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{{Refimprove|date=March 2009}}
The '''theory of tides''' is the application of [[continuum mechanics]] to interpret and predict the [[tide|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 [[ocean]]s.

==Origin of theory==
In 1616, [[Galileo Galilei]] wrote ''Discourse on the Tides'' (in Italian: ''Discorso del flusso e reflusso del mare''),<ref name="Galileo's Theory of the Tides">[[Rice University]] - [http://galileo.rice.edu/sci/observations/tides.html Galileo's Theory of the Tides] - by Rossella Gigli, retrieved 10 March 2010</ref> a paper in which he tried to explain the occurrence of the tides as the result of the Earth's rotation around the Sun. However, Galileo's theory was, in the later [[Newtonian]] terms, an error.<ref name="Galileo's Theory of the Tides"/>  In subsequent centuries, [[Tide#Analysis|further analysis]] led to the current tidal physics.

==Tidal physics==

{| border=0 cellpadding=1 cellspacing=0 align=right
|- align=center
|[[Image:Tideforcenw.jpg|thumb|150px| ''A.'' Lunar gravitational potential: this depicts the Moon directly over 30° N (or 30° S) viewed from above the Northern Hemisphere.]]
|[[Image:Tideforcese.jpg|thumb|150px| ''B.'' This view shows same potential from 180° from view ''A''. Viewed from above the Northern Hemisphere. Red up, blue down.]]
|}

===Laplace's tidal equations===<!-- Referenced here from [[Tide]] and [[Laplace's tidal equations]]-->
in 1776, [[Pierre-Simon Laplace]] formulated a single set of linear [[partial differential equation]]s, for tidal flow described as a [[barotropic]] two-dimensional sheet flow. [[Coriolis effect]]s 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''':<ref>{{cite web|url=  http://kiwi.atmos.colostate.edu/group/dave/pdf/LTE.frame.pdf|title= The Laplace Tidal Equations and Atmospheric Tides}}</ref>
:<math>
  \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}
</math>
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 tidal-forcing [[potential]].

[[William Thomson, 1st Baron Kelvin|William Thomson (Lord Kelvin)]] rewrote Laplace's momentum terms using the [[Curl (mathematics)|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 [[George Darwin]]: Darwin's work was based on the lunar theory current in his time. His symbols for the tidal harmonic constituents are still used. Darwin's harmonic developments of the tide-generating forces were later brought up to date with modern developments by [[Arthur Thomas Doodson|A T Doodson]] whose development of the tide generating potential (TGP) in harmonic form was carried out and published in 1921:<ref>A T Doodson (1921), "The Harmonic Development of the Tide-Generating Potential", Proceedings of the Royal Society of London. Series A, Vol. 100, No. 704 (Dec. 1, 1921), pp. 305-329.</ref> Doodson distinguished 388 tidal frequencies.<ref>S Casotto, F Biscani, "A fully analytical approach to the harmonic development of the tide-generating potential accounting for precession, nutation, and perturbations due to figure and planetary terms", AAS Division on Dynamical Astronomy, April 2004, vol.36(2), 67.</ref>  Doodson's analysis of 1921 was based on the then-latest [[lunar theory]] of [[Ernest William Brown|E W Brown]].<ref name=dec2001>D E Cartwright, "Tides: a scientific history", Cambridge University Press 2001, [http://books.google.com/books?id=sSesNjG96JYC&pg=PA163 at pages 163-4].</ref>

Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the [[Arthur Thomas Doodson#Doodson Numbers|Doodson Numbers]], a system still in use.<ref>See e.g. T D Moyer (2003), "Formulation for observed and computed values of Deep Space Network data types for navigation", vol.3 in Deep-space communications and navigation series, Wiley (2003), e.g. at pp.126-8.</ref>

Since the mid-twentieth 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 [[tide-predicting machine]], a special-purpose 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 endlessly-varying aggregate because of their different and incommensurable frequencies: the effect is visualized in an [http://www.ams.org/featurecolumn/archive/tidesIII3.html animation of the American Mathematical Society] illustrating the way in which the components used to be mechanically combined in the [[tide-predicting machine]].  Amplitudes of tidal constituents are given below for the following example locations:
:'''ME''' Eastport,<ref>{{cite web|last=NOAA|title=Eastport, ME Tidal Constituents|url=http://tidesandcurrents.noaa.gov/cgi-bin/co-ops_qry.cgi?stn=8410140%20Eastport,%20ME&dcp=1&ssid=WL&pc=P2&datum=NULL&unit=0&bdate=20130521&edate=20130522&date=1&shift=0&level=-4&form=0&data_type=har&format=View+Data|publisher=NOAA|accessdate=2012-05-22}}</ref> 
:'''MS''' Biloxi,
:'''PR''' San Juan,
:'''AK''' Kodiak,
:'''CA''' San Francisco, and
:'''HI''' Hilo.

{|class="wikitable" border:1; cellpadding:0; cellspacing:0; vertical-align:top; align:left"|
|bgcolor="lightblue"|

====Higher harmonics====
|bgcolor="lightblue"|
'''Darwin'''
|bgcolor="lightblue"|
'''Period'''
|bgcolor="lightblue"|
'''Speed'''
|bgcolor="lightblue" colspan="4"|
'''Doodson coefficients'''
|bgcolor="lightblue"|
'''Doodson'''
|bgcolor="lightblue" colspan="6"|
'''Amplitude at example location (cm)'''
|bgcolor="lightblue"|
'''NOAA'''
|-
|bgcolor="lightblue"|
'''Species'''
|bgcolor="lightblue"|
'''Symbol'''
|bgcolor="lightblue"|
'''(hr)'''
|bgcolor="lightblue"|
'''rate(°/hr)'''
|bgcolor="lightblue"|'''''n'''''<sub>'''1'''</sub> (''L'')
|bgcolor="lightblue"|''n''<sub>2</sub> (''m'')
|bgcolor="lightblue"|''n''<sub>3</sub> (''y'')
|bgcolor="lightblue"|''n''<sub>4</sub> (''mp'')
|bgcolor="lightblue"|
'''number'''
|bgcolor="lightblue"|
'''ME'''
|bgcolor="lightblue"|
'''MS'''
|bgcolor="lightblue"|
'''PR'''
|bgcolor="lightblue"|
'''AK'''
|bgcolor="lightblue"|
'''CA'''
|bgcolor="lightblue"|
'''HI'''
|bgcolor="lightblue"|
'''order'''
|-
|'''Shallow water overtides of principal lunar'''
|''M''<sub>4</sub>
|6.210300601
|57.9682084
|4
|
|
|
|455.555
|6.0
|0.6
|
|0.9
|2.3
|
|'''5'''
|-
|'''Shallow water overtides of principal lunar'''
|''M''<sub>6</sub>
|4.140200401
|86.9523127
|6
|
|
|
|655.555
|5.1
|0.1
|
|1.0
|
|
|'''7'''
|-
|'''Shallow water terdiurnal'''
|''MK''<sub>3</sub>
|8.177140247
|44.0251729
|3
|1
|
|
|365.555
|
|
|
|0.5
|1.9
|
|'''8'''
|-
|'''Shallow water overtides of principal solar '''
|''S''<sub>4</sub>
|6
|60
|4
|4
| -4
|
|491.555
|
|0.1
|
|
|
|
|'''9'''
|-
|'''Shallow water quarter diurnal '''
|''MN''<sub>4</sub>
|6.269173724
|57.4238337
|4
| -1
|
|1
|445.655
|2.3
|
|
|0.3
|0.9
|
|'''10'''
|-
|Shallow water overtides of principal solar
|''S''<sub>6</sub>
|4
|90
|6
|6
| -6
|
|*
|
|0.1
|
|
|
|
|12
|-
|Lunar terdiurnal
|''M''<sub>3</sub>
|8.280400802
|43.4761563
|3
|
|
|
|355.555
|
|
|
|
|0.5
|
|32
|-
|Shallow water terdiurnal
|2"MK''<sub>3</sub>
|8.38630265
|42.9271398
|3
| -1
|
|
|345.555
|0.5
|
|
|0.5
|1.4
|
|34
|-
|Shallow water eighth diurnal
|''M''<sub>8</sub>
|3.105150301
|115.9364166
|8
|
|
|
|855.555
|0.5
|0.1
|
|
|
|
|36
|-
|Shallow water quarter diurnal
|''MS''<sub>4</sub>
|6.103339275
|58.9841042
|4
|2
| -2
|
|473.555
|1.8
|
|
|0.6
|1.0
|
|37
|-
|bgcolor="lightblue"|

====Semi-diurnal====
|bgcolor="lightblue"|
'''Darwin'''
|bgcolor="lightblue"|
'''Period'''
|bgcolor="lightblue"|
'''Speed'''
|bgcolor="lightblue" colspan="4"|
'''Doodson coefficients'''
|bgcolor="lightblue"|
'''Doodson'''
|bgcolor="lightblue" colspan="6"|
'''Amplitude at example location (cm)'''
|bgcolor="lightblue"|
'''NOAA'''
|-
|bgcolor="lightblue"|'''Species'''
|bgcolor="lightblue"|
'''Symbol'''
|bgcolor="lightblue"|
'''(hr)'''
|bgcolor="lightblue"|
'''(°/hr)'''
|bgcolor="lightblue"|''n''<sub>1</sub> (''L'')
|bgcolor="lightblue"|''n''<sub>2</sub> (''m'')
|bgcolor="lightblue"|''n''<sub>3</sub> (''y'')
|bgcolor="lightblue"|''n''<sub>4</sub> (''mp'')
|bgcolor="lightblue"|
'''number'''
|bgcolor="lightblue"|
'''ME'''
|bgcolor="lightblue"|
'''MS'''
|bgcolor="lightblue"|
'''PR'''
|bgcolor="lightblue"|
'''AK'''
|bgcolor="lightblue"|
'''CA'''
|bgcolor="lightblue"|
'''HI'''
|bgcolor="lightblue"|
'''order'''
|-
|'''Principal lunar semidiurnal '''
|''M''<sub>2</sub>
|12.4206012
|28.9841042
|2
|
|
|
|255.555
|268.7
|3.9
|15.9
|97.3
|58.0
|23.0
|'''1'''
|-
|'''Principal solar semidiurnal '''
|''S''<sub>2</sub>
|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''<sub>2</sub>
|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''<sub>2</sub>
|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''<sub>2</sub>
|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''<sub>2</sub>
|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''<sub>2</sub>
|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''<sub>2</sub>
|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''<sub>2</sub>
|11.98359564
|30.0410667
|2
|2
| -1
|
|274.555
|0.9
|
|
|0.2
|0.1
|0.1
|28
|-
|Shallow water semidiurnal
|2''SM''<sub>2</sub>
|11.60695157
|31.0158958
|2
|4
| -4
|
|291.555
|0.5
|
|
|
|
|
|31
|-
|Smaller lunar elliptic semidiurnal
|''L''<sub>2</sub>
|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''<sub>2</sub>
|11.96723606
|30.0821373
|2
|2
|
|
|275.555
|11.6
|0.9
|0.6
|9.0
|4.0
|2.8
|35
|-
|bgcolor="lightblue"|

====Diurnal====
|bgcolor="lightblue"|
'''Darwin'''
|bgcolor="lightblue"|
'''Period'''
|bgcolor="lightblue"|
'''Speed'''
|bgcolor="lightblue" colspan="4"|
'''Doodson coefficients'''
|bgcolor="lightblue"|
'''Doodson'''
|bgcolor="lightblue" colspan="6"|
'''Amplitude at example location (cm)'''
|bgcolor="lightblue"|
'''NOAA'''
|-
|bgcolor="lightblue"|'''Species'''
|bgcolor="lightblue"|
'''Symbol'''
|bgcolor="lightblue"|
'''(hr)'''
|bgcolor="lightblue"|
'''(°/hr)'''
|bgcolor="lightblue"|'''''n'''''<sub>'''1'''</sub> (''L'')
|bgcolor="lightblue"|''n''<sub>2</sub> (''m'')
|bgcolor="lightblue"|''n''<sub>3</sub> (''y'')
|bgcolor="lightblue"|''n''<sub>4</sub> (''mp'')
|bgcolor="lightblue"|
'''number'''
|bgcolor="lightblue"|
'''ME'''
|bgcolor="lightblue"|
'''MS'''
|bgcolor="lightblue"|
'''PR'''
|bgcolor="lightblue"|
'''AK'''
|bgcolor="lightblue"|
'''CA'''
|bgcolor="lightblue"|
'''HI'''
|bgcolor="lightblue"|
'''order'''
|-
|'''Lunar diurnal '''
|''K''<sub>1</sub>
|23.93447213
|15.0410686
|1
|1
|
|
|165.555
|15.6
|16.2
|9.0
|39.8
|36.8
|16.7
|'''4
|-
|'''Lunar diurnal '''
|''O''<sub>1</sub>
|25.81933871
|13.9430356
|1
| -1
|
|
|145.555
|11.9
|16.9
|7.7
|25.9
|23.0
|9.2
|'''6'''
|-
|Lunar diurnal
|''OO''<sub>1</sub>
|22.30608083
|16.1391017
|1
|3
|
|
|185.555
|0.5
|0.7
|0.4
|1.2
|1.1
|0.7
|15
|-
|Solar diurnal
|''S''<sub>1</sub>
|24
|15
|1
|1
| -1
|
|164.555
|1.0
|
|0.5
|1.2
|0.7
|0.3
|17
|-
|Smaller lunar elliptic diurnal
|''M''<sub>1</sub>
|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''<sub>1</sub>
|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''<sub>1</sub>
|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
|2''Q''<sub>1</sub>
|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''<sub>1</sub>
|24.06588766
|14.9589314
|1
|1
| -2
|
|163.555
|5.2
|5.4
|2.9
|12.6
|11.6
|5.1
|30
|-
|bgcolor="lightblue"|

====Long period====
|bgcolor="lightblue"|
'''Darwin'''
|bgcolor="lightblue"|
'''Period'''
|bgcolor="lightblue"|
'''Speed'''
|bgcolor="lightblue" colspan="4"|
'''Doodson coefficients'''
|bgcolor="lightblue"|
'''Doodson'''
|bgcolor="lightblue" colspan="6"|
'''Amplitude at example location (cm)'''
|bgcolor="lightblue"|
'''NOAA'''
|-
|bgcolor="lightblue"|'''Species'''
|bgcolor="lightblue"|
'''Symbol'''
|bgcolor="lightblue"|
'''(hr)'''
|bgcolor="lightblue"|
'''(°/hr)'''
|bgcolor="lightblue"|''n''<sub>1</sub> (''L'')
|bgcolor="lightblue"|''n''<sub>2</sub> (''m'')
|bgcolor="lightblue"|''n''<sub>3</sub> (''y'')
|bgcolor="lightblue"|''n''<sub>4</sub> (''mp'')
|bgcolor="lightblue"|
'''number'''
|bgcolor="lightblue"|
'''ME'''
|bgcolor="lightblue"|
'''MS'''
|bgcolor="lightblue"|
'''PR'''
|bgcolor="lightblue"|
'''AK'''
|bgcolor="lightblue"|
'''CA'''
|bgcolor="lightblue"|
'''HI'''
|bgcolor="lightblue"|
'''order'''
|-
|Lunar monthly
|''M''<sub>m</sub>
|661.3111655
|0.5443747
|0
|1
|
| -1
|65.455
|
|
|0.7
|1.9
|
|
|20
|-
|Solar semiannual
|''S''<sub>sa</sub>
|4383.076325
|0.0821373
|0
|
|2
|
|57.555
|1.6
|
|2.1
|1.5
|3.9
|
|21
|-
|Solar annual
|''S''<sub>a</sub>
|8766.15265
|0.0410686
|0
|
|1
|
|56.555
|
|
|5.5
|7.8
|3.8
|4.3
|22
|-
|Lunisolar synodic fortnightly
|''M''<sub>sf</sub>
|354.3670666
|1.0158958
|0
|2
| -2
|
|73.555
|
|
|
|1.5
|
|
|23
|-
|Lunisolar fortnightly
|''M''<sub>f</sub>
|327.8599387
|1.0980331
|0
|2
|
|
|75.555
|
|
|1.4
|2.0
|
|0.7
|24
|}

==Dynamic theory of tides==

While [[Isaac Newton|Newton]] explained the tides by describing the tide generating forces, and [[Daniel Bernoulli|Bernoulli]] gave a description of the static reaction of the waters on earth to the tidal potential, the ''dynamic theory of tides'', first introduced by [[Pierre-Simon Laplace|Laplace]], 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 {{cn|date=December 2013}}). 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, couldn't explain the real ocean tides.<ref>[http://www.sanho.co.za/tides/tide_theory.pdf Tidal theory] website South African Navy Hydrographic Office</ref><ref>{{cite web|url=http://www.oberlin.edu/faculty/swojtal/SFWpage/161Stuff/161Lect17/sld012.htm |title=Dynamic theory for tides |publisher=Oberlin.edu |date= |accessdate=2012-06-02}}</ref><ref>{{cite web|url=   http://ffden-2.phys.uaf.edu/645fall2003_web.dir/ellie_boyce/dynamic.htm|title=  Dynamic Theory of Tides}}</ref><ref>{{cite web|url=http://web.vims.edu/physical/research/TCTutorial/dynamic.htm |title=Dynamic Tides – In contrast to "static" theory, the dynamic theory of tides recognizes that water covers only three-quarters o |publisher=Web.vims.edu |date= |accessdate=2012-06-02}}</ref><ref>{{cite web|url=http://www.coa.edu/stodd/oceanweb/oceanography/Oceanlectures02/Lecture8/sld014.htm |title=The Dynamic Theory of Tides |publisher=Coa.edu |date= |accessdate=2012-06-02}}</ref><ref>https://beacon.salemstate.edu/~lhanson/gls214/gls214_tides</ref><ref>{{cite web|url=http://www.waterencyclopedia.com/St-Ts/Tides.html |title=Tides - building, river, sea, depth, oceans, effects, important, largest, system, wave, effect, marine, Pacific |publisher=Waterencyclopedia.com |date=2010-06-27 |accessdate=2012-06-02}}</ref><ref>{{cite web|url=http://ocean.tamu.edu/wormuth/tidestxt.html |title=TIDES |publisher=Ocean.tamu.edu |date= |accessdate=2012-06-02}}</ref> 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.<ref>{{cite web|author=Floor Anthoni |url=http://www.seafriends.org.nz/oceano/tides.htm |title=Tides |publisher=Seafriends.org.nz |date= |accessdate=2012-06-02}}</ref>  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.<ref>{{cite web|url=   http://www.linz.govt.nz/hydro/tidal-info/tidal-intro/cause-nature|title=  The Cause & Nature of Tides}}</ref>
The theory was developed by [[Pierre-Simon Laplace]] in 1775.<ref>{{cite web|url=http://www.es.flinders.edu.au/~mattom/ShelfCoast/chapter05.html |title=Shelf and Coastal Oceanography |publisher=Es.flinders.edu.au |date= |accessdate=2012-06-02}}</ref> Satellite data has confirmed the Dynamic theory as accurate, and the tides worldwide are now measured to within a few centimeters.<ref>{{cite web|url=http://svs.gsfc.nasa.gov/stories/topex/tides.html |title=Scientific Visualization Studio TOPEX/Poseidon images |publisher=Svs.gsfc.nasa.gov |date= |accessdate=2012-06-02}}</ref><ref>{{cite web|url=http://archive.org/details/SVS-1333 |title=TOPEX/Poseidon Western Hemisphere: Tide Height Model : NASA/Goddard Space Flight Center Scientific Visualization Studio : Free Download & Streaming : Internet Archive |publisher=Archive.org |date= |accessdate=2012-06-02}}</ref> Measurements from the [[CHAMP (satellite)|CHAMP]] satellite closely matches the models based on the [[TOPEX]] data.<ref>http://www.geomag.us/info/Ocean/m2_CHAMP+longwave_SSH.swf</ref><ref>{{cite web|url=http://volkov.oce.orst.edu/tides/ |title=OSU Tidal Data Inversion |publisher=Volkov.oce.orst.edu |date= |accessdate=2012-06-02}}</ref>  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.<ref>{{cite web|url=   http://www.dgfi.badw.de/index.php?id=143|title=  Dynamic and residual ocean tide analysis for improved GRACE de-aliasing (DAROTA) }}</ref>

==See also==
*[[Kelvin wave]]
*[[Tide|Tides]]
*[[amphidromic point]]

==References and notes==
{{reflist}}

==External links==
*[http://www.esr.org/ptm_index.html Arctic and Antarctic Barotropic Tide Models] 
*[http://www.eoearth.org/article/Amphidrome Amphidrome]
*[http://www.waterencyclopedia.com/St-Ts/Tides.html#b Equilibrium Theory of Tides]  
*[http://web.vims.edu/physical/research/TCTutorial/dynamic.htm Dynamic Tides]  
*[http://www.mendeley.com/research/annual-amphidromes-a-common-feature-in-the-ocean/ Annual amphidromes: a common feature in the ocean?]
* http://www.aviso.oceanobs.com/en/news/idm/2000/oct-2000-sun-and-moon-shape-tides-on-earth/index.html
*[http://www.uwgb.edu/dutchs/EarthSC202Notes/TIDES.HTM Tides]
*[http://www.eas.slu.edu/GGP/BIM_Recent_Issues/bim134-2001/bim134_report_WG6_SLR.pdf  Contributions of satellite laser ranging to the studies of Earth tides]
*[http://www.austides.com/chap1_2011_febr.pdf Sun and Moon shape tides on Earth] 
*[http://www.agu.org/pubs/crossref/2003/2002JB001801.shtml Study of harmonic site position variations determined by very long baseline interferometry]
* http://www.coa.edu/stodd/oceanweb/oceanography/Oceanlectures02/Lecture8/sld014.htm
*[http://ffden-2.phys.uaf.edu/645fall2003_web.dir/Ellie_Boyce/dynamic.htm Dynamic Theory of Tides]
*[http://mysite.du.edu/~jcalvert/geol/tides.htm Tidal Observations]
*[http://www.jal.cc.il.us/~mikolajsawicki/tides_new2.pdf 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
* [http://tidesandcurrents.noaa.gov/pub.html Publications from NOAA's Center for Operational Oceanographic Products and Services]
** [http://tidesandcurrents.noaa.gov/publications/Understanding_Tides_by_Steacy_finalFINAL11_30.pdf Understanding Tides]
** [http://tidesandcurrents.noaa.gov/publications/150_years_of_tides.pdf 150 Years of Tides on the Western Coast]
** [http://tidesandcurrents.noaa.gov/restles1.html Our Relentless Tides]

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[[Category:Geophysics| ]]
[[Category:Oceanography|Oceanography]]
[[Category:Continuum mechanics]]
[[Category:Fluid dynamics| ]]
[[Category:Fluid mechanics| ]]
[[Category:Planetary science]]

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