Why there are two tides a day

One Moon, yet the sea rises twice a day. The Moon's pull is not the point. The difference in its pull across the Earth is.

The puzzle

Stand on a shore for a day and the water comes in and goes out twice. Each day high tide arrives a little later, by the same delay as the Moon's rising, so the Moon is behind it. But the obvious story — the Moon pulls the water up on the side facing it — gives one bulge, under the Moon. It says nothing about the second high tide, when the Moon is on the far side of the planet.

Why it is worth getting right

Every tide table starts from this two-bulge shape, then works out the details port by port. It also explains why the highest tides of the month come at both new moon and full moon, when the Sun and Moon sit on opposite sides of the Earth as often as the same side. Newton's theory of gravity was the first that could account for it.

Differences, not pulls

The Earth is not a point. Its near side is closer to the Moon than its centre, and its centre is closer than its far side. Gravity weakens with distance, so the near side is pulled toward the Moon more strongly than the centre, and the centre more strongly than the far side.

Now take away the pull on the centre, which is the pull the Earth as a whole feels. What is left over is called the differential force. On the near side, the leftover points toward the Moon: that side is pulled ahead of the centre. On the far side, the leftover points away from the Moon: that side is pulled less than the centre, so it lags behind. Both sides are stretched outward, and the planet is drawn into a slightly elongated shape, like an American football, with its long axis aimed at the Moon.

Solid rock does stretch, but only about 20 centimetres, roughly a third of what a planet made of water would do. The leftover force then shows up as tiny sideways tugs across the surface, far too weak to move a rock or a person, but the oceans are free to flow. Over several hours, water slides toward the two regions under and opposite the Moon and piles up there. The bulges are not water lifted off the Earth; they are water that has flowed sideways and gathered.

The Earth then turns once a day beneath both bulges, carrying an observer into deep water, out, into the second bulge, and out again: two high tides and two low tides.

Sun, spring and neap

The Sun raises tides too, by the same reasoning, though it is less than half as effective as the Moon. At new moon and full moon the Sun and Moon lie on one line, their bulge pairs coincide, and the tides are higher than normal: spring tides. Because every bulge comes as a pair, same side or opposite sides makes no difference. At first or last quarter they pull at right angles and partly cancel: neap tides. Coastlines, depth and friction reshape all of this locally, which is why each port needs its own table.

In short

The Moon pulls the near side of the Earth harder than the centre, and the centre harder than the far side. Subtract the centre's share and both sides are left stretched outward. Water gathers under the Moon and opposite it, and the turning Earth carries every shore through both piles daily.