Hull speed is not the hump

The angle that does not care finds that every ship on deep water leaves a wake inside the same wedge, and then turns to speed. Its account of hull speed is the one that every sailor learns: as a vessel goes faster its wave lengthens, and when one wavelength spans the hull, at a Froude number $U/\sqrt{gL}$ of $1/\sqrt{2\pi} = 0.399$, the vessel sits in its own trough and the resistance rises steeply. It adds that the resistance curve is bumpy, and places its humps and hollows by counting how many wavelengths fit between the bow and the stern.

The drag that is made of waves derives the wave resistance of a submerged two-dimensional body from first principles and ends by naming what a real hull needs: Michell’s thin-ship integral, in which the interference between the bow and stern systems produces the humps and hollows. This essay computes that integral for a standard hull and asks three things of it that the counting argument cannot answer. Where exactly are the humps and hollows? What is happening at hull speed? And what does the curve do beyond it?

The integral a mathematician published and naval architects ignored #

John Henry Michell, a mathematician in Melbourne, published it in 1898. He took a hull whose half-breadth $y = f(x, z)$ is small compared with its length — a thin ship — moving at speed $U$ over deep water, and replaced it by a sheet of sources on its centreplane, the same device by which a body in an unbounded stream is built out of sources and sinks and found to feel no drag at all, with a strength at each point proportional to the slope of the hull there: the bow, where the hull widens, pushes water aside and is a source; the stern, where it narrows, draws water in and is a sink. Linearising the free surface, as the two-dimensional essay does, he found the resistance as a sum over waves travelling at every angle $\theta$ to the track, each of them stationary relative to the ship:

$$R = \frac{4\rho g^2}{\pi U^2}\int_1^\infty |F(\lambda)|^2\,\frac{\lambda^2\,d\lambda}{\sqrt{\lambda^2 - 1}}, \qquad F(\lambda) = \iint f_x\,e^{\lambda^2 k_0 z}\,e^{i\lambda k_0 x}\,dx\,dz,$$

with $k_0 = g/U^2$ and $\lambda = \sec\theta$. The factor $F$ is the hull’s Kochin function: the amplitude with which it radiates a wave in each direction, formed by adding up every source on the hull with the phase its position gives it. Everything about humps and hollows is in that phase.

The phase is also why the waves in the integral are the ones they are. A ship moving steadily makes only the waves that stand still relative to it, and a wave travelling at angle $\theta$ to the track stands still when its phase speed equals $U\cos\theta$; on deep water that fixes its wavenumber at $k_0\sec^2\theta$, which is the $\lambda^2 k_0$ in the exponent. Adding those waves up with the phases the ship’s position gives them is what produced the wedge of the Kelvin wake, whose edge sits where their group velocity carries energy no further sideways. Michell’s integral is the same sum weighted by the energy each wave carries away, which is the drag.

The integral sat almost unused for thirty years. Evaluating it by hand for a real hull was forbidding, and it was Thomas Havelock and then W. C. S. Wigley, in the 1920s and 1930s, who made it a working tool. The hull used here is Wigley’s own test form, whose half-breadth is parabolic both along the length and down the draught,

$$f(x, z) = \frac{B}{2}\left(1 - \frac{4x^2}{L^2}\right)\left(1 - \frac{z^2}{T^2}\right),$$

at a length ten times its beam and sixteen times its draught. It is still a standard benchmark because both integrals in $F$ are elementary for it, which leaves one integral to do numerically — and that is also why the calculation can be checked. The closed-form Kochin function agrees with a brute-force double integral over the hull to 1.2 parts in $10^5$, the cosine part of the integral vanishes to $10^{-19}$ because the hull is symmetric fore and aft, and the wave-angle integral, after the substitution $\lambda = \cosh t$ that removes its endpoint singularity, is converged to six parts in $10^7$.

The curve, and where hull speed falls on it #

The first figure is the result, as the coefficient $C_w = R/\tfrac12\rho U^2 S$ with $S$ the wetted area. At low speed it ripples: humps at Froude numbers of 0.204, 0.238 and 0.299, hollows at 0.217, 0.260 and 0.346, each swing larger than the one before. Then it climbs steeply to the main hump, at Fr = 0.499, where $C_w = 4.52\times10^{-3}$, and falls away smoothly beyond.

Hull speed, 0.399, is not a feature of this curve at all. It sits halfway up the climb from the last hollow to the main hump. For a thirty-metre hull of this shape the last hollow is at 11.5 knots, hull speed at 13.3 and the main hump at 16.6: the resistance coefficient more than doubles across the first gap and rises by another two-thirds across the second.

The counting argument placed the features at 0.25, 0.28, 0.33, 0.40 and 0.56 — humps at $1/\sqrt{(2n+1)\pi}$ and hollows at $1/\sqrt{2n\pi}$, from a bow wave and a stern wave of opposite sign a hull’s length apart. For this hull every one of them is too fast, by between 5 and 15 per cent of the speed, and the one at 0.40 is labelled wrongly: in the counting argument it is a hollow, the speed at which the bow and stern transverse waves exactly cancel, not the speed at which the hull is sitting in its own trough.

A hollow is where the hull’s transverse wave switches off #

The reason for both the error and the label is in the Kochin function, and the second figure shows it by spreading the resistance over the directions of the waves that carry it.

At the hump near Fr 0.30 most of the resistance leaves in waves travelling nearly along the track — the transverse system, the crests square to the ship’s course that a stern-on photograph shows. At the hollow at 0.346 the resistance density at zero degrees has fallen to a fifth of the hump’s, and what remains is carried by the divergent waves, the diagonal crests along the edges of the Kelvin wedge, at angles beyond the cusp at 35.3°. A hollow is the transverse wave switched off.

When does the hull stop making transverse waves? The transverse wave is the one at $\lambda = 1$, and for the Wigley hull the source strength $f_x$ varies linearly along the length, from a strong source at the bow through nothing amidships to a strong sink at the stern. The Kochin function at $\lambda = 1$ is then proportional to $\sin c - c\cos c$ with $c = k_0 L/2$, and it vanishes when $\tan c = c$: at $c = 4.493$, 7.725 and 10.904, which are Froude numbers of 0.334, 0.254 and 0.214. The full integral’s hollows sit within four per cent of those, and the gap is the divergent waves, which fill in each hollow a little and push it to slightly higher speed.

The counting argument puts all the source at two points, the bow and the stern. Its transverse wave is then proportional to $\sin c$, which vanishes at $c = \pi$ — at $k_0 L = 2\pi$, when the wave is exactly as long as the hull. So hull speed is where two point sources’ transverse waves cancel, and the counting argument’s hollow. A real hull spreads its sources along its length, and where they are spread is what moves the hollows. The Wigley hull’s sources are strongest at the ends but not concentrated there, and its first hollow sits at 0.346 instead of 0.399. A hull with its volume pushed further towards its ends — what naval architects call a higher prismatic coefficient — moves its hollows back towards the two-point values, which is part of why the best prismatic coefficient for a hull depends on the speed it is designed for.

The third figure follows the transverse waves’ share of the resistance across the whole range. It swings from three-quarters at the hump near 0.30 to an eighth at the hollow, and it passes through its last hollow and last peak by Fr 0.42. Beyond the main hump it only falls: to a half at the hump, a fifth by 0.62 and a sixteenth by 0.78. The reason is the wavelength. At Fr 0.5 the transverse wave is $2\pi \times 0.25 = 1.57$ hull lengths long, and a hull shorter than the wave it would make cannot make much of it. The divergent waves, shorter and at an angle, take over the resistance, and they have no bow-to-stern interference to speak of. That is why the curve stops oscillating after the main hump: the waves that interfered are no longer the waves that matter.

Where the resistance climbs, and where it stops climbing #

The hull-speed rule, stated fully, says two things: that the resistance rises steeply at hull speed, and that beyond it the resistance keeps climbing faster than any plausible engine can supply. The fourth figure tests both.

The first half is true, and more than true. At Fr 0.399 the wave resistance is growing as the speed to the power 7.6, and per unit of the hull’s weight it multiplies about sixfold between Fr 0.30 and 0.50. A vessel trying to go from the last hollow to the main hump needs eleven times the wave-making power for a speed only 44 per cent higher. That is a wall in every practical sense, and it is what the rule is right about.

The second half is not. Past the main hump the exponent falls below two, where the resistance coefficient starts to decline, then below one: at Fr 0.6 the wave resistance grows as the speed to the power 0.75, at 0.8 to the power 0.5. Per unit weight it rises by only three-fifths while the speed doubles from 0.5 to 1.0. In thin-ship theory a hull that has reached the main hump has paid for most of the wave-making it will ever pay for; the wall is between the last hollow and the hump, and there is open water on the other side of it.

This is the regime fast displacement hulls live in. A destroyer at full power runs at a Froude number of 0.4 to 0.5, a racing single scull at about 0.55, and a fast ferry well beyond; their hulls are long and slender enough that the power to get over the hump is available, and once over it they gain speed cheaply in wave resistance. The rule is about heavy, full, short vessels, for which the power at the hump is out of reach — which is a statement about engines and weight, not about a resistance that keeps rising.

The things linear theory leaves out, and which of them the rule is really about #

A reader who has sailed a heavy displacement boat will object that it does not behave like this: push it past hull speed and it squats by the stern, its bow rises, and the power required keeps soaring. The objection is right, and Michell’s integral is silent about it, because the integral holds the hull at a fixed position in the water. A real hull sinks and trims as the pressure field of its own waves acts on it — at Fr 0.4 to 0.6 the stern drops into the trough the hull is making — and the hull that trims is effectively a different, fuller and blunter hull making larger waves. For a heavy boat that effect, not the linear wave resistance, is most of the wall.

The integral leaves out three other things. It is linear, so its waves have small slopes, never break and do not interact; the bow wave of a real hull near the hump does all three, and a breaking bow wave dissipates energy in the way a bore in a river does, which no linear theory can account for. It ignores viscosity, and the boundary layer thickens the stern’s effective shape and weakens the stern wave, so a measured hollow is shallower than Michell’s, because the stern no longer quite cancels the bow. Measured wave resistance on this hull form typically falls below Michell’s curve at the main hump for the same reason. And it treats the hull as thin; the Wigley hull, ten lengths to a beam, is thin enough, and a tug, three lengths to a beam, is not.

What survives is the structure: the ripple at low speed, the positions of the hollows set by where the hull’s sources are, the steep climb and the main hump, and the fall in the resistance coefficient beyond it. Those follow from the phase in the Kochin function and the wavelength against the hull, and neither of those is changed by the omitted effects.

How the humps were found before anyone could compute them #

The humps were measured long before they were computed. William Froude, towing models in his tank at Torquay in the 1870s, found that the resistance of a model and of its full-size ship could be compared only if the two ran at the same value of $U/\sqrt{gL}$ — his law of comparison, which is why the number bears his name — and that at matched Froude numbers the curves of wave resistance had the same humps in the same places. He also found that the other part of the resistance, the skin friction, does not scale that way, because it follows the Reynolds number instead. A model cannot match both at once; the model that cannot match follows that impossibility through, and Froude’s answer — measure the total, subtract a friction estimated from a flat plank of the same wetted area, and scale only the remainder — is still how a towing tank reports a ship.

So the remainder, the residuary resistance, is the measured counterpart of this essay’s curve, and it carries the humps and hollows because they belong to the wave-making, not the friction. Michell’s integral was the first explanation of why they sit where they do, and for thirty years it was an explanation nobody used, because a towing tank could measure a curve in an afternoon that took weeks to compute. What the computation offers that the tank does not is the decomposition — which waves, in which directions, from which part of the hull — and that is what the second and third figures are for.

A bulb is a second source, and a second source has a phase #

The angle that does not care describes a bulbous bow as a third wave source placed to cancel the hull’s, working at one speed and one draught and able to make things worse away from them. Michell’s integral makes that a calculation. A submerged sphere of radius $r_b$ in a stream is a doublet of moment $2\pi U r_b^3$, and it enters the Kochin function as one more term, with its own depth and its own position:

$$F \;\to\; F - i\pi r_b^3\,\lambda k_0\,e^{\lambda^2 k_0 z_b}\,e^{i\lambda k_0 x_b}.$$

The fifth figure places a sphere of radius two-fifths of the draught, centred at seven-tenths of the draught and two per cent of the length ahead of the stem, its depth and position chosen by a search for the least wave resistance at Fr 0.30 and its size kept smaller than that search’s optimum. There it cuts the wave resistance to 53 per cent of the bare hull’s. At Fr 0.6 it still saves 8 per cent. At slow speed the same bulb’s wave arrives in phase with the bow’s instead of against it, and at Fr 0.16 the wave resistance is 5.1 times the bare hull’s. In absolute terms that is a large multiple of a small number, since the wave resistance at 0.16 is a twentieth of its value at 0.30, but it is exactly the mechanism by which a bulb designed for one speed becomes a penalty at another.

That is not a hypothetical. When fuel prices rose, around 2008, many container ships designed with bulbs for service speeds near 25 knots were slowed to steam at 15 to 18. Their bulbs, now far from their design Froude number, were adding wave resistance rather than cancelling it, and a number of owners had the bulbs cut off and replaced with shapes designed for the new speed. The phase that makes the bulb work is the same phase that makes it fail, and the only thing that selects between them is the Froude number.

The ledger, and what the next hull would need #

The sixth figure is the ledger. The checks are of three kinds: the closed-form Kochin function against a direct integration over the hull; the wave-angle integral against a finer step and a cut-off four times further out, necessary because a hull that reaches the surface has a Kochin function that decays only algebraically at steep angles; and the hollows against an independent prediction, the transverse zeros from $\tan c = c$, which the full integral does not know about and matches to within four per cent.

It is worth being clear about what kind of result the essay’s title is. Hull speed is a useful rule: it marks, to within a few per cent for many hulls, the middle of the steepest part of the resistance curve, which is where a designer of a slow vessel should stop asking for more speed. It is not the hump, it is not a wall that the resistance keeps rising against, and its derivation from a wave the length of the hull is, taken literally, the condition for a hollow. The number that really is one in a channel is sharp because the physics changes type there. The Froude numbers in this essay are soft, set by where a hull’s sources lie, and every hull has its own.

Still open: the hull the integral prefers #

Michell’s integral can be turned round: instead of computing the resistance of a given hull, ask which hull of a given length, draught and displacement makes the least wave resistance at a given Froude number. The Kochin function is linear in the hull’s offsets and the resistance quadratic, so this is a quadratic minimisation, and it is known to be delicate — the unconstrained optimum is not a smooth hull, and the answer depends on what is held fixed. The next calculation is to pose it for this hull’s length, draught and volume at Fr 0.30 and at the main hump, with the curvature of the waterline bounded, and see whether the optimum hull is recognisably a ship: whether it moves volume to the ends as the prismatic coefficient rule says it should, and whether, at 0.30, it grows something at the bow that looks like a bulb.