The tide a lake makes by itself

Only some notes fit found that a string clamped at both ends can vibrate only at frequencies that fit whole numbers of half-waves into its length, and the same rule has since been followed into drums, boxes and rooms. The rule needs nothing but a wave, a speed and a boundary that reflects, and so it applies to anything that has those three — including a lake.

Water in a basin carries waves, and long waves, whose wavelength is much greater than the depth, travel at a speed set by the depth alone, $\sqrt{gh}$ — the regime in which a wave’s speed depends on its length no longer, because the whole column of water moves together. The shores reflect them. So a lake has standing waves, with periods set by its length and depth, and anything that tilts its surface — a gust of wind piling water at one end, a change in air pressure passing over, a small earthquake — leaves it ringing at those periods after the disturbance has gone.

The phenomenon has a name from the shore of Lake Geneva, where people had long noticed the water rising and falling by a few centimetres for no visible reason: a seiche. François-Alphonse Forel measured Geneva’s in the 1870s and 1890s, and found the water at the two ends moving in opposite directions with a period of seventy-three minutes, the whole lake tilting about a line across its middle.

The shapes a basin can hold #

In a closed lake the water cannot flow through the shores, so the flow is zero at both ends and the surface there rises and falls the most: the ends are antinodes of the surface. The slowest standing wave has half a wavelength in the lake, a single node across the middle, and the whole surface tilts back and forth about it like a seesaw. The next has two nodes and a full wavelength, the middle rising while both ends fall, and so on up. Their periods, for a rectangular basin of length $L$ and uniform depth $h$, are

$$T_n = \frac{2L}{n\sqrt{gh}},$$

which Rudolf Merian worked out in 1828, a formula with the same form as a string’s harmonics and the same reason.

A harbour is a basin open at one end to the sea, and the open end behaves differently. The sea is so large that a harbour cannot change its level at the mouth, so the mouth is held fixed: a node of the surface, where the water flows in and out the most but does not rise or fall. The closed head of the harbour is an antinode, as a lake’s shore is. So a harbour fits odd numbers of quarter-waves — like an organ pipe closed at one end — and its slowest period is $4L/\sqrt{gh}$, twice that of a closed basin of the same length.

Periods from a length and a depth #

The formula’s reach is its striking feature. A bath a metre and a half long and twenty-five centimetres deep sloshes with a period of two seconds, which is why a child rocking in it can build up a wave that slops onto the floor: the rocking finds the bath’s resonance. Loch Ness, thirty-six kilometres long and a hundred and thirty metres deep on average, should ring at thirty-four minutes; Edward Watson measured thirty-one and a half in 1904, from the rise and fall of the water at the loch’s ends. Lake Geneva, seventy-two kilometres long, should ring at an hour; it rings at an hour and a quarter. Lake Erie, nearly four hundred kilometres long and only nineteen metres deep on average, should ring at sixteen hours; it rings at fourteen and a half, and its seiches, driven by storms blowing along its length, can raise the water at one end by more than two metres — enough to flood the shore at Buffalo while the far end, at Toledo, is left with stranded boats.

The formula misses by up to a fifth, and the misses are informative. Real lakes are not rectangles of uniform depth. Lake Geneva is deep in its wide eastern part and shallow and narrow in its western arm, so waves travel slowly through the arm and the period is longer than a mean depth suggests. Ellipsoidal and parabolic basins have their own exact formulas, and for real lakes the periods are now computed numerically from depth charts — but the order of magnitude, from seconds to hours, comes from Merian’s two quantities.

The water goes sideways more than it rises #

A seiche looks like a rise and fall of the shore’s water level, a few centimetres on a calm lake. Most of the motion is horizontal. At the node in the middle of the lake the surface does not move at all, and the whole body of water there flows back and forth along the lake, first towards one end and then the other, as the node of a standing wave is where the flow is greatest. For a long wave the flow speed is the surface height times $\sqrt{g/h}$, so a seiche five centimetres high in a lake as deep as Geneva moves its water at about a centimetre and a half per second, and over half a seventy-minute period the water at the middle travels some twenty metres one way before turning back. Fishermen’s nets and floating debris swing to and fro with the seiche long after the wind that started it has died, and in shallow, long lakes like Erie the currents of a storm seiche are strong enough to move sediment along the bed.

Lakes are not one-dimensional, and they ring across their width as well as along their length, in two-dimensional patterns like those of a drum, which have no harmonics in the musical sense: the periods of a basin’s transverse and higher modes are not whole-number fractions of its slowest one. A tide gauge records a mixture of them, sorted out by analysing which periods appear and how they depend on where the gauge is.

The seiche inside the lake #

In summer a deep lake is layered: a warm, light upper layer a few tens of metres thick floats on cold, dense water below, separated by a thin thermocline. Fluid in a layered column resists being displaced, and the interface between the layers carries waves of its own. Their restoring force is gravity reduced by the small difference in density between the layers — a thousandth of $g$ or less — so they are slow, and a whole lake has an internal seiche on its thermocline whose period is days rather than hours.

For a lake like Loch Ness, with a density contrast of about a part in a thousand across a thermocline thirty metres down, the reduced gravity gives a wave speed of about half a metre a second and a slowest internal seiche of roughly two days. Its amplitude is far larger than the surface seiche’s: the thermocline at one end can rise and fall by tens of metres while the surface barely moves. Edward Wedderburn and his colleagues found this in Loch Ness in the first years of the twentieth century, by lowering thermometers and watching the cold water come and go. Internal seiches mix the lake, carry nutrients up from the deep water, and are the inland version of the waves that hold a ship back on a layered sea.

What sets a lake ringing #

A seiche needs a push that varies over the whole lake on a time scale not much shorter than its period. Wind blowing steadily along a lake drags the surface water downwind and tilts the surface, piling water at the downwind end; when the wind drops, the tilted surface swings back, overshoots, and rings. A band of high or low air pressure passing over a lake pushes the surface down or lets it rise beneath it, and if the band moves at about the speed of the lake’s long waves, it can pump energy into the seiche the way a child pumps a swing, which is why some of the largest seiches follow the passage of squall lines. Earthquakes, even distant ones, shake lakes and swimming pools into seiches; the great Alaskan earthquake of 1964 set water sloshing in wells and lakes across North America.

Joseph Proudman showed in 1929 why the speed matters. A pressure disturbance moving across water at exactly the speed of long waves keeps pace with the bump it raises, and adds to it continuously instead of leaving it behind, so the bump grows for as long as the match lasts. A jump in air pressure of a couple of hectopascals, the kind a line of thunderstorms carries, is worth only two centimetres of water held still; travelling at the right speed over a shallow shelf for an hour it can build a wave of a metre or more. These meteotsunamis strike coasts with no earthquake anywhere, and when one arrives at a bay or harbour tuned to its period, the harbour’s own resonance multiplies it again — the Adriatic, the Balearic harbours and the Great Lakes all have long records of them.

Once set going, a seiche decays only slowly, losing energy to friction on the bed and to the shores, and a large lake may ring for days after a storm. The same physics in the ocean’s bays and gulfs — basins open to the sea at one end and driven by the tide — makes some coasts’ tides much larger than the open ocean’s; that is a story of resonance with a periodic drive, and harbours show the most counter-intuitive part of it.

A harbour’s resonance, and its entrance #

A harbour is meant to be quiet water. Waves arriving from the sea at its own resonant period, though, are amplified rather than stopped, and long-period swell or the small, long waves that ride on storm surges can set a harbour’s water surging back and forth through its basins strongly enough to snap mooring lines and damage ships, on a calm day. The usual remedy for waves is to narrow the entrance with breakwaters.

The figure computes what happens for a simple harbour: a straight channel, closed at its head, joined to a much wider sea through a short entrance whose width can be narrowed. A long wave arriving from the sea is partly reflected and partly enters; inside, it reflects off the head, returns to the mouth, and is partly reflected back in. With the entrance at full width, the harbour resonates at the quarter-wave frequency and the water at its head swings 7.6 times as high as the wave would rise on a straight coast. Narrow the entrance and two things happen. The resonance moves to a lower frequency, because the water must now slosh through the narrow mouth, which adds inertia, and the harbour starts to behave like the air in a bottle that sings whatever its shape, a Helmholtz resonator. And the peak grows: 8.6 at a tenth of the width, 11.5 at three hundredths.

That second effect is the harbour paradox, worked out by John Miles and Walter Munk in 1961. A narrower entrance lets less wave energy in, but it also lets less out, and at resonance the second effect wins: energy accumulates in the harbour over more cycles before it can leak back to the sea, so the standing wave inside builds higher. The breakwater that protects the harbour from waves at most frequencies makes it more vulnerable at one.

The resonance also sharpens. Its width in frequency falls nearly tenfold as the mouth narrows to a hundredth of the harbour’s width, and a sharp resonance is one that takes many cycles to build up and many to die away. The real harbour therefore has a defence the steady-state picture hides.

A harbour with a narrow entrance responds slowly. A short burst of waves at the resonant period passes before the harbour has built up much motion, and the harbour is better protected against it than an open one would be. A long, steady train of waves at the resonant period — swell from a distant storm, or the regular long waves some coasts receive — has time to build the full, larger response. Which matters more depends on the waves a harbour actually gets, and harbour engineers model both before deciding where to put a breakwater.

The paradox has a limit, which the model leaves out but real harbours do not. Water squeezing through a narrow entrance at speed loses energy in eddies and jets, a loss that grows as the entrance narrows and the flow through it speeds up. Beyond some point the entrance losses dominate and the amplification falls again. Real harbours sit on both sides of that point, and some that were given narrower entrances to keep out storm waves turned out to surge worse at their resonant period than before.

Bays, gulfs and the tide #

The same reasoning, at a larger scale, explains why tides differ so much from coast to coast. A bay or gulf open to the ocean at one end is a harbour on a scale of hundreds of kilometres, and the ocean’s tide drives it at a period of about twelve and a half hours. If the bay’s own quarter-wave period is near that, the tide in the bay is amplified, and at its head can be many times the open-ocean range; if it is far from it, the bay’s tide is unremarkable. The Bay of Fundy, whose quarter-wave period is not far from the tide’s, has the largest tides in the world, and its long, deep shape is the reason. That the drive is the tide and the response a seiche is the whole of it, and the arithmetic is the harbour’s arithmetic, scaled up.

What the pictures cannot show #

The mode drawings are for basins of uniform depth and straight sides, and the harbour model is one-dimensional: a straight channel joined to a sea represented by a wider channel, with friction represented by a small damping of the wave. A real harbour is two-dimensional, its waves spread out of the mouth in all directions rather than along a channel, and Miles and Munk’s own calculation, for a rectangular harbour opening onto a straight coast, shows the paradox more strongly than the channel model. The model has the right physics — energy leaking out through the mouth less readily as it narrows — and the wrong geometry for any particular harbour.

The lake periods compare a formula with measurements of real lakes, and none of the lakes is a rectangle. The periods measured are those of the fundamental longitudinal seiche; lakes also ring across their width and in higher modes, and a tide gauge at one end records a mixture.

The domain of the argument is long waves — wavelengths much greater than the depth — in basins whose boundaries reflect them. Inside it, a basin rings at periods set by its length and the speed $\sqrt{gh}$, and a harbour’s entrance controls how strongly and how slowly it answers a wave train from outside.

Still open: what a harbour’s surging costs a ship #

The water in a resonating harbour moves mostly horizontally, sloshing back and forth with a period of minutes, and a ship moored in it moves with the water until its mooring lines stop it. The loads that result depend on the ship’s own response, the stretch of its lines and the fenders against the quay, and they can be large enough to break lines and stop cargo handling for days, as several ports have found. How to predict those loads from forecasts of the incoming long waves, which are themselves small — centimetres in the open sea — and generated by storms far away, is an active problem in port engineering, and the long waves that drive it, riding beneath ordinary swell, are only recently being forecast routinely.

The physics underneath is the oldest in this collection. A basin of length L and depth h holds standing waves of period $2L/(n\sqrt{gh})$ — two seconds for a bath, seventy-three minutes for Lake Geneva, fourteen hours for Lake Erie — and a harbour, whose mouth the sea holds level, rings at odd quarter-waves; narrowing its mouth lowers its resonance and raises it, from 7.6 to 11.5 times the incoming wave in the model drawn, because energy that gets in has a harder time getting out. The breakwater that keeps a storm out can make a calm-day swell worse.