How the Moon builds two bulges

In Discover you met the headline: the Moon's gravity stretches Earth's ocean into two bulges, and Earth turns through them. That sentence is true, and it is also doing a great deal of quiet work. This layer opens it up.

By the end you should be able to answer five questions properly, not just repeat them:

1. What exactly is rising when the tide comes in?
2. Why does a pull towards the Moon produce a bulge away from the Moon?
3. Where does the strange figure of 24 h 50 min come from?
4. Why does the Sun, which is 27 million times more massive than the Moon, raise a tide less than half as big?
5. Why is the tidal range 1 metre at Kochi and 10 metres at Bhavnagar, when the Moon pulls on both almost exactly the same?

When the tide comes in, nothing is being poured into the sea. The total amount of water on Earth does not change at all between high and low tide. What changes is where the water is — it has slid sideways from one part of the ocean to another.

That matters for a picture you may be carrying in your head. High tide is not water travelling all the way from another country to your beach. Over each stretch of ocean the water moves only a modest distance, but because the ocean is so wide, a small sideways drift everywhere adds up to a big rise at the edge — at the coast, where the water runs out of room and has to go up.

That is also why tides are so much bigger at the coast than in the middle of the ocean. Out in the deep mid-Pacific, far from any land, the tide is only about half a metre. Ships there do not notice it at all. Bring that same tide to a coast and squeeze it into a bay and it becomes the wall of water that empties the Bay of Fundy.

Every tide idea in this lesson comes from a single fact about gravity: it gets weaker the further away you are, and it does so quickly.

Newton's rule is that the pull falls off with the square of the distance. Double the distance and the pull is not half as strong but a quarter. Triple it and the pull is a ninth. Ten times further, a hundredth.

Now put Earth in that picture. Earth is 12,742 km wide, and the Moon is about 384,400 km away, measured centre to centre. So the near side of Earth is about 6,371 km closer to the Moon than the centre is, and the far side is about 6,371 km further away. That is only about 1.7% of the distance either way — small, but not zero.

And because it is not zero, the Moon does not pull on all of Earth equally. It pulls the near side slightly harder than the middle, and the middle slightly harder than the far side. Every single thing about tides follows from that little inequality.

Now put the three pieces of Earth side by side — near ocean, solid planet, far ocean — and let the Moon pull on each one by the amount it deserves.

• Near ocean: pulled hardest, about 3.4% more than average.
• Solid Earth: pulled by the average amount.
• Far ocean: pulled least, about 3.2% less than average.

Everything is moving towards the Moon. Nothing is moving away from it. But they are not moving equally, and that is the same as being stretched.

To see it, stop watching from outside and stand on Earth instead. From here, the average pull that everything shares is invisible — you are moving with it, so it feels like nothing. What is left over is the difference:

• The near ocean seems to creep towards the Moon, away from the ground beneath it: it heaps up.
• The far ocean seems to creep away from the Moon, because the solid Earth is being dragged out from under it: it heaps up too.

Two heaps, at opposite ends of the line pointing at the Moon. That is the whole trick, and it works for any object stretched by any gravity anywhere in the universe.

The bulges are not water racing round the planet. They are more like two standing heaps, lined up with the Moon, while the planet rotates underneath.

Stand on a coast and let Earth carry you round once. You pass through the near bulge — high tide. Six hours later you are in the shallow band between the bulges — low tide. Six hours after that you are in the far bulge — a second high tide, usually about the same size. Six hours later, the other low. Then you are back where you started.

That gives the pattern most of the world's coasts know: two highs and two lows a day, called a semidiurnal tide. Kochi, Mumbai, Bhavnagar, Kolkata, Chennai and almost every Indian port work this way.

Notice how the picture flips your intuition. The water is not coming to you. You are being carried into the water, at more than 1,600 kilometres an hour if you live near the equator.

If Earth simply spun and the Moon stayed put, tides would repeat every 12 hours exactly and high tide would be at the same clock time every day. It is not, and the reason is that the Moon keeps moving.

In the time Earth takes one turn, the Moon has gone about 12.2 degrees further round its orbit as seen against the Sun. Your coast comes back to where it was, but the Moon has left; Earth has to turn that extra bit to bring you back under it.

There is a very clean way to count this. The Moon goes from one new moon to the next in 29.53 days. In that same stretch of time, how many times does your coast come back under the Moon? Exactly one time fewer than it comes back under the Sun — because the Moon has gone all the way round once in the meantime, in the same direction as Earth's spin.

So 29.53 solar days contain only 28.53 lunar days. One lunar day must therefore be 29.53 ÷ 28.53 = 1.0351 days, which is 24 h 50 min.

The Sun raises tides too, and comparing them teaches you something important about how tidal forces work.

Start with the raw numbers. The Sun is about 27 million times more massive than the Moon. It is also about 390 times further away. If tides depended on plain gravitational pull, the Sun would win easily: its total pull on Earth really is about 180 times the Moon's.

But tides do not care about the pull. They care about how much the pull changes across Earth's width — and from very far away, a pull barely changes at all over a mere 12,742 km. From the Sun's distance, the near side and the far side of Earth are almost identical places.

Work it out properly and the Moon's tide-raising effect comes to about 2.2 times the Sun's. In other words the Sun's tide is roughly 46% of the Moon's: not the main player, but far too big to ignore. That 46% is exactly what makes spring and neap tides.

Two tides are being raised at every moment: a big one by the Moon and a smaller one, about 46% as strong, by the Sun. Each of them has two bulges. What you actually get is the two patterns added together.

When Sun, Earth and Moon are in a straight line — at new moon and at full moon — the Sun's bulges sit exactly on top of the Moon's. The heaps add: higher high water, lower low water, the biggest range of the fortnight. This is a spring tide. It happens at both new and full moon, because each body makes a pair of bulges at opposite ends; what matters is the straight line, not which end the Sun is at.

When the Moon is at first or last quarter, the Sun is off at 90°. The Sun's high water lands where the Moon's low water is and partly fills it in. The range shrinks to its smallest: a neap tide.

Compare them with the simplest possible sum. If the Moon's tide is 2.18 units and the Sun's is 1 unit: spring range goes as 2.18 + 1 = 3.18, neap range as 2.18 − 1 = 1.18. So a spring range is roughly 2.7 times a neap range — and that is about what real ports record.

How often do spring tides come round? The Moon takes 29.53 days from one new moon to the next, and it lines up with the Sun twice in that time: once at new moon and once at full moon. So spring tides come every 29.53 ÷ 2 = 14.77 days — about a fortnight.

There is a second, lovelier way to get the same answer, and it is the one tide scientists actually use. The Moon's tide repeats every 12 h 25 min and the Sun's every 12 h exactly. Two rhythms that are nearly but not quite the same drift in and out of step, the way two clocks ticking at slightly different rates drift from agreeing to disagreeing and back.

Work out how long they take to go from in-step to in-step again and you get 14.77 days — the same fortnight, arrived at from the clock instead of from the sky.

The Moon pulls on Kochi and on Bhavnagar by the same amount, to within a hair. Yet Kochi's tide is about a metre and Bhavnagar's about ten. Three things about a place turn the same sky-driven nudge into wildly different tides.

1. Funnelling. When a tide runs into a bay that narrows, the same volume of water is squeezed into a smaller and smaller width. It has nowhere to go but up. The Gulf of Khambhat is a textbook funnel: broad at the mouth, narrow at Bhavnagar.

2. Shallowing. A tide entering shallower water slows down, and slowing water piles up behind itself, exactly as a traffic jam builds when cars slow. Wide, shallow shelves — like the one off Gujarat and the one at the head of the Bay of Bengal — grow tides.

3. Resonance. This is the big one, and the most beautiful. Every basin of water has a natural rhythm at which it likes to slosh, set by its length and depth. If that natural rhythm happens to be close to 12 h 25 min, each new tide arrives just as the basin is ready for a push, and the sloshing builds enormously — exactly like pushing a child on a swing at the right moment. The Bay of Fundy's natural rhythm is close to the tidal rhythm, which is why it has the biggest tide on Earth.

Turn all three off and you get an open, straight, deep coastline: Kerala, with its one-metre tide.

A tide table gives you, for one port, the time and height of every high and low water. Everything else is arithmetic.

Three questions a coastal worker asks it every day:

• When can I get out and back? Find the high waters and work backwards from the hours when there is enough depth.
• How much water will there be at 3 pm? The tide is not between highs at a steady rate, so there is a rule for this, coming up next.
• Is this a big tide or a small one? Compare today's range with the fortnight around it, or look at the moon symbol.

For the middle question, sailors use the rule of twelfths. Divide the six hours of the rise into six parts. The water does not come in one sixth each hour; it comes in 1, 2, 3, 3, 2, 1 twelfths, slow at first, fast in the middle two hours, slow at the end. The ebb does exactly the same in reverse.

It is an approximation, not a law — but it is close enough that it has been used at sea for well over a century.