Theory of tides#Laplace's tidal equations

The tides received relatively little attention in the civilizations around the Mediterranean Sea, as the tides there are relatively small, and the areas that experience tides do so unreliably.{{cite journal |last1=Tabarroni |first1=G. |title=The tides and Newton |journal=Memorie della Società Astronomia Italiana |year=1989 |volume=60 |pages=770–777 |bibcode=1989MmSAI..60..769T |url=http://adsabs.harvard.edu/full/1989MmSAI..60..769T |access-date=27 December 2020}}{{cite book |last1=Pugh |first1=David T. |title=Tides, Surges and Mean Sea-Level |date=28 December 1987 |publisher=JOHN WILEY & SONS |url=https://eprints.soton.ac.uk/19157/1/sea-level.pdf |isbn=047191505X |pages=2–4 |access-date=27 December 2020}} A number of theories were advanced, however, from comparing the movements to breathing or blood flow to theories involving whirlpools or river cycles. A similar "breathing earth" idea was considered by some Asian thinkers. Plato reportedly believed that the tides were caused by water flowing in and out of undersea caverns. Crates of Mallus attributed the tides to "the counter-movement (ἀντισπασμός) of the sea” and Apollodorus of Corcyra to "the refluxes from the Ocean".{{Cite book |last=Heath |first=Thomas Little |url=http://archive.org/details/aristarchusofsam00heatuoft |title=Aristarchus of Samos, the ancient Copernicus |date=1913 |publisher=Oxford : Clarendon Press |others=Gerstein - University of Toronto |pages=306–307}} An ancient Indian Purana text dated to 400-300 BC refers to the ocean rising and falling because of heat expansion from the light of the Moon.{{efn|In all the oceans the water remains at all times the same in quantity, and never, increases or diminishes; but like the water in a caldron, which, in consequence of its combination with heat, expands, so the waters of the ocean swell with the increase of the moon. The waters, although really neither more nor less, dilate or contract as the moon increases or wanes in the light and dark fortnights. - [https://www.sacred-texts.com/hin/vp/vp062.htm The Vishnu Purana book II ch. IV]}}{{cite book |last1=Cartwright |first1=David Edgar |title=Tides: A Scientific History |page=6 |url=https://books.google.com/books?id=78bE5U7TVuIC&q=descartes+tides&pg=PA31 |access-date=28 December 2020|date=1999 |publisher=Cambridge University Press|isbn=9780521797467 }} The Yolngu people of northeastern Arnhem Land in the Northern Territory of Australia identified a link between the Moon and the tides, which they mythically attributed to the Moon filling with water and emptying out again.{{Cite web|title=Moon|url=http://www.aboriginalastronomy.com.au/content/topics/moon/|access-date=2020-10-08|website=Australian Indigenous Astronomy|language=en-US}}{{cite journal |title="Bridging the Gap" through Australian Cultural Astronomy. |journal=Archaeoastronomy & Ethnoastronomy – Building Bridges Between Cultures |date=2011 |pages=282–290}}

Ultimately the link between the Moon (and Sun) and the tides became known to the Greeks, although the exact date of discovery is unclear; references to it are made in sources such as Pytheas of Massilia in 325 BC and Pliny the Elder's Natural History in 77 AD. Although the schedule of the tides and the link to lunar and solar movements was known, the exact mechanism that connected them was unclear.{{cite journal |last1=Marmer |first1=H. A. |title=The Problems of the Tide |journal=The Scientific Monthly |date=March 1922 |volume=14 |issue=3 |pages=209–222 |bibcode=1922SciMo..14..209M |url=https://www.jstor.org/stable/6374}} Classicists Thomas Little Heath claimed that both Pytheas and Posidonius connected the tides with the moon, "the former directly, the latter through the setting up of winds". Seneca mentions in De Providentia the periodic motion of the tides controlled by the lunar sphere.Seneca, De Providentia, section IV Eratosthenes (3rd century BC) and Posidonius (1st century BC) both produced detailed descriptions of the tides and their relationship to the phases of the Moon, Posidonius in particular making lengthy observations of the sea on the Spanish coast, although little of their work survived. The influence of the Moon on tides was mentioned in Ptolemy's Tetrabiblos as evidence of the reality of astrology.{{cite journal |last1=Cartwright |first1=David E. |title=On the Origins of Knowledge of the Sea Tides from Antiquity to the Thirteenth Century |journal=Earth Sciences History |year=2001 |volume=20 |issue=2 |pages=105–126 |doi=10.17704/eshi.20.2.m23118527q395675 |jstor=24138749 |bibcode=2001ESHis..20..105C |url=https://www.jstor.org/stable/24138749 |access-date=27 December 2020}} Seleucus of Seleucia is thought to have theorized around 150 BC that tides were caused by the Moon as part of his heliocentric model.Lucio Russo, Flussi e riflussi, Feltrinelli, Milano, 2003, {{ISBN|88-07-10349-4}}.{{cite journal

| last = Van der Waerden

| first = B. L.

| author-link = Bartel Leendert van der Waerden

| title = The Heliocentric System in Greek, Persian and Hindu Astronomy

| journal = Annals of the New York Academy of Sciences

| volume = 500

| pages = 525–545

| date = 1987

| issue = 1

| bibcode = 1987NYASA.500..525V | doi = 10.1111/j.1749-6632.1987.tb37224.x | s2cid = 222087224

}}

Aristotle, judging from discussions of his beliefs in other sources, is thought to have believed the tides were caused by winds driven by the Sun's heat, and he rejected the theory that the Moon caused the tides. An apocryphal legend claims that he committed suicide in frustration with his failure to fully understand the tides. Heraclides also held "the sun sets up winds, and that these winds, when they blow, cause the high tide and, when they cease, the low tide". Dicaearchus also "put the tides down to the direct action of the sun according to its position". Philostratus discusses tides in Book Five of Life of Apollonius of Tyana (circa 217-238 AD); he was vaguely aware of a correlation of the tides with the phases of the Moon but attributed them to spirits moving water in and out of caverns, which he connected with the legend that spirits of the dead cannot move on at certain phases of the Moon.{{efn|Now I myself have seen among the Celts the ocean tides just as they are described. After making various conjectures about why so vast a bulk of waters recedes and advances, I have come to the conclusion that Apollonius discerned the real truth. For in one of his letters to the Indians he says that the ocean is driven by submarine influences or spirits out of several chasms which the earth afford both underneath and around it, to advance outwards, and to recede again, whenever the influence or spirit, like the breath of our bodies, gives way and recedes. And this theory is confirmed by the course run by diseases in Gadeira, for at the time of high water the souls of the dying do not quit the bodies, and this would hardly happen, he says, unless the influence or spirit I have spoken of was also advancing towards the land. They also tell you of certain phenomena of the ocean in connection with the phases of the moon, according as it is born and reaches fullness and wanes. These phenomena I verified, for the ocean exactly keeps pace with the size of the moon, decreasing and increasing with her. - Philostratus, The Life of Apollonius of Tyana, V}}

The Venerable Bede discusses the tides in The Reckoning of Time and shows that the twice-daily timing of tides is related to the Moon and that the lunar monthly cycle of spring and neap tides is also related to the Moon's position. He goes on to note that the times of tides vary along the same coast and that the water movements cause low tide at one place when there is high tide elsewhere.{{cite book |last=Bede |date=2004 |title=The Reckoning of Time |translator=Faith Wallis |publisher=Liverpool University Press |isbn=978-0-85323-693-1 |pages=64–65}} However, he made no progress regarding the question of how exactly the Moon created the tides.

Medieval rule-of-thumb methods for predicting tides were said to allow one "to know what Moon makes high water" from the Moon's movements.{{cite journal |last1=HUGHES |first1=PAUL |title=A STUDY IN THE DEVELOPMENT OF PRIMITIVE AND MODERN TIDE TABLES |journal=PhD Thesis, Liverpool John Moores University |url=http://researchonline.ljmu.ac.uk/id/eprint/5645/1/419741.pdf |access-date=27 December 2020}} Dante references the Moon's influence on the tides in his Divine Comedy.{{Cite web|url=https://dante.princeton.edu/cgi-bin/dante/campuscgi/mpb/GetCantoSection.pl?LANG=2&INP_POEM=Par&INP_SECT=16&INP_START=75&INP_LEN=15|title=Princeton Dante Project: Main Poem Browser (2.0)|website=dante.princeton.edu}}

Medieval European understanding of the tides was often based on works of Muslim astronomers that became available through Latin translation starting from the 12th century.{{Cite book| publisher = Routledge| isbn = 978-1135459321| editor1-last = Glick| editor1-first = Thomas F.| year= 2014| title=Geography, Chorography | page=188 | author=Marina Tolmacheva}} Abu Ma'shar al-Balkhi, in his Introductorium in astronomiam, taught that ebb and flood tides were caused by the Moon. Abu Ma'shar discussed the effects of wind and Moon's phases relative to the Sun on the tides. In the 12th century, al-Bitruji contributed the notion that the tides were caused by the general circulation of the heavens. Medieval Arabic astrologers frequently referenced the Moon's influence on the tides as evidence for the reality of astrology; some of their treatises on the topic influenced western Europe. Some theorized that the influence was caused by lunar rays heating the ocean's floor.

Simon Stevin in his 1608 De spiegheling der Ebbenvloet (The Theory of Ebb and Flood) dismisses a large number of misconceptions that still existed about ebb and flood. Stevin pleads for the idea that the attraction of the Moon was responsible for the tides and writes in clear terms about ebb, flood, spring tide and neap tide, stressing that further research needed to be made.[http://www.vliz.be/imisdocs/publications/224466.pdf Simon Stevin – Flanders Marine Institute] (pdf, in Dutch)Palmerino, [https://books.google.com/books?id=a5lkdlMPi1AC&dq=%22johannes+kepler%22+%22simon+stevin%22+ebb+flood&pg=PA200 The Reception of the Galilean Science of Motion in Seventeenth-Century Europe, pp. 200] op books.google.nl In 1609, Johannes Kepler correctly suggested that the gravitation of the Moon causes the tides,{{efn|"Orbis virtutis tractoriæ, quæ est in Luna, porrigitur utque ad Terras, & prolectat aquas sub Zonam Torridam, … Celeriter vero Luna verticem transvolante, cum aquæ tam celeriter sequi non possint, fluxus quidem fit Oceani sub Torrida in Occidentem, … " ("The sphere of the lifting power, which is [centered] in the moon, is extended as far as to the earth and attracts the waters under the torrid zone, … However the moon flies swiftly across the zenith; because the waters cannot follow so quickly, the tide of the ocean under the torrid [zone] is indeed made to the west, …")Johannes Kepler, Astronomia nova … (1609), p. 5 of the Introductio in hoc opus (Introduction to this work). [https://archive.org/stream/Astronomianovaa00Kepl#page/n24/mode/1up From page 5:]}} which he compared to magnetic attractionJohannes Kepler, Astronomia nova ... (1609), p. 5 of the Introductio in hoc opus{{cite web |last1=Popova |first1=Maria |url=https://www.brainpickings.org/2019/12/26/katharina-kepler-witchcraft-dream/ |website=Brain Pickings |access-date=27 December 2020|title=How Kepler Invented Science Fiction ... While Revolutionizing Our Understanding of the Universe|date=27 December 2019 }}{{cite journal |last1=Eugene |first1=Hecht |title=Kepler and the origins of the theory of gravity |journal=American Journal of Physics |year=2019 |volume=87 |issue=3 |pages=176–185 |doi=10.1119/1.5089751 |bibcode=2019AmJPh..87..176H |s2cid=126889093 |url=https://aapt.scitation.org/doi/10.1119/1.5089751}} basing his argument upon ancient observations and correlations.

In 1616, Galileo Galilei wrote Discourse on 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 He strongly and mockingly rejects the lunar theory of the tides, and tries to explain the tides as the result of the Earth's rotation and revolution around the Sun, believing that the oceans moved like water in a large basin: as the basin moves, so does the water.{{cite web|last=Tyson|first=Peter|title=Galileo's Big Mistake|url=https://www.pbs.org/wgbh/nova/earth/galileo-big-mistake.html|website=NOVA|date=29 October 2002 |publisher=PBS|access-date=2014-02-19}} But his contemporaries noticed that this made predictions that did not fit observations.{{cite journal|last1=Naylor|first1=Ron|title=Galileo's Tidal Theory|journal=Isis|volume=98|issue=1|pages=1–22|year=2007|doi=10.1086/512829|pmid=17539198|bibcode=2007Isis...98....1N|s2cid=46174715}}

René Descartes theorized that the tides (alongside the movement of planets, etc.) were caused by aetheric vortices, without reference to Kepler's theories of gravitation by mutual attraction; this was extremely influential, with numerous followers of Descartes expounding on this theory throughout the 17th century, particularly in France.{{cite journal |last1=Aiton |first1=E.J. |title=Descartes's theory of the tides |journal=Annals of Science |year=1955 |volume=11 |issue=4 |pages=337–348 |doi=10.1080/00033795500200335 |url=https://www.tandfonline.com/doi/abs/10.1080/00033795500200335?journalCode=tasc20}} However, Descartes and his followers acknowledged the influence of the Moon, speculating that pressure waves from the Moon via the aether were responsible for the correlation.{{Cite web |url=http://people.bu.edu/dklepper/RN242/voltaire.html |title=Voltaire, Letter XIV |access-date=28 December 2020 |archive-date=13 April 2021 |archive-url=https://web.archive.org/web/20210413042930/http://people.bu.edu/dklepper/RN242/voltaire.html |url-status=dead }}{{cite news |title=Understanding Tides—From Ancient Beliefs toPresent-day Solutions to the Laplace Equations |volume=33 |url=https://archive.siam.org/pdf/news/621.pdf |issue=2 |publisher=SIAM News}}{{cite book |last1=Cartwright |first1=David Edgar |title=Tides: A Scientific History |page=31 |url=https://books.google.com/books?id=78bE5U7TVuIC&q=descartes+tides&pg=PA31 |access-date=28 December 2020|date=1999 |publisher=Cambridge University Press|isbn=9780521797467 }}

File:Newton's three-body diagram.PNG

Newton, in the Principia, provides 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.{{cite web |url=http://web.vims.edu/physical/research/TCTutorial/static.htm |title=Static Tides – the equilibrium Theory |access-date=2014-04-14 |url-status=dead |archive-url=https://web.archive.org/web/20140410173511/http://web.vims.edu/physical/research/TCTutorial/static.htm |archive-date=10 April 2014}}

While Newton explained the tides by describing the tide-generating forces and Daniel Bernoulli gave a description of the static reaction of the waters on Earth to the tidal potential, the dynamic theory of tides, developed by Pierre-Simon Laplace in 1775,{{Cite web|url = http://www.preservearticles.com/2011112017524/short-notes-on-the-dynamical-theory-of-laplace.html|title = Short notes on the Dynamical theory of Laplace|date = 20 November 2011|access-date = 31 March 2015|archive-date = 2 April 2015|archive-url = https://web.archive.org/web/20150402155415/http://www.preservearticles.com/2011112017524/short-notes-on-the-dynamical-theory-of-laplace.html|url-status = dead}} describes the ocean's real reaction to tidal forces.{{Cite web | url=http://faculty.washington.edu/luanne/pages/ocean420/notes/tidedynamics.pdf | title=Tide Dynamics | website=faculty.washington.edu}} Laplace's theory of ocean tides takes into account friction, resonance and natural periods of ocean basins. It predicts the large amphidromic systems in the world's ocean basins and explains the oceanic tides that are actually observed.{{Cite journal | url=http://ocean.kisti.re.kr/downfile/volume/kess/JGGHBA/2009/v30n5/JGGHBA_2009_v30n5_671.pdf | title=An Astronomer's View on the Current College-Level Textbook Descriptions of Tides | journal=Journal of the Korean Earth Science Society | volume=30 | number=5 | date=September 2009 | pages=671–681| doi=10.5467/JKESS.2009.30.5.671 | last1=Ahn | first1=Kyung-Jin }}

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.Bryden, I.G. (2003). "Tidal Power Systems", in Meyers, R.A. (ed.) Encyclopedia of Physical Science and Technology. Aberdeen: Academic Press, p. 753. doi:[https://www.doi.org/10.1016/b0-12-227410-5/00778-x 10.1016/b0-12-227410-5/00778-x] Since measurements have confirmed the dynamic 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.{{cite web|author=Floor Anthoni |url=http://www.seafriends.org.nz/oceano/tides.htm |title=Tides |publisher=Seafriends.org.nz |access-date=2012-06-02}} 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.{{Cite web|url=https://www.linz.govt.nz/guidance/marine-information/tides/cause-and-nature-tides|title=The Cause & Nature of Tides}}

Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within a few centimeters.{{cite web|url=http://svs.gsfc.nasa.gov/stories/topex/tides.html |title=Scientific Visualization Studio TOPEX/Poseidon images |publisher=Svs.gsfc.nasa.gov |access-date=2012-06-02}}{{cite web|url=https://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|date=15 June 2000 }} Measurements from the CHAMP satellite closely match the models based on the TOPEX data.{{cite web|title=TOPEX data used to model actual tides for 15 days from the year 2000|date=15 June 2000|url=http://svs.gsfc.nasa.gov/vis/a000000/a001300/a001332/|access-date=14 September 2015|archive-date=18 September 2015|archive-url=https://web.archive.org/web/20150918082009/http://svs.gsfc.nasa.gov/vis/a000000/a001300/a001332/|url-status=dead}}http://www.geomag.us/info/Ocean/m2_CHAMP+longwave_SSH.swf{{cite web |url=http://volkov.oce.orst.edu/tides/ |title=OSU Tidal Data Inversion |publisher=Volkov.oce.orst.edu |access-date=2012-06-02 |archive-date=22 October 2012 |archive-url=https://wayback.archive-it.org/all/20121022041453/http://volkov.oce.orst.edu/tides/ |url-status=dead }} 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.{{cite web|url= http://www.dgfi.tum.de/en/news/baroclinic-tides/|title= Dynamic and residual ocean tide analysis for improved GRACE de-aliasing (DAROTA)|url-status= dead|archive-url= https://web.archive.org/web/20150402194935/http://www.dgfi.tum.de/en/news/baroclinic-tides/|archive-date= 2 April 2015|df= dmy-all}}{{clear right}}

====Laplace's tidal equations====

{{multiple image

|width=150

|image1=Tideforcenw.jpg|caption1=A. Lunar gravitational potential: this depicts the Moon directly over 30° N (or 30° S) viewed from above the Northern Hemisphere. Note however that the moon is never more than about 28.6° north of the equator.

|image2=Tideforcese.jpg|caption2=B. This view shows same potential from 180° from view A. Viewed from above the Northern Hemisphere. Red up, blue down.

}}

In 1776, Laplace formulated a single set of linear partial differential equations for tidal flow described as a barotropic two-dimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity. Laplace obtained these equations by simplifying the fluid dynamics 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 ζ, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations:{{cite web|url= https://hogback.atmos.colostate.edu/group/dave/pdf/LTE.frame.pdf|title= The Laplace Tidal Equations and Atmospheric Tides}}

:$$

\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 \, 2 \Omega \sin( \varphi )

+ \frac{1}{a \cos( \varphi )} \frac{\partial}{\partial \lambda} \left( g \zeta + U \right)

= 0,

\quad \text{and} \\[2ex]

\frac{\partial v}{\partial t}

&+ u \, 2 \Omega \sin( \varphi )

+ \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 tidal-forcing 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.

File:Tides Fourier Transform.png

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 Lord Kelvin's application of Fourier analysis to the tidal motions as harmonic analysis. Thomson's work in this field was further developed and extended by George Darwin, applying the lunar theory current in his time. {{vanchor|Darwin's symbols}} for the tidal harmonic constituents are still used, for example: M: moon/lunar; S: sun/solar; K: moon-sun/lunisolar.

Darwin's harmonic developments of the tide-generating forces were later improved when A.T. Doodson, applying the lunar theory of E.W. Brown,{{cite book |last1=Cartwright |first1=David Edgar |title=Tides: A Scientific History |pages=163–164 |url=https://books.google.com/books?id=sSesNjG96JYC&pg=PA163|date=1999 |publisher=Cambridge University Press|isbn=9780521797467 }} developed the tide-generating potential (TGP) in harmonic form, distinguishing 388 tidal frequencies.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. Doodson's work was carried out and published in 1921.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 (1 December 1921), pp. 305–329. Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the Doodson numbers, a system still in use.

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 fewer 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. More recently digital computers, using the method of matrix inversion, are used to determine the tidal harmonic constituents directly from tide gauge records.

{{further|Tide#Constituents}}

File:Tidal constituent sum.gif

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 (half of peak-to-peak amplitude) of tidal constituents are given below for six example locations:

Eastport, Maine (ME),{{cite web|last=NOAA|title=Eastport, ME Tidal Constituents|url=http://tidesandcurrents.noaa.gov/cgi-bin/co-ops_qry.cgi?stn=8410140%20Eastport,%20ME&pc=P2&unit=0&date=1&shift=0&level=-4&form=0&format=View+Data|publisher=NOAA|access-date=2012-05-22}} Biloxi, Mississippi (MS), San Juan, Puerto Rico (PR), Kodiak, Alaska (AK), San Francisco, California (CA), and Hilo, Hawaii (HI).

In order to specify the different harmonic components of the tide-generating potential, Doodson devised a practical system which is still in use, involving what are called the Doodson numbers based on the six Doodson arguments or Doodson variables. The number of different tidal frequency components is large, but each corresponds to a specific linear combination of six frequencies using small-integer multiples, positive or negative. In principle, these basic angular arguments can be specified in numerous ways; Doodson's choice of his six "Doodson arguments" has been widely used in tidal work. In terms of these Doodson arguments, each tidal frequency can then be specified as a sum made up of a small integer multiple of each of the six arguments. The resulting six small integer multipliers effectively encode the frequency of the tidal argument concerned, and these are the Doodson numbers: in practice all except the first are usually biased upwards by +5 to avoid negative numbers in the notation. (In the case that the biased multiple exceeds 9, the system adopts X for 10, and E for 11.)

The Doodson arguments are specified in the following way, in order of decreasing frequency:{{Cite journal | last1 = Melchior | first1 = P. | title = Precession-nutations and tidal potential | doi = 10.1007/BF01228823 | journal = Celestial Mechanics | volume = 4 | issue = 2 | pages = 190–212 | year = 1971 |bibcode = 1971CeMec...4..190M | s2cid = 126219362 }} and T D Moyer (2003) already cited.

:$\backslash beta\_1\; =\; \backslash tau\; =\; (\; \backslash theta\_M\; +\; \backslash pi\; -\; s\; )$ is mean Lunar time, the Greenwich hour angle of the mean Moon plus 12 hours.

:$\backslash beta\_2\; =\; s\; =\; (\; F\; +\; \backslash Omega\; )$ is the mean longitude of the Moon.

:$\backslash beta\_3\; =\; h\; =\; (\; s\; -\; D\; )$ is the mean longitude of the Sun.

:$\backslash beta\_4\; =\; p\; =\; (\; s\; -\; l\; )$ is the longitude of the Moon's mean perigee.

:$\backslash beta\_5\; =\; N\text{'}\; =\; (\; -\backslash Omega\; )$ is the negative of the longitude of the Moon's mean ascending node on the ecliptic.

:$\backslash beta\_6\; =\; p\_l$ or $p\_s\; =\; (\; s\; -\; D\; -\; l\text{'}\; )$ is the longitude of the Sun's mean perigee.

In these expressions, the symbols $l$, $l\text{'}$, $F$ and $D$ refer to an alternative set of fundamental angular arguments (usually preferred for use in modern lunar theory), in which:-

:$l$ is the mean anomaly of the Moon (distance from its perigee).

:$l\text{'}$ is the mean anomaly of the Sun (distance from its perigee).

:$F$ is the Moon's mean argument of latitude (distance from its node).

:$D$ is the Moon's mean elongation (distance from the sun).

It is possible to define several auxiliary variables on the basis of combinations of these.

In terms of this system, each tidal constituent frequency can be identified by its Doodson numbers. The strongest tidal constituent "M₂" has a frequency of 2 cycles per lunar day, its Doodson numbers are usually written 255.555, meaning that its frequency is composed of twice the first Doodson argument, and zero times all of the others. The second strongest tidal constituent "S₂" is influenced by the sun, and its Doodson numbers are 273.555, meaning that its frequency is composed of twice the first Doodson argument, +2 times the second, -2 times the third, and zero times each of the other three.See for example Melchior (1971), already cited, at p.191. This aggregates to the angular equivalent of mean solar time +12 hours. These two strongest component frequencies have simple arguments for which the Doodson system might appear needlessly complex, but each of the hundreds of other component frequencies can be briefly specified in a similar way, showing in the aggregate the usefulness of the encoding.

• Long-period tides
• Lunar node § Effect on tides
• Kelvin wave
• Tide table

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• [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] {{Webarchive|url=https://web.archive.org/web/20130728163045/http://www.eas.slu.edu/GGP/BIM_Recent_Issues/bim134-2001/bim134_report_WG6_SLR.pdf |date=28 July 2013 }}
• [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://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]
• [https://geomatix.net/tides/geotide.htm GeoTide Tidal Analysis System]

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{{Physical oceanography}}

{{DEFAULTSORT:Theory Of Tides}}

Category:Tides

Category:Geophysics

Category:Oceanography

Category:Continuum mechanics

Category:Fluid dynamics

Category:Fluid mechanics

Category:Planetary science