2h$ for an antenna at height h. Because the reflection is reversed, the first null lies exactly along the sea surface: whatever the radar’s power, its field at zero elevation is zero.

A target skimming the sea is therefore in the worst possible place, below the first lobe, where the field grows only in proportion to the target’s height. A radar sees by two such passages, out to the target and back, so its echo falls as the fourth power of the target’s height above the water at a given range. Ships’ radars are mounted as high as possible and use short wavelengths to bring the first lobe down towards the horizon, and missiles designed to approach a ship unseen fly a few metres above the waves for this reason. The lobing is visible in the data of radars over water as a pattern of fades and recoveries as an aircraft climbs at constant range through the lobes.

The ionosphere returns radio waves to the ground at a slant, as the frequency the sky returns only at a slant found, and those long-distance sky waves too arrive at the ground nearly grazing. A receiving antenna for them works best at a height where the direct and ground-reflected waves add rather than cancel, which for low elevation angles means high up, measured in wavelengths.

A cliff that became an interferometer #

The pattern can be read the other way: not as a nuisance in an antenna’s coverage but as an instrument. An antenna on a cliff overlooking the sea receives a radio source directly and by reflection from the water below, as though there were a second antenna at the image point, twice the cliff’s height below the first. As the source rises in the sky, the path difference changes and the received power passes through fringes, nearly from zero to four times one beam’s power for a point source.

A source with an angular size covers a range of path differences at once, and its fringes are partly washed out: one half a degree across, the width of the Sun, filling about half a fringe, swings with a visibility of only two-thirds. Measuring the visibility measures the size, which is exactly the principle of the stellar interferometer the fringe that measures a star described, with the sea providing the second mirror.

In 1946 radio astronomers at Dover Heights, on the cliffs south of Sydney Harbour, aimed an antenna out to sea at the rising Sun and saw fringes with nearly full visibility during the Sun’s strong radio bursts. The bursts came from a region far smaller than the Sun’s disc, associated with a large group of sunspots — the first resolved structure in the radio Sun. Two years later the same method, applied to the night sky, gave positions and upper limits on the sizes of the first discrete radio sources, one of them in Taurus at the position of the Crab Nebula. The cliff and the sea had made a two-element interferometer with a baseline of 160 metres, using the reversed reflection that makes a radar blind at the horizon.

Sound over a field #

Sound obeys the same limit, with one exception that proves the rule. A sound wave meeting the ground reflects according to how the ground’s acoustic impedance compares with the air’s, and grassland, snow and loose soil are porous and comparatively soft, with impedances a few times the air’s rather than the vastly larger impedance of concrete. At steep incidence such ground absorbs a good deal of sound. At grazing incidence its reflection coefficient, like the electromagnetic one, approaches −1, because the same cosine multiplies the air’s side of the comparison and goes to zero.

So a sound source and a listener both near a grassy field hear the direct sound partly cancelled by the ground’s reversed reflection, and the cancellation is worst at the frequencies whose half-wavelength matches the small path difference — usually a few hundred hertz for heights of a metre or two over tens of metres. Acousticians call it the ground effect: an excess attenuation, on top of the loss with distance that takes the treble out, that makes a road or a factory across open grass quieter at those frequencies than the same source across a car park. The exception is a perfectly rigid surface, whose impedance is infinite for sound as a perfect conductor’s is for one polarisation of light; there the grazing reflection is +1, the paths add, and sound over a frozen lake or a concrete plain carries unusually well near the ground.

Two limits that do not commute #

There is one surface for which the rule seems to fail, and it shows where the rule comes from. A perfect conductor reflects a wave polarised along its surface with a coefficient of −1 at every angle, as the method of images inside a conductor requires: the image of a horizontal antenna above a conducting plane is an antenna carrying the opposite current. But the image of a vertical antenna carries the same current, and the reflection of the p-polarised wave from a perfect conductor is +1 at every angle short of exactly ninety degrees. Over a perfect conductor a vertically polarised wave and its reflection add along the surface instead of cancelling.

The two answers come from taking two limits in different orders. For a real material with a large but finite index, the p coefficient stays near +1 until the grazing angle is about one over the index, and then swings through zero to −1 in the last sliver of angle. Make the index infinite first and the sliver vanishes; take the angle to grazing first and it is always reached. Sea water, which conducts well, has a large effective index at low radio frequencies, so its sliver is a fraction of a degree, and a vertically polarised wave travels along the sea surface without being cancelled over a wide range of distances. That is why long-wave and medium-wave broadcasting, which follows the curve of the Earth as a ground wave, uses vertical antennas, and why it carries much further over sea than over dry land.

Glare on a lake at sunset #

The same limit is visible without instruments. A still lake viewed from above shows the bottom near the shore; the same lake viewed along its surface, from a low bank, is a mirror reflecting the far shore and the sky with almost undiminished brightness. Water reflects two per cent of light at normal incidence and nearly all of it at grazing, and the change between happens mostly in the last thirty degrees. A road ahead on a sunny day looks wet for the same reason in part — the bright reflection of the sky from a smooth surface at a grazing angle — though the mirage of hot air over the road adds to it. Anything flat and smooth enough becomes a mirror when it is looked at along itself.

The reversed sign does not show to the eye, since an eye measures intensity. It shows only when the reflected light is made to interfere with light that was not reflected, which is what Lloyd’s arrangement, the mast and the cliff all do.

Where the flat-mirror picture stops #

The figures assume a reflecting surface that is flat and smooth on the scale of the wavelength and large enough to contain the region where the reflection happens, the first Fresnel zone, which for grazing radio paths is long and narrow, stretching over hundreds of metres. Real ground is rough: a surface whose height varies by more than a fraction of a wavelength over that zone scatters rather than reflects, and the criterion for smoothness, due to Rayleigh, depends on the grazing angle as well as the roughness — a surface rough at steep incidence can be smooth at grazing, which helps the reflection persist. The Earth is curved, which for long paths moves the reflection point and diverges the reflected wave. And the coefficient is −1 only in the limit; at small but finite grazing angles over a lossy ground the reflection is a little less than total and its phase a little less than half a cycle, which limits how deep the nulls are.

The two-ray picture also ignores diffraction over the horizon and the bending of radio waves by the atmosphere’s gradient of refractive index, both of which matter beyond a few tens of kilometres. Within its domain — a flat, smooth surface, a grazing path and a coherent source — the cancellation is exact in the limit and the consequences follow.

Still open: how well a rough sea reflects at the lowest angles #

For radars and radio links over the ocean, what matters most is the reflection at the smallest grazing angles, from a surface covered with waves of every size. Theories that treat the sea as a random rough surface predict how the coherent reflection, the part that interferes and makes the lobes, is replaced by diffuse scattering as the sea gets rougher; shadowing of the troughs by the crests at very low angles complicates them, and measurements at sea disagree with them by large factors in some conditions. How strong the reflection is at a few milliradians over a real sea, and so how much a low target is hidden or revealed, is still settled by measurement campaign by campaign rather than by any single model.

The habit worth carrying away is to look at a formula’s limits before its middle. As a wave’s angle of incidence approaches grazing, $\cos\theta_i$ goes to zero and Fresnel’s coefficients go to −1 for any surface, so a source and its image cancel along the surface: a dark fringe at Lloyd’s mirror, a $d^{-4}$ fade beyond $4h_t h_r/\lambda$ over flat ground, a radar null at the horizon, and a cliff that measures the size of the Sun’s radio bursts. Every material looks the same to a wave that only skims it.