A foil flies level through a short sea and follows a long one

A foil under the surface flies with a phantom found how deep a foiling dinghy should fly on flat water. At speed the surface above the foil cannot hold a pressure, its image is a biplane partner that takes lift, and the board that carries the foil down through the surface is reflected into a reversed copy that makes its induced drag two and a half times a keel’s. The board, not the foil, sets the depth: for a 110-kilogram boat at 8 metres a second the drag is least at 0.484 metres, where a shorter board would save profile drag and cost more induced.

That essay ended on the obvious objection. A boat at sea does not fly over flat water, and a boat whose best depth is half a metre is flying through the top half-metre of every passing wave. The surface above the foil rises and falls, the water under it moves in orbits, and the question it left was how much the average drag rises when the depth swings about its best value, and whether that pushes the boat deeper still. The answer is that it does, by less than one might expect — but the depth turns out to be the smaller half of the question. A boat in waves has to decide what to hold steady, and that choice matters more than any depth.

Two ways to fly through a wave #

A foiling boat’s height is held by a control, in the smallest classes a wand that trails on the surface ahead of the foil and moves a flap as the surface nears or recedes. How the control is set decides what the boat holds constant, and there are two limits.

Flying level — platforming, in the sailor’s word — holds the foil at a fixed height and lets the surface move over it. The foil’s depth below the local surface swings with the wave’s elevation: under a crest it is deeper, under a trough shallower, and the board’s wetted span swings with it. The boat itself does not move up and down, so the foil carries the boat’s weight and nothing else.

Following the surface — contouring — holds the foil at a fixed depth below the local surface, and the boat rides up every crest and down every trough. The depth never changes, so every part of the drag budget stays at its flat-water value except one: to lift the boat up a wave face the foil must carry more than the boat’s weight, and to let it fall into the trough, less.

The model is the phantom essay’s boat, unchanged, flown through a regular train of deep-water waves. The waves are linear, with the elevation a cos θ and the orbital velocity $a\omega e^{-kz}$ at depth z below the mean level, as in the drift in a wave that has none, which traced those orbits and the drift they leave behind. At every phase of the wave the boat is quasi-steady: the foil and the board are the flat-water calculation at the local depth and the local inflow speed, with the flap setting whatever incidence carries the lift that phase needs. That is fair when the wave is many chords long and is met slowly compared with the time the water takes to cross the chord, and here the shortest waves are 27 chords long and the fastest encounter takes twenty chord-transits.

Flying level pushes the foil deeper #

The first figure is the prediction tested. Each curve is the boat’s drag averaged over one passing wave, for a fixed mean depth, against that depth. With no wave it is the phantom essay’s curve, least at 0.484 metres and 66.1 newtons. With a wave of amplitude 0.2 metres — 0.4 metres from trough to crest — the least falls at 0.562 metres and costs 68.7 newtons: the boat flies eight centimetres deeper and pays 4 per cent more. With an amplitude of 0.3 metres it flies at 0.637 metres and pays 71.4.

The push downward is the shape of the flat-water curve at work. The drag climbs steeply on the shallow side, where the board’s wetted span shrinks and its induced drag grows as one over its square, and gently on the deep side, where it grows only as the profile drag of the extra wetted length. A depth that swings evenly about its mean samples the steep side harder than the gentle one, so the average rises, and it rises least if the mean moves away from the steep side. For a small wave the extra drag is exactly a quarter of the amplitude squared times the curve’s second derivative, which the calculation reproduces to a quarter of a per cent at an amplitude of two centimetres; the quarter is the half from the Taylor series times the half that is the average of cos² θ.

How much the depth matters is clearest by leaving it where it was. A boat that kept its flat-water depth of 0.484 metres through the 0.4-metre sea would average 69.7 newtons, a newton more than at its best depth for that sea. Through a 0.8-metre sea, with the trough reaching within eight centimetres of the foil, it would average 112 — the board’s induced drag, growing as one over its wetted span squared, runs away as the trough uncovers it.

Where in the wave the drag is paid #

The second figure opens up one wave. The expectation is that the trough is where the boat pays, since that is where the board is shortest. The board’s induced drag does climb there, from 3 newtons under the crest to 15.3 in the trough. But the total drag is largest under the crest, 74.4 newtons against 67.4 in the trough, for two reasons that both work at the crest. The board is 0.76 metres deep there, and its profile drag on the extra wetted length is 29.3 newtons against 11.8 in the trough. And heading into the waves, the orbital motion at the crest moves the water towards the boat, so the foil meets it faster and pays more of every drag that grows with speed.

The average sits above the calm boat’s best because the drag curve bends upward on both sides of its minimum and the swing samples both. The orbital inflow adds a little on top: with it left out, the best depth for the same sea is 0.554 metres and the average 68.1 newtons, so of the 2.6 newtons the sea costs, about 0.6 are the water’s motion and 1.9 the surface’s.

The flap has work to do as well. The orbital motion’s vertical component changes the foil’s incidence by its speed over the boat’s, here 0.34 metres a second against 8, or 2.4 degrees either way, and the depth and speed changes move the incidence the foil needs between 3.3 and 4.0 degrees. The flap has to take out a swing of the same size as the foil’s own flat-water incidence, once per wave, which is why a foiling boat in a chop is a control problem before it is a drag problem.

The trough is not what sets the depth #

The third figure asks the question a sailor would ask first: how deep must the foil fly to stay wet? A trough that reaches the foil ventilates it and drops the boat, so there is a hard limit, the amplitude plus a margin. The figure draws that limit with a five-centimetre margin beside the depth of least averaged drag, and the two never meet. The best depth grows roughly as the square of the amplitude — 0.506 metres at an amplitude of 0.1, 0.562 at 0.2, 0.722 at 0.4 — while the trough limit grows in proportion to it, and at every amplitude drawn the drag has already moved the foil deeper than the trough requires.

So the flat-water essay’s conclusion survives into a seaway, with an extension. On flat water the board sets the depth; in waves the board still sets it, through the curvature of its drag rather than through the danger of the trough. A boat flying level through a sea deep enough to be safe is also, in this model, deep enough to be efficient, and a boat flying shallower to save board drag is losing drag as well as margin.

Following the surface asks for acceleration #

The other strategy has a different bill. A boat following a surface that rises and falls by a at a frequency $\omega_e$ must accelerate vertically by up to $a\omega_e^2$, and the frequency is not the wave’s own but the one the boat meets it at. A deep-water wave of length λ has the frequency $\omega = \sqrt{gk}$, with $k = 2\pi/\lambda$ — the dispersion that also fixes the half-angle of a ship’s wake in the angle that does not care, and that makes long waves outrun short ones. A boat heading into it meets crests at ω + kU; one running with it meets them at |ω − kU|, which falls to nothing when the boat keeps pace with the wave — at 8 metres a second that is a wave 41 metres long, $2\pi U^2/g$, the same length that hull speed is not the hump tied to a displacement hull’s speed. The wave is the one the boat would make itself if it were a hull, and running with it the boat sits still on its face.

The fourth figure is $a\omega_e^2/g$. Heading into waves 12 metres long, the boat must accelerate at 0.85 g; into waves 8 metres long, 1.67 g; into 5 metres, 3.75 g. Above one g the foil would have to pull the boat down into each trough faster than gravity lets it fall, and a foil that carries a boat can only push up. Following the surface into a short head sea is not expensive; it is impossible. Running with the same waves the demand is a fraction of that — 0.25 g at 8 metres, 0.87 at 5 — because the boat overtakes each wave slowly.

What the acceleration costs is simple to state. The foil carries $m(g - a\omega_e^2\cos\theta)$, so its induced drag, which goes as the lift squared, averages to the flat-water value times $1 + \tfrac12(a\omega_e^2/g)^2$. The calculation reproduces that closed form to rounding, and with the orbital inflow added it is the only part of the budget that changes: at 0.85 g the boat pays 36 per cent more induced drag on its foil than it would on flat water.

Which strategy costs less #

The fifth figure sets the two side by side. Flying level costs about the same whatever the wavelength — 68.5 to 68.7 newtons heading into the waves, 67.6 running with them — because what it pays for is the surface’s elevation, which is the wave’s amplitude and not its length. Following costs the square of the heave, which falls steeply as the waves lengthen. Heading into the waves the two cost the same at 15.6 metres: into a shorter sea the boat should fly level, into a longer one follow. Running with the waves the dividing length is 5.9 metres, and a boat reaching or running before a sea of any ordinary length should follow it.

This is the refuted claim’s answer. The flat-water best depth is best only if the boat can hold it for free, and in a short head sea holding it is the most expensive thing the boat can do. The depth the boat should fly at is a consequence of the strategy, not the other way round.

The wave’s height hardly matters #

The sixth figure repeats the comparison at every speed from 6 to 14 metres a second, for two wave heights. The dividing wavelength into a head sea shortens as the boat speeds up, from 21 metres at 6 metres a second to 12 at 12, because a faster boat meets each wave sooner and pays more to follow it. Running with the waves the dividing length stays between five and seven metres at every speed.

The height of the waves hardly enters. Into a head sea at 8 metres a second, the dividing wavelength is 15.2 metres for an amplitude of 0.1 and 16.2 for 0.3. Both strategies’ costs are second order in the amplitude — the level boat’s as the curvature of its drag curve, the following boat’s as the square of its heave — so a bigger sea raises both and moves the balance between them very little. What decides how to fly is the wavelength and the heading, not the height of the sea. The height decides only how much either strategy costs.

What the calculation was checked against #

The seaway calculation is built on the phantom essay’s budget and adds only an average over the wave, so its checks are that the average reduces to what it should. With no wave, both strategies reproduce the flat-water budget to rounding at three depths. For a small wave with the orbital inflow left out, the level boat’s extra drag matches a quarter of the amplitude squared times the flat-water curve’s own second derivative, found by finite differences, to 0.27 per cent at an amplitude of two centimetres and 1.6 per cent at four, the error falling as the amplitude squared as it should. The contouring boat’s induced drag matches $1 + \tfrac12(a\omega_e^2/g)^2$ exactly. The checks refuse a wavelength of zero, a heading that is neither into the waves nor with them, and a negative amplitude.

What the calculation leaves out #

Real seas are not regular. A seaway is a spectrum of wavelengths and directions, and a boat meets short and long waves at once. A control that follows the long components and flies level through the short ones — a filter on the wand’s signal — would beat either pure strategy, and this calculation gives the frequency at which such a filter should cross over: the encounter frequency at the dividing wavelength, about 5 radians a second for this boat heading into the waves at 8 metres a second.

The boat is quasi-steady. The foil’s lift responds to a change of incidence over a few chord-lengths of travel, and the wave’s unsteadiness adds a lift lag and a wake the calculation does not have. The fastest encounter here takes twenty chord-transits, so the error is small; it grows with the encounter frequency, and it would be largest exactly in the short head seas where following is already impossible.

Beam seas and pitch. Waves met from the side roll the boat and tilt the board, which changes the board’s image as heel did in the keelboat essays, and a real boat pitches as well as heaves. Neither is in this calculation.

The phantom’s own limits. The surface’s image was worked out for a flat surface; under a wave the surface is curved and moving, and the image of a foil under a crest is not exactly the flat-water image at the local depth. For waves many foil depths long the correction is small, and for the shortest waves here, three metres at a depth of half a metre, it is not negligible.

The control, not the foil, is the answer #

The broader lesson is about what a design question is. Faster than the wind that drives it reduced a boat’s performance to two drag angles, and a keel flies wherever the course puts it and the rudder pays for the keel’s wake found how the underwater angle is set. On flat water those angles are properties of the foils. In a seaway they are properties of the foils and of the rule that flies them, and the rule’s best form depends on the wave the boat meets — its length and heading, not its height. A foiling boat is well designed for a sea only when its control knows which of the two strategies that sea calls for.

The same division turns up wherever a vehicle meets a moving boundary. A hydrofoil loses most lift on the way up found that the surface’s effect on a foil depends on the speed through the Froude number; here the choice of what to hold depends on the frequency the boundary is met at. A car’s suspension faces the same choice on a rough road — follow the long undulations and filter out the short ones — and the dividing frequency is set there, as here, by what each strategy costs.

Who worked it out #

The linear deep-water wave, with its orbital velocities decaying as $e^{-kz}$, is Airy’s of 1841 and Stokes’s of 1847. The encounter frequency is the naval architect’s tool for ship motion in waves, and the division between platforming and contouring is the language of the hydrofoil ships built from the 1950s on, whose control systems were designed to fly level through short waves and follow long ones. The mechanical wand that makes a small foiling dinghy fly itself, and whose gearing sets how much it follows the surface, came from the sailors of the International Moth class in the early 2000s.

Still open: a sea of many waves #

Every number here is for a regular wave of one length. A real sea is a spectrum, and a control can follow one part of it and fly level through another. The next calculation gives the boat a standard wind-sea spectrum and a control that follows the surface below a chosen encounter frequency and flies level above it, and asks where the crossover frequency should sit to make the averaged drag least, how much that filtered control saves over either pure strategy, and whether the answer still hardly depends on how rough the sea is — as it does for the single wave, where both costs are second order in the height.