The angle that does not care

Look down at any harbour from any height and every wake is the same shape. The little boats and the big ones, the fast ones and the slow, all trail a wedge that opens at the same angle behind them.

That is a strange thing for a physical system to do. Nothing else in this subject behaves that way: the Reynolds number changes everything about a flow, and here is a result that changes with nothing at all.

A second number for a second kind of flow #

Everything else on this site is a flow round a body in unbounded fluid, governed by the ratio of inertia to viscosity. A flow with a free surface has something else in it: gravity, which pulls the surface flat and which the fluid’s inertia resists.

The ratio of those two is the Froude number,

$$Fr = \frac{U}{\sqrt{gL}}$$

and it decides free-surface behaviour the way the Reynolds number decides viscous behaviour. Two quite different lengths get used in it — the vessel’s length, and the water’s depth — and they answer different questions, which is the same trap the Reynolds number sets.

The distinction between the two lengths is worth settling before anything else, because it causes the same confusion the Reynolds number’s does. The length Froude number uses the vessel and answers questions about wave-making resistance and hull speed. The depth Froude number uses the water and answers questions about whether waves can travel upstream at all. A vessel can be at 0.3 on one and 1.5 on the other simultaneously, and the two say completely different things about it.

Which waves can keep up #

The construction behind the wedge starts with a question about bookkeeping. A moving disturbance radiates waves in every direction, but almost all of them are left behind. Only those that can hold station with the vessel form a steady pattern.

A wave whose crests run at angle θ to the vessel’s track has to match the component of the vessel’s speed along its own normal:

$$c = U\cos\theta$$

Every θ from zero to a right angle gives a wave that satisfies this, with a different speed and therefore a different wavelength. So the wake is not one wave but a continuous family of them, and the pattern is what that family adds up to.

The factor of a half #

Now the crucial fact, and it is the whole result.

Deep-water gravity waves are dispersive: their speed depends on their wavelength, going as the square root of it. And for any dispersive system, the energy does not travel at the speed of the crests. It travels at the group velocity, and for deep-water waves the group velocity is exactly half the phase velocity.

So a packet of waves radiated at angle θ does not go where its crests go. It goes half as far, and the locus of everything radiated at one instant is built from that half rather than from the phase speed.

Working through the geometry — the vessel has moved Ut, the packet has moved (c/2)t in direction θ — gives a locus whose angle from the track is

$$\tan\phi = \frac{\sin\theta\cos\theta}{1 + \sin^2\theta}$$

and every U, every g and every wavelength has cancelled out.

What the solver computed, and how it was checked #

That expression is maximised numerically rather than differentiated on paper, by sweeping θ across a quadrant in three thousand steps and finding where the locus reaches furthest from the track.

The maximum occurs at θ = 35.27 degrees and gives a half-angle of 19.4712 degrees. The closed form is arcsin(1/3), which is 19.4712 degrees.

The check that matters is not that match. It is that the sweep is run at three different speeds — 2, 5 and 11 metres per second — and returns 19.471, 19.471 and 19.471. Speed-independence is the surprising half of the claim, and one figure at one speed could not demonstrate it however precise it was.

That check is also the one a wrong construction would fail. Building the locus from the phase speed instead of the group speed — an easy mistake, and the one most hand-drawn versions of this figure make — produces a wedge that widens with speed and looks entirely plausible. The gate refuses a computed angle more than 0.002 radians from arcsin(1/3) at any speed, and refuses a set of speeds that disagree with each other.

Why one third #

The number itself deserves an explanation rather than a decimal.

Setting the derivative of the locus angle to zero gives sin θ = 1/√3, and substituting back gives tan φ = √2/4, which is sin φ = 1/3. The one-third is the ratio between the two ways the geometry scales, and it survives because both scale the same way with everything that was cancelled.

There is a cleaner way to see it that is worth having. The wedge angle is set by a competition between two effects as θ increases: the packet is radiated more sideways, which widens the pattern, and it travels more slowly relative to the vessel, which narrows it. The competition has an interior maximum because the second effect is quadratic in cos θ and the first is linear.

The result is that a wake carries information about nothing. Measure the angle of a wake from a satellite photograph and it tells nothing whatever about the vessel’s speed or size — which is a genuine operational annoyance, and the reason wake analysis for ship detection uses the spacing of the transverse waves rather than the angle of the wedge.

The wavelength does carry information #

The spacing is a different quantity and it is not scale-free at all.

The transverse waves — the ones running across the track, at θ = 0 — have to travel at exactly the vessel’s speed, so their wavelength is fixed by the dispersion relation:

$$\lambda = \frac{2\pi U^2}{g}$$

That is the square of the speed. Doubling the speed makes the wave four times as long.

There is a second reason the wavelength is the useful quantity. It is measurable from a single photograph — count the transverse crests along the track and divide — whereas the angle is uninformative and the amplitude depends on the hull. So a wake does carry the vessel’s speed after all; it simply carries it in the spacing rather than in the shape, which is the opposite of where the eye looks first.

Hull speed #

That square law is why displacement vessels have a speed they cannot economically pass.

As the vessel goes faster its wave gets longer, and at some point one wavelength spans the whole hull. The bow wave’s crest is at the bow and the next crest is behind the stern, so the vessel is sitting in its own trough, bow-up, climbing continuously. The wave-making resistance rises very steeply there.

The condition λ = L gives U = √(gL/2π), which is a Froude number of 1/√(2π) = 0.399. For a thirty-metre hull that is 6.84 metres a second, about thirteen knots, and the figure marks it.

This is the hull speed, and it is a real constraint rather than a rule of thumb. A displacement vessel below it is cheap to drive; above it the resistance climbs faster than any plausible engine. Getting past it requires ceasing to be a displacement vessel — planing on top of the water rather than pushing through it, which is a different regime entirely.

It is also why long ships are fast ships. Hull speed goes as the square root of length, so a three-hundred-metre hull has a hull speed three times a thirty-metre one’s, and that is most of why large vessels are shaped the way they are.

The same reasoning applies to animals, and gives one of the sharper examples of a regime boundary in biology. A duck paddling is a displacement hull and is limited to a Froude number near 0.4 on a body perhaps twenty centimetres long, which is about ninety centimetres a second — and ducks do not exceed it. Going faster means leaving the water, which is what they do. The bird has the same options as the naval architect and takes them for the same reason.

Why the resistance curve is bumpy #

Hull speed is described above as though the wave-making resistance rose steadily to it, and it does not. The curve has humps and hollows in it, and their positions follow from the same wavelength.

A hull makes waves at more than one place — chiefly at the bow and at the stern — and the two transverse systems travel together and interfere. Whether they add or cancel depends on how many wavelengths fit between them, which is $L/\lambda = gL/2\pi U^2$: a quantity that falls as the square of the speed, so it sweeps down through the half-integers as the vessel accelerates.

Where the two systems reinforce, the vessel is dragging a large wave and the resistance has a hump. Where they cancel, it has a hollow. Converting to Froude number puts the features at roughly 0.25, 0.28, 0.33, 0.40 and 0.56, and the spacing widens as the speed rises — so the low-speed ripples are close together and shallow, and the last hump, near a Froude number of a half, is the large one that hull speed is really about.

Two consequences follow that are worth having.

A service speed is chosen in a hollow. Two vessels a few per cent apart in speed can differ by ten or twenty per cent in wave resistance, so a hull is designed and operated at a Froude number between humps rather than wherever the engine happens to be comfortable. That is a real reason ships cruise at oddly specific speeds.

And a bulbous bow is a third wave source, placed to cancel. It is not a fairing and it does not reduce the bow wave by being smooth. It generates its own wave system, submerged and therefore phased differently, chosen so that it interferes destructively with the hull’s — which works at one speed and one draught, and is why a bulb is sized for a service condition and can make things worse away from it.

Where the constant angle breaks #

The wedge is constant in deep water, and “deep” is a Froude number in disguise.

As the water shallows, long waves start to feel the bottom. Their speed stops depending on wavelength and approaches √(gh) for all of them — the water stops being dispersive — and the factor of a half that produced the whole result goes away, because the group velocity rises to meet the phase velocity.

The behaviour has three regimes and the figure shows all of them. Below about Fr = 0.7 the wedge is still essentially 19.5 degrees. Approaching Fr = 1 it opens rapidly, reaching a right angle exactly at the critical value — the wake spreads all the way across the channel. Above it the wedge closes again along a completely different law, sin φ = 1/Fr, which is an ordinary Mach cone. The site’s checks require both halves separately: the deep-water assertion refuses a shallow wake, the shallow assertion refuses a deep one, and a third check requires the two computed angles to be far apart — because a pair of tests that both happened to pass at 19.5 degrees would prove nothing about the distinction between them.

And that is this essay’s surprise. Above the critical Froude number a boat’s wake is a sonic boom. The mathematics is identical: a disturbance moving faster than the waves it makes, radiating a cone whose angle depends on the ratio of the two speeds, exactly as an aircraft past Mach one does. Shallow water is non-dispersive, non-dispersive is what sound is, and the two problems become the same problem.

What the picture cannot show #

The first figure draws the loci of the wave groups and not the waves themselves. Each curve joins everything radiated at one moment, wherever it has since got to; the actual crest pattern — the familiar herringbone of transverse and divergent waves — is a different construction over the same family, and it is not drawn.

Nothing here shows amplitude either. The wedge says where waves can be and says nothing about how big they are, and in a real wake the amplitude varies enormously across the pattern and vanishes at the edges rather than stopping abruptly. The sharp boundary drawn is a caustic, where the geometry piles waves up; the real pattern fades through it rather than ending at it.

And the whole construction is linear and inviscid. A real wake at high Froude number breaks, foams and dissipates, and none of that is in a stationary-phase argument.

Where the model stops #

Deep water means depth greater than about half a wavelength, which for a fast vessel is a substantial depth. A large ship in coastal water is routinely outside the deep-water regime, and the opened wake that results does real damage to riverbanks — which is why speed limits on inland waterways exist and why they are quoted in a way that amounts to a Froude number.

Surface tension is also absent. For waves shorter than about two centimetres, tension rather than gravity is the restoring force, dispersion works the opposite way round, and there is a second wave system ahead of a slow-moving object rather than behind it. A water strider’s wake is not a Kelvin wake at all.

And nothing here has viscosity in it, which is fine for the pattern and useless for the resistance. A ship’s drag is wave-making plus friction plus form, and only the first is what this essay is about — which is exactly the two-part budget an aircraft has, with a different first term.

That combination is what made ship testing hard, and Froude’s solution to it is worth knowing because it is unusual. Matching Froude number and Reynolds number at model scale is impossible, for the same reason matching Reynolds and Mach is. Froude’s answer was to match the Froude number in the tank, compute the friction separately from a flat-plate formula at both scales, subtract the model’s friction from the measured total, scale what was left, and add the full-scale friction back. It is a decomposition rather than a similarity, it rests on the two components being independent, and it is still how ship resistance is predicted.

Who found it, and when #

Kelvin derived the pattern in 1887, using the method of stationary phase which he had developed largely for the purpose. It is one of the more impressive pieces of nineteenth-century applied mathematics: an exact result about a nonlinear-looking phenomenon, obtained by an argument about where the phase of an integral stops varying.

The method has had a much larger career than the result. Stationary phase is now standard in optics, quantum mechanics and signal processing, and it arrived because Kelvin wanted to know the shape of a ship’s wake.

The result also has an unusually clean status among nineteenth-century fluid mechanics, which is not a field with many of those. It is exact within its assumptions, it makes a prediction with no free parameters at all, and the prediction is checkable by anybody standing on a bridge. Very little else in the subject can be verified from a photograph without a single measurement.

The Froude number is older and comes from a different tradition. William Froude was a naval architect who in the 1860s established that model ship testing works if the models are run at matched Froude number, and built the first ship model tank at Torquay to do it. That is model similarity fifty years before the same argument was made about aircraft, and Froude had to fight the Admiralty for it.

Where the ladder goes next #

Next rungs on this anchor: the crest pattern itself, and the two wave systems — transverse and divergent — that make up the herringbone; wave-making resistance and its humps and hollows, where bow and stern waves interfere and the resistance curve is not monotonic; the bulbous bow, which is a deliberate second wave source arranged to cancel the first; and the hydraulic jump, which is what a free-surface flow does when it crosses Fr = 1 in the other direction.

Then across to the Reynolds number, the other dimensionless group a ship must worry about, and to matching a model, where trying to satisfy both at once turns out to be impossible.