Moon and tides

What it is

The tide is the slow rise and fall of the sea, twice a day on most coasts, driven by the moon and the sun. It is the one part of the water level that can be predicted years ahead, to the minute, at any gauge that has been measured long enough.

The moon and the sun do not simply pull the ocean toward them. They pull on the whole earth, and they pull harder on the side nearer to them than on the far side. The difference between the pull at a point on the earth and the pull at its centre is the tidal force. It stretches the ocean along the line to the moon, toward the moon on the near side and away from it on the far side. That is why there are two tides a day and not one.

What it is not

It is not the storm surge, and it is not a coastal flood. The tide is the base the weather builds on: a coastal flood is the surge added to the tide, and it comes at high tide, Coastal flood FB-EVT-105.

The two bulges of Fig. A are not the real ocean. They are the equilibrium tide, the shape the ocean would take if it could follow the moon instantly across a planet covered in water. The real ocean cannot: the continents block it and the tide wave cannot travel fast enough. So the tide moves around each ocean basin as rotating waves, circling points where the range is near zero, and each coast has its own timing and its own range. The equilibrium cartoon explains the rhythm; the gauge measures the tide.

Lookalikes

Storm surge: The part of the water level the weather adds on top of the tide. The gauge shows it as observed less predicted. That is Storm tide, storm surge and wave runup FB-EVT-112.
Seiche: A basin sloshing at its own period after a storm or a pressure jump, in a lake or a bay. That is Seiche FB-EVT-110.
Tidal wave: An old name for a tsunami, which has nothing to do with the tide. The Book does not use it.

The machine

The tidal force from a body grows with its mass and falls with the cube of its distance, as the equation below says. The sun is 27 million times heavier than the moon but 389 times farther away, and the cube wins: the sun's tidal force is 0.46 of the moon's.

The daily rhythm. The earth turns under the moon's pull once every lunar day, 24 h 50 min, because the moon moves along its orbit as the earth turns. On a coast with a semidiurnal tide there are two highs and two lows in that time, one every 12 h 25 min: the main lunar constituent, M2.

The fortnightly rhythm. Near new and full moon the sun and the moon are in line and their tidal forces add: the range is largest, the spring tides, which come a day or two after the phase. At the first and last quarters they pull at right angles and partly cancel: the neap tides. From spring to spring is 14.77 days, half the month of phases. Fig. B draws a month of it at one gauge.

The distance. The moon's orbit is not a circle. At its closest, its tidal force is about a quarter stronger than at its mean distance. A spring tide that falls near perigee is a perigean spring tide, the king tide, and it is the highest of the year on many coasts even in calm weather.

The declination. The moon moves north and south of the equator each month. When it is far from the equator the two daily tides differ in height: the diurnal inequality. Where the ocean's response favours the daily constituents over the twice daily ones, a coast has one tide a day. The form number below sorts the kinds. Fig. C draws them: most of the Atlantic coast is semidiurnal, the Pacific coast mixed, and parts of the Gulf coast diurnal.

A chart of thirty days of tide at a semidiurnal gauge: two highs and two lows every day, the range swelling to spring tides a day or two after new and full moon and shrinking to neap tides after each quarter, with the four phases of the moon marked above on their days, heights in feet with metres beside them. — **Fig. B** A month at a gauge. Thirty days of predicted tide, two a day, swelling to spring tides after new and full moon.Maximize The drawing, to scale

Three painted shores, each with a tide staff, over a two day tide curve: a semidiurnal tide of the Atlantic coast with two nearly equal highs a day, form number 0.15; a mixed tide of the Pacific coast with two unequal highs a day, form number 1.0; and a diurnal tide of parts of the Gulf of America coast with one high and one low a day, form number 4; the form number written below each as the sum of K1 and O1 over the sum of M2 and S2. — **Fig. C** Three kinds of tide. Semidiurnal, mixed and diurnal, each curve computed over two days from its constituents.Maximize The drawing, to scale

The drawings

Each figure drawn as an engineering sheet, its parts numbered, to print at 11 by 17.

Fig. A, the drawing Spring and neap, the equilibrium cartoon. The earth, moon and sun seen from above the North Pole; the equilibrium cartoon, not the dynamical ocean NOT TO SCALEMaximize The sheet, SVG, 11 by 17 The painting

The earth, and its bulges
The moon, at new moon
The moon, at full moon
The moon, at a quarter
The sun, far out of the drawing
Spring tide, the bulges added
Neap tide, the bulges opposed

Fig. B, the drawing A month at a gauge. The predicted tide at a semidiurnal gauge over 30 days, from its four main constituents TIME TO SCALE, HEIGHT TO SCALEMaximize The sheet, SVG, 11 by 17 The painting

The tide, twice a day
Spring tides
Neap tides
The phases of the moon

Fig. C, the drawing Three kinds of tide. Two days of tide at three kinds of coast, each computed from its constituents, with its form number TIME TO SCALE, HEIGHT NTSMaximize The sheet, SVG, 11 by 17 The painting

Semidiurnal
Mixed
Diurnal

Ingredients

The moon, whose tidal force on the earth is about twice the sun's
The sun, adding to the moon near new and full moon, and working against it at the quarters
The ocean basins, which turn the forces into waves that rotate around points of no tide and are amplified on wide shelves and in long bays

Scales

time: 12 h 25 min for the main lunar tide; 14.8 days from one spring tide to the next
horizontal: ocean basins, thousands of kilometres; each coast has its own
vertical: under 3 ft (1 m) of range on some coasts; about 30 ft (9 m) at Anchorage
orlanski: planetary

Equations

The tidal acceleration

a_t \approx \frac{2\,G M r}{d^3}

$a_t$: the difference between the pull on the near side of the earth and the pull at its centre, m s⁻²
$G$: the gravitational constant, 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻²
$M$: the mass of the moon (7.35 × 10²² kg) or the sun (1.99 × 10³⁰ kg)
$r$: the radius of the earth, 6.371 × 10⁶ m
$d$: the distance from the earth to the moon or the sun

Assumes The distance is much greater than the earth's radius, so only the first term of the difference matters. It is the force; the height of the tide on a coast depends on the ocean's response to it.

Working form For the moon, about 1.1 × 10⁻⁶ m s⁻², a nine millionth of gravity. The sun's is 0.46 of the moon's: its mass is 27 million times greater, but it is 389 times farther away, and the force falls with the cube of the distance.

The form number, which sorts a coast's tide

F = \frac{K_1 + O_1}{M_2 + S_2}

$K_1, O_1$: the amplitudes of the two main daily constituents at the gauge
$M_2, S_2$: the amplitudes of the main lunar and solar twice daily constituents

Assumes Amplitudes from the gauge's harmonic analysis. Under 0.25 the tide is semidiurnal; 0.25 to 1.5 mixed, mainly semidiurnal; 1.5 to 3 mixed, mainly diurnal; over 3 diurnal.

Signatures

gauge: two highs and two lows a day on a semidiurnal coast; one of each on a diurnal coast; the range swelling toward spring and shrinking toward neap every two weeks
surface: [object Object]

The numbers

Quantity	Value, and the kind of number it is
Main lunar constituent M2	12.42 hoursStandard, Pugh 2014
Main solar constituent S2	12.00 hoursStandard, Pugh 2014
Daily constituents K1 and O1	23.93 and 25.82 hoursStandard, Pugh 2014
Spring to spring	14.77 days, half the 29.53 day cycle of the moon's phasesStandard, Pugh 2014
Sun's tidal force against the moon's	0.46Textbook, Pugh 2014
Moon's distance	About 221,500 mi (356,500 km) at a close perigee to 252,700 mi (406,700 km) at a far apogee; the tidal force about a quarter stronger at a close perigee than at the mean distanceTextbook, Pugh 2014
Form number	Under 0.25 semidiurnal, as on most of the Atlantic coast; 0.25 to 3 mixed, as on the Pacific coast; over 3 diurnal, as on parts of the Gulf coastTextbook, Pugh 2014
Largest range in the United States	Cook Inlet, Alaska: about 30 ft (9 m) at AnchorageTypical

How the station sees it

Tide gauges in the instrument atlas measure the water level every 6 minutes. Beside each measurement is the tide predicted for that minute, computed from dozens of constituents found in years of that gauge's record. The two lines normally lie on each other within centimetres. When they part, the difference is the weather: the wind and the pressure of a storm raising or lowering the sea. That difference, observed less predicted, is how the surge is measured.

Tide gauges: the water level every 6 minutes, against the tide predicted for that minute

How it is warned

The tide is not warned; it is predicted, and published a year ahead for every gauge. The warnings it enters are the Coastal Flood Watch, Warning, Advisory and Statement, whose text names the times of high tide because that is when the water will be highest. A perigean spring tide is a common reason for an advisory on a calm day.

Sources

American Meteorological Society. Glossary of Meteorology.
Pugh, D. and P. Woodworth. Sea-Level Science (2014).
NOAA National Ocean Service. Tides and Water Levels, an education tutorial.
NOAA National Ocean Service. Tidal Datums.

Definition after the Glossary of Meteorology. Plate FB-CLK-003, revision 1, 2026-09-24. The number is permanent; cite it.