Stage 1 · San Francisco Bay
The sea breathes
Twice a day the water in San Francisco Bay rises and falls by a meter or two. A gauge at the Golden Gate has measured it since 1854. Could you predict it years ahead?
San Francisco Bay, built from real terrain and depth soundings. The water stands where it really stood in the first week of June 2025.
Just inside the Golden Gate a tide gauge has measured the water since 1854. Its six-minute readings for that week set the level of the bay here.
The water rises and falls about twice a day. The two highs are not the same height, and neither are the two lows.
Each rise carries ocean water in through the Gate and out across the bay. Each fall draws it back out to sea.
At the south end the bay is shallow. When the tide falls, square kilometers of mud come out of the water. When it rises, they disappear again.
The swing changes through the week: close to two meters on June 1, little more than one by midweek, and growing again by the weekend.
Nothing on Earth pushes this water around. The Moon and the Sun do it, from hundreds of thousands and millions of kilometers away.
The pattern repeats, but never exactly. Could you say, years ahead, where the water will stand at 3 p.m. on some Tuesday? Sailors needed to know.
Stage 2 · Two bulges
The Moon pulls unevenly
The Moon pulls the near side of Earth harder than the middle, and the middle harder than the far side. The ocean is stretched into two bulges, and Earth turns beneath them.
The Moon pulls on every part of Earth, but not equally. Gravity weakens with distance, so the near side is pulled harder than the middle, and the far side least.
Earth as a whole falls toward the Moon (the two circle a shared center) at the rate set by the pull on its middle. A pull every part shares moves Earth without changing its shape.
Take that shared pull away and what is left stretches Earth along the line to the Moon: the near side pulled ahead, the far side left behind, the sides squeezed in.
An ocean free to follow would stand in two bulges, one under the Moon and one opposite: 36 cm above the average, and 18 cm below it in between. Drawn here a million times higher.
Earth turns beneath the bulges. San Francisco rides into one, out, into the other and out again: two high tides and two low tides.
Not quite once a day. While Earth turns, the Moon moves on about 12° along its orbit, and San Francisco needs 50 more minutes to come back under it: a lunar day is 24 h 50 min.
This is Newton’s equilibrium tide, and it is too simple. The Moon’s twice-daily tide at San Francisco is 0.58 m, three times this picture’s, and the ocean sets its timing. Next: why.
Stage 3 · Co-tidal lines
The bulge is a wave
The ocean cannot keep up with the Moon. The tide travels as a wave, wheeling around points where there is almost none, and sweeps up the coast from San Diego to Seattle.
This is the Moon’s main tide, M2, in every ocean at once, measured from space. Satellites have tracked the height of the sea surface since 1992.
The bright line is high water, right now. It does not race around the planet under the Moon. It turns around a few points in the ocean.
At those amphidromic points there is almost no tide. The water there barely rises or falls while high water goes around it like the hand of a clock.
The ocean is too shallow to keep up. A tide wave in water 4 km deep travels about 700 km an hour; to stay under the Moon at the equator it would need 1,600. Continents block the way and Earth’s spin steers it.
Each thin line joins places that have high water at the same moment. Neighboring lines are 62 minutes apart.
Along the west coast of North America, the crest runs north.
High water reaches San Diego first, Los Angeles 5 minutes later and San Francisco 2 hours 14 minutes after San Diego.
Then Newport, Oregon, after 3 hours 2 minutes, and Seattle, deep in Puget Sound, after 7 hours 51 minutes.
Every coast has its own tide because every coast sits at a different place on this wave. Next: the wave squeezes through the Golden Gate.
Stage 4 · Shallow water
Into the bay
Through the Golden Gate the wave slows in shallow water, grows in the South Bay and arrives late in Suisun Bay. Thirty-five gauges inside the bay show it happening.
First, the waves you can see. Out at sea an 11-second swell has crests 189 meters apart and travels at 17 meters a second.
At Ocean Beach the bottom comes up. In 10 meters of water the same swell is 103 meters long and moves at 9.4 m/s. The crests crowd together, grow taller and swing round to face the beach.
When a wave’s height reaches about four-fifths of the depth, it breaks.
The tide is a wave too, hundreds of kilometers long, so it feels the bottom everywhere. Here the water glows brighter the higher it stands above its average. Lines mark every 10 centimeters.
High water reaches the Golden Gate at 9:47 a.m. on December 23, 2026. It takes 59 more minutes to reach Redwood City, and 2 hours 12 minutes to reach Port Chicago in Suisun Bay.
A long wave travels at the square root of gravity times depth: about 700 km an hour in the deep Pacific, 36 km an hour in 10 meters of water. Inside the bay the tide moves at the pace of a bicycle.
The South Bay is a dead end. The wave runs in, reflects off the far end and adds to itself: the range grows from 2.72 meters at the Gate to 3.48 at Redwood City and 3.59 at Coyote Creek.
To the northeast the water has to squeeze through the narrow Carquinez Strait and loses energy on the way. Port Chicago sees a range of only 1.99 meters, more than two hours late.
In the afternoon the water at the south end drops 3.5 meters. The flats drain into their channels.
Then the next tide comes in and floods them again, twice a day, every day.
Every level here comes from the harmonic constants of 35 gauges inside the bay. Next: why some weeks the range is huge and others small.
Stage 5 · Springs and neaps
The Sun joins in
The Sun makes its own, smaller tide. Twice a month the two line up and the range swells; in between they fight and it shrinks.
San Francisco’s biggest rhythm is the Moon’s, M2: 0.58 m, once every 12 h 25 min. Draw it as an arm on a dial. As it turns, its height is the tide, written on the chart.
The Sun raises bulges too, 0.46 times the Moon’s. Its rhythm here, S2, is 0.14 m and turns once every 12 hours exactly. Hang it on the end of the Moon’s arm: the tip draws the sum.
S2 turns about one degree an hour faster than M2, so it keeps gaining. Every 14.77 days it laps M2 once.
Ride along with M2 and only S2’s gain is left. Pointing the same way, the two add up to about 0.7 m: spring tides. Opposed, about 0.4 m is left: neap tides.
The beat is set by the sky. M2 and S2 line up when Sun, Earth and Moon do, at new and full moon, and oppose at the quarters. At San Francisco the springs come about 8 hours later.
So spring tides come twice a month, all year; the name is about water springing up. And no two are alike here: the Moon’s tilt and changing distance add rhythms of their own.
Stage 6 · Constituents
Many rhythms
The Moon’s tilt, its elliptical orbit, the seasons and the shallow bay each add a rhythm with a speed fixed by astronomy. A port’s tide is their sum.
Each line is one rhythm in San Francisco’s tide. Where it stands is its speed, fixed by astronomy; its height is how strong it is here, measured. Most crowd near once and twice a day.
Twice a day: M2 from the Moon’s bulges, S2 from the Sun’s. N2 comes from the Moon’s elliptical orbit: it pulls harder when near, and that 27.6-day swelling is a rhythm of its own.
Once a day: the Moon and Sun are rarely over the equator, so the bulges tilt and one daily tide outdoes the other. Here K1 and O1 together rival M2: the two lows of a day differ by 0.7 m.
Slow rhythms, drawn eight times taller. Sa, once a year, is not gravity but the seasons: warm water expands, winds and rivers change. It is fitted like a tide anyway: 4.4 cm.
Three, four, six a day: nothing in the sky moves this fast. In shallow water a crest runs faster than a trough and friction drags, so the wave distorts. M4 here is 2.2 cm.
NOAA lists 37 rhythms. Every speed is a sum of six cycles: the day, the month, the year, the 8.85-year turn of the Moon’s perigee, the 18.6-year turn of its orbit, and Earth’s perihelion.
Stage 7 · Harmonic analysis
Taking the tide apart
Fit a year of measurements with those rhythms and each one comes out with its own height and timing. Those numbers are all you need to predict the tide.
Every hourly reading from NOAA’s San Francisco gauge in 2025: 8,760 numbers. The fast flicker is the daily tide; the slow swelling is the two-week beat.
Up close, December: two tides a day, unequal, swelling and shrinking. All the rhythms of the last stage are in this tangle.
Each rhythm’s speed is known. Its height and timing here are not, so try a wave of each speed against the whole year, all 31 at once, and keep the best fit: Kelvin’s harmonic analysis of 1867.
Each rhythm comes out as an arm: its length is the height, its angle sets the timing, and it turns at its own speed. Unwound, each arm is a wave: M2, K1, O1, S2, N2…
Our fit lands on NOAA’s published values. M2: 0.568 m at 209.2°, against NOAA’s 0.576 m at 208.2°. NOAA averages five years; one year’s weather moves ours by a centimeter.
Add the 31 waves back together and the sum lies on the record.
Almost. What is left is weather: wind and air pressure move the bay by 7 cm in a typical hour of 2025. On December 24 the water stood 47 cm above the tide.
Those 63 numbers are San Francisco’s tide. With them it can be worked out for any hour, past or future, by adding waves. Next: a machine that adds.
Stage 8 · Kelvin, 1872
A machine that adds
William Thomson built a machine to do the sum: one pulley per rhythm, each on a crank geared to its own speed, and one wire threading them all. A pen on the end of the wire draws the tide.
Start with one crank. A pin on a turning disc drives a slotted yoke up and down, and the yoke carries a pulley. Trace the pulley’s height over time and you get a sine wave.
How far the pin sits from the center sets the height of the wave: the crank throw is the amplitude. Where the disc starts sets the timing: the crank angle is the phase.
Gears turn the disc at one fixed speed. This crank is M2, the Moon’s half-day rhythm, so it turns once every 12 hours 25 minutes. The gear ratio is the speed.
Add a second crank for S2, the Sun’s half-day. Then thread a wire: fixed at one end, under one pulley, over the next, up over two guides and down to a pen.
When a pulley moves, it takes up or lets out wire on both sides of it, and the pen moves twice as far. With every pulley on the same wire, the pen moves by the sum of them all.
M2 and S2 run at slightly different speeds, so they drift in and out of step every 14.8 days. In step, the tides are big (spring tides). Out of step, they are small (neap tides).
Thomson’s machine had ten cranks. Here they go in his order from the pen end: M2, S2, N2, K1, O1, K2, L2, P1, M4, MS4. Each pin slides out to San Francisco’s amplitude as it joins.
With all ten, the line on the paper is San Francisco’s tide: two highs and two lows a day, one pair much bigger than the other.
Thomson designed the machine in 1872. A. Légé & Co. of London built it, and it was shown at the British Association’s meeting in Bradford in 1873. Turned by hand, it drew a year of tides for a port in about four hours.
Nothing inside it knows about the Moon. The astronomy is in the gear ratios, the port is in the throws and angles, and the wire does the adding.
Stage 9 · Gear trains
Gears that keep astronomical time
Each crank turns at its rhythm’s speed through a train of gears. Teeth come in whole numbers and the Moon does not, so every ratio is a little off. How far off, after a year?
Every crank has to turn at its own rhythm’s speed, and all of them are driven from shafts the operator turns. In our drawing of Kelvin’s machine the driving shaft turns once a day.
M2 must turn 1.9322736 times for each turn of the shaft. Gears only make ratios of whole numbers of teeth: 176/169 × 154/83 gives 1.9322735. Kelvin’s tooth counts don’t survive, so these are the closest pairs of 10 to 200 teeth we could find.
The Sun’s rhythms are easy. S2 turns exactly twice a day: 54 teeth driving 27. The Moon’s rhythms are not whole-number fractions of a day, so every lunar crank runs a little fast or a little slow.
To see how much, run a year and compare each crank with the sky. Each rung is one crank of the US machine of 1912, Old Brass Brains, geared with the tooth counts its makers published. The center line is where the sky says the crank should point.
The errors grow steadily. ρ1 falls 0.75° behind, so by the end of the year its peaks come 3.4 minutes late. K1 and K2 run about a minute fast. M2, the biggest rhythm, is off by 3 seconds.
At San Francisco, all of those errors together move the predicted tide by less than 3 millimeters. Kelvin’s ten, with our gear trains (the white ticks), barely leave the line.
And no error builds up for longer than a year. Before each port and each year, the operator set every crank again from astronomy, so the gears only had to keep time for one run.
The astronomy is cut into the brass: two numbers of teeth per stage, chosen once, good for a year at a time.
Stage 10 · US Coast and Geodetic Survey, 1912
Old Brass Brains
America’s tide machine summed 37 rhythms with one steel chain. It weighed about 2,500 pounds, turned out a year of tides for a port in a working day, and made the official US predictions from 1912 to 1965.
The United States built its own. Tide Predicting Machine No. 2 of the Coast and Geodetic Survey was designed by Rollin Harris and built by E. G. Fischer, starting in 1896. It was finished in 1910 and first used for the tide tables of 1912.
It sums 37 rhythms instead of ten. Start at the fixed end of its chain, with Sa: the sea’s yearly rise and fall as it warms and cools.
Next come the slow ones: the Moon’s monthly and fortnightly terms. Then dozens of smaller daily and twice-daily rhythms, and overtides such as M6 and M8 that shallow water adds to the wave.
Instead of a wire it uses a fine steel chain, running under one pulley and over the next. The pulleys are two inches across, and each crank turns through gears whose tooth counts the Survey published.
The biggest rhythms sit at the front, next to the pen: M4, O1, K1, N2, S2 and M2. On the real machine, M2, S2 and K1 each had two pulleys, at half amplitude.
Setting the machine up for a new port took two or three hours. Running and copying a year of predictions took another seven to twelve. Dials gave the time and height of every high and low water, and a pen drew the curve.
Is 37 worth it? For San Francisco, Kelvin’s ten rhythms and these 37 differ by 6.5 cm RMS over a year, and by up to 20 cm at the worst hour.
It made the official US tide predictions from 1912 to 1965. During the Second World War its work was classified, and it predicted the tides for the landings in Normandy and the Pacific. It is on display at NOAA in Silver Spring, Maryland.
Stage 11 · Machine and bay
Turn the crank
Set the machine to San Francisco, turn the crank, and watch the pen and the water move together. Then compare with NOAA’s official prediction.
Old Brass Brains, set for San Francisco on New Year’s Day 2026. Beside it, the bay, with the water where 35 tide stations put it at the same moment.
Day 1: Turn the crank. The pen rises and falls, and so does the water at the Golden Gate: the same number, one in brass, one in the bay.
The week: These are the king tides of the new year. The range reaches 2.7 m, low water drops half a meter below chart datum, and the flats of the bay drain.
The blue dots are NOAA’s official prediction for the same hours, computed today from the same constants.
Over the week’s 169 hourly values, the machine and NOAA never differ by more than 2 cm. A century-old mechanism and a modern computer agree to the width of a finger.
The machine never knew about this bay. It only knew 37 rhythms and how strongly each one moves the sea at Fort Point.
Stage 12 · Hurricane Sandy, 2012
When the sea disagrees
The machine knows the Moon and the Sun, not the weather. On 29 October 2012 the water at the Battery rose almost three meters above its prediction.
New York Harbor on October 29, 2012, as the tide tables saw it. The water follows the predictions for NOAA’s harbor gauges: an ordinary day, with high water of 1.48 meters at the Battery at 8:48 p.m.
That prediction is everything this story has built so far: the Moon and the Sun, summed, with constants measured at the Battery over many years.
Now the water that actually came. Hurricane Sandy’s winds had been pushing the sea toward the coast for two days.
At 7 p.m. the Battery gauge read 3.47 meters, already 2.26 meters above the prediction, and rising.
The gold line is where the tide tables put the shore. Everything past it is water no prediction expected.
At 9:24 p.m. the gauge read 4.28 meters, the highest since its record began in 1920. The prediction for that minute was 1.45 meters.
Seawater poured into subway and road tunnels. Much of Lower Manhattan lost power.
The difference, 2.84 meters at the peak, was weather: wind and low air pressure. A tide machine cannot know it. It knows the Moon and the Sun, and nothing else.
The timing made it worse. The surge peaked 36 minutes after the predicted high tide, on the day of a full moon.
That is why forecasters add storm-surge models to the tide tables. The tide is the part that can be known years ahead.
Stage 13 · Your tide, now
Any coast on Earth
Pick a coast. If NOAA has a gauge nearby, the machine uses its constants; anywhere else, a global ocean tide model. Here is the tide there, right now.
The amber points are NOAA’s 1,374 tide stations. Each has years of records, and so its own harmonic constants: there, the tide is measured.
Everywhere else, satellites have measured the sea surface for three decades. The EOT20 model turns that into constants for every ocean on Earth. The brighter the sea, the bigger its twice-daily tide.
Your turn. Drag the globe and tap a coast, search for a place, or pick one of the suggestions.