The Tide the Textbook Got Wrong

Almost everyone is taught the tides as two bulges: the Moon pulls the ocean into an egg, the Earth turns under it, and you get two highs a day. The picture is tidy, it is famous, and in three separate ways it is wrong — or rather, it is a careful idealization that the real ocean ignores. Each correction is a number you can check, so let's compute them, starting with the one that sounds impossible.

It out-pulls the Moon 179 to 1 — and loses the tide

Newton's law of gravity is an inverse-square law: the force between two masses falls off as 1/d². Put in the real masses and distances and ask which body pulls the whole Earth harder. It isn't close.

Here is the whole secret. The force of gravity goes as 1/d², but a tide is not the force — it is the difference in the force across the width of the Earth, the near side pulled a little harder than the far side. Differencing an inverse-square law turns it into an inverse-cube one: the tide-raising effect goes as M/d³. The Sun is about 389 times farther than the Moon, so that extra factor of distance divides its enormous pull right back down. 179 ÷ 389 ≈ 0.46. The Sun wins the pull and loses the tide, by exactly the margin the third power demands.

The far bulge is not a fudge

The near-side bulge is easy: the Moon pulls the water nearest it hardest, and it heaps up. But why a bulge on the far side, pointing away from the Moon? The popular answers — "centrifugal force flings the water off," "the Earth gets pulled out from under the far ocean" — are at best bookkeeping and at worst nonsense. The honest answer is that the tidal force is the gradient of gravity, and a gradient points outward on both ends. Drag the Moon and watch the actual field.

Subtract the pull on the Earth's centre from the pull at each point of the surface; that residual is the tide. At the point nearest the Moon the residual points toward the Moon (outward). At the point farthest away, gravity is weaker than at the centre, so relative to the centre the water there is left behind, and "left behind on the far side" means heaped outward, away from the Moon. Computed exactly from the inverse-square law, the far bulge comes out at 1.07×10⁻⁶ m/s² against the near side's 1.13×10⁻⁶: real, outward, and only about 5% smaller. Not absent. Not centrifugal. Just the difference of a force across about 12,700 kilometres of rock.

Why the Sun shows up anyway — spring and neap

The Sun's tide is 0.46 of the Moon's: small, but not nothing. When the two line up (new Moon and full Moon) their bulges add; when they pull at right angles (the quarter Moons) the Sun's bulge sits in the Moon's trough and partly cancels it. That is the two-week swing on semidiurnal coasts: the big spring tides and the meek neap ones. Pick a phase.

And the bulges aren't the ocean

Now the honest part, the one most pages skip. Everything above is equilibrium theory — Newton's idealization, in which the ocean has time to settle into the static egg the Moon's gravity wants. Compute how big that egg is, and the textbook's own picture quietly falls apart.

The equilibrium high tide comes to about 0.36 m for the Moon and 0.16 m for the Sun; a combined spring range, high to low, of about 0.78 m. Three-quarters of a metre. The Bay of Fundy gets sixteen metres. The equilibrium bulge is off by more than a factor of twenty, and no refinement of Newton's static ocean will ever close that gap, because the static ocean is the wrong model. Compare the two speeds in a mean-depth model:

The real ocean's daily tide needs a dynamic account. It behaves as a forced wave sloshing in the basins, the way water in a carried bowl heaps at the walls: Laplace's dynamic theory (Paris Académie, 1775–76), the equations we still solve. In a basin the tide runs around the edges as a rotary wave, pivoting about still points called amphidromic points where the range falls to nearly zero. William Whewell predicted such a no-tide point in the North Sea in 1836; the surveyor William Hewett sailed out and confirmed it in 1840. Around an amphidromic system, high-water timing varies from place to place along the shore.

Which kills the last piece of the cartoon: at most places, high tide does not happen when the Moon is overhead. The common establishment of the port, the average interval from the Moon's transit to high water on days of new and full Moon, is a fixed local number you can read off a chart, and it has nothing to do with a bulge sitting under the Moon, because there is no such bulge. Individual high-water intervals vary. The Moon and Sun supply the forcing; basin shape, depth, rotation and friction shape the local response.

The two-bulge story isn't a lie so much as a first chord. It gets the cause right (gravity's gradient), the semidiurnal cadence right (two highs per lunar day, springs about every fortnight), and the actual water almost entirely wrong. The honest version is better: a forced wave shaped by ocean basins, pivoting around invisible still points, piling sixteen metres into one Canadian bay and leaving the principal lunar tide with almost no range near North Sea amphidromic points, all of it forced by a difference in gravity of about a ten-millionth of Earth's surface gravity.