The hull Michell's integral prefers

Hull speed is not the hump computed Michell’s integral for the Wigley hull — the parabolic test hull of every wave-resistance comparison — and found the resistance curve’s humps and hollows where the bow’s and stern’s wave systems reinforce and cancel. Its last section turned the integral round. The Kochin function inside it is linear in the hull’s offsets and the resistance is its square, so the resistance is a quadratic form in the hull’s shape, and the hull of least resistance at a given speed is the minimum of a quadratic. The problem is known to be delicate — the unconstrained optimum is not a smooth hull, and the answer depends on what is held fixed — and the essay asked whether, posed with the Wigley hull’s length, draught and displacement and a bound on the waterline’s curvature, the optimum is recognisably a ship: whether it moves volume to the ends as the designer’s prismatic-coefficient rule says it should, and whether at a Froude number of 0.30 it grows something at the bow that looks like a bulb.

This essay poses it, and the three answers are yes, no, and no — each for a reason that says something about linear wave theory.

Twelve shapes, and a quadratic #

The hull’s half-breadth, $f(x, z)$, is written as a sum of twelve separable pieces. Along the hull, $(1 - \xi^2)\xi^i$ for $i$ from 0 to 5, with $\xi$ running from $-1$ at the stern to $+1$ at the bow: each closes at both ends, and the odd powers let the hull be different fore and aft. Down the hull, $(1 - \zeta^2)$ and $(1 - \zeta^2)\zeta^2$, with $\zeta$ the depth over the draught: the first is the Wigley hull’s own depth profile, the second puts volume towards the keel. The Wigley hull is one coefficient, the first.

Michell’s Kochin function — the amplitude of the waves the hull sends out at each angle — is an integral of the hull’s slope times a wave, and with separable pieces it separates too: a Fourier transform of each shape along the hull times a Laplace transform of each profile down it, the second being the rate at which a wave’s influence decays with depth. Both transforms have exact recurrences, so the resistance becomes $c^{\mathsf T}Qc$ for a twelve-by-twelve matrix $Q$ at each Froude number. For the Wigley coefficients the separable Kochin function matches the earlier essay’s closed form, and the resistance its result, to two parts in $10^{14}$.

The displacement is linear in the coefficients, so minimising the resistance at fixed displacement is one linear solve. The waterline’s bending, $\int\!\!\int f_{xx}^2\,dx\,dz$, is added with a small weight: the integral on its own prefers hulls with sharp features, and a bound on curvature is what the essay’s question asked for. The weight used is a thousandth of the Wigley hull’s resistance scale; the waterline optima barely change between a hundredth and a ten-thousandth.

Why the resistance is a quadratic, and what that buys #

It is worth stating why the problem is so tractable, because the tractability is the whole of the method. Michell’s theory replaces the hull by sources spread over its centreplane, each of strength proportional to the hull’s slope there, and the waves the hull leaves behind are the sum of the waves each source makes. That sum is the Kochin function, and because the sources simply add, it is linear in the hull’s shape: double every offset and every wave doubles. The resistance is the energy the waves carry away, which is the square of their amplitude summed over every direction a wave can travel — the Kelvin pattern’s angles, each carried at its own speed — and a sum of squares of linear functions is a quadratic.

A quadratic with a linear constraint has one minimum and it is found in one step. Nothing iterative is needed, no search can get stuck, and the answer is the answer. What the method cannot do is tell whether the quadratic describes the right physics; that is what the rest of the essay is about. The linearity is also the root of both its failures below: it lets volume sink without limit, because a deep source is simply a weak one, and it makes the bow and stern interchangeable, because a sum of waves does not know which was made first.

Fine ends slow, full ends fast #

The first figure holds the depth profile at the Wigley hull’s and frees only the waterline. At a Froude number of 0.25 the optimum pulls its volume into the middle and leaves long, fine ends; at 0.50 it pushes volume out towards the ends and makes them full. At 0.30 it cuts the wave resistance to 0.057 of the Wigley hull’s, which is not a small improvement: the Wigley hull at that speed sits between humps, and the optimum finds the shape whose bow and stern waves cancel over most of the spectrum of angles at once.

The second figure reduces each optimum to the number designers use for exactly this, the prismatic coefficient: how much of the cylinder of the hull’s greatest section, over its length, the hull fills. A fine-ended hull has a low one and a full-ended hull a high one. The optimum’s rises steadily with its design speed, from 0.47 at a Froude number of 0.22 to 0.76 at 0.70, crossing the Wigley hull’s 0.667 near 0.38.

The same freedom explains why the resistance at 0.30 can be cut so far. At that speed the transverse waves are a little over half the hull’s length, the Wigley hull sits between a hollow at 0.28 and a hump at 0.33, and the divergent waves that run off towards the Kelvin angle carry much of the energy. Fining the ends spreads the bow’s and stern’s sources over a longer stretch of the hull, and a source spread over more than a wavelength makes waves that interfere with themselves; the optimum finds the spreading that cancels most of the divergent system at once.

That is the rule ship designers have followed since the nineteenth century, when Froude and Taylor tabulated models — a slow cargo ship wants fine ends and a fast one full ends, with the prismatic coefficient chosen from the design speed — and here it comes out of the integral with nothing put in. The reason is the wave the ends make. The resistance is dominated by the two places the hull’s slope is largest, its bow and its stern, and the hull’s middle, with no slope, makes no waves at all. A hull’s wave system is therefore mainly two systems, one from each end, separated by the hull’s length, and whether they add or cancel depends on how many of their own wavelengths fit into that length — which is the Froude number, and which is why the humps and hollows exist. The optimum’s freedom is in how abruptly each end begins. A slow hull makes short waves, and a short wave sees a full end as an abrupt source; spreading the ends out makes the bow’s waves gentle. A fast hull makes waves longer than itself, and then what matters is where along the hull the volume sits relative to the wave’s crest, and pushing volume to the ends moves the bow and stern systems into cancellation at the design speed.

What the number means for a ship #

The Froude number is the whole of the scaling, and it is worth turning it into ships. At 0.30 a hull of 100 metres moves at 9.4 metres a second, about 18 knots — a fast cargo ship or a ferry; at 0.25, 15 knots, a slower bulk carrier; at 0.50, 30 knots, a warship or a fast ferry near its hump. A ship’s Froude number is the number that really is one in the shallow-water sense only when it runs in water as shallow as its length is long; in deep water it is the ratio of the ship’s length to its own wave’s, and it places the ship on the humps and hollows once and for all.

So the figures above read directly as design practice. A 15-knot, 100-metre ship wants a prismatic coefficient near 0.50; a 31-knot one near 0.73; the same hull cannot be both, and a ship designed for one speed and operated at another — slow steaming, which cargo lines adopted to save fuel — sits on the wrong side of its own optimum. The resistance the optimum saves at its design speed is not saved at the other, which is the third figure’s lesson in ships rather than curves.

Each optimum is best at one speed #

The third figure runs the two optima across the whole range of speeds. The hull optimised at 0.30 beats the Wigley hull by a factor of seventeen at its design speed and loses to it above about 0.34, badly near the main hump. The hull optimised at 0.50 cuts the resistance by a fifth there and loses below 0.39. A hull has one length, and every hump and hollow of its resistance is a function of that length’s Froude number; an optimum is a choice of which humps to sit between. That is why hull speed is not the hump mattered: the hollows are where a designer wants the service speed to fall, and the optimum shape widens and deepens the hollow at the speed it is told.

Set free in depth, it is not a ship #

The fourth figure frees the second depth profile, the one that puts volume towards the keel. The integral takes it and runs. It cuts the resistance to 0.01 of the Wigley hull’s by moving the displacement down, pinching the waterline to nothing and, over parts of the hull, giving it negative breadth. It is the delicacy the earlier essay warned of, and it has a plain cause, the one that makes a hydrofoil deep under the surface make almost no waves and a foil at speed fly with only its phantom partner to answer for: the drag that is made of waves falls off exponentially with the depth of whatever makes them, because a wave’s motion decays with depth, so volume placed deep makes almost no waves. The linear theory has nothing to stop it — no ship needs a waterline in Michell’s integral — and its optimum is a submarine with a knife on top.

Real hull forms solve this with constraints the integral does not know about: a waterline wide enough to float stably, a section shape that can be built and loaded, a draught the harbour allows. With the depth profile held, as in the first three figures, the optimum is a ship; with it free, the constraints have to be written in, and the answer is theirs rather than the integral’s.

It cannot tell the bow from the stern #

The fifth figure answers the essay’s last question, and the answer is the most instructive. A bulb at the bow is a second source whose waves, at the right speed, cancel the bow’s; the earlier essay found one cutting the Wigley hull’s wave resistance to 0.53 of its bare value at 0.30. Put the same bulb at the stern and the resistance is — to every digit — the same. The two curves in the figure are one.

The reason is a symmetry of the integral. Reversing a hull, bow for stern, turns its Kochin function into its complex conjugate, and the resistance is the Kochin function’s squared magnitude, which does not change. So Michell’s theory gives a stern bulb the same benefit as a bow bulb at every speed, and every optimum it produces is symmetric fore and aft: in the waterline optima above, the odd, asymmetric coefficients came out zero to eighteen decimal places. Asked whether the optimum hull at 0.30 grows something at the bow, the integral cannot say bow; whatever it grows, it grows at both ends.

The same symmetry says something about the Kelvin wake itself. A ship moving forward and the same ship moving backward, at the same speed, leave wakes with identical energy in every direction; only the phases of the waves differ, and the phases are invisible to the resistance. Linear theory is time-reversible in this sense, and a hull’s shape enters it only through the magnitude of its Kochin function at each angle.

What makes a bow bulb better than a stern bulb in practice is outside the theory. The stern sits in the hull’s own boundary layer and wake — a layer that is an integral of everything upstream, a hundred metres of it by the time it reaches the stern of a large ship — which reduce the effective speed and the wave-making there, and the flow along a real hull separates at the stern and not at the bow. A thin-ship integral with no viscosity has neither, and so its reason for a bulb is a phase, not a location.

What was checked #

The ledger holds three checks: the separable Kochin function and the resistance for the Wigley coefficients against the earlier essay’s closed form, to two parts in $10^{14}$; a deliberately asymmetric hull and its reverse giving the same resistance to rounding; and the optimum’s odd coefficients vanishing.

What the thin-ship integral leaves out #

Viscosity. The wave resistance is a fraction of a displacement ship’s resistance at service speed — small for a slow, full ship and large for a fast, fine one; the rest is skin friction, which the prismatic coefficient also affects, through wetted area. A real optimum trades the two.

Thin ship. Michell’s theory treats the hull as thin, its slopes small; the Wigley hull’s beam is a tenth of its length, and bluff bows are outside it.

Sinkage and trim. A real ship squats and trims at speed, changing its own shape relative to the water; the integral holds the hull fixed.

The basis. Twelve shapes are enough to show the trend and the symmetry, and not enough to find sharp-shouldered optima; more shapes lower the resistance further and make the bending bound more important.

The convention: Froude number on length #

The Froude number is $U/\sqrt{gL}$ with $L$ the hull length. The Wigley hull has length ten times its beam and sixteen times its draught. Resistances are Michell’s, in the units of the earlier essay, and compared as ratios to the Wigley hull’s at the same speed. The prismatic coefficient is the volume over the product of the greatest sectional area and the length.

Who worked it out #

Michell published the integral in 1898. Its use for optimising hull form began with Weinblum in the 1930s, who found the optimum sectional-area curves and their symmetry; Krein and Sretensky studied the minimisation in the 1950s, and the recognition that the unconstrained problem has no smooth solution, and must be constrained, runs through that work. Taylor’s standard series and Froude’s model tests had fixed the prismatic-coefficient rule empirically long before.

Still open: an integral that knows which end is which #

The symmetry that forbids a bow bulb is a property of an integral with no wake. The simplest way to break it is to let the stern’s waves be made by a hull whose effective speed is reduced by its own boundary layer: a wake fraction that multiplies the stern’s source strength. The next calculation adds that to the Kochin function, re-solves the optimum with a stated wake fraction, and asks whether the optimum then becomes asymmetric in the right direction — fuller aft, with its bow and stern waves re-timed — and whether a bulb placed by it goes to the bow, which would say that the bulbous bow is, in the end, a consequence of the stern’s viscosity rather than of the bow’s waves.