A foil in a random sea follows it up to its peak

A foil flies level through a short sea and follows a long one put the phantom-foil dinghy into a regular wave and found that a boat in a seaway has to choose what to hold steady. It can fly level, holding its height while the surface rises and falls over the foil, and pay for the swinging depth; or it can follow the surface, heaving with every wave, and pay for the lift the heave demands. For a 110-kilogram boat at 8 metres a second the two cost the same at a wavelength of about sixteen metres heading into the waves and six running with them, and the wave’s height hardly moved either number, because both costs grow as the square of the height.

That essay ended on the obvious objection. No sea is a single wave. A wind sea is a spectrum, a superposition of many wavelengths with random phases, and the boat meets all of them at once. A foiling dinghy’s height is sensed by a wand trailing on the surface, and the wand’s signal can be filtered: the slow part passed to the flap, so the boat follows the long swells, and the fast part ignored, so it flies level through the chop. The question it left was where that filter should cross over, how much it saves against either pure strategy, and whether the answer still hardly depends on how rough the sea is.

The answers are, in order: close to the frequency at which the boat meets the sea’s peak; little in a moderate sea and a great deal in a rough one; and no — the crossover moves with the roughness, because in a random sea the two strategies stop being two smooth costs that cross and become one cost and one hard limit.

A sea that is many waves at once #

The sea in this calculation is the Pierson–Moskowitz spectrum, the form oceanographers fitted in 1964 to fully developed wind seas in the North Atlantic. Its energy density in frequency rises steeply from nothing, peaks at a frequency set by the wind, and falls away as the fifth power of the frequency above it. Two numbers fix it: the significant height, four times the standard deviation of the surface’s elevation, which is roughly the height an observer on deck would report; and the peak period. Three seconds is a short, steep sea of the kind a small boat meets on open water inside a few hours of a fresh breeze, and its peak wavelength, fourteen metres in deep water, sits right on the regular-wave essay’s dividing line heading into the waves.

The spectrum is sampled by eighty components at equal steps in frequency from 0.6 to four times the peak, each a linear deep-water wave whose amplitude carries its strip of the spectrum’s variance and whose phase is drawn at random from a seeded generator. At the foil, which travels with the boat, each component is met at its encounter frequency, which heading into the sea is its own frequency raised by the boat’s speed times its wavenumber. Because deep-water waves have a wavenumber that grows as the square of their frequency, the encounter frequency spreads the spectrum out: the longest waves sampled are met at about twice the rate they pass a fixed point, and the shortest at nearly eight times.

Each component contributes to the surface’s elevation above the foil and to the orbital velocity at the foil’s depth, and the controller divides them at a crossover encounter frequency. Below it the boat follows: it heaves with those components, so its depth below them stays fixed and the foil must supply the acceleration the heave needs. Above it the boat flies level: the surface’s fast part sweeps up and down over the foil, the depth swings, and the foil meets the orbital velocity as a change of inflow. At every instant the drag is the phantom essay’s budget at the instantaneous depth and speed, with the induced part scaled by the square of the lift the heave asks for. Averaged over five minutes of sea it is the number the figure plots.

The figure is the whole argument in one picture. At its right-hand end the crossover is high enough that the boat follows everything, and the drag is large: 199 newtons heading into a sea 0.4 metres high, three times the calm-water value of 66.1. At its left-hand end the boat follows nothing and flies level through all of it, and in that sea the drag is 67.9 newtons, barely above calm water. Between them there is a shallow minimum, at a crossover of 2.92 times the peak frequency, where the drag is 67.6. In the moderate sea the divided control saves a third of a newton against flying level and more than 130 against following. In the roughest of the three seas the picture changes completely, and that is where the rest of the essay goes.

Where the crossover sits in the sea #

The best crossover has a natural reading in terms of the sea itself. The second figure plots the eighty components of the 0.4-metre sea by their encounter frequency, each by its share of the elevation’s variance, with the best crossover marked. The sea’s energy is concentrated in a narrow band, and the boat, heading into the sea at 8 metres a second, meets the peak at 2.71 times its own frequency, because the speed adds the wave’s own frequency to the rate at which the boat runs into its crests.

The crossover sits just above that: the control follows the peak and everything longer, and flies level through what is shorter. Translated back from encounter frequency to the wave itself, the shortest wave the best control follows heading into the moderate sea has a frequency about 1.05 times the peak’s, a wavelength of about thirteen metres. The regular-wave essay put the dividing wavelength heading into the waves at 15.6 metres. The two are not the same number, and they should not be, because they answer different questions: a regular wave asks which strategy is cheaper for that one wave, while a spectrum asks where to cut a sum of waves whose costs interact.

They interact because each strategy’s cost depends on the whole of what it handles, not on each component separately. The heave’s induced-drag penalty is the mean square of the lift ratio, and that is the sum of each followed component’s contribution — so far additive. But the level-flight penalty is the drag curve’s response to the depth’s swing, and the depth’s swing is the sum of every level-flown component’s elevation. The curve’s curvature makes a small swing nearly free and a large one expensive, so a short wave flown level costs more when the boat is also flying level through the peak than when it is following the peak. A control that follows the peak removes most of the swing, and the components it leaves to fly level through are then cheap. The best crossover is placed where the marginal costs balance across the whole sea rather than wave by wave.

Running with the sea #

The third figure repeats the calculation with the boat running with the sea. Now the boat overtakes the waves: at 8 metres a second it is faster than every wave in the spectrum, whose phase speeds run from 7.8 metres a second at the long end down to just over one at the short. It meets each wave from behind, at an encounter frequency equal to its own frequency’s square times the speed over gravity minus its frequency, and the spectrum is met more slowly than heading into it.

Meeting the waves more slowly makes following them cheaper, since the heave acceleration grows as the square of the encounter frequency. Following everything in the 0.4-metre sea costs 94.4 newtons running with it against 199 heading into it. The level flight costs 67.4, about the same as heading into the sea, because the depth’s swing is the elevation’s swing whatever the heading. The best crossover sits at 2.77 times the peak frequency in encounter and gives 66.4 newtons, within a third of a newton of calm water, so running with a moderate sea the divided control recovers almost all of what the waves cost.

Translated back to the waves themselves, the shortest wave followed running with the sea has a frequency about 1.6 times the peak’s — the boat follows further into the spectrum than it does heading into it, as the regular-wave essay found when its crossover was six metres running and sixteen heading. The sea’s peak is followed in either direction. What changes is how much of the short side of the peak is worth following too.

A foil cannot pull #

The pure following strategy hides an impossibility, which the regular-wave essay met in its shortest head seas and which the random sea makes routine. To heave with a wave the boat must accelerate downward into each trough, and the foil supplies only the part of the boat’s weight the downward acceleration does not take away. When the downward acceleration exceeds gravity, the foil would have to carry a negative lift: pull the boat down. A foil can be made to do that — at a negative incidence or with its flap raised — but not a foiling dinghy’s, whose wand and flap are rigged to give lift, not to take it, and which would in any case have to shed the boat’s weight in a fraction of a second.

The fourth figure counts the share of time the heave would need negative lift. Following everything in the 0.4-metre sea, it is 38 per cent of the time heading into the sea and 26 per cent running with it. The drag figures above charge the heave’s induced drag as the square of the lift ratio, so a lift of minus a third of the weight costs as much as one of a third, and they count the impossible moments as if the foil could deliver them. So the cost of following everything is understated, not overstated: a real boat would fall off the waves, slam and lose its speed. At the best crossover the share is zero in the moderate sea and a tenth of a per cent in the rough one. The divided control asks the foil only for what a foil can give.

This is one place where a spectrum is harsher than a single wave. A regular wave either needs negative lift or does not, and the regular-wave essay could simply mark the wavelengths below which following was impossible. A random sea has a distribution of heave accelerations, and the shortest components in it, met at the highest encounter frequencies, contribute to the acceleration as the square of their encounter frequency even when they carry little of the elevation. Following them is not merely expensive but impossible for a fraction of the time that grows with the sea, which is why the following curves rise so steeply at the figures’ right-hand ends.

The best crossover moves with the roughness #

The fifth figure answers the essay’s last question directly. It plots the best crossover, expressed as the frequency of the shortest wave the boat follows, against the sea’s significant height from 0.15 to 0.6 metres. Heading into the sea it rises from 1.0 times the peak frequency in a sea 0.2 metres high to 1.17 in one 0.6 metres high; running with it, from 1.6 to 1.64. The regular-wave essay found its crossover within a metre of wavelength for amplitudes from 0.1 to 0.3 metres. In the random sea the crossover moves, and heading into the sea it moves by a sixth of the peak frequency across the range.

The reason is the depth. In the regular-wave essay the level-flight strategy could always fly deeper: the mean depth was the calm-water best depth or enough to clear the trough, whichever was greater, and a deeper foil costs a little more board drag but is otherwise safe. In a random sea there is no trough to clear, only a distribution of elevations with tails. This calculation sets the mean depth to the calm-water best, 0.484 metres, or to three standard deviations of the level-flown elevation plus five centimetres of margin, whichever is greater. In the 0.2 and 0.4-metre seas the calm depth is already deep enough. In the 0.6-metre sea, flying level through everything leaves a standard deviation of fifteen centimetres in the depth, and three of those plus the margin asks for half a metre, deeper than the calm best.

Both effects push the rough-sea crossover up. A taller sea’s short waves are taller in absolute terms, so flying level through them swings the depth further and its cost rises faster than the heave penalty of following them. The control that is right for a 0.2-metre sea leaves too much of a 0.6-metre sea to the level-flight half of the filter, and the regular-wave essay’s insensitivity to height, which held because both costs were second order in amplitude and so scaled together, is broken by a constraint that is not second order at all.

A rough sea punishes the level boat #

That constraint shows itself in the sixth figure. It plots the shallowest the foil came below the local surface in five minutes of each sea, against the crossover, heading into the sea. In the 0.2 and 0.4-metre seas the foil always stays well under: at worst fifteen centimetres down, flying level through the 0.4-metre sea. In the 0.6-metre sea, flying level through everything, the foil reaches the surface in the sea drawn for the figures, and in other draws of the same spectrum it comes within two centimetres of it or breaks through.

Three standard deviations of a Gaussian are exceeded about once in seven hundred samples, and five minutes of a sea with a peak period of three seconds holds about a hundred waves met at encounter frequencies two or three times the peak, so several hundred crests pass over the foil in the run. The tail of the distribution is visited, and once is enough: of four seeds for the random phases, two breach and two pass within 1.6 centimetres, which no sailor would call flying. The calculation has no model of a breach — the foil ventilating, the lift collapsing, the boat dropping onto its hull — so it charges the drag of the board and foil at a depth of one millimetre as a stand-in, and the mean drag of the level strategy in the 0.6-metre sea becomes 653 newtons heading into it and 1446 running with it. Those numbers are not predictions: they mark the point where the model stops being a model of a flying boat. In the two draws that do not breach, flying level costs 72.6 and 74.9 newtons, still more than the divided control, because a foil a few centimetres down is deep inside the surface’s grip. What is a prediction is that the level boat comes to the surface or within a hand’s breadth of it, in a sea a small boat sails in routinely.

The divided control in the same sea follows the long waves, which carry most of the variance, and leaves only a standard deviation of eleven centimetres to fly level through. The foil comes no closer than eleven centimetres to the surface, the drag is 70.1 newtons heading into the sea and 66.8 running with it, and the foil asks for negative lift a tenth of a per cent of the time. In a rough sea the divided control is not a refinement of the two pure strategies. It is the only one of the three that flies.

The scale of the saving depends on how rough the sea is, and that is the honest summary of the comparison. In the 0.2-metre sea the divided control and the level strategy give the same drag to a tenth of a newton heading into the sea, 66.5 against a calm 66.1, so a boat that simply flew level would lose nothing worth measuring. In the 0.4-metre sea the divided control saves a third of a newton heading into the sea and one newton running with it — one and a half per cent, which is a race’s margin but not a design’s. In the 0.6-metre sea it is the difference between flying and not.

What the calculation was checked against #

The sea and the time average were checked separately, since each could be wrong without the other. The sampled spectrum with four hundred components carries a variance within 0.49 per cent of the significant height’s square over sixteen, the integral of the Pierson–Moskowitz form; the shortfall is the energy outside the sampled band, above four times the peak frequency and below 0.6 times it. A sea a micrometre high gives the calm-water drag at the calm best depth to one part in a hundred billion, which checks that the averaging adds nothing when there is nothing to average. Halving the time step to a hundredth of a second and doubling the run to ten minutes moves the mean drag by seven thousandths of a per cent.

The checks also refuse a sea of zero height, a heading that is neither into the sea nor with it, and a negative crossover. The drag budget underneath every number is the phantom essay’s, unchanged, and its own checks carry over: the image of a foil under a flat surface, the reversed board, and the budget’s least at 0.484 metres on flat water.

What the calculation leaves out #

The sea is long-crested. Every component here travels in the same direction, the boat’s or its opposite. A real wind sea spreads its energy over directions about the wind, and a boat on a reach meets components from the side, which are met more slowly and add roll to heave. Spreading leaves the elevation’s statistics at a point unchanged, so the breach is not softened, but it lowers the encounter frequencies of the components met obliquely, which makes following them cheaper and would move the best crossover.

The filter is ideal. A real wand-and-flap linkage, or a sensor and actuator, filters with a finite slope and a phase lag, and the lag matters near the crossover, where the control is neither following nor flying level and can be doing something between them at the wrong moment. A wandering flock passes no error down the V found that two first-order filters summing to one share their error without losing any; the boat’s version is a complementary pair on the wand and an accelerometer, and the lag of a real pair would add a cost this calculation does not carry.

The boat is quasi-steady. As in the regular-wave essay the foil’s lift responds instantly. Heading into the sea the shortest components are met at about thirty times the peak frequency, a period of a tenth of a second, which is under ten chord-transits of the foil. They carry almost none of the variance, but the quasi-steady assumption is weakest exactly in the level-flown part of the spectrum, and which way the unsteady lift would move the level strategy’s cost this calculation does not settle.

One sample of one sea. The eighty phases are one draw. Across four seeds the moderate sea’s best crossover does not move at all and its least drag moves by three hundredths of a newton. The rough sea is touchier: its best crossover ranges from 3.3 to 3.7 times the peak frequency and its least drag from 69.9 to 70.4 newtons, and the level strategy’s breach comes and goes with the draw, as a tail event should. A longer run would make the breach more certain, not less.

A filter set by the sea it meets #

The broader point is that a control’s best setting is a property of the disturbance, not only of the vehicle. A wing averages a gust through its own loading found that a wing in turbulence is a spatial filter whose weighting is its own span loading, so the gust it feels is the atmosphere’s spectrum seen through the wing’s shape. The foiling dinghy is the same picture in time: the sea’s spectrum seen through the wand’s filter, and the filter is something the sailor can set.

The regular-wave essay’s answer was a property of the boat and the heading — a dividing wavelength — and it would have suggested setting the filter once. The spectrum’s answer is a property of the sea as well, and of how close the sea’s tails come to the foil. A foil under the surface flies with a phantom found the best depth on flat water from the board’s drag; a hydrofoil loses most lift on the way up found that the surface’s grip on a foil is a matter of Froude number; and the drift in a wave that has none measured the orbital motion the foil flies through. In a seaway all three are read through the filter, and the filter is read through the sea.

The analysis also closes the sailing essays’ account of the underwater angle. Faster than the wind that drives it reduced a boat’s performance to two drag angles; a keel flies wherever the course puts it and the rudder pays for the keel’s wake set the underwater one on flat water. In a seaway the underwater angle is the foil’s drag under its control in that sea, and the control has one number that matters most — where it stops following.

Who worked it out #

The spectrum is Willard Pierson and Lionel Moskowitz’s of 1964, fitted to fully developed seas measured by weather ships; the JONSWAP form that refines it for fetch-limited seas came nine years later. Representing a sea as a sum of components with random phases is Longuet-Higgins’s practice of the 1950s, and the encounter spectrum, the sea seen from a moving ship, is St Denis and Pierson’s of 1953, which founded the study of ship motion in irregular waves. Hydrofoil ships of the 1960s and 70s had automatic control systems built around the same division between platforming and contouring, and their designers published crossover frequencies chosen against measured sea spectra. The small foiling dinghy’s mechanical wand, whose gearing and damping are its filter, is the International Moth sailors’ contribution of the early 2000s.

Still open: a filter with a lag #

Every number here is for an ideal filter that divides the sea cleanly at one frequency. A real wand is a mechanical linkage with its own inertia and damping, and a real electronic controller a first- or second-order filter with a phase lag that is largest exactly at the crossover. The next calculation replaces the ideal cut by a complementary pair of first-order filters, follows the sea through the low-pass and flies level through the high-pass, and asks how much the lag at the crossover costs against the ideal cut, whether the best crossover moves when the cut is soft, and whether a second-order pair, which costs more lag but cuts more sharply, is worth its price in a rough sea where the tails come close to the foil.