Why There Are Two Tides a Day

Spend a day on a harbor wall and you can watch the sea climb up the stones and slide back down. On most of the world's coasts it does this twice a day: two high tides and two low tides, each high about 12 hours and 25 minutes after the one before.

Here is the water against a harbor wall over two days, at a place where the tide is the twice-daily kind, as it is along most of the Atlantic coast of Europe. Drag the time through the two days, or jump to the first high tide and the first high on day two, and keep an eye on the clock times in the plot:

The water climbs about 2.6 meters from low to high and drains away again, and the boat rides up and down with it. The high tides come at 03:12 and 15:37 on the first day, then at 04:02 and 16:28 on the second. Two a day, and each day they arrive about 50 minutes later than the day before.

Fifty minutes is also, on average, how much later the Moon rises each day, and that's no coincidence: the tides are raised by the Moon, with some help from the Sun.Not everyone believed it. Galileo argued that the tides came from Earth's own motions, its spin and its path around the Sun, and dismissed the Moon's influence as an occult idea. What's strange is the count. The Moon is on one side of Earth, yet there are two high tides a day, as if the sea also piled up on the side facing away from it. It does. To see why, we have to look at how the Moon's pull changes from one side of Earth to the other.

The Moon's pull

The Moon pulls on everything on Earth: the oceans, the rock, you. Like all gravity, its pull weakens with distance, as one over the distance squared. The side of Earth facing the Moon is closer to it than Earth's center is, by one Earth radius of 6,371 km, and the far side is farther by the same amount.

The purple arrows show the Moon's pull at Earth's center and at eight points around its surface. The real Moon is about 60 Earth radii away, which doesn't fit beside a readable Earth, so I've started it at . Drag that number, or move the Moon to its real distance, and compare the arrows:

With the Moon close, the difference is plain. The arrow on the near side is much longer than the one at the center, and the one on the far side is shorter. Every arrow points at the Moon, so the ones at the top and bottom also lean slightly inward. Move the Moon away and all the arrows shrink toward the same length. At the real distance the near side is pulled only 3.4% harder than the center and the far side 3.2% less, and the arrows look almost identical.

That few percent is the whole story of the tides. The part of the pull that's the same everywhere does nothing to the shape of the oceans, because Earth as a whole is free to move in answer to it.

Falling together

Earth and the Moon are falling toward each other all the time. They never meet because they're also moving sideways, fast enough to keep missing, and so they circle each other once a month. That's what an orbit is. Whether a body falls sideways or straight in makes no difference to how the pull varies across it, so picture the simplest case: Earth dropped straight toward the Moon.

Imagine Earth made of loose beads that don't hold on to each other, all starting at rest. Each bead falls toward the Moon, pulled as hard as the Moon pulls at its own spot. Watch the ring from outside on the left, and from the middle bead on the right:

From outside, the whole ring simply falls toward the Moon. Riding with the middle bead, we see something else. The near bead pulls ahead, because the Moon pulls it harder, and the far bead drops behind, because the Moon pulls it less. The beads at the top and bottom drift inward, since they're all falling toward the same point. The ring stretches out along the line to the Moon. Here it comes out egg-shaped, because the Moon is so close that the near side feels much more of the difference; at the real distance both ends stretch almost equally.

The far bead moves away from the middle just as the near one does. Nothing pushes it away from the Moon. It's pulled toward the Moon too, only less than the middle, so it gets left behind, and from the middle that looks exactly like being pulled outward.

I've put the Moon 8 Earth radii away instead of 60 and sped up time, so the stretching shows within a few seconds. Real Earth isn't loose beads, either: its own gravity holds it together far more strongly than this stretch pulls it apart. But Earth is falling freely, and on a freely falling Earth only the differences in the Moon's pull are left over.

The stretch

That leftover is called the tidal force. Here it is in red all around Earth, computed for the Moon at its real distance. Drag the probe around the surface, or turn the knob, and read the numbers under the figure:

The tidal force points away from Earth's center on the side facing the Moon and on the side facing away, and inward around the middle, half as strongly. It is tiny. At its largest it's about 1.1 millionths of a meter per second squared, roughly one nine-millionth of Earth's gravity. Pulling straight up against gravity, it can't do much: it makes the water under the Moon weigh one part in nine million less.

What moves the ocean is the sideways part. Everywhere except directly under the Moon, directly opposite it, and on the circle halfway between, the tidal force leans along the surface, and nothing resists water sliding sideways. Show only the sideways parts and notice where they point: toward the spot under the Moon and toward the spot opposite it. The sideways push is strongest 45° from those spots, at about 0.8 millionths of a meter per second squared.

Along the line to the Moon there's a short formula for the stretch. With \purple{M} the Moon's mass, \purple{d} its distance, R Earth's radius, and G the gravitational constant, the tidal force on the near and far sides is close to

The pull itself falls off as 1/\purple{d}^2. The difference across Earth falls off one power faster, as 1/\purple{d}^3, because the farther away the Moon is, the more alike its pulls on the two sides become. That extra power matters when we get to the Sun.

Two bulges

Pushed toward two spots, the ocean piles up there until the slope of its surface balances the sideways push. The result is an ocean stretched into an oval, with two bulges, one under the Moon and one opposite it, and low water in a ring between them. This is the equilibrium tide, the shape the oceans would take if they had all the time they needed to settle.

The real heights are tiny next to Earth, so the figure draws them 2,000,000 times taller. The Moon's orbit is slightly oval: compare it at its nearest, 363,300 km with its farthest, 405,500 km, and watch the two gauges:

At the Moon's average distance the open-ocean water stands about 36 cm higher under the bulges than it would with no Moon, and about 18 cm lower in the ring between them, a little over half a meter from low to high. Bring the Moon in to three quarters of its distance and its pull on Earth grows 1.8 times, but the tide grows 2.4 times, because it follows 1/\purple{d}^3. Even the Moon's real wobble between nearest and farthest changes the tide by about a third.

A day under the bulges

The two bulges stay lined up with the Moon while Earth turns underneath them. A town on the coast is carried through one bulge, then the low ring, then the other bulge, then the low ring again: two high tides and two low tides for every turn.

Here is that turning Earth, with a town on the equator and the height of the water there plotted alongside. Let it run or drag the time, and drag the globe to look at it from above:

Each time the town passes under the Moon, or directly opposite it on the far side of Earth, the water reaches a high, and the purple lines mark those moments. But the highs aren't 12 hours apart. While Earth turns, the Moon moves along its orbit in the same direction, about 12° a day measured against the Sun, and Earth needs about 50 extra minutes to catch up with it. So the Moon comes back overhead every 24 hours and 50 minutes, a span called the lunar day, and the two highs within it come 12 hours and 25 minutes apart.

Two unequal highs

So far the Moon has sat over the equator. It doesn't stay there. Its orbit is tilted to Earth's equator, so over each month the Moon swings north to as much as about 28° of latitude, back over the equator, south to the same latitude, and back again. The bulges swing with it: one tilts north, the other south.

A town away from the equator then passes through the two bulges at different places: one near its middle, where it's tall, the other near its edge. Move the Moon north to and put the town at :

With the Moon over the equator the two highs match. Tilt the bulges and they don't: at 45° north with the Moon 20° north, one high reaches 26 cm above the average level while the other is so weak it stays 8 cm below it. This is the diurnal inequality, and at high latitudes it goes further. Put the town above about 62° and the Moon at its farthest north, and the smaller high disappears: the town gets one tide a day.

On real coasts the inequality is often much stronger than this, and on some, such as parts of the Gulf of Mexico, there is usually just one high tide a day. That isn't in the equilibrium picture. It's the oceans themselves, which respond to the once-a-day part of the push and the twice-a-day part very differently, and we'll come to them soon.

The Sun's share

The Sun raises tides too. It pulls on Earth about 180 times harder than the Moon does, which might suggest enormous solar tides. But tides come from the difference in pull across Earth, which falls off as 1/d^3, and the Sun is 390 times farther away than the Moon. Its mass is 27 million times the Moon's, and 27 million divided by 390 cubed comes to 0.46: the Sun's tide is a little under half the Moon's.

So the oceans carry two pairs of bulges at once, one lined up with the Moon and one with the Sun, and their heights add. Here Earth is seen from above its north pole, with the Sun far off to the left. Turn the knob through a month, or jump to new moon, first quarter, and full moon, and watch the shape of the ocean:

At new moon the Moon sits between Earth and the Sun, and at full moon it's on the far side. Either way the Moon's bulges and the Sun's bulges point along the same line and stack, so the highs are higher and the lows are lower. These are the spring tides, about one and a half times the range of the Moon's tide alone; the name has nothing to do with the season. At the first and last quarter the two pairs sit at right angles, the Sun's bulges partly fill the Moon's lows, and the range drops to about half of the Moon's alone. These are the neap tides.

The Moon's changing distance adds a second rhythm. It comes nearest every 27.55 days, a little faster than the 29.53 days from one new moon to the next, so the nearest point drifts through the phases. When it lands on a new or full moon, the biggest springs of all arrive, the ones sometimes called king tides. Scrub through 14 months:

The springs pulse every 14.8 days, and their peaks rise and fall over about seven months: the biggest, about 88 cm on the equator in this model, when the Moon is nearest at new or full moon, every 206 days. The smallest neaps are about a quarter of that. Coastal flood warnings follow the same calendar, because a king tide plus a storm surge is what tops sea walls.

A wave that can't keep up

Everything so far has been the equilibrium tide: the shape the oceans would take if they could settle under the bulges. The rhythm it predicts is right, two highs every lunar day, springs and neaps twice a month. But it gets the size and the timing badly wrong at most harbors, and the reason is speed.

To stay under the Moon, a bulge on the equator has to sweep around Earth once every lunar day, at about 1,600 km/h.Earth is 40,075 km around the equator, and the Moon comes back overhead every 24.84 hours: 40,075 ÷ 24.84 ≈ 1,613 km/h. But a bulge of water isn't a solid thing that can be dragged along. It's a wave, and a wave as long as an ocean travels at a speed set by the depth: c = \sqrt{g\,\blue{h}}, faster in deeper water. Try a few depths:

The oceans average about 3.7 km deep, which carries a long wave at about 690 km/h. To keep pace with the Moon on the equator the ocean would have to be about 20 km deep, twice as deep as the deepest trench. Nearer the poles the point under the Moon moves more slowly, but nowhere in the open ocean does the water get close.

Here is that thought experiment as a simulation: a canal of water running all the way around the equator, with nothing in its way. The dashed line is the equilibrium tide, sliding west with the Moon, and the water is computed from the equations for long waves, pushed by the real tidal force. Set the canal's depth to :

At the real depth the water is high where the equilibrium tide is low: low water sits under the Moon, high water a quarter of the way around. The push still makes the water rise and fall twice a lunar day, but the wave it drives can't travel fast enough to stay with it, and settles out of step. Deepen the canal past 20 km and the water flips over and follows the Moon. In between, near 20 km, the wave travels at almost exactly the Moon's speed, every push lands on the crest, and the tide grows to many times the equilibrium height until friction stops it growing. That's resonance.

A canal around the equator is still a cartoon. Real oceans are broken up by continents, so the tide can't run around the world at all. It sloshes inside each basin instead, and how big the sloshing gets depends on the basin.

Sloshing in a bay

Push a child on a swing in time with the swing and a small push builds a big motion. A bay works the same way. Water in a bay has a natural rhythm: a wave runs in from the mouth, reflects off the head, and runs back out. If that round trip takes about as long as the tide's 12 hours 25 minutes, every push from the ocean arrives in step and the sloshing builds.

Here is a straight bay of even depth, open to the ocean on the left, where the ocean tide raises and lowers the water by 1 m. The water in the bay is simulated with the same long-wave equations as the canal. Change the bay's length and depth, or try a bay 270 km long and 60 m deep:

A short bay simply rises and falls with the ocean: the water has time to spread out evenly, and the head of the bay sees about the same tide as the mouth. As the bay gets longer the tide at its head grows, and when the bay is about a quarter of a tidal wavelength long, the round trip takes just about one tidal cycle and the head of the bay sees six times the tide at the mouth. Make the bay longer still and the sloshing falls out of step again. In deeper water waves travel faster, so the resonant bay is longer: the quarter-wavelength rule is L = \tfrac{1}{4}\,T\sqrt{g\,h}.

This is the Bay of Fundy's secret. The bay together with the Gulf of Maine outside it has a natural period of about 13 hours, close to the tide's 12.4 , and at its head the water rises and falls by up to about 16 meters twice a day, the largest tides in the world. The Mediterranean is the opposite case: a basin whose natural sloshing doesn't match the tide, fed only through the narrow Strait of Gibraltar, with tides of a few tens of centimeters at most of its coasts.

Points with no tide

One more thing turns the sloshing into something stranger. A large basin sits on a spinning Earth, and moving water is deflected by the spin: to the right in the northern hemisphere, to the left in the southern. The tide wave entering a long gulf hugs one coast, reflects at the far end, and comes back hugging the other. The two waves together don't make a simple seesaw. They make the tide rotate.

Here is that pair of waves in a gulf 400 km wide and 60 m deep, the size of the North Sea. Blue is water above average, gray below. Let it run, and change the latitude, which sets how strongly the spin deflects the water:

With the spin on, the high water doesn't slosh from one end of the gulf to the other. It wheels around two amphidromic points, places in the middle where the water barely rises or falls at all, counterclockwise in the northern hemisphere. The thin co-tidal lines radiate out from each one like the hands of a clock, and the tide gets bigger toward the coasts. Switch off the spin and the rotation goes: the tide just rocks up and down, with still lines straight across the gulf.

The pattern is real. G. I. Taylor worked out the full version for a closed rectangular gulf in 1922 , and the real North Sea has three amphidromic points, with the tide wheeling around each of them twice a day.One sits in the southern North Sea between England and the Netherlands, one off southern Norway, and one near the middle; the real ones are pulled off-center by friction and the coastline. On maps of the world's main twice-daily tide, the oceans are covered with these wheels, and the size of the tide at any harbor depends on where it sits: near a wheel's hub, almost nothing; at the rim, meters.

Tide tables

With all this sloshing, nobody predicts a harbor's tide from first principles. Instead, the tide is taken apart into a handful of steady rhythms, each one tied to a cycle of the Moon or the Sun, and each harbor's measurements tell how big each rhythm is there and how late it arrives . Then the rhythms are added back together for any date in the future.

Five rhythms do most of the work. Each is called a constituent. Switch them on one at a time, and compare an Atlantic-style coast with a once-a-day coast:

The M₂ wave alone is the tide of the earlier figures: two equal highs every lunar day. The S₂ wave has a period of exactly 12 hours, so it drifts in and out of step with M₂ every 14.8 days, and their sum swells and shrinks: springs and neaps, from two plain sine waves. N₂ carries the Moon's changing distance. The once-a-day waves K₁ and O₁ carry the tilt of the bulges, and when they're strong, the two daily highs become unequal, or merge into one. Every coast's tide is some mix of these, with dozens of smaller ones for precise work.

The brake

All this moving water rubs against the seafloor, especially over shallow continental shelves, and turns the tide's energy into heat. The friction does more than wear out the waves. It drags the bulges along with Earth's spin, so they sit a little ahead of the line to the Moon instead of right on it, and that small lead has two large consequences. Set the bulges' lead:

With no lead, the Moon pulls both bulges straight toward itself, along lines through Earth's center, and nothing twists. With a lead, the nearer bulge is pulled back against the direction of spin, harder than the far bulge is pulled forward, and Earth's spin slows. Earth's day is getting longer. Ancient eclipse records show by how much: over the last 2,700 years the day has lengthened by about 1.8 thousandths of a second per century .Tidal friction alone would lengthen the day by about 2.3 thousandths of a second per century. Earth is also still slowly rebounding from the weight of the last ice age's ice sheets, which speeds its spin up a little. That's tiny, but it adds up: the time of an eclipse seen in Babylon in 136 BC is off by hours if we assume the day has always been as long as today's.

The bulge pulls back on the Moon, too. The nearer bulge tugs the Moon slightly forward along its orbit, and a Moon pushed forward climbs into a wider orbit. The spin Earth loses, the Moon gains as orbital motion. Laser pulses fired at mirrors left on the Moon by Apollo astronauts and Soviet rovers measure its distance to within a few centimeters, and show it moving away by about 3.8 cm a year .

The same thing has already run to completion on the Moon. Earth raises tides in the Moon's rock, about 20 times stronger than the Moon raises on Earth, and they have braked the Moon's spin until it turns exactly once per orbit, so it always shows us the same face. That end state is called tidal locking. Spin a young Moon up and watch it settle:

While the Moon spins faster than it orbits, its own bulge is carried ahead of the line to Earth, Earth's pull on it twists back against the spin, and the spin slows. It stops slowing only when the spin matches the orbit, the bulge sits exactly on the line to Earth, and nothing twists any more. From then on the dark mark keeps its place as seen from Earth: the Moon shows us the same face forever. Most large moons in the solar system have locked this way. Earth is headed the same way, much more slowly: in billions of years, if nothing else intervened, its day would stretch toward the length of the month.

The next time the sea creeps up a beach that faces the Moon, it's because the Moon pulls the water there a little harder than it pulls the middle of Earth. Twelve and a half hours later the sea comes back up the same beach, now on the far side of Earth, because the Moon pulls it a little less. How high it comes depends on the shape and depth of the sea around that beach, and every tide a little of Earth's spin is handed to the Moon.

References

1. David Pugh and Philip Woodworth, Sea-Level Science: Understanding Tides, Surges, Tsunamis and Mean Sea-Level Changes, Cambridge University Press, 2014.
2. Chris Garrett, “Tidal resonance in the Bay of Fundy and Gulf of Maine”, Nature 238 (1972), 441–443.
3. G. I. Taylor, “Tidal oscillations in gulfs and rectangular basins”, Proceedings of the London Mathematical Society s2-20 (1922), 148–181.
4. David Edgar Cartwright, Tides: A Scientific History, Cambridge University Press, 1999.
5. F. R. Stephenson, L. V. Morrison and C. Y. Hohenkerk, “Measurement of the Earth's rotation: 720 BC to AD 2015”, Proceedings of the Royal Society A 472 (2016), 20160404.
6. J. O. Dickey and others, “Lunar Laser Ranging: A Continuing Legacy of the Apollo Program”, Science 265 (1994), 482–490.

Further reading

• NOAA's tutorial Tides and Water Levels, from the U.S. National Ocean Service, for a gentle tour of springs, neaps, and tide tables.
• Theory of tides and Amphidromic point on Wikipedia.