The Mathematics of Tides: Why the Sea Has Two Bumps a Day

Most people learn that the tide is caused by the moon's gravity, and most people stop there. The version that survives the rest of the lesson goes something like this: the moon attracts the ocean toward it; the water nearest the moon is pulled into a bulge; the Earth rotates under the bulge; coastlines pass through the bulge twice a day, generating two high tides separated by twelve hours.

The story is plausible and widespread and almost entirely misleading. Almost everything in it is wrong or misleading except for the conclusion that there are two high tides a day. The actual mechanism, the actual mathematics, and the actual history are all stranger.

The wrong picture

Start with the obvious problem: if the moon's gravity attracts water toward the moon, why is there a high tide on the side of the Earth facing away from the moon? The schoolroom story has no answer. Some textbooks add a hand-wave about centrifugal force from the Earth-moon orbit, which is closer to the truth but still confused.

The right answer is differential gravity. The moon does not pull water toward itself; it pulls everything toward itself, including the Earth. What matters for tides is not the moon's pull on the ocean but the difference between the moon's pull on the near side of the Earth, the moon's pull on the center of the Earth, and the moon's pull on the far side. The near-side ocean is pulled toward the moon harder than the center; the far-side ocean is pulled toward the moon less than the center. Relative to the Earth as a whole, the near-side water rises toward the moon and the far-side water rises away from it. Two bulges. Twelve hours apart, given Earth's rotation.

The mathematics is a Taylor expansion of the gravitational potential. The leading term (the moon pulls everything) cancels out because it accelerates everything equally. The next term, the gradient of the gravitational field, is the tidal force. It scales with the inverse cube of distance, not the inverse square; this is why the moon's tidal effect is twice the sun's even though the sun's gravity at Earth is much stronger. The sun is much further away in tidal-force terms than in gravity terms.

Newton's static theory

Isaac Newton derived the differential-gravity story in the Principia in 1687. His framework was static: imagine the Earth covered with an idealized ocean of uniform depth, no continents, no friction, no rotation effects beyond the Earth-moon orbit. The static tide is the equilibrium shape this ocean would take under the differential gravity of the moon and sun. It is an ellipsoid, slightly elongated along the Earth-moon axis, the elongation amounting to about 50 centimeters.

The static theory is correct as a first approximation. Real tides are about 50 centimeters in deep ocean, matching Newton's prediction. Where it fails is everywhere except the deep open ocean. Coastal tides are larger, sometimes much larger, sometimes by a factor of twenty (the Bay of Fundy can have a 16-meter tidal range). The static theory has nothing to say about why some coastlines have huge tides and others have almost none, why high tide on one coast can be in the middle of the day while a few hundred kilometers away it is at midnight, or why some places have one tide per day instead of two.

Laplace's dynamic theory

The fix came from Laplace in 1775. Laplace replaced the static-equilibrium picture with a dynamic one: the tides are forced waves traveling around the ocean, driven by the moon's and sun's gravitational potential. The wave speed is set by ocean depth (shallow water, slow waves; deep water, fast waves). The wave shape is set by coastlines, which act as boundaries.

The dynamic theory predicts everything the static theory missed. Tidal range varies by location because waves resonate or interfere with the basin's natural frequencies. The Bay of Fundy has huge tides because its natural period is close to twelve hours; the tide forcing is in resonance with the bay. The Mediterranean has small tides because its connection to the Atlantic is too narrow for the tide wave to enter strongly.

Laplace's equations are the foundation of all modern tide prediction. They are also unsolvable in closed form for any realistic ocean. For two centuries, tide prediction proceeded by approximation, by tabulation of past observations, and by the kind of mathematical machinery that ended up being more useful than the equations it approximated.

Doodson's harmonics

In 1921, the British oceanographer Arthur Doodson at the Liverpool Tidal Institute published a method for predicting tides based on harmonic analysis. The idea: the gravitational potential at any point on Earth is a sum of periodic functions. The moon's contribution has a period of roughly 12.4 hours (half its orbit), but with sub-periods related to the moon's elliptical orbit (29.5 days), the orbital plane's wobble (18.6 years), and a few dozen smaller effects. The sun's contribution has a 12-hour period and similar sub-periods. Each periodic component, when fed into the local response of a coastline (amplitude and phase shift), produces a contribution to the local tide.

Doodson identified 388 tidal constituents, each a periodic function with a known frequency derivable from astronomy and the local amplitude and phase determined by observation. The tide at any location is, to within a few centimeters, the sum of these 388 sinusoids weighted by the local response.

The brilliance of this approach was that it converted tide prediction into a problem of measuring local constants. Once you had a year of tide-gauge readings at a port, you could solve for the local amplitude and phase of each constituent and then predict the tide indefinitely. Doodson's machinery made tide prediction routine for the first time in history. The British Admiralty's tide tables, used by every ship sailing British waters, were generated by Doodson's method for sixty years.

The moon recedes

One small consequence of tides that took two centuries to confirm: the moon is moving away from Earth. The tidal bulge does not point exactly at the moon, because Earth's rotation drags the bulge slightly forward. The bulge, slightly ahead of the moon, exerts a torque on the moon, accelerating it in its orbit. An accelerated orbit is a higher orbit, so the moon spirals slowly outward.

The number is small but measurable: 3.8 centimeters per year, confirmed by laser ranging to retroreflectors left on the moon by Apollo astronauts. Over a billion years, the moon will have moved tens of thousands of kilometers further away. Over four billion years, total solar eclipses will become impossible because the moon will appear too small to cover the sun.

The reverse effect operates on Earth: the moon's pull on the bulge slows Earth's rotation. Days are getting longer at about 1.7 milliseconds per century. This is also confirmed observationally; ancient eclipse records from Babylon, properly interpreted, are consistent with a slowly lengthening day.

Tides on other bodies

The mathematics generalizes. Any body in orbit around another experiences tidal forces; what differs is whether they matter. Mars has tides from Phobos but the moon is small enough that the effect is microscopic. Jupiter's moons have enormous tides from Jupiter; Io's interior is heated to volcanic temperatures by tidal flexing. Saturn's rings exist because Mimas's tidal forces shred any small body that tries to coalesce inside the Roche limit.

Many close binary stars are tidally locked, meaning each star always shows the same face to the other. The Earth-moon system is on the way to this state: the moon already shows the same face to Earth, and Earth is slowly being braked toward the same outcome. In about 50 billion years (much longer than the lifetime of either body, given the sun's expansion), the day and the month would be equal and Earth would show the same face to the moon. The system runs out of time before reaching its asymptote.

What is impressive

The impressive thing about tides is how much of the story was worked out before the relevant tools existed. Newton in 1687 derived the differential-gravity argument with no satellites, no global tide gauges, and no understanding of resonant ocean basins. Laplace in 1775 wrote the dynamic equations correctly with no computer. Doodson in 1921 produced 388 constituents by hand, with a slide rule and printed astronomical ephemerides.

The current best tide-prediction software runs on Doodson's framework with finer-grained constituents (more than a thousand) and finite-element ocean models that respect coastline geometry to a kilometer. The leading edge of the science is climate-driven sea-level change as it interacts with the tide constituents over decades, and the small but real effects of solid-Earth tides (the planet's crust flexes by 30 centimeters as the moon passes overhead). The mathematics has not changed since Laplace. The astronomy has not changed since Newton. What has improved is the quality of the inputs, which is, on the long arc, what improvement in any mature science looks like.