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Tides

Why the water level rises and falls

You're not standing where the tide is. You're standing at a point the tide passes through. The Earth spins while its envelope of water stays turned towards the Moon — here the Earth has been stopped and set orbiting the Moon instead, so you can see what passes overhead in the course of a day.

View from the North Pole · observer fixed at the top · lunar day 24 h 50 min

Earth, the orbiting Moon and the tidal bulges Earth at the centre with its envelope of water stretched into two bulges. The Moon orbits along a clock-face circle, and the observer is fixed at the top of the Earth. Sun Moon port
water envelope observer and tide gauge direction of the Sun
level above mean
+0,00m
spring tide
00:00 · range 0,00 m
0,0 h
new moon
384 400
45°
1,0×

Tide gauge over 24 hours

Curve of water level over 24 hours with two peaks and two troughs 00 06 12 18 24

Two bulges, not one

The tide isn't caused by the Moon's gravity itself, but by its difference across the Earth. The near side is pulled more strongly than the Earth's centre and bulges out towards the Moon. The far side is pulled more weakly than the centre, so the centre slips out from under it — and the water there lags behind. The result is two bulges on opposite sides.

So in a single rotation you pass through both: two high tides and two low tides per lunar day, roughly every 12 hours 25 minutes. Move the time slider in the model and watch the tide gauge — a peak comes both when the Moon is directly overhead and when it's exactly on the opposite side.

Difference in pull across the Earth stronger on the right, weaker on the left

What changes the height

Phase of the Moon
At new moon and full moon, the Moon's and Sun's bulges line up and add together — a spring tide. At the quarters they pull at right angles and partly cancel — a neap tide. The range varies roughly twofold between the two.
Distance
Tidal force falls off with the cube of distance. That's why the Moon loses to the Sun on mass but wins on effect, by a ratio of roughly 2,2 : 1. The difference between perigee and apogee is about 20 %.
Latitude
The semidiurnal component weakens with cos²φ. It's strongest at the equator, and little remains near the poles. The Moon's declination adds a diurnal inequality on top of that — the two daily peaks tend to differ in height.
Shape of the coastline
The strongest factor in practice. The open ocean has around 0,5 m. A narrowing bay with a natural period close to 12,4 h can amplify the wave as much as tenfold. Moving the last slider shows that astronomy sets the rhythm and the coastline sets the amplitude.

The same Moon, a fiftyfold difference

Both places receive almost the same forcing from the Atlantic — a swing of around one metre at the entrance. The difference isn't in the astronomy, but in what the shape of the basin does with that forcing. The charts above share a common scale in metres.

00:00

Bay of Fundy

16,3 m · resonance The Bay of Fundy with a resonantly amplified wave A chart of water level with a range of over 16 metres, and a cross-section of the bay in which the wave grows towards its head. +8 0 −8 mouth, 90 km head, 10 km

The natural period of the Gulf of Maine and Bay of Fundy system is about 13 hours, almost exactly the period of M2. The basin responds like a tuned string: a node at the mouth, an antinode at the head. Resonance accounts for about a fivefold gain, and the narrowing funnel and shoaling account for the rest — the same flow of energy has to pass through a smaller cross-section.

Mediterranean

0,3 m · closed basin The Mediterranean beyond the narrow Strait of Gibraltar A chart of water level that's almost flat at zero, and a cross-section of the basin that the narrow strait can't push enough water into in time. +8 0 −8 the same curve magnified 20× Atlantic Gibraltar 2,5 million km²

A strait 14 km wide with a sill 300 m deep acts as a throttle valve: in six hours not enough water can flow through it for a basin covering 2,5 million km² to rise noticeably. Nor is there anything to amplify the wave, because the natural periods of the main basins don't sit at 12,42 h.

Why a few hours matter

Curve of a basin's resonant response Amplification of the tidal wave as a function of the basin's natural period, peaking at a period of 12,42 hours. basin's natural period (h) amplification

A basin behaves like a resonator. When its natural period approaches 12,42 h, the response spikes; a few hours either side and you get an ordinary coastline. The Bay of Fundy sits almost at the peak. So does the northern Adriatic — which is why Venice gets around a metre while the rest of the Mediterranean gets centimetres.

Eight ports, eight rhythms

So far we've had one model and one curve. Real coastlines don't just differ in height — they also differ in how many tides they get per day and in the shape of the curve. Below are curves built up from harmonic constituents, the way real tide tables do it. Each thumbnail has its own scale so the shape is visible; the actual range is shown next to the name.

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Water level over seven days at the selected port
Amplitudes of the four main constituents

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The model computes the equilibrium theory of the tide. Real oceans behave as a system of amphidromic systems rotating around nodal points under the Coriolis force, and the peak lags the Moon's culmination by anywhere from hours to half a day. The declination of the Moon and Sun isn't included, so here both daily peaks come out the same height.