Deepen: the mathematics and history behind a tide table

Understand gave you the mechanism, and Investigate let you test it. This layer goes further still: into the history of how people worked all this out, the mathematics that actually predicts a real tide, and a few calculations that would satisfy a genuinely sceptical scientist.

Expect it to feel more like proper physics than the earlier layers. That is the point: tides are one of the oldest problems in science, and the story of solving them properly took over two hundred years.

The first person to explain tides with gravity was Isaac Newton, in the Principia of 1687. His equilibrium theory imagined the ocean as a layer of water thin enough, and responding fast enough, to sit permanently in balance with the Moon's and Sun's pull — always instantly bulging exactly where the two-bulge picture in Understand says it should.

It was a triumph: for the first time, tides were connected to the same gravity that holds the Moon in its orbit and drops an apple to the ground. It correctly predicts spring and neap tides, the rough size of the effect, and the existence of two bulges.

It also makes a testable, and wrong, prediction: that every ocean, everywhere, should have very nearly the same small tidal range (a few tens of centimetres), always at the moment the Moon crosses overhead. You already know that is false — Kochi's metre and the Bay of Fundy's sixteen metres are nothing alike, and high water at most ports arrives hours after the Moon passes overhead, not at that instant.

The fix came from Pierre-Simon Laplace, who published a dynamic theory of the tides in 1775. Laplace's key move was to stop asking "where would the water settle if it had time to reach equilibrium" and start asking "how does a real ocean, with real depth and real coastlines, actually respond when a tidal force pushes on it, moment by moment?"

That turns the problem from simple geometry into the physics of waves in water, which is exactly the physics you already used for tidal bores. Laplace's equations describe how a tide sloshes, reflects and resonates inside an ocean basin — precisely the behaviour that explains why the Bay of Fundy is not like Kochi, and why high water is late almost everywhere.

Here is a fact that sounds backwards at first: for the purposes of a tide, every ocean on Earth counts as shallow.

The shallow-water wave-speed rule from Investigate, v = √(g × depth), applies to any wave whose length is much greater than the water's depth — and a tidal 'wave' is roughly half the width of an ocean basin long, thousands of kilometres, which utterly dwarfs even the deepest trench. So even the open Pacific, averaging about 3700 m deep, counts as "shallow" to a tide, and the same simple formula applies.

Using that average depth: v = √(9.8 × 3700) ≈ 191 m/s, which is about 686 km/h — roughly the cruising speed of a jet airliner. A tidal bulge does not creep across the ocean; it races across it, and still takes many hours to cross a whole basin, because the basins are so enormous.

You met the idea of harmonic analysis in Investigate: a real tide is the sum of many simple, steady waves. Here are the four biggest, with the periods oceanographers actually use.

• M2, the principal lunar semidiurnal wave: period 12 h 25 min — this is simply the half-lunar-day you calculated in Understand, given a name.
• S2, the principal solar semidiurnal wave: period exactly 12.0 hours, because it is tied to the ordinary solar day, not the Moon.
• K1, the lunisolar diurnal wave: period 23.9345 hours — precisely one sidereal day (Earth's spin relative to the stars), reflecting the combined pull of the Moon and Sun once each rotation.
• O1, the principal lunar diurnal wave: period 25.8193 hours.

Notice something important: O1's period is not the same as the lunar (tidal) day of 24 h 50 min from Understand, even though both come from the Moon. They arise from different parts of the Moon's motion (M2 and O1 both depend on the Moon, but at different harmonics of its orbit), and only when you add several constituents together do you get the familiar 24 h 50 min pattern of two highs a day.

Investigate asked you to classify tides as semidiurnal, diurnal or mixed by eye. Oceanographers have a precise number for it, the tidal form factor:

F = (K1 + O1) ÷ (M2 + S2)

— the combined size of the two big daily waves, divided by the combined size of the two big twice-daily waves. A port dominated by M2 and S2 gets a small F and looks semidiurnal; a port dominated by K1 and O1 gets a large F and looks diurnal; anything in between is mixed. The standard boundaries, used worldwide, are:

Investigate showed that the Bay of Fundy's real size lands close to a calculated "quarter wavelength". This chapter explains why a quarter, not a half or a whole wavelength.

A bay like the Bay of Fundy is closed at its head (the land) and open at its mouth (the ocean). That is exactly the shape of a musical instrument you may already know: a closed organ pipe, or a bottle you blow across — closed at one end, open at the other. Such a pipe resonates most strongly when it is a quarter of a wavelength long, because the closed end must be a point where the wave's motion is zero (water cannot slosh through solid land) while the open end is free to move as much as possible — and a quarter of a full wave is the shortest length that fits a zero at one end and a maximum at the other.

A bay open at both ends (imagine a strait connecting two seas) instead resonates like a pipe open at both ends, at a half wavelength. The Bay of Fundy, closed at its head, is the quarter-wavelength kind — which is exactly the formula Investigate used.

Investigate showed that water in 4 m of depth outruns water in 1 m of depth by about 11.3 km/h. Here is the harder question: given a head start, how long does it actually take for the deep-water front to catch up completely and turn a gentle slope into a bore?

It is not just the ocean that flexes under the Moon's pull. The solid rock of Earth rises and falls too, by tens of centimetres, twice a day, everywhere — including under dry land far from any coast. You cannot feel it, because everything around you (the ground, the building, your own body) rises and falls together, but precise instruments detect it clearly: sensitive gravimeters and GPS stations record the ground itself moving by roughly the same order of magnitude as the open-ocean tide, a scale of tens of centimetres.

This matters for real engineering, not just curiosity. Particle accelerators and gravitational wave detectors are sensitive enough that Earth tides have to be corrected for in their measurements, and precise satellite positioning (the kind that underlies modern surveying and map-making) must account for the ground itself shifting under the receiver.

Understand's 'age of the tide' — the day or two's delay between a new or full moon and the biggest tides that follow — has a proper name in physics: phase lag, and it is a general property of any system that is pushed rhythmically rather than dragged instantly into place.

Push a playground swing exactly in time with its own natural rhythm and it swings almost directly under your hand. Push it at a rate quite different from its natural rhythm and its highest point lags noticeably behind your push, arriving late. An ocean basin driven by the Moon's tidal force behaves the same way: it responds like a mass on a spring being pushed by an outside rhythm, not like a puppet moved exactly in step with the puppeteer's hand.

This is also why resonance and lag are two sides of the same coin. A basin driven very close to its own natural period (like the Bay of Fundy) both resonates strongly and shows a particular, fairly extreme phase relationship; a basin driven far from its natural period responds weakly and with a different lag. Every real port's 'age of the tide' and its typical range are both consequences of the same underlying physics: how well its own natural rhythm matches the rhythm the Moon and Sun are pushing it at.

These problems each need more than one idea from this layer or an earlier one. Work through them properly before checking the steps.