How do tides work? Spring and neap tides simulator • The Science 360

By The Science 360 · Updated October 2026 · 8 min read

How do tides work, and why are there two high tides a day?

Tides are the ocean's response to the Moon's gravity changing from one side of the Earth to the other. The Moon pulls the water facing it harder than it pulls the Earth's centre, and pulls the water on the far side less. Seen from the Earth, that difference stretches the oceans into two bulges, one under the Moon and one on the opposite side. The Earth turns once under this pattern, so a coast passes through two highs and two lows every lunar day of 24 h 50 min, which is why high tides come about every 12 h 25 min.

The Sun raises tides too, but only 46% as strong as the Moon's. When the Sun and Moon line up (new moon and full moon) the two tides add and we get spring tides. A week later, at the quarter moons, they pull at right angles and partly cancel, giving neap tides. In this model the tide range for an ideal ocean runs from about 0.29 m at neap to 0.78 m at spring. Real coasts can be far larger or smaller, see below. Move the day slider above and watch the bulges, the curve and the monthly chart change together.

Key results from the simulation

Differential gravity: the stretching acceleration at the near and far points is 2GMR ÷ d³. For the Moon that is 1.10 millionths of a metre per second squared, about a ten-millionth of the gravity you feel standing on the ground.
The Sun's tide is 46% of the Moon's: the Sun pulls on the Earth about 179 times harder than the Moon does, but it is 389 times farther away, and the stretch falls with the cube of distance.
Spring and neap: the combined range swings between about 0.29 m and 0.78 m, a ratio of roughly 2.7 to 1, twice a month.
Two highs a day: the lunar tide repeats every 12 h 25 min, because the Moon moves on in its orbit while the Earth turns, so a place needs 24 h 50 min to come back under it.

Six days of the lunar month

Each section gives the numbers the simulation shows for that day, explains the picture, and has a button that sets the day slider above. The tide range is twice the height of the combined wave, the sum of a Moon wave and a Sun wave, which is the same sum that draws the curve.

New moon: the Moon between Earth and Sun

Spring tide, range 0.78 m

At new moon the Moon sits on the Sun side of the Earth, so both bulges line up along one axis. The Moon's wave (0.54 m from low to high) and the Sun's wave (0.25 m) peak at the same moment and add to about 0.78 m.

This is a spring tide. The name has nothing to do with the season: the water "springs up". High tide arrives near noon and midnight at this phase in the simulation because the bulge points at the Sun. Real coasts run a day or two behind, see "what the model leaves out".

Moon age 0 days Moon wave 0.54 m Sun wave 0.25 m Range 0.78 m

Waxing crescent: the Moon 45° from the Sun

Between spring and neap, range 0.59 m

Three and a half days after new moon the Moon has moved 45° around the sky from the Sun. The two bulges no longer share an axis, so the waves are out of step by 90° of tidal phase and only partly add. The range falls to about 0.59 m.

Look at the dashed lines in the curve. The Moon's wave and the Sun's wave drift apart by about 12.2° of Moon angle every day, which is why the combined curve slowly changes shape from one day to the next.

Moon age 3.7 days Angle to Sun 45° Alignment 71% Range 0.59 m

First quarter: Sun and Moon at right angles

Neap tide, range 0.29 m

A week after new moon the Moon is 90° from the Sun. The Moon's bulge now sits where the Sun's tide would be at its lowest, so the Sun's trough partly fills the Moon's high and the Sun's high partly fills the Moon's low. The waves subtract: 0.54 m minus 0.25 m leaves 0.29 m.

This is a neap tide, the weakest of the month. Boats at a harbour with a big tide have only about 37% of the spring range to work with. The ratio of spring to neap, 2.7 to 1, comes straight from the 46% figure: (1 + 0.46) ÷ (1 − 0.46).

Moon age 7.4 days Angle to Sun 90° Moon wave 0.54 m Range 0.29 m

Waxing gibbous: the tides grow again

Between neap and spring, range 0.59 m

At 11.1 days the Moon is 135° from the Sun, which is 45° short of opposition. The alignment is as good as at the waxing crescent, just from the other side, so the range is again about 0.59 m.

The monthly chart at the bottom of the stage shows the whole pattern: two peaks (spring) and two dips (neap) in a 29.53-day month, because the tide cares about the Moon's line, not which end of the line it is on. New moon and full moon give the same tide.

Moon age 11.1 days Angle to Sun 135° Alignment 71% Range 0.59 m

Full moon: Sun and Moon on opposite sides

Spring tide, range 0.78 m

At full moon the Earth is between the Sun and the Moon. You might expect the opposite pulls to cancel, but tides stretch rather than push one way: both bodies raise a bulge on the side facing them and one on the far side, so the bulges again share an axis. The range is 0.78 m, the same as at new moon.

This is why very high tides cluster around both new and full moon. The full moon is also easy to see, so people notice the link with it more, but the new moon has the same effect.

Moon age 14.8 days Angle to Sun 180° Sun share 31% Range 0.78 m

Last quarter: the second neap

Neap tide, range 0.29 m

Three weeks into the month the Moon is again 90° from the Sun, on the other side, and the waves cancel to about 0.29 m. Press "Moon only" to remove the Sun and the range jumps to 0.54 m: the Sun's wave is what is subtracting.

Press "Sun only" and the range is 0.25 m with highs exactly 12 hours apart. The 12 h 25 min rhythm of the real tides belongs to the Moon alone. For more on how the Moon moves, see Kepler's laws simulator.

Moon age 22.1 days Moon only 0.54 m Sun only 0.25 m Range 0.29 m

Tide range across the lunar month

Moon age	Phase	Angle to Sun	Moon wave	Sun wave	Tide range
0 days	New moon	0°	0.54 m	0.25 m	0.78 m
3.7 days	Waxing crescent	45°	0.54 m	0.25 m	0.59 m
7.4 days	First quarter	90°	0.54 m	0.25 m	0.29 m
11.1 days	Waxing gibbous	135°	0.54 m	0.25 m	0.59 m
14.8 days	Full moon	180°	0.54 m	0.25 m	0.78 m
22.1 days	Last quarter	270°	0.54 m	0.25 m	0.29 m

Values are for an ideal ocean that covers the whole Earth and can respond instantly (the equilibrium tide), at the mean Moon and Sun distances, with the Moon in the equator plane. Open-ocean tides are about this size; coasts differ a lot. See Top 20 interesting facts about our solar system for more about our neighbourhood.

How the simulation works

A body of mass M at distance d pulls with acceleration GM ÷ d² at the Earth's centre, a little more at the near side and a little less at the far side. In the frame that falls freely with the Earth, only the difference is left. At the points on the line to the body it points outward, and its size is a = 2GMR ÷ d³, where R is the Earth's radius. Across the line, at 90°, it points inward with half that size. For the Moon, a = 1.10 × 10⁻⁶ m/s². For the Sun it is 0.51 × 10⁻⁶ m/s². The Sun's ordinary pull is about 179 times the Moon's, but the cube in the distance reverses the ranking for tides.

An ocean in equilibrium rises until its own gravity balances this stretching. The surface height at angle θ from the body is ζ = (GMR² ÷ g d³) × (3cos²θ − 1) ÷ 2. For a point on the equator, with θ changing as the Earth turns, that becomes a wave of amplitude A = ¾ GMR² ÷ (g d³) around the mean sea level: 0.268 m for the Moon and 0.123 m for the Sun. It goes through two cycles per turn of the Earth, because cos(2θ) repeats every 180°.

The tide at the coast is the sum h(t) = A_moon cos 2(θ − ε) + A_sun cos 2θ. θ is the angle of the coast from the Sun, growing 15° per hour, and ε is the angle of the Moon from the Sun, growing 12.19° per day. The difference between the two rates makes the lunar tide repeat every 12 h 25 min. The Sun's wave repeats every 12 hours, so the spacing of highs in the combined curve drifts between about 12 h 15 min and 12 h 50 min over the month, and averages out to the Moon's 12 h 25 min. The slider sets the day of the month, which sets ε. The drawing, the 2-day curve, the monthly chart and every number in the panel are all computed from this one function. Time-lapse is marked on screen: the Earth turns 3 hours per second, and "Run the month" covers about 2 days per second.

What the model leaves out

Real oceans are not an even shell. Continents block the water, and each basin has its own natural sloshing period. Tides in the Bay of Fundy reach about 16 m because the bay resonates near the tidal period, and the Mediterranean has tides of only a few tens of centimetres.
Delay. The ocean takes time to respond, so real spring tides arrive about one to two days after new and full moon, and high tide is not exactly when the Moon is overhead.
Changing distance. The Moon's orbit is slightly elliptical. Its distance changes between about 356,500 and 406,700 km and the tide scales as 1 ÷ d³, so tides near the closest point are up to about 1.5 times as strong as near the farthest point. The simulation uses the mean distance.
Latitude and tilt. The Moon and Sun are not always above the equator, so many places have a bigger tide on one of the two daily cycles. The model puts the coast on the equator with both bodies in the equator plane.
Weather and the solid Earth. Wind, pressure and storm surge add or subtract height. The land also rises and falls by a few tens of centimetres in the Moon's pull, which is not shown.

Common misconceptions

"The tide is caused by the Moon pulling the water towards it, so there should be one bulge." A single pull would give one bulge. The second bulge appears because the Earth itself is pulled towards the Moon more than the far water is, leaving that water behind. The key is the difference in pull across the planet.

"The Sun's tide is negligible." It is 46% of the Moon's, which is large enough to change the range by a factor of 2.7 between spring and neap. The Earth and Moon also orbit a common centre about 4,671 km from the Earth's centre, inside the planet, but that is not what raises the far-side bulge. Tides come from the difference in pull.

Frequently asked questions

Why are there two high tides a day?

Because the oceans have two bulges, one facing the Moon and one on the far side. The Earth turns under them, so a coast crosses a bulge about every 12 h 25 min. The simulation shows this as two peaks in each 24-hour stretch of the curve.

What is the difference between a spring tide and a neap tide?

A spring tide happens at new and full moon, when the Sun and Moon line up and their tides add, giving the biggest range (0.78 m in the model). A neap tide happens at the quarter moons, when they are at right angles and partly cancel, giving the smallest range (0.29 m). Both happen twice a month.

Why is the Sun's tide smaller than the Moon's if the Sun is so much heavier?

Tides depend on how much gravity changes across the Earth, which falls with the cube of distance, not the square. The Sun is 389 times farther away than the Moon and about 27 million times more massive, which gives a ratio of 0.46.

Why is high tide not at the same time every day?

The Moon moves along its orbit while the Earth turns, so a place needs 24 h 50 min to come back under it. High tides therefore come about 50 minutes later each day. See Top 20 interesting facts about our solar system for more on the Earth-Moon pair.

Do tides happen in lakes and small seas?

They do, but they are tiny. The stretching acts on any body of water, but it depends on the size of the basin, so the Great Lakes and the Mediterranean have tides of only a few centimetres to a few tens of centimetres.

Do tides slow the Earth down?

Yes, slightly. The tidal bulge is pulled a little ahead of the Earth-Moon line by the planet's spin, and the Moon's pull on it acts as a brake. Laser ranging shows the Moon moving away by about 3.8 cm a year and the length of the day growing by a couple of milliseconds per century. For more extreme worlds, see 10 cosmic objects that will amaze you.