The Moon pulls on all of Earth, but it pulls harder on the side facing it and more weakly on the far side. Tides come from that difference in pull. It stretches the oceans into two bulges, it falls off as the cube of distance, and once the Sun adds its own stretch the result is a sum of a few cosines that you can predict years ahead.
Earth falls freely toward the Moon, so what matters at any point is how much the Moon's pull there differs from its pull at Earth's center. Subtract the two and the arrows point away from Earth on the near side and on the far side, and inward around the middle. Water collects where the arrows point out: two bulges.
How to read this: Earth seen from above the north pole, with the Moon off to the side. The arrows show how the Moon's pull differs from place to place, and longer means stronger. The blue outline is the ocean, stretched into two bulges (hugely exaggerated).
The small chart in the corner compares how fast two things weaken as the Moon moves away, both measured against the Moon's average distance (1 = today's average). "∝" means "grows along with", so the tide weakens like 1/d³ while the plain pull weakens more slowly, like 1/d².
Try this: click Raw pull to see how nearly equal the pulls are, then Difference again.
With G = 6.674×10⁻¹¹, M = 7.342×10²² kg, d = 3.844×10⁸ m and r = 6.371×10⁶ m, that is about 1.1×10⁻⁶ m/s², roughly a nine-millionth of g. The pull itself goes as 1/d². Its change across Earth's width goes as one more power of d, so 1/d³. The small chart shows both.
On the far side the Moon still pulls toward itself, but less than it pulls Earth's center. Earth's center is pulled away from that water, so relative to Earth the water is left behind. Switch to raw pull to see how nearly equal the arrows are: tides come from a 3% spread in them.
The equilibrium bulge here, about on the Moon's line, assumes an ocean covering the whole planet that adjusts instantly. Real oceans don't, which Lab 3 comes back to.
The Sun pulls on Earth 179 times harder than the Moon, yet its tide is less than half the Moon's. Tides go as M/d³, and the Sun is 390 times farther away. Both bodies stretch the ocean into an oval, and the two ovals add. Lined up at new and full moon they make spring tides. At right angles, at the quarter moons, they partly cancel into neap tides.
How to read this: the green dot is a beach on the equator, carried around once a day as Earth spins; the length of its stick is the water height there. When it passes through a bulge of the blue oval, it has high tide.
Try this: press Play and watch the gold and amber ovals line up (spring tides) and cross (neap tides).
How to read this: a month of water level at the green beach above. Each wiggle is one tide. The gold dashed outline shows the tides growing (spring) and shrinking (neap). Drag across the chart to move through the month.
The Sun's bulge returns every 12 hours. The Moon moves along its orbit while Earth turns, so a point needs 24 h 50 min to face the Moon again and the lunar tide repeats every 12.42 h. That lunar constituent is called M2 and the solar one S2.
Two waves with close periods beat. They fall back into step after 1/(1/12 − 1/12.4206) = 354.4 h, or 14.77 days, which is half a lunar month. The gold dashed curve is that envelope.
Tide strength scales as M/d³. Sun over Moon: (1.989×10³⁰ / 7.342×10²²) × (3.844×10⁸ / 1.496×10¹¹)³ = 0.46. The amplitudes here are the equilibrium tide on the equator with both bodies over the equator: for M2 and for S2. On real coasts the observed spring tides often arrive a day or two after new or full moon.
Real tides are shaped by ocean basins. Water can't keep up with the bulges, it sloshes at the basin's own resonant periods, and in many basins the tide rotates around amphidromic points where the range is nearly zero. The periods still come from the sky, though. Measure each constituent's amplitude and phase at a port for a year, then add the cosines forward in time. The values below are illustrative examples, not a real station.
How to read this: the bright green line is the predicted water level, built by adding the four faint waves. Dots mark each high tide: gold if it's higher than the one before, amber if lower. Each slider pair sets one wave's size and its "phase", which shifts that wave earlier or later.
Try this: click Semidiurnal, Mixed, then Diurnal, and count high tides per day.
M2 = 12.4206 h and S2 = 12.0000 h are the twice-daily terms from Lab 2. K1 = 23.9345 h and O1 = 25.8193 h are once-daily terms. They exist because the Moon and Sun are usually north or south of the equator, which tilts the bulges so one high tide of the day is bigger than the other.
F = (K1 + O1)/(M2 + S2) sorts tides: below 0.25 semidiurnal, 0.25 to 1.5 mixed but mainly semidiurnal, 1.5 to 3 mixed but mainly diurnal, above 3 diurnal.
In the 1870s William Thomson, later Lord Kelvin, built a tide-predicting machine. Each constituent turned a pulley on a crank at its own rate, one cord ran over all the pulleys, and the cord's end traced the sum. Machines of that kind predicted tides until digital computers took over. Real stations use dozens of constituents, and the result still describes only the astronomical tide. Wind and pressure add surge on top, as in W00 · Hurricanes.