The speed that depends on the length

Drop a stone in a pond and the ring that spreads is not one ring. Watch closely and the longer waves run out ahead while the shorter ones lag, so the disturbance sorts itself as it travels. Nothing about the stone chose that; it is a property of water.

An earlier rung on this anchor established that a wave group can travel at a different speed from its crests. This one supplies the relation that decides both speeds, and shows that the familiar rules for water waves are its limits rather than competitors.

The relation #

For waves on the surface of water of depth $h$, with gravity as the restoring force, the frequency and wavenumber are tied by

$$\omega^2 = gk \tanh(kh),$$

with $k = 2\pi/\lambda$. That single equation is the subject of this page. The hyperbolic tangent is what makes it interesting, because $\tanh$ is almost linear for small arguments and almost exactly one for large ones — so the relation has two clean limits and a transition between them.

The restoring force is gravity: water pushed up by a passing wave is heavier than its surroundings and falls back, and the fall overshoots. What sets the speed is how much water has to move, and that depends on the wavelength — because a wave disturbs the water to a depth of roughly its own wavelength and no further.

Deep water: the long waves win #

When $kh \gg 1$ — that is, when the water is much deeper than the wavelength — $\tanh(kh) \to 1$ and

$$\omega^2 = gk, \qquad c = \frac{\omega}{k} = \sqrt{\frac{g}{k}} = \sqrt{\frac{g\lambda}{2\pi}}.$$

Speed rising as the square root of wavelength. A 100-metre swell travels at 12.5 m/s; a 10-metre wave at 4.0 m/s; a 1-metre ripple at 1.25 m/s. There is no single speed for water waves and there never was.

The depth has dropped out entirely, which is the definition of “deep”: once the water is deeper than about half a wavelength, the wave cannot tell how deep it is. The orbital motion of the water particles decays exponentially with depth, and by half a wavelength down it is about four per cent of the surface value.

That decay is why a submarine at modest depth is untroubled by a surface storm, and why the seabed under deep water is undisturbed by waves while the seabed in shallow water is worked over by every swell.

Shallow water: everything travels together #

When $kh \ll 1$ — wavelength much longer than the depth — $\tanh(kh) \to kh$ and

$$\omega^2 = g h k^2, \qquad c = \sqrt{gh}.$$

The wavelength has dropped out. Every wave travels at the same speed, set by the depth alone, and the medium is non-dispersive: a packet holds its shape indefinitely because all its components move together.

For four metres of water that is 6.26 m/s. For a four-kilometre ocean it is 198 m/s — about 710 km/h, the speed of an airliner.

That last number is the tsunami. Its wavelength is hundreds of kilometres, so even a deep ocean is shallow by the criterion that matters, and it travels at $\sqrt{gh}$ across an entire basin. Its amplitude at sea is often less than a metre and a ship will pass over it without noticing. What makes it dangerous is arriving: as the depth falls the speed falls, the front of the wave slows before the back, the energy is compressed into a shorter length, and the amplitude climbs.

Over time, dispersion pulls a packet apart. Components at different wavelengths travel at different speeds, so a compact disturbance spreads and its crests move through it — arriving somewhere else as a long train rather than as a bump. A tsunami does not do this, because in shallow water there are no different speeds: the whole disturbance travels at $\sqrt{gh}$, and a wave that left as a pulse arrives as a pulse.

The group, and the factor of a half #

The speed a wave packet travels at is the group velocity $c_g = \mathrm{d}\omega/\mathrm{d}k$, and differentiating the relation gives a result worth stating twice.

In deep water, $\omega = \sqrt{gk}$ gives $c_g = \tfrac12\sqrt{g/k} = c/2$. Exactly half. In shallow water, $\omega = k\sqrt{gh}$ gives $c_g = \sqrt{gh} = c$. Exactly equal.

The deep-water half is directly observable and is the classic demonstration of the distinction. Watch a group of waves moving across open water and individual crests appear at the back of the group, travel forward through it, grow as they reach the middle, shrink toward the front, and vanish. The crests move at twice the group’s speed, so each one spends a limited time in the group and then runs out of it.

Energy travels with the group, not with the crests, which is why the group speed is the one that matters for anything being delivered — and it is why a distant storm’s energy takes twice as long to arrive as watching the crests would suggest.

Why the depth decides which limit applies #

The transition between the two limits is not sharp, and it is worth being precise about where a wave sits, because the same body of water is deep for one wave and shallow for another at the same instant.

The parameter is $h/\lambda$. Above about a half, the wave is in deep water and the depth is irrelevant. Below about a twentieth, it is in shallow water and the wavelength is irrelevant. Between them is the intermediate regime where the full $\tanh$ is needed and neither simple rule works.

So the words deep and shallow are not descriptions of the water at all — they are descriptions of the relationship between a wave and the water. The Pacific Ocean is shallow for a tsunami and deep for a swell, simultaneously. A puddle is deep for a capillary ripple.

That is the same species of statement as a quantity belonging to an observer rather than to a thing, and it is worth noticing because the vocabulary actively obscures it. Nothing about four kilometres of ocean is shallow in any ordinary sense; it is shallow only with respect to a wave three hundred kilometres long.

Sorting a storm #

Put the two together and something practical falls out. A storm at sea generates waves across a broad band of wavelengths at once. In deep water they travel at different speeds, so by the time they reach a distant coast they have separated: the longest arrive first, and the period of the arriving swell falls steadily over the following days.

This is measurable and was measured. The rate at which the observed period drops gives the distance to the storm, because the spread in arrival times is the spread in group speeds times the distance travelled. Storms thousands of kilometres away have been located from a single tide-gauge record this way, and surf forecasting is built on it.

It is a good example of dispersion being useful rather than a nuisance. A non-dispersive medium delivers the storm as one confused arrival and says nothing about where it came from; a dispersive one performs a spectral analysis in transit.

A disturbance contains many wavelengths because anything that is not a pure sinusoid is a sum of them. Add two of nearly equal amplitude with a phase between them and the sum is a third shape, and in a dispersive medium the components of that sum travel at different speeds — so what arrives somewhere else is a different sum, in a different order, with the long waves at the front.

What the water actually does #

The wave travels and the water does not, which was established on the first rung of the wave ladder. What is worth adding here is the shape of the motion, because it differs between the two limits and explains both.

In deep water each parcel moves in a circle, whose radius decays exponentially with depth. Forward under a crest, backward under a trough, back where it started after one period. Nothing is carried along, to first order.

In shallow water the circles are flattened by the bottom into ellipses, and at the bed itself the motion is purely horizontal — a back-and-forth sloshing. That horizontal motion at the bed is what moves sand, which is why shallow-water waves shape beaches and deep-water ones do not.

The exponential decay also explains the depth criterion. A wave “feels the bottom” when its orbital motion still has appreciable amplitude there, which happens once the depth falls below about half a wavelength — and that is precisely where $\tanh(kh)$ departs from one.

The energy, and where it sits #

One more quantity is worth extracting from the relation, because it explains why a swell that looks harmless is not.

The energy per unit area of a linear surface wave is $\tfrac18 \rho g H^2$ with $H$ the crest-to-trough height, and it does not depend on the wavelength at all — only on the amplitude. Half of it is potential, in the raised and lowered water, and half kinetic, in the orbital motion. The equality of those halves is exactly the equipartition an oscillator always shows, and it holds at every instant when averaged over a wavelength.

The flux of energy, however, is the energy density times the group speed — and the group speed does depend on wavelength. So a long swell of a given height carries its energy along much faster than a short wave of the same height, and delivers more power per metre of coast.

For a two-metre swell of ten-second period the flux is about 40 kilowatts per metre. That figure is the whole basis of wave-power engineering, and it is also why a modest-looking ocean swell is capable of moving armour blocks weighing tens of tonnes. What is being delivered is not the height; it is the height squared times a speed that grows with period.

The same reasoning explains why swell is more destructive than locally generated wind waves of equal height: it has a longer period, so a higher group speed, so a larger flux.

Everywhere under a wave the water is performing one trade: potential energy in displaced water against kinetic energy in orbital motion, exchanging twice per period and averaging equal. That equality is what makes the total expressible in terms of the amplitude alone — $\tfrac18\rho g H^2$ per unit area, with no wavelength in it — which is why a wave’s energy is a property of its height and its speed is a property of its length.

Where surface tension takes over #

Gravity is not the only restoring force available. At short enough wavelengths, surface tension dominates and the relation gains a term:

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

The two contributions are equal at $\lambda = 2\pi\ell_c \approx 1.7$ cm for water, where the phase speed has a minimum of about 23 cm/s. Below that wavelength the waves are capillary ripples and their speed rises as the wavelength gets shorter — the opposite of the gravity behaviour.

Two consequences follow. There is a slowest possible wave on water, and nothing can travel more slowly than 23 cm/s as a surface wave. And an object moving through still water more slowly than that produces no wave pattern at all, which is why a very slowly moving insect leaves an undisturbed surface while a faster one leaves a wake.

The same structure elsewhere #

The reason this page belongs on the wave-packets ladder rather than in a corner of fluid mechanics is that the dispersion relation is a portable idea, and water is the case where it can be watched with the naked eye.

A dispersion relation is a statement tying frequency to wavenumber, and everything about how a disturbance propagates follows from it: the phase speed is $\omega/k$, the group speed is $\mathrm{d}\omega/\mathrm{d}k$, and a medium is dispersive exactly when those differ. Water supplies both cases in one substance, which is what makes it such a good teacher.

Light in glass has a dispersion relation, and its wavelength dependence is why a prism separates colours and why a rainbow has an angle. A wave on a stretched string is non-dispersive, which is why a plucked string keeps its shape and sounds a definite set of frequencies. An electron has $\omega \propto k^2$, giving a group speed exactly half the phase speed at every wavelength — the same factor of a half as deep water, from a different exponent, and it is a coincidence of the arithmetic rather than a shared mechanism.

The habit worth taking from this page is to ask, of any wave at all, what its dispersion relation is — and to treat “the wave speed” as a question rather than a quantity until the answer comes back flat.

Two waves of nearly equal wavelength added together are the simplest case of a group, and where the question comes from at all. The sum has an envelope, and how fast that envelope travels compared with the crests inside it is exactly the group-versus-phase question. The dispersion relation is what answers it: the envelope goes at $d\omega/dk$ and the crests at $\omega/k$, and only a straight relation makes those the same.

What the shoaling actually does #

The essay says a tsunami’s amplitude climbs as the depth falls, which is right and leaves out how much and what the result looks like. Both come out of the relation on this page.

Energy travels at the group speed and is not created or destroyed as the wave crosses a gently sloping shelf, so the energy flux — energy density times group speed — is constant. The density goes as the square of the height and the shallow-water group speed goes as the square root of the depth, so the height goes as the depth to the power minus a quarter.

That is Green’s law, and it is a weak dependence. Coming from four kilometres of ocean into ten metres of water is a factor of four hundred in depth and only four and a half in height. A tsunami half a metre high at sea arrives as a wave a couple of metres high at the ten-metre contour, which does not sound like the event.

Two further effects supply the rest, and one is in the same law. Green’s law has a second factor for a wave funnelled into a narrowing channel, with the height rising as the inverse square root of the width — so a bay, an inlet or a river mouth concentrates what a straight coast would spread. Beyond that the wave refracts over the bathymetry and can be focused onto a headland or a submarine ridge, and the near-shore run-up is typically two to four times the offshore height. Between them the factors multiply into the tens of metres that are observed.

The other consequence of the same arithmetic is more useful for recognising one. The period does not change as the wave shoals, so the wavelength shortens in proportion to the speed — as the square root of the depth. A wave two hundred kilometres long in the open ocean is still ten kilometres long in ten metres of water.

Ten kilometres is not a wave anybody sees as a wave. What arrives is a rise in sea level lasting several minutes: the water comes up steadily, keeps coming, floods inland, and then drains. Eyewitness descriptions consistently report a rapidly rising tide rather than a breaking crest, which is exactly what a wave that long has to look like from the shore, and which is where the old and misleading name came from.

If the seafloor displacement dropped the water on the landward side, the leading edge is a trough rather than a crest, and the sea withdraws before it returns — which is the warning that has saved lives and which happens for roughly half of events, depending on which side of the fault the coast is.

One measurement closed the loop on the whole argument. A satellite altimeter happened to cross the Indian Ocean two hours after the 2004 earthquake and recorded the open-ocean wave directly: an amplitude of about half a metre, at the position the shallow-water speed puts it. The speed, the amplitude and the shoaling factor were all confirmed in one pass.

The buoy that measures a spectrum by bobbing #

The dispersion relation is what makes it possible to measure a whole directional sea state from a single point, which is not obviously possible at all.

A wave buoy is a float moored so that it follows the surface. It carries accelerometers, and integrating its vertical acceleration twice gives the surface elevation as a function of time. Transforming that record gives the frequency spectrum — how much energy is present at each period — which already tells a forecaster a good deal.

Getting the direction out of a point measurement is the interesting part, and it works because the orbital motion is tied to the wavelength. In deep water a water parcel moves in a circle whose plane contains the direction of travel, so the buoy’s horizontal displacement at any frequency is in quadrature with its vertical displacement, and the ratio of the two horizontal components gives the compass bearing.

Measuring heave together with pitch and roll — or with two horizontal displacements — therefore yields the direction of every frequency component separately, from one instrument on one mooring. Hundreds of such buoys report continuously and their spectra are assimilated into the models that produce every wave forecast.

The same relation is what makes a sensor on the seabed usable. Pressure under a passing wave is attenuated with depth by the same hyperbolic function that appears in the dispersion relation, so converting a bottom-mounted pressure record into a surface elevation means dividing that attenuation out frequency by frequency. It works well for long waves and badly for short ones, because the attenuation is exponential in the wavenumber: a bottom sensor in twenty metres of water is essentially blind to anything with a period under about four seconds, and there is no gain setting that recovers what never reached it.

Where the model stops #

The amplitude is small. Everything here is linearised in wave height, valid while the height is small compared with both the wavelength and the depth. Real waves steepen, their crests sharpen, and their speed rises slightly with amplitude — and eventually they break, which is entirely outside a linear theory.

The water is inviscid and the flow is irrotational. Both are excellent for surface waves over the relevant timescales; viscosity damps short waves noticeably and long ones hardly at all.

The depth is uniform. Where it varies, the wave speed varies with it, and the wave refracts — bending toward shallower water, which is why waves arrive nearly parallel to a beach whatever direction they came from, and why a headland concentrates wave energy onto itself while a bay is sheltered — the wave fronts wrap around the shallows in front of the headland and converge there. That is the same refraction that bends light, by the same mechanism, with depth in place of refractive index.

Nothing is moving but the wave. A current alters the relation, and waves against a current steepen — which is what makes some tidal races and river mouths dangerous out of proportion to the wind.

Gravity is the only restoring force considered until the last section. Rotation matters for waves long enough to feel the Earth’s spin, which brings in a whole further family.

One interface. Two fluid layers of different density support internal waves at the interface between them, with a much smaller effective gravity and correspondingly slower and larger waves. Those exist throughout the ocean and are invisible from the surface.

The ladder from here #

Later rungs on this anchor: the ship’s wake and the fixed angle it makes, which comes out of this relation and a stationary-phase argument. Wave refraction over a sloping bottom, and why swell arrives parallel to the shore. Breaking, and the two ways it happens. Capillary–gravity waves properly, with the minimum speed derived. Nonlinear waves, including the solitary wave that keeps its shape by balancing dispersion against steepening. And the same mathematics with a different restoring force, which is where a quantum particle’s dispersion turns out to have the same structure and a different exponent.