Why are there two tides a day?

On most coasts the sea comes in and goes out twice a day. High tide arrives roughly every 12 hours and 25 minutes. If the Moon simply pulled the water towards itself, there would be one heap of water under the Moon and a coast would pass under it once a day. That is not what happens.

The answer is not that the Moon pulls the sea. The answer is that the Moon pulls unevenly on different parts of the Earth, and the sea is free to respond to that unevenness.

The Moon pulls everything, just not equally

Gravity weakens with distance: double the distance and the pull drops to a quarter. The side of the Earth facing the Moon is about 6,400 km closer to it than the Earth’s centre, and the far side is 6,400 km farther away. The Moon is about 384,000 km off, so these differences are small, but they are not zero.

Figure 1 shows the Moon’s pull at sixteen points on the Earth’s surface and at its centre. At first, with the Moon at its real distance, the arrows look identical: the near side is pulled only about 3% harder than the centre. Now tick Subtract the pull at the centre. The whole Earth is falling towards the Moon together (that is what an orbit is), so what matters for the sea is how each point’s pull differs from the pull on the Earth as a whole.

What is left is the tidal force: a stretch along the Earth–Moon line and a squeeze across it. On the near side the leftover arrow points towards the Moon, because that side is pulled harder than the centre. On the far side it points away from the Moon, because that side is pulled less hard than the centre: the Earth is pulled out from under it, so to speak. At the sides, the leftover arrows point slightly inward, because the pulls there all aim at the Moon and so converge a little.

This is two bulges, not one. Nothing about the Earth’s spin is needed to make the far-side bulge; it comes purely from the pull being weaker over there.

How strong is it?

The Moon’s pull at the Earth’s centre is about 3.3 × 10⁻⁵ m/s². The tidal stretch at the point directly under the Moon is about 1.1 × 10⁻⁶ m/s², roughly one ten-millionth of the 9.8 m/s² gravity you feel. That is far too small to lift water straight up. What it does is push water sideways along the surface, over thousands of kilometres of ocean, towards the two points under and opposite the Moon.

Now drag the Moon to half its distance and watch the two “vs today” readouts. The pull grows 4 times, but the stretch grows 8 times. Move it out to double the distance: the pull falls to a quarter, the stretch to an eighth. The pull follows 1/d², the stretch follows 1/d³. In words: the tidal force is the difference in pull across the Earth’s width, and the pull’s change over a fixed width shrinks one power of distance faster than the pull itself. For a body of mass M at distance d, on a planet of radius R, the stretch at the near point is about

tidal acceleration ≈ 2GMR / d³

where G is the gravitational constant. For the Moon this gives 1.10 × 10⁻⁶ m/s², matching the exact value in the figure to within 3%. That small gap is also why the readout says ×8.2 at half the distance, not exactly ×8: the figure uses the exact difference at the near side, which sits a little closer to the Moon than the formula assumes. The 1/d³ rule is the main term, and it gets more exact the farther away the Moon is.

Two bulges, one turning Earth

If the ocean could settle into the shape these forces ask for, it would form an egg stretched towards and away from the Moon. This idealised shape is called the equilibrium tide. Working it out, the water directly under the Moon would stand about 36 cm above average, and the water at the sides about 18 cm below: a range of about half a metre.

The Earth turns inside this shape once a day. In Figure 2 you are looking down on the North Pole; the gold dot is a spot on the coast at the equator, and the graph on the right traces its water level. Notice that the dot passes under two bulges per turn, and read the time between high tides on the graph.

Why 12 hours 25 minutes and not exactly 12 hours? Because the Moon does not wait. While the Earth turns once, the Moon moves about 13° further along its orbit in the same direction, so the coast needs about 50 extra minutes to catch up with it. One full cycle, from the Moon overhead to the Moon overhead again, takes 24 hours 50 minutes: a lunar day. Two bulges per lunar day gives high tides 12 hours 25 minutes apart, and this is why high tide comes about 50 minutes later each day.

The Sun does it too

The Sun pulls on the Earth about 180 times harder than the Moon does. So why do we hear mostly about the Moon? Because the tide depends on the difference in pull across the Earth, and that falls off as 1/d³. The Sun is 27 million times more massive than the Moon, but 389 times farther away, and 389³ is about 59 million. The table puts the numbers side by side.

27 million ÷ 59 million ≈ 0.46. The Sun makes its own pair of bulges, a little under half the size of the Moon’s, always pointing at the Sun. The Moon’s pair goes round once a month. The two add up.

In Figure 3 the Sun is off to the left. The dashed gold outline is the Sun’s tide on its own; the green shape is the total. Drag the Moon round its orbit (or use the slider) and watch the tidal range, the difference between high and low water, in the graph.

At new moon and full moon, Sun, Earth and Moon lie in a line and the two sets of bulges stack: these are spring tides (nothing to do with the season; the water “springs up”). The equilibrium range is about 78 cm. At first and last quarter the Sun’s bulges sit where the Moon’s low water is, and partly fill it in: these are neap tides, with a range of about 29 cm, roughly a third of spring. Full and new moon come every 29.5 days, so spring tides come every 14.8 days, and real tide tables show the same fortnightly rhythm.

So, why two tides a day?

Because the Moon pulls the near side of the Earth more than the centre, and the centre more than the far side. Seen from the Earth, that stretches the oceans both ways along the line to the Moon. A coast turning under two bulges meets high water twice per lunar day of 24 hours 50 minutes. The stretch fades as the cube of distance, which is why the nearby Moon beats the enormous, distant Sun, and why the Sun can only strengthen or weaken the Moon’s tides.

Real coasts aren’t the textbook

Everything above is the equilibrium tide, and real tides are not that. The bulges would have to sweep round the Earth at about 1,600 km/h at the equator, but a wave as long as an ocean, in water 4 km deep, travels at only about 700 km/h; and continents stand in the way. Instead, each ocean basin sloshes in response to the rhythmic push. The tide rotates around points of nearly zero range, called amphidromic points. Some coasts, such as parts of the Gulf of Mexico, get only one high tide a day. And where a basin’s natural sloshing period is close to 12 hours 25 minutes, the tide resonates: in the Minas Basin at the head of the Bay of Fundy, Canada, the water can rise as much as 16 m between low and high tide, thirty times the Moon’s equilibrium range. The forcing is the one in this exhibit; the response is shaped by the sea floor and the shore.