A swell lifts a ground-effect wing more at its crests than it takes in its troughs

A wing over the sea barely touches it asked whether the sea gives way under a ground-effect craft, and found that it hardly does. A wing presses on the water with its own lift, and the water’s density is so much greater than the air’s that the dent it makes and the waves it trails are millimetres high; to the wing the sea is as good as rigid. That settled the question of the sea’s response to the wing. It left the other question, the one that decides whether a ground-effect craft can operate at all, which is the wing’s response to the sea. No sea is calm. A craft flying a metre above the water over a swell sees its gap to the surface open and close with every wave.

The cushion that is not there made the essential point about where ground-effect lift comes from: not from air trapped and compressed under the wing, but from the ground’s reflection of the wing’s own vortex system, which reduces the downwash the wing makes and so raises the lift at a given angle. The reflection is a function of the gap, and it changes steeply as the gap closes. Over a swell, then, the lift is a function of the wave’s phase, and the questions are how large the swing is, how fast it comes and what it costs to escape it.

The law the swell samples #

The wing here is a flat plate of five metres chord at four degrees, flying at 50 metres a second with its leading edge 1.2 metres above the mean sea and its trailing edge 0.35 metres lower — the proportions of a small passenger ground-effect craft. It is modelled as a row of forty bound vortices, each with an image vortex of opposite sign mirrored about the water’s surface directly beneath it. Over a flat sea that is the same panel wing earlier essays used, and the calculation reproduces it to machine precision.

The first figure is the calm wing’s lift coefficient against the height of its leading edge. Far from the sea it tends to 0.438, the free-air value. At 1.2 metres it is 0.684, more than half as much again: that is the cushion. And the curve steepens as the wing descends. Half a metre lower, it is 0.965; half a metre higher, only 0.583.

That asymmetry is the whole of this essay’s first result. A swell of half a metre’s amplitude moves the gap half a metre up and half a metre down, but the lift does not move equally. Coming half a metre closer gains 0.281 in lift coefficient; going half a metre further away loses 0.101. The law the swell samples is convex, and any convex law sampled symmetrically gives an average above its value at the middle.

The swing, through one encounter #

The second figure follows the wing’s lift coefficient through one encounter with a 50-metre swell, heading into it, for amplitudes of a quarter of a metre, half a metre and seven tenths. Phase zero is a crest under the leading edge. At a quarter of a metre the lift runs from 0.600 to 0.803 and averages 0.694, against the calm sea’s 0.684. At half a metre it runs from 0.534 to 1.00, averaging 0.729. At seven tenths, with the trailing edge passing fifteen centimetres above the crests, it runs from 0.489 to 1.34: the crest nearly doubles the lift, and the trough takes off less than a third.

In force rather than coefficient, the half-metre swell is easier to feel. At 50 metres a second each metre of span of a five-metre chord carries 7.7 kilonewtons for every unit of lift coefficient, so the calm wing holds up 5.2 kilonewtons a metre, and over the swell the same wing swings between 4.1 and 7.7 — the weight of nearly four hundred kilograms coming and going on every metre of span, once a second.

Heading into a 50-metre swell at 50 metres a second the craft meets a crest every 0.85 seconds. The swell itself would pass a fixed point every 5.7 seconds; the craft’s own speed adds a crest every metre it flies against the swell’s motion, and multiplies the frequency nearly seven times. The swing comes at a little over one cycle a second, which is in the band a passenger feels as a hard, quick jolting rather than a slow ride over the waves, and in the band the craft’s structure and its pitch dynamics respond to.

The third figure measures the swing against the swell’s amplitude, as a root-mean-square fraction of the calm lift, beside the rise of the mean; the dial moves the cruise height.

At 1.2 metres the swing grows faster than the swell does. At small amplitudes it is about 40 per cent of the calm lift for every metre of amplitude, but at 0.75 metres it has reached 48 per cent where proportion would have given 30, and the mean has risen 20 per cent. Both curves bend upward because the gap law does: the closer the crests come, the larger the share of the swing they make, and the larger the rectified rise of the mean.

The rise of the mean matters as much as the swing. A craft trimmed to fly level at 1.2 metres over a calm sea carries a lift coefficient of 0.684. Over a half-metre swell, at the same attitude and height, its wing offers 0.729 on average. It will climb until the mean comes back to its weight — the sea pushes a ground-effect craft up away from itself. In a sense this is the cushion in its most literal form, but it is a statistical cushion: the craft rises because the crests give more than the troughs take, not because air is trapped under it.

The dial shows what that climb buys. At 2.4 metres the swing at the largest swell shown falls from 48 per cent to 16 and the mean’s rise from 20 per cent to 2, but the calm lift coefficient has fallen to 0.52, two thirds of the way back to free air. Every metre of climb that steadies the lift spends cushion.

The air the waves move #

There is a second contribution to the swing, and its size is not what the popular picture of a craft skimming over waves suggests. A wing flying over a sloping surface might be expected to see the slope as an upwash: on the rising face of a wave the air should be deflected upward, on the falling face downward. For a 60-metre swell of half a metre’s amplitude, at 50 metres a second, that slope upwash would be 2.6 metres a second at the steepest point — a change of incidence of three degrees.

The air does not do that, because the air over the sea is still and the waves are moving through it. In the earth’s frame, the only air motion the swell makes is the motion its moving surface forces on the air above it, and that is the water’s own vertical speed at the surface, the wave’s amplitude times its frequency, decaying upward over a fraction of a wavelength. In the wing’s frame the same statement appears as a cancellation: the upwash of the slope swept past at the craft’s speed is cancelled by the motion of the wave pattern itself, down to the water’s vertical speed. For the 60-metre swell that is 0.51 metres a second — a fifth of the slope’s naive upwash. The ledger records the cancellation, to machine precision.

What remains still matters, and it matters more for shorter swells. The fourth figure plots the swing against wavelength, with the air’s motion and without it. For a 200-metre swell the two are 20 and 19 per cent: the gap does nearly all of it. For a 40-metre swell they are 25 and 19, and for a 20-metre swell 29 and 18. The water’s vertical speed grows as the square root of the wave’s frequency for a given amplitude, and its contribution comes a quarter of a cycle after the gap’s — greatest over the wave’s faces rather than at its crest — so the two add in quadrature rather than directly. The drift in a wave that has none measured that orbital motion beneath the surface; above it, in the air, the same motion is mirrored and decays as fast.

The gap-alone swing falls slightly for the shortest swells, as a wave of twenty metres under a five-metre chord begins to raise one part of the wing’s image while lowering another. A wing averages a gust through its own loading found the same averaging for a wing in turbulence; here it is the chord averaging the ground rather than the span averaging the air.

Climbing out costs the cushion #

A ground-effect craft’s designers face a choice the fifth figure draws: fly high enough that the sea does not shake the lift, or low enough that the sea still reflects the wing. It plots, against the swell’s amplitude, the leading-edge height at which the root-mean-square swing is a tenth of the calm lift, and the cushion left there — the calm lift’s excess over free air.

In a swell a tenth of a metre high the wing can fly at 0.64 metres and keep a cushion of 137 per cent: it more than doubles its lift. At 0.3 metres it must climb to 1.5 metres, keeping 42 per cent. At half a metre, 2.6 metres and 16 per cent. At 0.75 metres, 4.7 metres and 5 per cent. At a metre of amplitude — a sea about two metres from trough to crest, which is an ordinary day on open water — the wing has to fly at 6.8 metres, more than a chord up, and keeps 2 per cent: it is no longer a ground-effect craft in any useful sense.

The climb runs out for a reason that is not the gap. At large heights the gap’s swing has faded, but the air the waves move has not faded as fast, because it decays over a fraction of the wavelength rather than of the chord. A craft high enough to escape its own reflection’s sensitivity is still flying through the wave-induced air of a long swell. That is why the required height jumps between 0.5 and 1 metre of amplitude: the second contribution, not the first, sets the ceiling.

This is the honest form of the claim the essay refutes. A ground-effect craft does not ride on a cushion that smooths the waves; its cushion is precisely the thing the waves modulate. The sea states in which it flies steadily are the ones where the waves are small against its chord, and the craft that has been built for open water — the large Soviet ekranoplans, with chords approaching twenty metres — were large in part for that reason: the chord sets the scale against which a sea is calm.

Below the wing, and below the water #

Two neighbouring cases make the result sharper by contrast. The cushion that is there after all found that a wing flown very low, with a gap of a few per cent of its chord, does trap a ram pressure under itself, the one place the popular picture of a cushion is right. That regime is worse over a swell, not better: in the thin-gap limit a flat plate’s lift grows as the inverse of the gap, steeper still than the reflection does, so a craft flying in it is the one the crests shake hardest: halve the gap and a given swell is twice as large a share of it, on a law that is itself twice as steep. The steadier the cushion over calm water, the more violently a sea modulates it.

A hydrofoil loses most lift on the way up turned the problem over and put the wing under the surface. A foil near the surface loses lift as the gap to the surface closes, the opposite sign to the wing above it, because the free surface gives way rather than reflecting. Over a swell the foil’s gap too opens and closes, and its lift falls steeply as the gap closes while hardly changing at depth, so the same rectification that raises the wing’s mean lift would lower the foil’s, and a foiling boat over a swell would settle deeper rather than rise; the dinghy’s wand control in a wand that lags its sea makes flying level the better bet is a way of managing that gap actively. The ground-effect craft has no such control on its gap except its own height and pitch, which is why its operating limit is a sea state rather than a setting.

What was checked #

Over a calm sea the wing reproduces the panel wing with its ground image to machine precision. The crest of a swell a hundred kilometres long — a raised floor, to the wing — gives the calm wing’s lift at the reduced height to a part in a billion, which checks the mirrored images against the flat-ground answer they must reduce to. The still air’s upwash at the surface equals the water’s vertical speed to machine precision, and the slope term alone would have been 5.2 times larger. The calculation refuses a sea that reaches the wing and a swell of no length.

What the calculation leaves out #

The craft is held at a fixed height. The essay computes the lift the wing is offered, not the motion it would make in answer. A real craft heaves and pitches under the swing, and the motion feeds back on the gap; with the pitch stiffness ground effect provides, the response may amplify or attenuate depending on the craft’s natural frequencies against the encounter frequency of about one per second.

Quasi-steady. The encounter’s reduced frequency, the wing’s chord against the distance it flies in one encounter, is 0.37 here, large enough that the lift lags its quasi-steady value. The lift that arrives late computed that lag in free air; in ground effect the wake’s memory is changed by its own image, and the attenuation of the swing would be somewhat smaller than in free air, not removed.

Two dimensions, head seas. A real wing has a span, and a swell met at an angle reaches one tip before the other, adding roll. Most of the swing would remain, since each spanwise strip sees the same gap law.

The images are mirrored, not reflected. Each image sits vertically below its vortex across the local surface, which is the long-wave form and is right to the square of the wave’s slope; the 20-metre swell, with a slope of 0.16, is at the edge of where that holds.

Who found it, and when #

The image method for a wing in ground effect is Wieselsberger’s of 1922. Alexeyev’s ekranoplans of the 1960s, and the American and German studies that followed, met the sea-state limit in service: the craft were fast and economical over calm water and could not operate in seas of more than a metre or two, which is the practical reason ground-effect craft remained a niche. Later studies of wing-in-ground flight over waves computed the lift over regular waves with panel and vortex-lattice methods, of which this essay’s is the simplest two-dimensional form.

Still open: a craft that answers the swing #

Every number here holds the wing still while the sea moves under it. A real craft has a mass, a pitch inertia and a tailplane, and ground effect gives it a height stiffness of its own. The next calculation would free the craft in heave and pitch over the same swell, with the lift’s quasi-steady gap law and its lag, and ask whether the craft follows the long swell and flies level through the short — as a foil in a random sea follows it up to its peak found a foiling boat should — and whether its own height stiffness, which rises as the gap closes, makes the following automatic or makes the motion unstable near the crests.