A wand that lags its sea makes flying level the better bet

A foil in a random sea follows it up to its peak flew the phantom-foil dinghy, 110 kilograms at 8 metres a second, through a Pierson–Moskowitz sea and let its height control divide the sea in two. Below a crossover frequency the boat followed the surface, heaving with the long waves so that its foil stayed at a fixed depth beneath them; above it the boat flew level, holding its height while the short waves swept over the foil. The best crossover sat just above the frequency at which the boat met the sea’s peak. In a sea 0.4 metres high it saved a third of a newton against flying level, and in one 0.6 metres high it was the only one of the three strategies that kept the foil under the surface at all.

That essay divided the sea with an ideal cut: every component below the crossover followed entirely, every component above it flown through entirely, and the change from one to the other instantaneous. No wand does that. A foiling dinghy senses its height with a wand trailing on the surface, a rod on a pivot whose angle sets the main foil’s flap through a pushrod; the linkage has inertia, the wand’s float has drag in the water, and the whole system responds to a change of surface height over a time that is the boat’s own. An electronic controller on a larger foiling boat does the same thing explicitly, with a filter whose order and crossover are settings. Both are filters with a finite slope and, inseparably from the slope, a phase lag. The question this essay asks is what the lag costs.

The answer is: everything the division gained. In a moderate sea a real filter does no better than flying level. In a rough sea it cannot be set to keep the foil both under the surface and pushing upward, and a sharper filter of the second order, which looks as though it should come closer to the ideal cut, does worse than the first-order one.

The divided control as a pair of filters #

Write the surface’s elevation at the foil as a sum of components, each met at its own encounter frequency. The control splits each component in two. One part the boat follows, heaving with it, and the foil has to supply the acceleration that heave needs. The other part the boat flies through, so the surface rises and falls over the foil and the foil’s depth swings with it. The two parts must add to the whole wave, because the boat cannot both follow a crest and fly through it, nor do neither: a filter pair whose two responses sum to one at every frequency is called complementary, and that constraint is the physics here, not a design choice.

The ideal cut is the complementary pair whose follow filter is one below the crossover and zero above it. The simplest real pair is first order: the follow part is one over one plus i times the frequency ratio, the response of a mass on a damper or of a single resistor and capacitor, and the fly-through part is what remains. The next is the second-order Butterworth pair, whose follow part falls off as the square of the frequency ratio above the crossover rather than its first power, with the flattest response below the crossover that a second-order filter can have.

The first figure shows the price of that slope. Its left panel plots the share of a wave the boat flies through, against frequency over the crossover. The ideal cut is a step. The first-order pair flies through a share that rises smoothly from nothing to all of the wave, 0.707 of it at the crossover itself, and the followed part lags the surface by 45 degrees there, as the right panel shows. A particle is a low-pass filter found exactly that corner for a seeding particle in a flow, one over the root of two in amplitude and 45 degrees of lag, and it is the same algebra.

The second-order pair is the surprise. Its follow part lags by 90 degrees at the crossover, and since the two parts must sum to the wave, a follow part a quarter-cycle late leaves a fly-through part that is larger than the wave: at the crossover it is 1.22 times the wave’s height, and its peak, 1.27 times, falls at 1.27 times the crossover, both equal to the square root of the golden ratio. A sharper cut buys its slope with a band in which the boat does worse than nothing — neither following the waves nor flying level through them, but heaving out of step so that the surface moves over the foil further than it would if the boat held still.

What a soft filter does with the sea #

The second figure turns the filters’ responses into the sea’s terms. Heading into the 0.4-metre sea at 8 metres a second, its left panel gives the root-mean-square height of the part of the sea the boat flies through, as a share of the sea’s own, and its right panel the root-mean-square heave acceleration of the part the boat follows, in units of gravity, both against the crossover in units of the waves’ peak frequency.

The two panels separate two different ways a soft filter leaks. At a crossover three times the peak frequency, close to where the ideal cut was best, the ideal cut flies through 0.78 of the sea and follows the rest with a heave acceleration of 0.17 g. The first-order pair flies through almost the same share, 0.76, so on depth it is no worse. But it follows with 0.66 g, nearly four times as much, because its follow part falls away only as the first power of frequency above the crossover while a heave’s acceleration rises as the square: a component met at ten times the crossover is followed at a tenth of its height and with ten times the crossover’s acceleration. Heading into the sea the boat meets its shortest components at thirty times the peak frequency, and the first-order pair heaves to every one of them.

The second-order pair follows the short sea more modestly, 0.31 g, because its follow part falls as the square and so the acceleration it asks for levels off rather than growing. Its leak is on the other side: it flies through 1.16 of the sea, more than the sea’s own height, because the sea’s most energetic components, at the peak, sit inside the band where its fly-through part exceeds one. The boat’s foil swings further below the surface than it would if the boat flew level through everything.

The induced drag the heave costs grows as one plus the mean square of the heave acceleration over gravity, and the board and foil drag the depth swing costs grows with the square of that swing. A first-order pair therefore pays heavily in induced drag, and a second-order pair pays in depth. Neither leak is a defect of a particular filter that a better design would remove. Both are what a complementary pair with a finite slope must do, because a slope means a band where the wave is shared, and in that band the shared wave is out of phase with itself.

In a moderate sea the gain is gone #

The third figure is the drag against the crossover, averaged over three seeded draws of the sea. Its dial moves the sea’s height through 0.2, 0.4 and 0.6 metres. In the 0.4-metre sea, flying level costs 67.85 newtons against 66.1 in calm water; the ideal cut, at 2.92 times the peak frequency, costs 67.56. Those numbers are the earlier essay’s, re-averaged over three seeds rather than one, and the gain from dividing the sea is the same third of a newton.

The two filter pairs have no minimum at all. Both curves start at level flight’s drag when the crossover is far below the sea’s peak, because then nothing is followed, and rise from there. At a crossover equal to the peak frequency the first-order pair costs 68.42 and the second-order pair 68.21; at twice the peak, 70.10 and 69.10; at the ideal cut’s best, 72.4 and close to 69.8. The best setting for either is to set the crossover so low that the filter follows nothing — that is, to fly level. In the 0.2-metre sea the picture is the same at a smaller scale: level flight 66.49, the ideal cut 66.45, and both filters best at level flight.

This is the essay’s first result, and it changes the earlier one’s reading. The ideal cut’s third of a newton was a property of the cut’s sharpness. The control was gaining by following the peak exactly and flying exactly level through everything a little shorter. Any lag at all spoils the components either side of the cut, and in a moderate sea those are where all the gain was, because the peak is where the sea’s energy is. A real wand set to divide a moderate sea at its peak is worse than a wand stiffened until the boat holds its height.

The ledger’s two numbers for the moderate sea say how small the margins are. The ideal cut’s third of a newton is half a per cent of the drag, and the first-order pair at the ideal cut’s crossover gives back seven per cent. The asymmetry is the leak in the second figure: the cut’s saving comes from the depth swing it removes, which is a small cost in a moderate sea, while the first-order pair’s loss comes from the heave it adds, which enters the induced drag through its square and is not small at all. A sailor tuning a wand on a moderate day by feel would find the stiffest setting fastest and would be right.

The 0.4-metre sea is a shade too gentle for the heave to matter much: a foil following the long waves of a 0.4-metre sea is hardly working harder than one flying level. That is why level flight there is already within two newtons of calm water, and why there is so little for the filter to lose. The rougher sea is where the choice is forced.

In a rough sea no setting is safe #

In the 0.6-metre sea, flying level brings the foil to the surface, as the earlier essay found: in all three seeded seas, the shallowest the foil comes below the surface is below zero, and the model’s stand-in for a breach charges 582 newtons of drag. Following everything costs 366 newtons and asks the foil to pull the boat downward more than a third of the time. The ideal cut is the only strategy that flies cleanly: 70.0 newtons at 3.32 times the peak frequency, the foil never shallower than six centimetres, and no moment at which it must pull. Its window of crossovers that neither breach nor pull more than one per cent of the time runs from about 3.3 to 3.9 times the peak frequency in the scan.

The drag curves in that sea, drawn again here without the dial, have the shape that tells the rest of the story. The ideal cut’s curve sits at the breach until its crossover reaches about three times the peak frequency, then drops in one step to 70 newtons and rises only slowly beyond, as it begins to follow the short sea. The first-order pair leaves the breach sooner, between two and two and a half times the peak frequency, because its soft follow part already takes in a share of the peak’s components there, but it lands on a floor that is already rising: at 2.5 times the peak frequency it costs 79 newtons and at four times 89. The second-order pair stays at the breach longest of all, through the band where its overshoot sits on the sea’s peak, and comes off it only at six to seven times the peak frequency, where it is following so much of the sea that the heave dominates.

The fifth figure plots, for each filter, the share of time its heave would need negative lift, but only at crossovers where the foil stays under the surface in every seeded sea. The ideal cut’s points lie along the bottom across its window. The first-order pair’s start a little further left, at about 2.4 times the peak frequency; at a crossover of 2 the foil still breaks the surface in one seed out of three. But where it stays under it has to pull at least 11.7 per cent of the time, because the same soft slope that lets it follow the peak’s components early has it heaving to the short sea as well. The second-order pair is worse on both counts: its overshoot near the crossover makes it breach in every seed at every crossover up to five times the peak frequency, and where it finally stays under, its pull share is more than a fifth.

So the rough sea has no setting for either real filter that does both of the things the earlier essay’s divided control did. Asked for its least drag, the first-order pair settles at 2.39 times the peak frequency, 79.1 newtons, pulling 11 per cent of the time; the second-order at 6.55, 90.5 newtons, pulling 22 per cent. Those drags are not predictions any more than level flight’s 582 was, because the model charges an impossible negative lift as if the foil delivered it. What they predict is that a boat with either filter, in that sea, alternates between dropping off the backs of waves it cannot follow down and holding its height through waves it should have followed.

Why sharper is worse #

The refuted claim deserves a direct answer, because the intuition behind it is strong. A higher-order filter is closer to an ideal filter in its magnitude: well above and well below its crossover it does almost exactly what the step does. In signal processing, where a filter’s job is to pass one band and reject another, that is the whole of the story, and a fourth-order filter beats a second-order one.

A complementary pair is different, because its two outputs are not a wanted signal and an unwanted one; they are two halves of one wave that must add back to it. The magnitude of each half near the crossover is fixed by the phase of the other. A first-order pair splits the wave so that the two halves’ squared magnitudes add to exactly one at every frequency. A wandering flock passes no error down the V used that same identity to show that a bird re-timing to its leader through one first-order lag shares the leader’s wander between following and error without losing any. A second-order pair cannot keep it: a quarter-cycle lag at the crossover makes the two halves’ squared magnitudes add to two, so the pair spends more than the whole wave between following and flying through. The sharper the roll-off, the larger the lag at the crossover, and the larger the overshoot in whichever half is not lagging.

The sea’s energy sits in a narrow band around its peak, and the crossover that makes sense for an ideal cut sits on that band’s shoulder. That is exactly where a complementary pair is worst. A sharper filter moves the trouble into a narrower band and makes it deeper, which helps only if the sea’s energy can be kept out of that band — if the crossover can sit well clear of the peak on one side or the other. Heading into a rough sea, neither side is available: below the peak the boat flies through the long waves and breaches, above it the boat follows the short waves and pulls. What works is not a sharper filter but a smaller lag, and a lag is a property of the hardware, not a setting.

What the calculation was checked against #

The filter pairs were checked in three ways. With the ideal cut, the calculation reproduces the earlier essay’s drag exactly — the same sea, the same crossover, the same number to the last digit — which checks that the new code adds nothing when there is nothing to add. Each follow filter was checked against its own defining equation at five frequency ratios, to a residual below a part in 10¹⁵, and each was checked to pass everything far below its crossover and nothing far above it. The lag at the crossover is 45 degrees for the first-order pair and 90 for the second-order, as each filter’s algebra requires. The calculation refuses a filter order it does not have and a crossover of zero.

The results are averaged over three seeded draws of the sea because the rough sea’s breach is a tail event, and one draw can hide or invent it. The earlier essay found the ideal cut’s best crossover in the 0.6-metre sea ranging from 3.3 to 3.7 times the peak frequency across four seeds; three seeds averaged put it at 3.32. The rough sea’s verdict is stated draw by draw rather than on the average: a crossover counts as keeping the foil under only if it does so in all three seas, and neither filter pair has one that does that without pulling.

What the calculation leaves out #

The wand is only its filter. A real wand also has a geometric gain — how many degrees of flap per centimetre of height — and a mechanical one adds friction and backlash, which are not linear and do not have a transfer function at all. The first-order pair is the most generous reading of a simple linkage: a smooth, linear response with the least lag a single time constant can have.

The flap’s own response is instant. The foil’s lift follows the flap and the inflow with no delay. The lift that arrives late showed that a wing’s circulation takes tens of chords to settle after a change, and the lag that makes flutter possible put that delay in the frequency domain as Theodorsen’s lift deficiency. On the dinghy’s foil, eleven centimetres in chord at 8 metres a second, twenty chords pass in about a quarter of a second, a quarter of the period at which the boat meets the sea’s peak. That lag would add to the filter’s, not cancel it.

The filter has no lead. A controller on a larger boat can add an accelerometer or a forward-looking height sensor to anticipate the wave rather than lag it, and complementary filters in aircraft navigation combine a lagging sensor with a leading one for exactly this reason. A lead could take back some of the lag at the crossover; this calculation does not price one.

The sea is long-crested and the boat heads straight into it, as in the earlier essay. Spreading the sea over directions lowers the encounter frequencies of components met obliquely, which would move the trouble band lower and may let a crossover sit clear of the peak.

The lesson for the divided control #

A foil flies level through a short sea and follows a long one found that a boat in a regular wave must choose what to hold steady, and its random-sea successor found that a boat with an ideal cut need not choose at all. The lagged control puts the choice back. A real filter cannot divide a sea at its peak without paying, near that peak, for the half it does badly; in a moderate sea the payment exceeds the saving, and in a rough sea it turns into a breach or a pull. The divided control is only as good as the sharpness of the division, and its sharpness is not free.

The picture is the same one a wing averages a gust through its own loading drew for a wing in turbulence: the disturbance’s spectrum seen through a filter the vehicle cannot escape. For the wing the filter was its span loading, fixed by its planform; for the foil it is the wand, chosen by its designer, and the lesson is that choosing one does not mean choosing the step that would be best. A foil under the surface flies with a phantom gave the drag budget every number here is charged against, and the budget is unchanged; what changed is that the control which spends it cannot see the sea in time.

Who worked it out #

Complementary filtering is older than the foiling dinghy by decades: it was used in the 1960s and 70s to blend a barometric altimeter’s slow, lagging reading with an accelerometer’s fast, drifting one into an aircraft’s height, and the hydrofoil ships of the same years blended a height sensor with a vertical accelerometer for the same reason. Butterworth’s maximally flat filter is of 1930. The observation that a complementary pair of order higher than one must overshoot near its crossover is a direct result of the sum constraint, familiar to designers of loudspeaker crossover networks, who meet it as a hump in the summed power where two drivers share a band.

Still open: a lead from the boat’s own acceleration #

The calculation here priced a wand alone. A boat that also measures its own vertical acceleration can combine the two: the wand’s height, filtered for the long waves, and the integrated acceleration, filtered for the short, so that the heave is anticipated rather than lagged. That is the scheme hydrofoil ships used, and it is the obvious next question for a small foiling boat with a controller. The calculation would replace the follow filter by one built from both sensors, add the accelerometer’s noise and drift, and ask whether the lead recovers the ideal cut’s window in the 0.6-metre sea, and how much sensor drift it takes to close it again.