The sea
keeps time
High water is not the moon overhead. The tide at any port is a chord — a handful of pure tones at frequencies fixed by astronomy, with amplitudes and phases the port alone decides — and a machine made of pulleys and cranks could sum it years ahead.
The picture everyone carries is a heap of water directly beneath the moon, dragged along as the Earth turns. It is wrong three times over. It predicts one tide a day; most coasts get two. It predicts high water when the moon is highest; at Eastport, Maine, high water trails the moon's crossing by 3.4 hours, every time, and the harbour pilots have known their own lag for centuries — they called it the establishment of the port. And it predicts the same tide everywhere, when Eastport swings through 6.1 metres on the day this plate was inked while Pensacola, Florida — same moon, same day — drifts through less than half a metre, once.
Newton put the real mechanism in the Principia. The moon does not pull the ocean up; it pulls the whole Earth, and it pulls the near side harder than the centre, and the centre harder than the far side. Subtract the average — the orbit everything shares — and what remains at the surface is a residue of differences.
I · The pull-apart
The far side is not an afterthought: the centre of the Earth falls toward the moon faster than the far water does, so the far water is left behind — outward, in the Earth's frame, by almost the same residue as the near side. The stretched surface is the equilibrium tide, 0.53 m of it from the moon and 0.25 m from the sun, drawn here 4.2 million times too large. Two bulges, fixed under the moon; one planet turning beneath them; two tides a lunar day.
A lunar day is 24.84 hours — the moon has moved on by the time the Earth comes back around, so the double tide slips 50 minutes later each day. That single fact gives the dominant tone: a rise and fall every 12.42 hours, which tide tables call M2.
If the moon's orbit were a circle in the equator's plane and the sun did not exist, M2 would be the whole story. None of that holds. The sun raises its own pair of bulges at exactly twelve hours (S2). The moon's orbit is an ellipse, so its pull swells and slackens over the 27.3-day month (N2). Both orbits are tilted, so the two daily highs go unequal on a once-a-day beat (K1, O1). Each imperfection does not smear the spectrum — it splits it, cleanly, into another pure tone. The tide is not a wave. It is a chord.
Five clocks generate every note: the lunar day τ, the moon's slow circuit s, the sun's h, the creep of the moon's closest approach p, and the 18.6-year swing of its orbital crossing N. A constituent is nothing but an integer recipe over those clocks, and its frequency follows by arithmetic — which is why the whole table below carries seven decimal places without a single measurement in it.
| tone | recipe | speed °/h | period h | character |
|---|---|---|---|---|
| M2 | 2τ | 28.9841042 | 12.42 | principal lunar, twice a lunar day |
| S2 | 2τ + 2s − 2h | 30.0000000 | 12.00 | principal solar, twice a solar day |
| N2 | 2τ − s + p | 28.4397295 | 12.66 | the moon's ellipse, monthly modulation |
| K1 | τ + s | 15.0410686 | 23.93 | declinational, sun and moon together |
| O1 | τ − s | 13.9430356 | 25.82 | declinational, the moon alone |
| P1 | τ + s − 2h | 14.9589314 | 24.07 | declinational, the sun alone |
| K2 | 2τ + 2s | 30.0821373 | 11.97 | declinational overtone, semidiurnal |
| Q1 | τ − 2s + p | 13.3986609 | 26.87 | elliptic declinational |
Twenty-nine fainter tones follow the same arithmetic; NOAA publishes thirty-seven per station. The recipes are exact integers — the speeds inherit their precision from the astronomy, not from any tide gauge.
Astronomy fixes the frequencies for every ocean on Earth. It fixes nothing else. Each basin takes the same forcing and answers with its own amplitude and its own delay at every frequency — a shallow shelf drags the wave, a funnel concentrates it, a basin whose natural period sits near a forcing period resonates. The bay does not receive the tide. It plays it, the way a violin body plays a string.
So a port is characterised by a short table: for each tone, how loud here (H) and how late here (g). Those pairs are the harmonic constants, measured once from a year or two of gauge records, good for decades. Eastport's answer to M2 is 2.65 m of amplitude — 10 times what equilibrium theory offers; Pensacola's is seventeen millimetres. The ratio of diurnal to semidiurnal loudness — the form factor — tells you what kind of day a coast keeps:
| station | F | regime | highs today | range m |
|---|---|---|---|---|
| Eastport, Maine | 0.09 | semidiurnal | 2 | 6.1 |
| San Francisco, California | 0.84 | mixed, mainly semidiurnal | 2 | 2.4 |
| Pensacola, Florida | 11.23 | diurnal | 1 | 0.6 |
F = (K1+O1)/(M2+S2), computed from each station's constants. Below 0.25 a coast keeps semidiurnal time; above 3 the once-a-day tones govern alone.
II · The tide-predicting machine
One crank per constituent: radius fH from the station's constants, angle V + u − g from the clocks — the eight loudest drawn large, the other twenty-nine summed all the same. The vertical parts of the arms add up to the height of the sea, inked on the roll. This is not a picture of a Kelvin machine; it is one, in the only sense that mattered — the sum is the prediction.
The Gulf of Maine and the Bay of Fundy form a basin whose natural sloshing period sits close to the lunar half-day, so M2 arrives at a bay already swinging in sympathy. Eastport's constants answer with an M2 amplitude of 2.65 m — ten times the equilibrium theory's offer, the difference being geometry.
| event | local time | height m MLLW |
|---|---|---|
| low water | 02:52 | 0.39 |
| high water | 08:58 | 6.02 |
| low water | 15:33 | -0.11 |
| high water | 21:35 | 5.55 |
Computed in this page from thirty-seven pairs of published numbers and five clock rates — nothing here was looked up. Hold it against NOAA's official prediction for the same station and day; the residue between the two curves is the part of the sea that is weather, not clockwork.
The chord has beats. M2 and S2 drift in and out of step every 14.77 days — aligned at new and full moon into spring tides, opposed at the quarters into neaps. The moon's ellipse beats against both on the month; and underneath everything, the 18.6-year wander of the lunar node swells and shrinks every lunar amplitude by a few per cent — this page corrects M2 by a factor between 0.963 and 1.038 depending on the year you read it in.
Thirty days at Eastport from the day this page is read, 6-minute steps. The envelope is the spring–neap beat: |speed of S2 − speed of M2| works out to one cycle per 14.77 days, and the arithmetic is visible from across the room.
Because the tide is a chord of known frequencies, prediction does not require understanding the ocean — only resolving the constants and re-summing the tones. William Thomson saw that the summing could be done by machinery: one crank per tone, geared to its speed, a cord passing over every crank-driven pulley in turn, a pen at the end. Turn the handle and a year of tides spools out in a few hours. The United States built its own in 1910 — Old Brass Brains, 2,500 pounds of gears that predicted for every American port into the 1960s — and it summed exactly thirty-seven constituents. That is why NOAA still publishes thirty-seven numbers per station: the list is the machine's gear train, fossilised into a data format.
The constants know nothing of wind, pressure, or a river in flood; a storm surge rides on top of the prediction and always will. What the machine promises is narrower and stranger: that the astronomical part of the sea — the part that is pure gravity and geometry — is knowable years ahead, to minutes and centimetres, by adding up circles.
The moon sets the beat. The shore chooses the voicing. The sea keeps time.