"The greater circle represents the orbit of the Earth around the Sun...while the smaller one is the rotating Earth itself" (Palmeri 1998, p. 229).
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 (especially Earth's Ocean) under the gravitational loading of another astronomical body or bodies (especially the Moon).
Contents

Kepler 1

Galileo's attempt to explain the tides 2

Newton 3

Laplace's dynamic theory 4

Laplace's tidal equations 5

Tidal analysis and prediction 6

Harmonic analysis 6.1

Tidal constituents 6.2

Higher harmonics 6.2.1

Semidiurnal 6.2.2

Diurnal 6.2.3

Long period 6.2.4

See also 7

References and notes 8

External links 9
Kepler
In 1609 Johannes Kepler correctly suggested that the gravitation of the Moon causes the tides,^{[1]} basing his argument upon ancient observations and correlations. It was originally mentioned in Ptolemy's Tetrabiblos as having derived from ancient observation.
Galileo's attempt to explain the tides
Justus Sustermans  Portrait of Galileo Galilei, 1636
In 1616, Galileo Galilei wrote Discourse on the Tides (Discorso sul flusso e il reflusso del mare),^{[2]} in a letter to Cardinal Orsini. In this discourse, he tried to explain the occurrence of the tides as the result of the Earth's rotation and revolution around the Sun. Galileo believed that the oceans moved like water in a large basin: as the basin moves, so does the water.^{[3]} Therefore, as the Earth revolves, the force of the Earth's rotation causes the oceans to "alternately accelerate and retardate".^{[4]} His view on the oscillation and "alternately accelerated and retardated" motion of the Earth's rotation is a "dynamic process" that deviated from the previous dogma, which proposed "a process of expansion and contraction of seawater." ^{[5]} However, Galileo's theory was erroneous.^{[2]} In subsequent centuries, further analysis led to the current tidal physics. Galileo rejected Kepler's explanation of the tides.
Newton
Newton, in the Principia, provided a correct explanation for the tidal force, which can be used to explain tides on a planet covered by a uniform ocean, but which takes no account of the distribution of the continents or ocean bathymetry. ^{[6]}
Laplace's dynamic theory
While Newton explained the tides by describing the tidegenerating forces and Bernoulli gave a description of the static reaction of the waters on earth to the tidal potential, the dynamic theory of tides, developed by PierreSimon Laplace in 1775,^{[7]} describes the ocean's real reaction to tidal forces. Laplace's theory of ocean tides took into account friction, resonance and natural periods of ocean basins. It predicted amphidromic circulation (the presence of large amphidromic systems in the world's ocean basins was long ago theorized, to explain the oceanic tides that are actually observed ). The equilibrium theory, based on the gravitational gradient from the sun and moon but ignoring the earth's rotation, the effects of continents, and other important effects, could not explain the real ocean tides.^{[8]}^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]}^{[15]} Since measurements have confirmed the theory, many things have possible explanations now, like how the tides interact with deep sea ridges and chains of seamounts give rise to deep eddies that transport nutrients from the deep to the surface.^{[16]} The equilibrium tide theory calculates the height of the tide wave of less than half a meter, while the dynamic theory explains why tides are up to 15 meters.^{[17]} Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within a few centimeters.^{[18]}^{[19]} Measurements from the CHAMP satellite closely match the models based on the TOPEX data.^{[20]}^{[21]} Accurate models of tides worldwide are essential for research since the variations due to tides must be removed from measurements when calculating gravity and changes in sea levels.^{[22]}
Laplace's tidal equations
A. Lunar gravitational potential: this depicts the Moon directly over 30° N (or 30° S) viewed from above the Northern Hemisphere.

B. This view shows same potential from 180° from view A. Viewed from above the Northern Hemisphere. Red up, blue down.

In 1776, PierreSimon Laplace formulated a single set of linear partial differential equations, for tidal flow described as a barotropic twodimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity. Laplace obtained these equations by simplifying the fluid dynamic equations. But they can also be derived from energy integrals via Lagrange's equation.
For a fluid sheet of average thickness D, the vertical tidal elevation ς, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations:^{[23]}

\begin{align} \frac{\partial \zeta}{\partial t} &+ \frac{1}{a \cos( \varphi )} \left[ \frac{\partial}{\partial \lambda} (uD) + \frac{\partial}{\partial \varphi} \left(vD \cos( \varphi )\right) \right] = 0, \\[2ex] \frac{\partial u}{\partial t} & v \left( 2 \Omega \sin( \varphi ) \right) + \frac{1}{a \cos( \varphi )} \frac{\partial}{\partial \lambda} \left( g \zeta + U \right) =0 \qquad \text{and} \\[2ex] \frac{\partial v}{\partial t} &+ u \left( 2 \Omega \sin( \varphi ) \right) + \frac{1}{a} \frac{\partial}{\partial \varphi} \left( g \zeta + U \right) =0, \end{align}
where Ω is the angular frequency of the planet's rotation, g is the planet's gravitational acceleration at the mean ocean surface, a is the planetary radius, and U is the external gravitational tidalforcing potential.
William Thomson (Lord Kelvin) rewrote 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
Laplace's improvements in theory were substantial, but they still left prediction in an approximate state. This position changed in the 1860s when the local circumstances of tidal phenomena were more fully brought into account by William Thomson's application of Fourier analysis to the tidal motions.
Thomson's work in this field was then further developed and extended by A T Doodson, applying the lunar theory of E W Brown,^{[24]} developed the tidegenerating potential (TGP) in harmonic form, distinguishing 388 tidal frequencies.^{[25]} Doodson's work was carried out and published in 1921.^{[26]}
Doodson devised a practical system for specifying the different harmonic components of the tidegenerating potential, the Doodson Numbers, a system still in use.^{[27]}
Since the midtwentieth century further analysis has generated many more terms than Doodson's 388. About 62 constituents are of sufficient size to be considered for possible use in marine tide prediction, but sometimes many less even than that can predict tides to useful accuracy. The calculations of tide predictions using the harmonic constituents are laborious, and from the 1870s to about the 1960s they were carried out using a mechanical tidepredicting machine, a specialpurpose form of analog computer now superseded in this work by digital electronic computers that can be programmed to carry out the same computations.
Tidal constituents
Tidal constituents combine to give an endlesslyvarying aggregate because of their different and incommensurable frequencies: the effect is visualized in an animation of the American Mathematical Society illustrating the way in which the components used to be mechanically combined in the tidepredicting machine. Amplitudes of tidal constituents are given below for six example locations: Eastport, Maine (ME),^{[28]} Biloxi, Mississippi (MS), San Juan, Puerto Rico (PR), Kodiak, Alaska (AK), San Francisco, California (CA), and Hilo, Hawaii (HI).
Higher harmonics

Darwin

Period

Speed

Doodson coefficients

Doodson

Amplitude at example location (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

Semidiurnal

Darwin

Period

Speed

Doodson coefficients

Doodson

Amplitude at example location (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 secondorder

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

Speed

Doodson coefficients

Doodson

Amplitude at example location (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

Speed

Doodson coefficients

Doodson

Amplitude at example location (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

See also
References and notes

^ Johannes Kepler, Astronomia nova … (1609), p. 5 of the Introductio in hoc opus

^ ^{a} ^{b} Rice University: Galileo's Theory of the Tides, by Rossella Gigli, retrieved 10 March 2010

^ Tyson, Peter. "Galileo's Big Mistake". NOVA. PBS. Retrieved 20140219.

^ Palmieri, Paolo (1998). Reexamining Galileo's Theory of Tides. SpringerVerlag. p. 229.

^ Palmeri, Paolo (1998). Reexamining Galileo's Theory of Tides. SpringerVerlag. p. 227.

^ http://web.vims.edu/physical/research/TCTutorial/static.htm

^ "Shelf and Coastal Oceanography". Es.flinders.edu.au. Retrieved 20120602.

^ Tidal theory website South African Navy Hydrographic Office

^ "Dynamic theory for tides". Oberlin.edu. Retrieved 20120602.

^ "Dynamic Theory of Tides".

^ "Dynamic Tides – In contrast to "static" theory, the dynamic theory of tides recognizes that water covers only threequarters o". Web.vims.edu. Retrieved 20120602.

^ "The Dynamic Theory of Tides". Coa.edu. Retrieved 20120602.

^ https://beacon.salemstate.edu/~lhanson/gls214/gls214_tides

^ "Tides  building, river, sea, depth, oceans, effects, important, largest, system, wave, effect, marine, Pacific". Waterencyclopedia.com. 20100627. Retrieved 20120602.

^ "TIDES". Ocean.tamu.edu. Retrieved 20120602.

^ Floor Anthoni. "Tides". Seafriends.org.nz. Retrieved 20120602.

^ "The Cause & Nature of Tides".

^ "Scientific Visualization Studio TOPEX/Poseidon images". Svs.gsfc.nasa.gov. Retrieved 20120602.

^ "TOPEX/Poseidon Western Hemisphere: Tide Height Model : NASA/Goddard Space Flight Center Scientific Visualization Studio : Free Download & Streaming : Internet Archive". Archive.org. Retrieved 20120602.

^ http://www.geomag.us/info/Ocean/m2_CHAMP+longwave_SSH.swf

^ "OSU Tidal Data Inversion". Volkov.oce.orst.edu. Retrieved 20120602.

^ "Dynamic and residual ocean tide analysis for improved GRACE dealiasing (DAROTA)".

^ "The Laplace Tidal Equations and Atmospheric Tides".

^ D E Cartwright, "Tides: a scientific history", Cambridge University Press 2001, at pages 1634.

^ S Casotto, F Biscani, "A fully analytical approach to the harmonic development of the tidegenerating potential accounting for precession, nutation, and perturbations due to figure and planetary terms", AAS Division on Dynamical Astronomy, April 2004, vol.36(2), 67.

^ A T Doodson (1921), "The Harmonic Development of the TideGenerating Potential", Proceedings of the Royal Society of London. Series A, Vol. 100, No. 704 (Dec. 1, 1921), pp. 305329.

^ See e.g. T D Moyer (2003), "Formulation for observed and computed values of Deep Space Network data types for navigation", vol.3 in Deepspace communications and navigation series, Wiley (2003), e.g. at pp.1268.

^ NOAA. "Eastport, ME Tidal Constituents". NOAA. Retrieved 20120522.
External links

Arctic and Antarctic Barotropic Tide Models

Amphidrome

Equilibrium Theory of Tides

Dynamic Tides

Annual amphidromes: a common feature in the ocean?

http://www.aviso.oceanobs.com/en/news/idm/2000/oct2000sunandmoonshapetidesonearth/

Tides

Contributions of satellite laser ranging to the studies of Earth tides

Sun and Moon shape tides on Earth

Study of harmonic site position variations determined by very long baseline interferometry

http://www.coa.edu/stodd/oceanweb/oceanography/Oceanlectures02/Lecture8/sld014.htm

Dynamic Theory of Tides

Tidal Observations

Myths about Gravity and Tides

http://books.google.com/books?id=3mGZUHDuEdwC&lpg=PA150&ots=soMAFKsRkc&dq=number%20of%20amphidromes&pg=PA150#v=onepage&q=amphidrome&f=false

Publications from NOAA's Center for Operational Oceanographic Products and Services

Understanding Tides

150 Years of Tides on the Western Coast

Our Relentless Tides

U. Manna, J. L. Menaldi and S. S. Sritharan: Stochastic Analysis of Tidal Dynamics Equation in Infinite Dimensional Stochastic Analysis, edited by A. N. Sengupta and P. Sundar, World Scientific Publishers, 2008.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.