The theory of tides is the application of continuum mechanics to interpret and predict the tidal deformations of planetary and satellite bodies and their atmospheres and oceans, under the gravitational loading of another astronomical body or bodies. It commonly refers to the fluid dynamic motions for the Earth's oceans.

The forces discussed here apply to body (Earth tides), oceanic and atmospheric tides. Atmospheric tides on Earth, however, tend to be dominated by forcing due to solar heating.

On the planet (or satellite) experiencing tidal motion consider a point at latitude φ and longitude λ at distance a from the center of mass, then point can written in cartesian coordinates as where

Let δ be the declination and α be the right ascension of the deforming body, the Moon for example, then the vector direction is

and r_{m} be the orbital distance between the center of masses and M_{m} the mass of the body. Then the force on the point is

where
For a circular orbit the angular momentum ω centripetal acceleration balances gravity at the planetary center of mass

where r_{cm} is the distance between the center of mass for the orbit and planet and M is the planetary mass.
Consider the point in the reference fixed without rotation, but translating at a fixed translation with respect to the center of mass of the planet. The body's centripetal force acts on the point so that the total force is

Substituting for center of mass acceleration,
and reordering

In ocean tidal forcing, the radial force is not significant, the next step is to rewrite the coefficient. Let then

where is the inner product determining the angle z of the deforming body or Moon from the zenith. This means that

if ε is small. If particle is on the surface of the planet then the local gravity is
and
set μ = M_{a} / M.

which is a small fraction of g. Note also that force is attractive toward the Moon when the z < π / 2 and repulsive when z > π / 2.

This can also be used to derive a tidal potential.

Laplace summarized the work to his time with a single set of linear partial differential equations simplified from the fluid dynamic equations, but can also be derived via Lagranges equation from energy integrals. It summarizes tidal flow as a barotropic two-dimensional sheet flow, where Coriolis effects are introduced a fictious lateral force.

Thomson rewote Laplace's momentum terms using the curl to find an equation for vorticity. Under certain conditions this can be further rewritten as a conservation of vorticity.

Tidal analysis and prediction

Harmonic analysis

There are about 62 constituents that could be used, but many less are needed to predict tides accurately.

Tidal constituents

Higher harmonics

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude (cm)

NOAA

Species

Symbol

(hr)

rate(°/hr)

n_{1} (L)

n_{2} (m)

n_{3} (y)

n_{4} (mp)

number

ME

MS

PR

AK

CA

HI

order

Shallow water overtides of principal lunar

M_{4}

6.210300601

57.9682084

4

455.555

6.0

0.6

0.9

2.3

5

Shallow water overtides of principal lunar

M_{6}

4.140200401

86.9523127

6

655.555

5.1

0.1

1.0

7

Shallow water terdiurnal

MK_{3}

8.177140247

44.0251729

3

1

365.555

0.5

1.9

8

Shallow water overtides of principal solar

S_{4}

6

60

4

4

-4

491.555

0.1

9

Shallow water quarter diurnal

MN_{4}

6.269173724

57.4238337

4

-1

1

445.655

2.3

0.3

0.9

10

Shallow water overtides of principal solar

S_{6}

4

90

6

6

-6

*

0.1

12

Lunar terdiurnal

M_{3}

8.280400802

43.4761563

3

355.555

0.5

32

Shallow water terdiurnal

2"MK_{3}

8.38630265

42.9271398

3

-1

345.555

0.5

0.5

1.4

34

Shallow water eighth diurnal

M_{8}

3.105150301

115.9364166

8

855.555

0.5

0.1

36

Shallow water quarter diurnal

MS_{4}

6.103339275

58.9841042

4

2

-2

473.555

1.8

0.6

1.0

37

Semi-diurnal

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude (cm)

NOAA

Species

Symbol

(hr)

(°/hr)

n_{1} (L)

n_{2} (m)

n_{3} (y)

n_{4} (mp)

number

ME

MS

PR

AK

CA

HI

order

Principal lunar semidiurnal

M_{2}

12.4206012

28.9841042

2

255.555

268.7

3.9

15.9

97.3

58.0

23.0

1

Principal solar semidiurnal

S_{2}

12

30

2

2

-2

273.555

42.0

3.3

2.1

32.5

13.7

9.2

2

Larger lunar elliptic semidiurnal

N_{2}

12.65834751

28.4397295

2

-1

1

245.655

54.3

1.1

3.7

20.1

12.3

4.4

3

Larger lunar evectional

ν_{2}

12.62600509

28.5125831

2

-1

2

-1

247.455

12.6

0.2

0.8

3.9

2.6

0.9

11

Variational

MU_{2}

12.8717576

27.9682084

2

-2

2

237.555

2.0

0.1

0.5

2.2

0.7

0.8

13

Lunar elliptical semidiurnal second-order

2"N_{2}

12.90537297

27.8953548

2

-2

2

235.755

6.5

0.1

0.5

2.4

1.4

0.6

14

Smaller lunar evectional

λ_{2}

12.22177348

29.4556253

2

1

-2

1

263.655

5.3

0.1

0.7

0.6

0.2

16

Larger solar elliptic

T_{2}

12.01644934

29.9589333

2

2

-3

272.555

3.7

0.2

0.1

1.9

0.9

0.6

27

Smaller solar elliptic

R_{2}

11.98359564

30.0410667

2

2

-1

274.555

0.9

0.2

0.1

0.1

28

Shallow water semidiurnal

2SM_{2}

11.60695157

31.0158958

2

4

-4

291.555

0.5

31

Smaller lunar elliptic semidiurnal

L_{2}

12.19162085

29.5284789

2

1

-1

265.455

13.5

0.1

0.5

2.4

1.6

0.5

33

Lunisolar semidiurnal

K_{2}

11.96723606

30.0821373

2

2

275.555

11.6

0.9

0.6

9.0

4.0

2.8

35

Diurnal

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude (cm)

NOAA

Species

Symbol

(hr)

(°/hr)

n_{1} (L)

n_{2} (m)

n_{3} (y)

n_{4} (mp)

number

ME

MS

PR

AK

CA

HI

order

Lunar diurnal

K_{1}

23.93447213

15.0410686

1

1

165.555

15.6

16.2

9.0

39.8

36.8

16.7

'4

Lunar diurnal

O_{1}

25.81933871

13.9430356

1

-1

145.555

11.9

16.9

7.7

25.9

23.0

9.2

6

Lunar diurnal

OO_{1}

22.30608083

16.1391017

1

3

185.555

0.5

0.7

0.4

1.2

1.1

0.7

15

Solar diurnal

S_{1}

24

15

1

1

-1

164.555

1.0

0.5

1.2

0.7

0.3

17

Smaller lunar elliptic diurnal

M_{1}

24.84120241

14.4920521

1

155.555

0.6

1.2

0.5

1.4

1.1

0.5

18

Smaller lunar elliptic diurnal

J_{1}

23.09848146

15.5854433

1

2

-1

175.455

0.9

1.3

0.6

2.3

1.9

1.1

19

Larger lunar evectional diurnal

ρ

26.72305326

13.4715145

1

-2

2

-1

137.455

0.3

0.6

0.3

0.9

0.9

0.3

25

Larger lunar elliptic diurnal

Q_{1}

26.868350

13.3986609

1

-2

1

135.655

2.0

3.3

1.4

4.7

4.0

1.6

26

Larger elliptic diurnal

2Q_{1}

28.00621204

12.8542862

1

-3

2

125.755

0.3

0.4

0.2

0.7

0.4

0.2

29

Solar diurnal

P_{1}

24.06588766

14.9589314

1

1

-2

163.555

5.2

5.4

2.9

12.6

11.6

5.1

30

Long period

Darwin

Period

Phase

Doodson coefs

Doodson

Amplitude (cm)

NOAA

Species

Symbol

(hr)

(°/hr)

n_{1} (L)

n_{2} (m)

n_{3} (y)

n_{4} (mp)

number

ME

MS

PR

AK

CA

HI

order

Lunar monthly

M_{m}

661.3111655

0.5443747

0

1

-1

65.455

0.7

1.9

20

Solar semiannual

S_{sa}

4383.076325

0.0821373

0

2

57.555

1.6

2.1

1.5

3.9

21

Solar annual

S_{a}

8766.15265

0.0410686

0

1

56.555

5.5

7.8

3.8

4.3

22

Lunisolar synodic fortnightly

M_{sf}

354.3670666

1.0158958

0

2

-2

73.555

1.5

23

Lunisolar fortnightly

M_{f}

327.8599387

1.0980331

0

2

75.555

1.4

2.0

0.7

24

Example amplitudes from Eastport, ME; Biloxi, MS; San Juan, PR; Kodiak, AK; San Francisco, CA; and Hilo HI;.