The speed below which nothing on water leaves a wake

The speed that depends on the length found that waves on deep water travel at a speed set by their wavelength: long waves faster than short ones when gravity drives them, short ones faster than long when surface tension does. Between the two families there is a wavelength, about a centimetre and three-quarters on clean water, where the speed is least: 23 centimetres a second. The packet that moves at another speed found that the energy of a group of such waves travels at a different speed from their crests, half the crest speed for long gravity waves.

Put those two facts together with one more — that anything moving steadily across the water can feed only waves that keep pace with it — and a striking conclusion follows. There is a speed below which nothing crossing clean water leaves any wake at all. Above it, the same object makes two quite different wave trains at once, one in front of it and one behind. This essay is about that threshold and the two trains, and about why the same arithmetic appears in a superfluid, a supersonic aircraft and a fishing line.

The slowest wave #

For waves on deep water restored by both gravity and surface tension, the frequency and wavenumber are related by

$$\omega^2 = gk + \frac{\gamma}{\rho}k^3,$$

with $g$ the acceleration of gravity, $\gamma$ the surface tension and $\rho$ the density. The phase speed, $\omega/k$, is the square root of $g/k + \gamma k/\rho$: the first term is large for long waves, the second for short ones, and their sum has a minimum where they are equal,

$$c_{\min} = \left(\frac{4g\gamma}{\rho}\right)^{1/4} \approx 23.1\ \text{cm/s}, \qquad \lambda_{\min} = 2\pi\sqrt{\frac{\gamma}{\rho g}} \approx 1.71\ \text{cm}.$$

The wavelength of the minimum is $2\pi$ times the capillary length, the distance over which gravity and surface tension are equally important — the length that also set the size of a meniscus against a wall and the height a drop can stand on a table. The speed of the minimum combines all three of the liquid’s surface properties into one number. For clean water at room temperature it is 23.1 cm/s, about the speed of a slow walk’s foot as it swings forward. No wave on clean water, of any wavelength, travels more slowly. The group speed has a minimum too, 17.7 centimetres a second at a wavelength of 4.4 centimetres, and it shows itself every time a pebble is dropped into a pond. The splash sends out waves of every wavelength, each group travelling outward at its own group speed; no energy travels slower than 17.7 centimetres a second, so after a second the region within about 18 centimetres of the splash is calm, a smooth disc inside the expanding rings, with short ripples racing out in front and long swells following in the middle. The calm centre grows at the slowest group speed, which is why it is there at all.

The two restoring forces trade places for a plain reason. When the surface is pushed into a wave, gravity pulls each raised crest back with a force that depends on its height but not on how sharply it is curved, so its pull per unit displacement is the same for every wavelength. Surface tension pulls the surface flat with a force set by its curvature, which for a given height grows as the inverse square of the wavelength. A long, gently curved wave hardly feels the tension; a short, sharply curved ripple hardly notices gravity by comparison. The skin that is not a skin found that surface tension is not a membrane but the cost of exposing molecules at the surface, and here that cost acts as a stiffness, just as how high water will climb found it competing with gravity over the same capillary length.

A moving object can only feed waves that keep up #

Watch a stick held still in a stream, or towed slowly through still water. If it makes a steady pattern of waves around itself — a pattern that looks the same from moment to moment in the stick’s frame — then each wave in the pattern must stand still relative to the stick. A wave stands still relative to an object moving at $V$ only if its crests travel at $V$ through the water. So the stationary waves of a moving object are exactly those whose phase speed equals the object’s speed.

Draw a horizontal line at the object’s speed across the curve of phase speed against wavelength. Where the line crosses the curve, there is a wave the object can feed. Below the curve’s minimum there is no crossing. An object moving slower than 23 centimetres a second has no wave it can feed, and it leaves the surface behind it flat. It still pushes water aside and still feels viscous drag, but it radiates no energy into waves, and its wave drag is exactly zero.

At exactly the threshold the line touches the curve at one point, and the object feeds one wave, of 1.71 centimetres. Above it the line crosses twice. A pebble towed at 30 centimetres a second can feed a capillary ripple of 5.6 millimetres and a gravity wave of 5.2 centimetres; at 50, a ripple of 1.6 millimetres and a gravity wave of 15 centimetres. Elie Raphaël and Pierre-Gilles de Gennes worked out in 1996 that the wave drag on a small disturbance should therefore switch on abruptly at the threshold, and experiments with a small magnetised needle pulled across a water surface found that it does — the drag jumps by a large factor as the speed passes 23 centimetres a second.

The same argument, made for an excitation in a superfluid, is Landau’s criterion. The speed below which nothing can be made found that a body moving through superfluid helium slower than the minimum of $\varepsilon(p)/p$ over all the liquid’s excitations cannot create any, and so feels no drag at all. The minimum of the water waves’ phase speed is exactly the minimum of $\omega/k$ over the surface’s excitations, and the threshold is the same theorem applied to a different spectrum. On water, viscosity and the push of displaced water still act below the threshold, so nothing on water becomes frictionless; but the waves, which are what a boat’s hull pays most for at speed, vanish.

Ripples ahead, waves behind #

The two waves an object feeds above the threshold are born together, at the same wavelength, and part as the speed rises: the ripple gets shorter, the gravity wave longer. At a metre a second the ripple is under half a millimetre and the gravity wave nearly two-thirds of a metre. And they appear on opposite sides of the object.

The reason is the group speed. A wave pattern standing still around a moving object must be fed continuously, and the energy fed into it must be carried away from the object at the wave’s group speed relative to the water. A gravity wave’s energy travels at half its crest speed, so, relative to an object moving with the crests, the energy falls behind: gravity waves trail the object as a wake. A capillary ripple’s energy travels at up to one and a half times its crest speed, so relative to the object the energy runs ahead, and the ripples stand in front of it, upstream.

That is the pattern seen around a fishing line or a reed standing in a gentle stream. Upstream of it, close in, the surface is crinkled with fine, short ripples, closely spaced and steady; downstream, a coarser train of longer waves spreads out behind in a V. In a stream slower than 23 centimetres a second, neither is there, and the line stands in a smooth surface broken only by the bulge of water flowing round it.

The gravity-wave wake behind a fast object is the one the cone the source leaves behind treated in its general form: a source outrunning its own waves leaves an envelope behind it. For water the envelope is not a Mach cone, because the waves are dispersive, but a wedge of fixed half-angle, about $19.5°$ for deep water, independent of speed. The capillary ripples ahead form an envelope of their own, a set of arcs bowed forward round the object’s nose, shrinking as the speed rises.

A threshold that light and sound do not have, and one they do #

Sound in air and light in a vacuum travel at a single speed whatever their wavelength, so a source moving more slowly than that speed can never keep pace with any of its waves, and a source moving faster keeps pace with all of them at once along a cone. The river that sound cannot swim up found the sound version, and the light version is Cherenkov radiation: a charged particle moving through glass or water faster than light travels there emits a cone of light, and slower than that emits nothing. Why the glow of a fast charge is blue described it, and the light a charge makes by changing medium found that a charge below the threshold still radiates where the medium changes under it — the counterpart of the duck’s bobbing feet, which make waves by changing rather than by speed.

Water waves sit between. Their dispersion gives them a minimum phase speed rather than a single one, so the threshold is a minimum rather than a fixed speed, and above it the source is matched to two waves rather than to a whole cone of one. The gravity branch’s wake has the same origin as the Cherenkov cone and the Mach cone — a moving source in step with its own waves — and differs from them only because each wavelength travels at its own speed.

The same matching sets the speed a displacement hull can reach. A boat’s own gravity-wave wake has a wavelength set by its speed, $2\pi V^2/g$; when that wavelength approaches the length of the hull, the boat sits in a trough of its own making with its bow climbing the next crest, and the power needed to go faster climbs steeply — the hull speed, the gravity branch’s version of the threshold at the other end of the curve. A layered sea adds a slower family of waves on the buried boundary between fresh and salt water, and the wave that holds a ship back found a ship trapped near the speed of those internal waves, feeding a wake no one could see on the surface. Each interface has its own curve of wave speeds, and a hull is held wherever its speed matches one of them.

A wind that cannot raise waves #

The threshold also shows up in the familiar fact that a light breeze leaves a lake glassy. Wind raises waves by passing over the surface faster than the waves it can couple to; in the simplest account, waves can be fed only once the wind at the surface, or the speed of the pressure fluctuations it carries, exceeds the slowest wave’s speed. The first patches of ripples — “cat’s paws” — appear on a calm lake when the wind exceeds about one metre a second at a height of a few metres, which corresponds to a speed near the surface of a few tens of centimetres a second: the threshold again, blurred by the turbulence of the wind and the slow drift of the water beneath it.

A film of oil or soap on the water lowers the surface tension and lowers the threshold with it, so that weaker winds might be expected to raise ripples on a contaminated surface. The opposite is observed — oil calms water — because a surface film does something the threshold does not account for: it resists being stretched, and the extra damping that elastic film produces kills short ripples far faster than the lowered tension helps make them.

Other liquids, other worlds #

Because the threshold goes as the fourth root of surface tension over density, it varies little among ordinary liquids. Ethanol and soapy water, with a third of water’s surface tension, bring it down to about 18 centimetres a second; mercury, with nearly seven times water’s tension but thirteen times its density, sits at 19. Liquid helium, whose surface tension is two hundred times smaller than water’s, has a threshold under 10 centimetres a second, so a helium surface is much more easily set rippling.

Gravity enters as the same fourth root. On Titan, Saturn’s largest moon, whose lakes are liquid methane and ethane at minus 180 °C and whose gravity is a seventh of the Earth’s, the threshold is about 12 centimetres a second; the capillary length, which goes as the inverse square root of gravity, is longer, and the slowest wave is about two and a half times longer than on Earth. Radar images of Titan’s lakes have shown them remarkably smooth, and whether that is because Titan’s winds are too weak to pass even this lowered threshold, or because something on the surface damps the ripples, has been argued since the first images came back.

What the threshold leaves out #

Depth and walls. The formulas are for deep water — deeper than a wavelength — which for the ripples and short gravity waves near the threshold means a few centimetres. In a shallow tray the gravity branch changes, and the long waves are slowed by the bottom, as the speed that depends on the length found for shallow water.

Viscosity. Short ripples are strongly damped by viscosity: a ripple of a millimetre on water dies within a few centimetres of travel. The upstream ripples of a fast object are therefore confined close to it, and at high speeds, where they are very short, they are barely seen.

The object’s size. A disturbance can only feed waves whose wavelength is comparable to or longer than its own size; a broad, smooth object barely couples to ripples shorter than itself, and a boat’s hull feeds mostly the long gravity waves of its wake. The sharp threshold is clearest for small objects — a needle, an insect’s leg, a fishing line — that couple to wavelengths near a centimetre.

Steadiness. The argument is for an object moving at constant speed. Anything that starts, stops, wobbles or bobs radiates waves of every wavelength the change contains, whatever its average speed, which is why a slowly paddling duck still leaves ripples: its feet are not moving steadily.

Still open: how the smallest walkers on water move #

The threshold produced a celebrated puzzle in animal locomotion. Water striders walk on water, propelling themselves by rowing with their middle legs against the surface. In 1993 Mark Denny pointed out that if their propulsion came from making waves — pushing against the surface the way a paddle pushes against water — a newly hatched strider, whose legs move more slowly than 23 centimetres a second, could make no waves at all and so could not move. Yet baby striders move as well as adults.

The resolution, found by David Hu, Brian Chan and John Bush in 2003 by filming the flow beneath a strider with dye and particles, was that striders do not rely on waves. Each stroke sheds a pair of small vortices into the water below the surface, like the vortices shed by an oar, and the momentum carried away by those vortices is what pushes the insect forward. The capillary waves a strider makes are a by-product, and below the threshold they are absent without the insect noticing. How smaller and faster surface-walking animals divide their momentum between waves and vortices, and how robots built to imitate them should be designed, is still being worked out.

The threshold itself is simple. A steady disturbance on water can only feed waves that keep pace with it, and there is a slowest one — so below 23 centimetres a second the surface carries nothing away, and above it the surface carries two trains at once, one in front and one behind, sorted by whether their energy runs faster or slower than their crests.