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id="toc-Tidal_constituents" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tidal_constituents"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Tidal constituents</span> </div> </a> <ul id="toc-Tidal_constituents-sublist" class="vector-toc-list"> <li id="toc-Semi-diurnal" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Semi-diurnal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Semi-diurnal</span> </div> </a> <ul id="toc-Semi-diurnal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diurnal" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Diurnal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Diurnal</span> </div> </a> <ul id="toc-Diurnal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Long_period" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Long_period"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Long period</span> </div> </a> <ul id="toc-Long_period-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Short_period" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Short_period"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.4</span> <span>Short period</span> </div> </a> <ul id="toc-Short_period-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Doodson_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Doodson_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Doodson numbers</span> </div> </a> <ul id="toc-Doodson_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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decoding="async" width="400" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/54/Bay_of_Fundy.jpg 1.5x" data-file-width="600" data-file-height="450" /></a><figcaption>High and low tide in the <a href="/wiki/Bay_of_Fundy" title="Bay of Fundy">Bay of Fundy</a></figcaption></figure> <p>The <b>theory of tides</b> is the application of <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a> to interpret and predict the <a href="/wiki/Tide" title="Tide">tidal</a> deformations of planetary and satellite bodies and their atmospheres and <a href="/wiki/Ocean" title="Ocean">oceans</a> (especially Earth's oceans) under the gravitational loading of another astronomical body or bodies (especially the <a href="/wiki/Moon" title="Moon">Moon</a> and <a href="/wiki/Sun" title="Sun">Sun</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Australian_Aboriginal_astronomy">Australian Aboriginal astronomy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=2" title="Edit section: Australian Aboriginal astronomy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Yolngu" title="Yolngu">Yolngu people</a> of northeastern <a href="/wiki/Arnhem_Land" title="Arnhem Land">Arnhem Land</a> in the <a href="/wiki/Northern_Territory" title="Northern Territory">Northern Territory</a> 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.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Classical_era">Classical era</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=3" title="Edit section: Classical era"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The tides received relatively little attention in the civilizations around the <a href="/wiki/Mediterranean_Sea" title="Mediterranean Sea">Mediterranean Sea</a>, as the tides there are relatively small, and the areas that experience tides do so unreliably.<sup id="cite_ref-Tabarroni_3-0" class="reference"><a href="#cite_note-Tabarroni-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SciMonthly_4-0" class="reference"><a href="#cite_note-SciMonthly-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Pugh_5-0" class="reference"><a href="#cite_note-Pugh-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A number of theories were advanced, however, from comparing the movements to breathing or blood flow to theories involving <a href="/wiki/Whirlpool" title="Whirlpool">whirlpools</a> or river cycles.<sup id="cite_ref-SciMonthly_4-1" class="reference"><a href="#cite_note-SciMonthly-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A similar "breathing earth" idea was considered by some Asian thinkers.<sup id="cite_ref-Siam_6-0" class="reference"><a href="#cite_note-Siam-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Plato" title="Plato">Plato</a> reportedly believed that the tides were caused by water flowing in and out of undersea caverns.<sup id="cite_ref-Tabarroni_3-1" class="reference"><a href="#cite_note-Tabarroni-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Crates_of_Mallus" title="Crates of Mallus">Crates of Mallus</a> attributed the tides to "the counter-movement (ἀντισπασμός) of the sea” and <a href="/w/index.php?title=Apollodorus_of_Corcyra&action=edit&redlink=1" class="new" title="Apollodorus of Corcyra (page does not exist)">Apollodorus of Corcyra</a> to "the refluxes from the Ocean".<sup id="cite_ref-:0_7-0" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> An ancient Indian <a href="/wiki/Puranas" title="Puranas">Purana</a> text dated to 400-300 BC refers to the ocean rising and falling because of <a href="/wiki/Thermal_expansion" title="Thermal expansion">heat expansion</a> from the light of the Moon.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Ultimately the link between the <a href="/wiki/Moon" title="Moon">Moon</a> (and <a href="/wiki/Sun" title="Sun">Sun</a>) and the tides became known to the <a href="/wiki/Ancient_Greece" title="Ancient Greece">Greeks</a>, although the exact date of discovery is unclear; references to it are made in sources such as <a href="/wiki/Pytheas_of_Massilia" class="mw-redirect" title="Pytheas of Massilia">Pytheas of Massilia</a> in 325 BC and <a href="/wiki/Pliny_the_Elder" title="Pliny the Elder">Pliny the Elder</a>'s <a href="/wiki/Natural_History_(Pliny)" title="Natural History (Pliny)"><i>Natural History</i></a> 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.<sup id="cite_ref-SciMonthly_4-2" class="reference"><a href="#cite_note-SciMonthly-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Classicists <a href="/wiki/Thomas_Little_Heath" class="mw-redirect" title="Thomas Little Heath">Thomas Little Heath</a> claimed that both Pytheas and <a href="/wiki/Posidonius" title="Posidonius">Posidonius</a> connected the tides with the moon, "the former directly, the latter through the setting up of winds".<sup id="cite_ref-:0_7-1" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Seneca_the_Younger" title="Seneca the Younger">Seneca</a> mentions in <i><a href="/wiki/De_Providentia" title="De Providentia">De Providentia</a></i> the periodic motion of the tides controlled by the lunar sphere.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Eratosthenes" title="Eratosthenes">Eratosthenes</a> (3rd century BC) and Posidonius (1st century BC) both produced detailed descriptions of the tides and their relationship to the <a href="/wiki/Lunar_phase" title="Lunar phase">phases of the Moon</a>, 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 <a href="/wiki/Ptolemy" title="Ptolemy">Ptolemy</a>'s <i>Tetrabiblos</i> as evidence of the reality of <a href="/wiki/Astrology" title="Astrology">astrology</a>.<sup id="cite_ref-Tabarroni_3-2" class="reference"><a href="#cite_note-Tabarroni-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Origins_11-0" class="reference"><a href="#cite_note-Origins-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Seleucus_of_Seleucia" title="Seleucus of Seleucia">Seleucus of Seleucia</a> is thought to have theorized around 150 BC that tides were caused by the Moon as part of his <a href="/wiki/Heliocentrism" title="Heliocentrism">heliocentric</a> model.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, 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.<sup id="cite_ref-Tabarroni_3-3" class="reference"><a href="#cite_note-Tabarroni-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Heraclides_Ponticus" title="Heraclides Ponticus">Heraclides</a> also held "the <a href="/wiki/Sun" title="Sun">sun</a> sets up winds, and that these winds, when they blow, cause the high tide and, when they cease, the low tide".<sup id="cite_ref-:0_7-2" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Dicaearchus" title="Dicaearchus">Dicaearchus</a> also "put the tides down to the direct action of the sun according to its position".<sup id="cite_ref-:0_7-3" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Philostratus" title="Philostratus">Philostratus</a> discusses tides in Book Five of <i><a href="/wiki/Life_of_Apollonius_of_Tyana" title="Life of Apollonius of Tyana">Life of Apollonius of Tyana</a></i> (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.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Medieval_period">Medieval period</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=4" title="Edit section: Medieval period"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Venerable_Bede" class="mw-redirect" title="Venerable Bede">Venerable Bede</a> discusses the tides in <i><a href="/wiki/The_Reckoning_of_Time" title="The Reckoning of Time">The Reckoning of Time</a></i> 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.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> However, he made no progress regarding the question of how exactly the Moon created the tides.<sup id="cite_ref-SciMonthly_4-3" class="reference"><a href="#cite_note-SciMonthly-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Medieval <a href="/wiki/Rule_of_thumb" title="Rule of thumb">rule-of-thumb</a> methods for predicting tides were said to allow one "to know what Moon makes high water" from the Moon's movements.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Dante_Alighieri" title="Dante Alighieri">Dante</a> references the Moon's influence on the tides in his <i><a href="/wiki/Divine_Comedy" title="Divine Comedy">Divine Comedy</a></i>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tabarroni_3-4" class="reference"><a href="#cite_note-Tabarroni-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Medieval European understanding of the tides was often based on works of <a href="/wiki/Astronomy_in_the_medieval_Islamic_world" title="Astronomy in the medieval Islamic world">Muslim astronomers</a> that became available through <a href="/wiki/Latin_translations_of_the_12th_century" title="Latin translations of the 12th century">Latin translation</a> starting from the 12th century.<sup id="cite_ref-Tolmacheva_18-0" class="reference"><a href="#cite_note-Tolmacheva-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Abu_Ma%27shar_al-Balkhi" title="Abu Ma'shar al-Balkhi">Abu Ma'shar al-Balkhi</a>, in his <i>Introductorium in astronomiam</i>, taught that ebb and flood tides were caused by the Moon.<sup id="cite_ref-Tolmacheva_18-1" class="reference"><a href="#cite_note-Tolmacheva-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Abu Ma'shar discussed the effects of wind and Moon's phases relative to the Sun on the tides.<sup id="cite_ref-Tolmacheva_18-2" class="reference"><a href="#cite_note-Tolmacheva-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In the 12th century, <a href="/wiki/Nur_ad-Din_al-Bitruji" title="Nur ad-Din al-Bitruji">al-Bitruji</a> contributed the notion that the tides were caused by the general circulation of the heavens.<sup id="cite_ref-Tolmacheva_18-3" class="reference"><a href="#cite_note-Tolmacheva-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-Origins_11-1" class="reference"><a href="#cite_note-Origins-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Tabarroni_3-5" class="reference"><a href="#cite_note-Tabarroni-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Some theorized that the influence was caused by lunar rays heating the ocean's floor.<sup id="cite_ref-Pugh_5-1" class="reference"><a href="#cite_note-Pugh-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Modern_era">Modern era</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=5" title="Edit section: Modern era"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a> in his 1608 <i>De spiegheling der Ebbenvloet (The Theory of Ebb and Flood</i>) 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.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> In 1609, <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> correctly suggested that the gravitation of the Moon causes the tides,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> which he compared to <a href="/wiki/Magnetism" title="Magnetism">magnetic</a> attraction<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SciMonthly_4-4" class="reference"><a href="#cite_note-SciMonthly-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-BrainPick_24-0" class="reference"><a href="#cite_note-BrainPick-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> basing his argument upon ancient observations and correlations. </p><p>In 1616, <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> wrote <i><a href="/wiki/Discourse_on_the_Tides" title="Discourse on the Tides">Discourse on the Tides</a>.</i><sup id="cite_ref-Galileo's_Theory_of_the_Tides_26-0" class="reference"><a href="#cite_note-Galileo's_Theory_of_the_Tides-26"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> He strongly and mockingly rejects the lunar theory of the tides,<sup id="cite_ref-BrainPick_24-1" class="reference"><a href="#cite_note-BrainPick-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SciMonthly_4-5" class="reference"><a href="#cite_note-SciMonthly-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and tries to explain the tides as the result of the <a href="/wiki/Earth" title="Earth">Earth</a>'s rotation and <a href="/wiki/Heliocentrism" title="Heliocentrism">revolution around the Sun</a>, believing that the oceans moved like water in a large basin: as the basin moves, so does the water.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Therefore, as the Earth revolves, the force of the Earth's rotation causes the oceans to "alternately accelerate and retardate".<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> 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."<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> However, Galileo's theory was erroneous.<sup id="cite_ref-Galileo's_Theory_of_the_Tides_26-1" class="reference"><a href="#cite_note-Galileo's_Theory_of_the_Tides-26"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> In subsequent centuries, <a href="/wiki/Tide#Analysis" title="Tide">further analysis</a> led to the current tidal physics. Galileo tried to use his tidal theory to prove the movement of the Earth around the Sun. Galileo theorized that because of the Earth's motion, borders of the oceans like the Atlantic and Pacific would show one high tide and one low tide per day. The Mediterranean Sea had two high tides and low tides, though Galileo argued that this was a product of secondary effects and that his theory would hold in the Atlantic. However, Galileo's contemporaries noted that the Atlantic also had two high tides and low tides per day, which led to Galileo omitting this claim from his 1632 <i>Dialogue</i>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> theorized that the tides (alongside the movement of planets, etc.) were caused by <a href="/wiki/Mechanical_explanations_of_gravitation#Vortex" title="Mechanical explanations of gravitation">aetheric vortices</a>, 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.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> However, Descartes and his followers acknowledged the influence of the Moon, speculating that pressure waves from the Moon via the <a href="/wiki/Aether_(classical_element)" title="Aether (classical element)">aether</a> were responsible for the correlation.<sup id="cite_ref-Pugh_5-2" class="reference"><a href="#cite_note-Pugh-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Siam_6-1" class="reference"><a href="#cite_note-Siam-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Newton%27s_three-body_diagram.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Newton%27s_three-body_diagram.PNG/350px-Newton%27s_three-body_diagram.PNG" decoding="async" width="350" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/79/Newton%27s_three-body_diagram.PNG 1.5x" data-file-width="471" data-file-height="297" /></a><figcaption>Newton's three-body model</figcaption></figure> <p><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>, in the <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i>, provides a correct explanation for the <a href="/wiki/Tidal_force" title="Tidal force">tidal force</a>, 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 <a href="/wiki/Bathymetry" title="Bathymetry">bathymetry</a>.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Dynamic_theory">Dynamic theory</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=6" title="Edit section: Dynamic theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While Newton explained the tides by describing the tide-generating forces and <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> gave a description of the static reaction of the waters on Earth to the tidal potential, the <i>dynamic theory of tides</i>, developed by <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> in 1775,<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> describes the ocean's real reaction to tidal forces.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Laplace's theory of ocean tides takes into account <a href="/wiki/Friction" title="Friction">friction</a>, <a href="/wiki/Resonance" title="Resonance">resonance</a> and natural periods of ocean basins. It predicts the large <a href="/wiki/Amphidromic_point" title="Amphidromic point">amphidromic</a> systems in the world's ocean basins and explains the oceanic tides that are actually observed.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>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.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> <br /> </p><p> Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within a few centimeters.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Measurements from the <a href="/wiki/CHAMP_(satellite)" title="CHAMP (satellite)">CHAMP</a> satellite closely match the models based on the <a href="/wiki/TOPEX" class="mw-redirect" title="TOPEX">TOPEX</a> data.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup></p><div style="clear:right;" class=""></div> <div class="mw-heading mw-heading4"><h4 id="Laplace's_tidal_equations"><span id="Laplace.27s_tidal_equations"></span>Laplace's tidal equations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=7" title="Edit section: Laplace's tidal equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:308px;max-width:308px"><div class="trow"><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Tideforcenw.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Tideforcenw.jpg/150px-Tideforcenw.jpg" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Tideforcenw.jpg/225px-Tideforcenw.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Tideforcenw.jpg/300px-Tideforcenw.jpg 2x" data-file-width="360" data-file-height="359" /></a></span></div><div class="thumbcaption"><i>A.</i> 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.</div></div><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Tideforcese.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Tideforcese.jpg/150px-Tideforcese.jpg" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Tideforcese.jpg/225px-Tideforcese.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Tideforcese.jpg/300px-Tideforcese.jpg 2x" data-file-width="360" data-file-height="359" /></a></span></div><div class="thumbcaption"><i>B.</i> This view shows same potential from 180° from view <i>A</i>. Viewed from above the Northern Hemisphere. Red up, blue down.</div></div></div></div></div> <p>In 1776, Laplace formulated a single set of linear <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a> for tidal flow described as a <a href="/wiki/Barotropic_fluid" title="Barotropic fluid">barotropic</a> two-dimensional sheet flow. <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis effects</a> are introduced as well as lateral forcing by <a href="/wiki/Gravity" title="Gravity">gravity</a>. Laplace obtained these equations by simplifying the <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a> equations, but they can also be derived from energy integrals via <a href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Lagrange's equation</a>. </p><p>For a fluid sheet of average thickness <i>D</i>, the vertical tidal elevation <i>ζ</i>, as well as the horizontal velocity components <i>u</i> and <i>v</i> (in the <a href="/wiki/Latitude" title="Latitude">latitude</a> <i>φ</i> and <a href="/wiki/Longitude" title="Longitude">longitude</a> <i>λ</i> directions, respectively) satisfy <b>Laplace's tidal equations</b>:<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\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{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.16em 1.16em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ζ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mi>D</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mi>D</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>−</mo> <mi>v</mi> <mspace width="thinmathspace" /> <mn>2</mn> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mi>ζ</mi> <mo>+</mo> <mi>U</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>+</mo> <mi>u</mi> <mspace width="thinmathspace" /> <mn>2</mn> <mi mathvariant="normal">Ω</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>a</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mi>ζ</mi> <mo>+</mo> <mi>U</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\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{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fa08ee6fd9993a354e58f22257c7847c90666d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:53.642ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}{\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{aligned}}}"></span></dd></dl> <p>where <i>Ω</i> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of the planet's rotation, <i>g</i> is the planet's <a href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a> at the mean ocean surface, <i>a</i> is the planetary radius, and <i>U</i> is the external gravitational tidal-forcing <a href="/wiki/Potential" title="Potential">potential</a>. </p><p><a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson (Lord Kelvin)</a> rewrote Laplace's momentum terms using the <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> to find an equation for <a href="/wiki/Vorticity" title="Vorticity">vorticity</a>. Under certain conditions this can be further rewritten as a conservation of vorticity. </p> <div class="mw-heading mw-heading2"><h2 id="Tidal_analysis_and_prediction">Tidal analysis and prediction<span class="anchor" id="Tidal_analysis"></span><span class="anchor" id="Tidal_prediction"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=8" title="Edit section: Tidal analysis and prediction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Harmonic_analysis">Harmonic analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=9" title="Edit section: Harmonic analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Tides_Fourier_Transform.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Tides_Fourier_Transform.png/300px-Tides_Fourier_Transform.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Tides_Fourier_Transform.png/450px-Tides_Fourier_Transform.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Tides_Fourier_Transform.png/600px-Tides_Fourier_Transform.png 2x" data-file-width="1200" data-file-height="900" /></a><figcaption>Spectrum of tides measured at Ft. Pulaski in 2012. Data downloaded from <a rel="nofollow" class="external free" href="http://tidesandcurrents.noaa.gov/datums.html?id=8670870">http://tidesandcurrents.noaa.gov/datums.html?id=8670870</a> Fourier transform computed with <a rel="nofollow" class="external free" href="https://sourceforge.net/projects/amoreaccuratefouriertransform/">https://sourceforge.net/projects/amoreaccuratefouriertransform/</a></figcaption></figure> <p>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 <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson</a>'s application of <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> to the tidal motions as <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>. Thomson's work in this field was further developed and extended by <a href="/wiki/George_Darwin" title="George Darwin">George Darwin</a>, applying the lunar theory current in his time. <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="Darwin's_symbols"></span><span class="vanchor-text">Darwin's symbols</span></span></b> for the tidal harmonic constituents are still used, for example: <i>M</i>: moon/lunar; <i>S</i>: sun/solar; <i>K</i>: moon-sun/lunisolar. </p><p>Darwin's harmonic developments of the tide-generating forces were later improved when <a href="/wiki/Arthur_Thomas_Doodson" title="Arthur Thomas Doodson">A.T. Doodson</a>, applying the <a href="/wiki/Lunar_theory" title="Lunar theory">lunar theory</a> of <a href="/wiki/Ernest_William_Brown" title="Ernest William Brown">E.W. Brown</a>,<sup id="cite_ref-dec2001_48-0" class="reference"><a href="#cite_note-dec2001-48"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> developed the tide-generating potential (TGP) in harmonic form, distinguishing 388 tidal frequencies.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Doodson's work was carried out and published in 1921.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> Doodson devised a practical system for specifying the different harmonic components of the tide-generating potential, the <a href="/wiki/Doodson_numbers" class="mw-redirect" title="Doodson numbers">Doodson numbers</a>, a system still in use. </p><p>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 <a href="/wiki/Tide-predicting_machine" title="Tide-predicting machine">tide-predicting machine</a>, a special-purpose form of <a href="/wiki/Analog_computer" title="Analog computer">analog computer</a>. More recently digital computers, using the method of matrix inversion, are used to determine the tidal harmonic constituents directly from tide gauge records. </p> <div class="mw-heading mw-heading3"><h3 id="Tidal_constituents">Tidal constituents</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=10" title="Edit section: Tidal constituents"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Tide#Constituents" title="Tide">Tide § Constituents</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Tidal_constituent_sum.gif" class="mw-file-description"><img alt="Graph showing one line each for M 2, S 2, N 2, K 1, O 1, P 1, and one for their summation, with the X axis spanning slightly more than a single day" src="//upload.wikimedia.org/wikipedia/commons/0/0e/Tidal_constituent_sum.gif" decoding="async" width="253" height="322" class="mw-file-element" data-file-width="253" data-file-height="322" /></a><figcaption>Tidal prediction summing constituent parts.</figcaption></figure> <p>Tidal constituents combine to give an endlessly varying aggregate because of their different and incommensurable frequencies: the effect is visualized in an <a rel="nofollow" class="external text" href="http://www.ams.org/featurecolumn/archive/tidesIII3.html">animation of the American Mathematical Society</a> illustrating the way in which the components used to be mechanically combined in the tide-predicting machine. Amplitudes (half of <a href="/wiki/Peak-to-peak_amplitude" class="mw-redirect" title="Peak-to-peak amplitude">peak-to-peak amplitude</a>) of tidal constituents are given below for six example locations: <a href="/wiki/Eastport,_Maine" title="Eastport, Maine">Eastport, Maine</a> (ME),<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Biloxi,_Mississippi" title="Biloxi, Mississippi">Biloxi, Mississippi</a> (MS), <a href="/wiki/San_Juan,_Puerto_Rico" title="San Juan, Puerto Rico">San Juan, Puerto Rico</a> (PR), <a href="/wiki/Kodiak,_Alaska" title="Kodiak, Alaska">Kodiak, Alaska</a> (AK), <a href="/wiki/San_Francisco,_California" class="mw-redirect" title="San Francisco, California">San Francisco, California</a> (CA), and <a href="/wiki/Hilo,_Hawaii" title="Hilo, Hawaii">Hilo, Hawaii</a> (HI). </p> <div class="mw-heading mw-heading4"><h4 id="Semi-diurnal">Semi-diurnal</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=11" title="Edit section: Semi-diurnal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="vertical-align:top; align:left"> <tbody><tr> <th rowspan="2" style="background-color:lightblue;">Species </th> <th rowspan="2" style="background-color:lightblue;">Darwin <br />symbol </th> <th rowspan="2" style="background-color:lightblue;">Period <br />(h) </th> <th rowspan="2" style="background-color:lightblue;">Speed <br />(°/h) </th> <th colspan="4" style="background-color:lightblue;">Doodson coefficients </th> <th rowspan="2" style="background-color:lightblue;">Doodson <br />number </th> <th colspan="6" style="background-color:lightblue;">Amplitude at example location (cm) </th> <th rowspan="2" style="background-color:lightblue;">NOAA <br />order </th></tr> <tr> <th style="background-color:lightblue;"><i>n</i><sub>1</sub> (<i>L</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>2</sub> (<i>m</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>3</sub> (<i>y</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>4</sub> (<i>mp</i>) </th> <th style="background-color:lightblue;">ME </th> <th style="background-color:lightblue;">MS </th> <th style="background-color:lightblue;">PR </th> <th style="background-color:lightblue;">AK </th> <th style="background-color:lightblue;">CA </th> <th style="background-color:lightblue;">HI </th></tr> <tr> <td><b>Principal lunar semidiurnal </b> </td> <td><i>M</i><sub>2</sub> </td> <td>12.4206012 </td> <td>28.9841042 </td> <td>2 </td> <td> </td> <td> </td> <td> </td> <td>255.555 </td> <td>268.7 </td> <td>3.9 </td> <td>15.9 </td> <td>97.3 </td> <td>58.0 </td> <td>23.0 </td> <td><b>1</b> </td></tr> <tr> <td><b>Principal solar semidiurnal </b> </td> <td><i>S</i><sub>2</sub> </td> <td>12 </td> <td>30 </td> <td>2 </td> <td>2 </td> <td>−2 </td> <td> </td> <td>273.555 </td> <td>42.0 </td> <td>3.3 </td> <td>2.1 </td> <td>32.5 </td> <td>13.7 </td> <td>9.2 </td> <td><b>2</b> </td></tr> <tr> <td><b>Larger lunar elliptic semidiurnal</b> </td> <td><i>N</i><sub>2</sub> </td> <td>12.65834751 </td> <td>28.4397295 </td> <td>2 </td> <td>−1 </td> <td> </td> <td>1 </td> <td>245.655 </td> <td>54.3 </td> <td>1.1 </td> <td>3.7 </td> <td>20.1 </td> <td>12.3 </td> <td>4.4 </td> <td><b>3</b> </td></tr> <tr> <td>Larger lunar evectional </td> <td><i>ν</i><sub>2</sub> </td> <td>12.62600509 </td> <td>28.5125831 </td> <td>2 </td> <td>−1 </td> <td>2 </td> <td>−1 </td> <td>247.455 </td> <td>12.6 </td> <td>0.2 </td> <td>0.8 </td> <td>3.9 </td> <td>2.6 </td> <td>0.9 </td> <td>11 </td></tr> <tr> <td>Variational </td> <td><i>μ</i><sub>2</sub> </td> <td>12.8717576 </td> <td>27.9682084 </td> <td>2 </td> <td>−2 </td> <td>2 </td> <td> </td> <td>237.555 </td> <td>2.0 </td> <td>0.1 </td> <td>0.5 </td> <td>2.2 </td> <td>0.7 </td> <td>0.8 </td> <td>13 </td></tr> <tr> <td>Lunar elliptical semidiurnal second-order </td> <td>2<i>N</i><sub>2</sub> </td> <td>12.90537297 </td> <td>27.8953548 </td> <td>2 </td> <td>−2 </td> <td> </td> <td>2 </td> <td>235.755 </td> <td>6.5 </td> <td>0.1 </td> <td>0.5 </td> <td>2.4 </td> <td>1.4 </td> <td>0.6 </td> <td>14 </td></tr> <tr> <td>Smaller lunar evectional </td> <td><i>λ</i><sub>2</sub> </td> <td>12.22177348 </td> <td>29.4556253 </td> <td>2 </td> <td>1 </td> <td>−2 </td> <td>1 </td> <td>263.655 </td> <td>5.3 </td> <td> </td> <td>0.1 </td> <td>0.7 </td> <td>0.6 </td> <td>0.2 </td> <td>16 </td></tr> <tr> <td>Larger solar elliptic </td> <td><i>T</i><sub>2</sub> </td> <td>12.01644934 </td> <td>29.9589333 </td> <td>2 </td> <td>2 </td> <td>−3 </td> <td> </td> <td>272.555 </td> <td>3.7 </td> <td>0.2 </td> <td>0.1 </td> <td>1.9 </td> <td>0.9 </td> <td>0.6 </td> <td>27 </td></tr> <tr> <td>Smaller solar elliptic </td> <td><i>R</i><sub>2</sub> </td> <td>11.98359564 </td> <td>30.0410667 </td> <td>2 </td> <td>2 </td> <td>−1 </td> <td> </td> <td>274.555 </td> <td>0.9 </td> <td> </td> <td> </td> <td>0.2 </td> <td>0.1 </td> <td>0.1 </td> <td>28 </td></tr> <tr> <td>Shallow water semidiurnal </td> <td>2<i>SM</i><sub>2</sub> </td> <td>11.60695157 </td> <td>31.0158958 </td> <td>2 </td> <td>4 </td> <td>−4 </td> <td> </td> <td>291.555 </td> <td>0.5 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>31 </td></tr> <tr> <td>Smaller lunar elliptic semidiurnal </td> <td><i>L</i><sub>2</sub> </td> <td>12.19162085 </td> <td>29.5284789 </td> <td>2 </td> <td>1 </td> <td> </td> <td>−1 </td> <td>265.455 </td> <td>13.5 </td> <td>0.1 </td> <td>0.5 </td> <td>2.4 </td> <td>1.6 </td> <td>0.5 </td> <td>33 </td></tr> <tr> <td>Lunisolar semidiurnal </td> <td><i>K</i><sub>2</sub> </td> <td>11.96723606 </td> <td>30.0821373 </td> <td>2 </td> <td>2 </td> <td> </td> <td> </td> <td>275.555 </td> <td>11.6 </td> <td>0.9 </td> <td>0.6 </td> <td>9.0 </td> <td>4.0 </td> <td>2.8 </td> <td>35 </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Diurnal">Diurnal</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=12" title="Edit section: Diurnal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="vertical-align:top; align:left;"> <tbody><tr> <th rowspan="2" style="background-color:lightblue;">Species </th> <th rowspan="2" style="background-color:lightblue;">Darwin <br />symbol </th> <th rowspan="2" style="background-color:lightblue;">Period <br />(h) </th> <th rowspan="2" style="background-color:lightblue;">Speed <br />(°/h) </th> <th colspan="4" style="background-color:lightblue;">Doodson coefficients </th> <th rowspan="2" style="background-color:lightblue;">Doodson <br />number </th> <th colspan="6" style="background-color:lightblue;">Amplitude at example location (cm) </th> <th rowspan="2" style="background-color:lightblue;">NOAA <br />order </th></tr> <tr> <th style="background-color:lightblue;"><i>n</i><sub>1</sub> (<i>L</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>2</sub> (<i>m</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>3</sub> (<i>y</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>4</sub> (<i>mp</i>) </th> <th style="background-color:lightblue;">ME </th> <th style="background-color:lightblue;">MS </th> <th style="background-color:lightblue;">PR </th> <th style="background-color:lightblue;">AK </th> <th style="background-color:lightblue;">CA </th> <th style="background-color:lightblue;">HI </th></tr> <tr> <td><b>Lunisolar diurnal </b> </td> <td><i>K</i><sub>1</sub> </td> <td>23.93447213 </td> <td>15.0410686 </td> <td>1 </td> <td>1 </td> <td> </td> <td> </td> <td>165.555 </td> <td>15.6 </td> <td>16.2 </td> <td>9.0 </td> <td>39.8 </td> <td>36.8 </td> <td>16.7 </td> <td><b>4</b> </td></tr> <tr> <td><b>Lunar diurnal </b> </td> <td><i>O</i><sub>1</sub> </td> <td>25.81933871 </td> <td>13.9430356 </td> <td>1 </td> <td>−1 </td> <td> </td> <td> </td> <td>145.555 </td> <td>11.9 </td> <td>16.9 </td> <td>7.7 </td> <td>25.9 </td> <td>23.0 </td> <td>9.2 </td> <td><b>6</b> </td></tr> <tr> <td>Lunar diurnal </td> <td><i>OO</i><sub>1</sub> </td> <td>22.30608083 </td> <td>16.1391017 </td> <td>1 </td> <td>3 </td> <td> </td> <td> </td> <td>185.555 </td> <td>0.5 </td> <td>0.7 </td> <td>0.4 </td> <td>1.2 </td> <td>1.1 </td> <td>0.7 </td> <td>15 </td></tr> <tr> <td>Solar diurnal </td> <td><i>S</i><sub>1</sub> </td> <td>24 </td> <td>15 </td> <td>1 </td> <td>1 </td> <td>−1 </td> <td> </td> <td>164.555 </td> <td>1.0 </td> <td> </td> <td>0.5 </td> <td>1.2 </td> <td>0.7 </td> <td>0.3 </td> <td>17 </td></tr> <tr> <td>Smaller lunar elliptic diurnal </td> <td><i>M</i><sub>1</sub> </td> <td>24.84120241 </td> <td>14.4920521 </td> <td>1 </td> <td> </td> <td> </td> <td> </td> <td>155.555 </td> <td>0.6 </td> <td>1.2 </td> <td>0.5 </td> <td>1.4 </td> <td>1.1 </td> <td>0.5 </td> <td>18 </td></tr> <tr> <td>Smaller lunar elliptic diurnal </td> <td><i>J</i><sub>1</sub> </td> <td>23.09848146 </td> <td>15.5854433 </td> <td>1 </td> <td>2 </td> <td> </td> <td>−1 </td> <td>175.455 </td> <td>0.9 </td> <td>1.3 </td> <td>0.6 </td> <td>2.3 </td> <td>1.9 </td> <td>1.1 </td> <td>19 </td></tr> <tr> <td>Larger lunar evectional diurnal </td> <td><i>ρ</i> </td> <td>26.72305326 </td> <td>13.4715145 </td> <td>1 </td> <td>−2 </td> <td>2 </td> <td>−1 </td> <td>137.455 </td> <td>0.3 </td> <td>0.6 </td> <td>0.3 </td> <td>0.9 </td> <td>0.9 </td> <td>0.3 </td> <td>25 </td></tr> <tr> <td>Larger lunar elliptic diurnal </td> <td><i>Q</i><sub>1</sub> </td> <td>26.868350 </td> <td>13.3986609 </td> <td>1 </td> <td>−2 </td> <td> </td> <td>1 </td> <td>135.655 </td> <td>2.0 </td> <td>3.3 </td> <td>1.4 </td> <td>4.7 </td> <td>4.0 </td> <td>1.6 </td> <td>26 </td></tr> <tr> <td>Larger elliptic diurnal </td> <td>2<i>Q</i><sub>1</sub> </td> <td>28.00621204 </td> <td>12.8542862 </td> <td>1 </td> <td>−3 </td> <td> </td> <td>2 </td> <td>125.755 </td> <td>0.3 </td> <td>0.4 </td> <td>0.2 </td> <td>0.7 </td> <td>0.4 </td> <td>0.2 </td> <td>29 </td></tr> <tr> <td>Solar diurnal </td> <td><i>P</i><sub>1</sub> </td> <td>24.06588766 </td> <td>14.9589314 </td> <td>1 </td> <td>1 </td> <td>−2 </td> <td> </td> <td>163.555 </td> <td>5.2 </td> <td>5.4 </td> <td>2.9 </td> <td>12.6 </td> <td>11.6 </td> <td>5.1 </td> <td>30 </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Long_period">Long period</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=13" title="Edit section: Long period"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="vertical-align:top; align:left;"> <tbody><tr> <th rowspan="2" style="background-color:lightblue;">Species </th> <th rowspan="2" style="background-color:lightblue;">Darwin <br />symbol </th> <th rowspan="2" style="background-color:lightblue;">Period <br />(h)<br />(days) </th> <th rowspan="2" style="background-color:lightblue;">Speed <br />(°/h) </th> <th colspan="4" style="background-color:lightblue;">Doodson coefficients </th> <th rowspan="2" style="background-color:lightblue;">Doodson <br />number </th> <th colspan="6" style="background-color:lightblue;">Amplitude at example location (cm) </th> <th rowspan="2" style="background-color:lightblue;">NOAA <br />order </th></tr> <tr> <th style="background-color:lightblue;"><i>n</i><sub>1</sub> (<i>L</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>2</sub> (<i>m</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>3</sub> (<i>y</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>4</sub> (<i>mp</i>) </th> <th style="background-color:lightblue;">ME </th> <th style="background-color:lightblue;">MS </th> <th style="background-color:lightblue;">PR </th> <th style="background-color:lightblue;">AK </th> <th style="background-color:lightblue;">CA </th> <th style="background-color:lightblue;">HI </th></tr> <tr> <td>Lunar monthly </td> <td><i>M</i><sub>m</sub> </td> <td>661.3111655<br />27.554631896 </td> <td>0.5443747 </td> <td>0 </td> <td>1 </td> <td> </td> <td>−1 </td> <td>65.455 </td> <td> </td> <td> </td> <td>0.7 </td> <td>1.9 </td> <td> </td> <td> </td> <td>20 </td></tr> <tr> <td>Solar semiannual </td> <td><i>S</i><sub>sa</sub> </td> <td>4383.076325<br />182.628180208 </td> <td>0.0821373 </td> <td>0 </td> <td> </td> <td>2 </td> <td> </td> <td>57.555 </td> <td>1.6 </td> <td> </td> <td>2.1 </td> <td>1.5 </td> <td>3.9 </td> <td> </td> <td>21 </td></tr> <tr> <td>Solar annual </td> <td><i>S</i><sub>a</sub> </td> <td>8766.15265<br />365.256360417 </td> <td>0.0410686 </td> <td>0 </td> <td> </td> <td>1 </td> <td> </td> <td>56.555 </td> <td> </td> <td> </td> <td>5.5 </td> <td>7.8 </td> <td>3.8 </td> <td>4.3 </td> <td>22 </td></tr> <tr> <td>Lunisolar synodic fortnightly </td> <td><i>MS</i><sub>f</sub> </td> <td>354.3670666<br />14.765294442 </td> <td>1.0158958 </td> <td>0 </td> <td>2 </td> <td>−2 </td> <td> </td> <td>73.555 </td> <td> </td> <td> </td> <td> </td> <td>1.5 </td> <td> </td> <td> </td> <td>23 </td></tr> <tr> <td>Lunisolar fortnightly </td> <td><i>M</i><sub>f</sub> </td> <td>327.8599387<br />13.660830779 </td> <td>1.0980331 </td> <td>0 </td> <td>2 </td> <td> </td> <td> </td> <td>75.555 </td> <td> </td> <td> </td> <td>1.4 </td> <td>2.0 </td> <td> </td> <td>0.7 </td> <td>24 </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Short_period">Short period</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=14" title="Edit section: Short period"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="vertical-align:top; align:left;"> <tbody><tr> <th rowspan="2" style="background-color:lightblue;">Species </th> <th rowspan="2" style="background-color:lightblue;">Darwin <br />symbol </th> <th rowspan="2" style="background-color:lightblue;">Period <br />(h) </th> <th rowspan="2" style="background-color:lightblue;">Speed <br />(°/h) </th> <th colspan="4" style="background-color:lightblue;">Doodson coefficients </th> <th rowspan="2" style="background-color:lightblue;">Doodson <br />number </th> <th colspan="6" style="background-color:lightblue;">Amplitude at example location (cm) </th> <th rowspan="2" style="background-color:lightblue;">NOAA <br />order </th></tr> <tr> <th style="background-color:lightblue;"><i>n</i><sub>1</sub> (<i>L</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>2</sub> (<i>m</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>3</sub> (<i>y</i>) </th> <th style="background-color:lightblue;"><i>n</i><sub>4</sub> (<i>mp</i>) </th> <th style="background-color:lightblue;">ME </th> <th style="background-color:lightblue;">MS </th> <th style="background-color:lightblue;">PR </th> <th style="background-color:lightblue;">AK </th> <th style="background-color:lightblue;">CA </th> <th style="background-color:lightblue;">HI </th></tr> <tr> <td><b>Shallow water overtides of principal lunar</b> </td> <td><i>M</i><sub>4</sub> </td> <td>6.210300601 </td> <td>57.9682084 </td> <td>4 </td> <td> </td> <td> </td> <td> </td> <td>455.555 </td> <td>6.0 </td> <td>0.6 </td> <td> </td> <td>0.9 </td> <td>2.3 </td> <td> </td> <td><b>5</b> </td></tr> <tr> <td><b>Shallow water overtides of principal lunar</b> </td> <td><i>M</i><sub>6</sub> </td> <td>4.140200401 </td> <td>86.9523127 </td> <td>6 </td> <td> </td> <td> </td> <td> </td> <td>655.555 </td> <td>5.1 </td> <td>0.1 </td> <td> </td> <td>1.0 </td> <td> </td> <td> </td> <td><b>7</b> </td></tr> <tr> <td><b>Shallow water terdiurnal</b> </td> <td><i>MK</i><sub>3</sub> </td> <td>8.177140247 </td> <td>44.0251729 </td> <td>3 </td> <td>1 </td> <td> </td> <td> </td> <td>365.555 </td> <td> </td> <td> </td> <td> </td> <td>0.5 </td> <td>1.9 </td> <td> </td> <td><b>8</b> </td></tr> <tr> <td><b>Shallow water overtides of principal solar </b> </td> <td><i>S</i><sub>4</sub> </td> <td>6 </td> <td>60 </td> <td>4 </td> <td>4 </td> <td>−4 </td> <td> </td> <td>491.555 </td> <td> </td> <td>0.1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td><b>9</b> </td></tr> <tr> <td><b>Shallow water quarter diurnal </b> </td> <td><i>MN</i><sub>4</sub> </td> <td>6.269173724 </td> <td>57.4238337 </td> <td>4 </td> <td>−1 </td> <td> </td> <td>1 </td> <td>445.655 </td> <td>2.3 </td> <td> </td> <td> </td> <td>0.3 </td> <td>0.9 </td> <td> </td> <td><b>10</b> </td></tr> <tr> <td>Shallow water overtides of principal solar </td> <td><i>S</i><sub>6</sub> </td> <td>4 </td> <td>90 </td> <td>6 </td> <td>6 </td> <td>−6 </td> <td> </td> <td>* </td> <td> </td> <td>0.1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>12 </td></tr> <tr> <td>Lunar terdiurnal </td> <td><i>M</i><sub>3</sub> </td> <td>8.280400802 </td> <td>43.4761563 </td> <td>3 </td> <td> </td> <td> </td> <td> </td> <td>355.555 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>0.5 </td> <td> </td> <td>32 </td></tr> <tr> <td>Shallow water terdiurnal </td> <td>2<i>MK</i><sub>3</sub> </td> <td>8.38630265 </td> <td>42.9271398 </td> <td>3 </td> <td>−1 </td> <td> </td> <td> </td> <td>345.555 </td> <td>0.5 </td> <td> </td> <td> </td> <td>0.5 </td> <td>1.4 </td> <td> </td> <td>34 </td></tr> <tr> <td>Shallow water eighth diurnal </td> <td><i>M</i><sub>8</sub> </td> <td>3.105150301 </td> <td>115.9364166 </td> <td>8 </td> <td> </td> <td> </td> <td> </td> <td>855.555 </td> <td>0.5 </td> <td>0.1 </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>36 </td></tr> <tr> <td>Shallow water quarter diurnal </td> <td><i>MS</i><sub>4</sub> </td> <td>6.103339275 </td> <td>58.9841042 </td> <td>4 </td> <td>2 </td> <td>−2 </td> <td> </td> <td>473.555 </td> <td>1.8 </td> <td> </td> <td> </td> <td>0.6 </td> <td>1.0 </td> <td> </td> <td>37 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Doodson_numbers">Doodson numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=15" title="Edit section: Doodson numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 <b>Doodson numbers</b> based on the six <i>Doodson arguments</i> 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.)<sup id="cite_ref-melch1971_52-0" class="reference"><a href="#cite_note-melch1971-52"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p>The Doodson arguments are specified in the following way, in order of decreasing frequency:<sup id="cite_ref-melch1971_52-1" class="reference"><a href="#cite_note-melch1971-52"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{1}=\tau =(\theta _{M}+\pi -s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>τ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>+</mo> <mi>π</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{1}=\tau =(\theta _{M}+\pi -s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2736f6d885c50ae377a5d7b61f2243cc0633c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.731ex; height:2.843ex;" alt="{\displaystyle \beta _{1}=\tau =(\theta _{M}+\pi -s)}"></span> is mean Lunar time, the Greenwich hour angle of the mean Moon plus 12 hours.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{2}=s=(F+\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>s</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo>+</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{2}=s=(F+\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b434380ee15c466f60ca9853c91825ed308af6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.726ex; height:2.843ex;" alt="{\displaystyle \beta _{2}=s=(F+\Omega )}"></span> is the mean longitude of the Moon.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{3}=h=(s-D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>h</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{3}=h=(s-D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5999c8faf114a031f5a5826a55198e220fad46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.57ex; height:2.843ex;" alt="{\displaystyle \beta _{3}=h=(s-D)}"></span> is the mean longitude of the Sun.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{4}=p=(s-l)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{4}=p=(s-l)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea3432dec17ed358f4eb88b511e3be342e6ee77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.17ex; height:2.843ex;" alt="{\displaystyle \beta _{4}=p=(s-l)}"></span> is the longitude of the Moon's mean perigee.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{5}=N'=(-\Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>N</mi> <mo>′</mo> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{5}=N'=(-\Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4235ccd75a3664ee006573bde42dfd721435460b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.67ex; height:3.009ex;" alt="{\displaystyle \beta _{5}=N'=(-\Omega )}"></span> is the negative of the longitude of the Moon's mean <a href="/wiki/Longitude_of_the_ascending_node" title="Longitude of the ascending node">ascending node</a> on the ecliptic.</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{6}=p_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{6}=p_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a216608b409bd7abc59bbf1f5063c716be045982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.36ex; height:2.509ex;" alt="{\displaystyle \beta _{6}=p_{l}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{s}=(s-D-l')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mi>D</mi> <mo>−</mo> <msup> <mi>l</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{s}=(s-D-l')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb13db1913cf38d6259d258e2378d58fe6af2f0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.243ex; height:3.009ex;" alt="{\displaystyle p_{s}=(s-D-l')}"></span> is the longitude of the Sun's mean perigee.</dd></dl> <p>In these expressions, the symbols <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>l</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efb2e54e32519bf534e146b9140c25b67e558d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.378ex; height:2.509ex;" alt="{\displaystyle l'}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> refer to an alternative set of fundamental angular arguments (usually preferred for use in modern lunar theory), in which:- </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> is the mean anomaly of the Moon (distance from its perigee).</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>l</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efb2e54e32519bf534e146b9140c25b67e558d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.378ex; height:2.509ex;" alt="{\displaystyle l'}"></span> is the mean anomaly of the Sun (distance from its perigee).</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the Moon's mean argument of latitude (distance from its node).</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is the Moon's mean elongation (distance from the sun).</dd></dl> <p>It is possible to define several auxiliary variables on the basis of combinations of these. </p><p>In terms of this system, each tidal constituent frequency can be identified by its Doodson numbers. The strongest tidal constituent "M<sub>2</sub>" 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 "S<sub>2</sub>" 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.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> 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. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Long-period_tides" title="Long-period tides">Long-period tides</a></li> <li><a href="/wiki/Lunar_node#Effect_on_tides" title="Lunar node">Lunar node § Effect on tides</a></li> <li><a href="/wiki/Kelvin_wave" title="Kelvin wave">Kelvin wave</a></li> <li><a href="/wiki/Tide_table" title="Tide table">Tide table</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=17" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">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. - <a rel="nofollow" class="external text" href="https://www.sacred-texts.com/hin/vp/vp062.htm">The Vishnu Purana book II ch. IV</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">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. - <a href="/wiki/Philostratus" title="Philostratus">Philostratus</a>, <i>The Life of <a href="/wiki/Apollonius_of_Tyana" title="Apollonius of Tyana">Apollonius of Tyana</a></i>, V</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><i>"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, … "</i> ("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, …")<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Theory_of_tides&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.aboriginalastronomy.com.au/content/topics/moon/">"Moon"</a>. <i>Australian Indigenous Astronomy</i><span class="reference-accessdate">. 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(1989). <a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/full/1989MmSAI..60..769T">"The tides and Newton"</a>. <i>Memorie della Società Astronomia Italiana</i>. <b>60</b>: <span class="nowrap">770–</span>777. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989MmSAI..60..769T">1989MmSAI..60..769T</a><span class="reference-accessdate">. Retrieved <span class="nowrap">27 December</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Memorie+della+Societ%C3%A0+Astronomia+Italiana&rft.atitle=The+tides+and+Newton&rft.volume=60&rft.pages=%3Cspan+class%3D%22nowrap%22%3E770-%3C%2Fspan%3E777&rft.date=1989&rft_id=info%3Abibcode%2F1989MmSAI..60..769T&rft.aulast=Tabarroni&rft.aufirst=G.&rft_id=http%3A%2F%2Fadsabs.harvard.edu%2Ffull%2F1989MmSAI..60..769T&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheory+of+tides" class="Z3988"></span></span> </li> <li id="cite_note-SciMonthly-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-SciMonthly_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SciMonthly_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-SciMonthly_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-SciMonthly_4-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-SciMonthly_4-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-SciMonthly_4-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarmer1922" class="citation journal cs1">Marmer, H. 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(March 1922). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/6374">"The Problems of the Tide"</a>. <i>The Scientific Monthly</i>. <b>14</b> (3): <span class="nowrap">209–</span>222.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Scientific+Monthly&rft.atitle=The+Problems+of+the+Tide&rft.volume=14&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E209-%3C%2Fspan%3E222&rft.date=1922-03&rft.aulast=Marmer&rft.aufirst=H.+A.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F6374&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheory+of+tides" class="Z3988"></span></span> </li> <li id="cite_note-Pugh-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Pugh_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Pugh_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Pugh_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPugh1987" class="citation book cs1">Pugh, David T. (28 December 1987). <a rel="nofollow" class="external text" href="https://eprints.soton.ac.uk/19157/1/sea-level.pdf"><i>Tides, Surges and Mean Sea-Level</i></a> <span class="cs1-format">(PDF)</span>. JOHN WILEY & SONS. pp. <span class="nowrap">2–</span>4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/047191505X" title="Special:BookSources/047191505X"><bdi>047191505X</bdi></a><span class="reference-accessdate">. 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Gerstein - University of Toronto. Oxford : Clarendon Press. pp. <span class="nowrap">306–</span>307.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aristarchus+of+Samos%2C+the+ancient+Copernicus&rft.pages=%3Cspan+class%3D%22nowrap%22%3E306-%3C%2Fspan%3E307&rft.pub=Oxford+%3A+Clarendon+Press&rft.date=1913&rft.aulast=Heath&rft.aufirst=Thomas+Little&rft_id=http%3A%2F%2Farchive.org%2Fdetails%2Faristarchusofsam00heatuoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheory+of+tides" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartwright1999" class="citation book cs1">Cartwright, David Edgar (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=78bE5U7TVuIC&q=descartes+tides&pg=PA31"><i>Tides: A Scientific History</i></a>. 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Cambridge University Press. pp. <span class="nowrap">163–</span>164. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521797467" title="Special:BookSources/9780521797467"><bdi>9780521797467</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tides%3A+A+Scientific+History&rft.pages=%3Cspan+class%3D%22nowrap%22%3E163-%3C%2Fspan%3E164&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=9780521797467&rft.aulast=Cartwright&rft.aufirst=David+Edgar&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsSesNjG96JYC%26pg%3DPA163&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheory+of+tides" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">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.</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text">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.</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNOAA" class="citation web cs1">NOAA. <a rel="nofollow" class="external text" href="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">"Eastport, ME Tidal Constituents"</a>. NOAA<span class="reference-accessdate">. Retrieved <span class="nowrap">22 May</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Eastport%2C+ME+Tidal+Constituents&rft.pub=NOAA&rft.au=NOAA&rft_id=http%3A%2F%2Ftidesandcurrents.noaa.gov%2Fcgi-bin%2Fco-ops_qry.cgi%3Fstn%3D8410140%2520Eastport%2C%2520ME%26pc%3DP2%26unit%3D0%26date%3D1%26shift%3D0%26level%3D-4%26form%3D0%26format%3DView%2BData&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheory+of+tides" class="Z3988"></span></span> </li> <li id="cite_note-melch1971-52"><span class="mw-cite-backlink">^ <a href="#cite_ref-melch1971_52-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-melch1971_52-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMelchior1971" class="citation journal cs1">Melchior, P. (1971). "Precession-nutations and tidal potential". <i>Celestial Mechanics</i>. <b>4</b> (2): <span class="nowrap">190–</span>212. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1971CeMec...4..190M">1971CeMec...4..190M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01228823">10.1007/BF01228823</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:126219362">126219362</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Celestial+Mechanics&rft.atitle=Precession-nutations+and+tidal+potential&rft.volume=4&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E190-%3C%2Fspan%3E212&rft.date=1971&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A126219362%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01228823&rft_id=info%3Abibcode%2F1971CeMec...4..190M&rft.aulast=Melchior&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATheory+of+tides" class="Z3988"></span> and T D Moyer (2003) already cited.</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text">See for example Melchior (1971), already cited, at p.191.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a 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href="https://web.archive.org/web/20130728163045/http://www.eas.slu.edu/GGP/BIM_Recent_Issues/bim134-2001/bim134_report_WG6_SLR.pdf">Archived</a> 28 July 2013 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://ffden-2.phys.uaf.edu/645fall2003_web.dir/Ellie_Boyce/dynamic.htm">Dynamic Theory of Tides</a></li> <li><a rel="nofollow" class="external text" href="http://mysite.du.edu/~jcalvert/geol/tides.htm">Tidal Observations</a></li> <li><a rel="nofollow" class="external text" href="http://tidesandcurrents.noaa.gov/pub.html">Publications from NOAA's Center for Operational Oceanographic Products and Services</a> <ul><li><a rel="nofollow" class="external text" href="http://tidesandcurrents.noaa.gov/publications/Understanding_Tides_by_Steacy_finalFINAL11_30.pdf">Understanding Tides</a></li> <li><a rel="nofollow" class="external text" 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<li><a href="/wiki/Kelvin_wave" title="Kelvin wave">Kelvin wave</a></li> <li><a href="/wiki/Kinematic_wave" title="Kinematic wave">Kinematic wave</a></li> <li><a href="/wiki/Longshore_drift" title="Longshore drift">Longshore drift</a></li> <li><a href="/wiki/Luke%27s_variational_principle" title="Luke's variational principle">Luke's variational principle</a></li> <li><a href="/wiki/Mild-slope_equation" title="Mild-slope equation">Mild-slope equation</a></li> <li><a href="/wiki/Radiation_stress" title="Radiation stress">Radiation stress</a></li> <li><a href="/wiki/Rogue_wave" title="Rogue wave">Rogue wave</a> <ul><li><a href="/wiki/Draupner_wave" title="Draupner wave">Draupner wave</a></li></ul></li> <li><a href="/wiki/Rossby_wave" title="Rossby wave">Rossby wave</a></li> <li><a href="/wiki/Rossby-gravity_waves" title="Rossby-gravity waves">Rossby-gravity waves</a></li> <li><a href="/wiki/Sea_state" title="Sea state">Sea state</a></li> <li><a href="/wiki/Seiche" title="Seiche">Seiche</a></li> <li><a href="/wiki/Significant_wave_height" title="Significant wave height">Significant wave height</a></li> <li><a href="/wiki/Soliton" title="Soliton">Soliton</a></li> <li><a href="/wiki/Stokes_drift" title="Stokes drift">Stokes drift</a></li> <li><a href="/wiki/Stokes_problem" title="Stokes problem">Stokes problem</a></li> <li><a href="/wiki/Stokes_wave" title="Stokes wave">Stokes wave</a></li> <li><a href="/wiki/Swell_(ocean)" title="Swell (ocean)">Swell</a></li> <li><a href="/wiki/Trochoidal_wave" title="Trochoidal wave">Trochoidal wave</a></li> <li><a href="/wiki/Tsunami" title="Tsunami">Tsunami</a> <ul><li><a href="/wiki/Megatsunami" title="Megatsunami">megatsunami</a></li></ul></li> <li><a href="/wiki/Undertow_(water_waves)" title="Undertow (water waves)">Undertow</a></li> <li><a href="/wiki/Ursell_number" title="Ursell number">Ursell number</a></li> <li><a href="/wiki/Wave_action_(continuum_mechanics)" title="Wave action (continuum mechanics)">Wave action</a></li> <li><a href="/wiki/Wave_base" title="Wave base">Wave base</a></li> <li><a href="/wiki/Wave_height" title="Wave height">Wave height</a></li> <li><a href="/wiki/Wave_nonlinearity" title="Wave nonlinearity">Wave nonlinearity</a></li> <li><a href="/wiki/Wave_power" title="Wave power">Wave power</a></li> <li><a href="/wiki/Wave_radar" title="Wave radar">Wave radar</a></li> <li><a href="/wiki/Wave_setup" title="Wave setup">Wave setup</a></li> <li><a href="/wiki/Wave_shoaling" title="Wave shoaling">Wave shoaling</a></li> <li><a href="/wiki/Wave_turbulence" title="Wave turbulence">Wave turbulence</a></li> <li><a href="/wiki/Wave%E2%80%93current_interaction" title="Wave–current interaction">Wave–current interaction</a></li> <li><a href="/wiki/Waves_and_shallow_water" title="Waves and shallow water">Waves and shallow water</a> <ul><li><a href="/wiki/One-dimensional_Saint-Venant_equations" class="mw-redirect" title="One-dimensional Saint-Venant equations">one-dimensional Saint-Venant equations</a></li> <li><a href="/wiki/Shallow_water_equations" title="Shallow water equations">shallow water equations</a></li></ul></li> <li><a href="/wiki/Wind_fetch" title="Wind fetch">Wind fetch</a></li> <li><a href="/wiki/Wind_setup" title="Wind setup">Wind setup</a></li> <li><a href="/wiki/Wind_wave" title="Wind wave">Wind wave</a> <ul><li><a href="/wiki/Wind_wave_model" title="Wind wave model">model</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="10" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Upwelling.svg" class="mw-file-description" title="Upwelling"><img alt="Upwelling" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/120px-Upwelling.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/180px-Upwelling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/240px-Upwelling.svg.png 2x" data-file-width="365" data-file-height="242" /></a></span><br /><br /><br /><br /><br /><br /><span typeof="mw:File"><a href="/wiki/File:Antarctic_bottom_water.svg" class="mw-file-description" title="Antarctic bottom water"><img alt="Antarctic bottom water" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/120px-Antarctic_bottom_water.svg.png" decoding="async" width="120" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/180px-Antarctic_bottom_water.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/240px-Antarctic_bottom_water.svg.png 2x" data-file-width="745" data-file-height="470" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ocean_current" title="Ocean current">Circulation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atmospheric_circulation" title="Atmospheric circulation">Atmospheric circulation</a></li> <li><a href="/wiki/Baroclinity" title="Baroclinity">Baroclinity</a></li> <li><a href="/wiki/Boundary_current" title="Boundary current">Boundary current</a></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Coriolis%E2%80%93Stokes_force" title="Coriolis–Stokes force">Coriolis–Stokes force</a></li> <li><a href="/wiki/Craik%E2%80%93Leibovich_vortex_force" title="Craik–Leibovich vortex force">Craik–Leibovich vortex force</a></li> <li><a href="/wiki/Downwelling" title="Downwelling">Downwelling</a></li> <li><a href="/wiki/Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">Eddy</a></li> <li><a href="/wiki/Ekman_layer" title="Ekman layer">Ekman layer</a></li> <li><a href="/wiki/Ekman_spiral" title="Ekman spiral">Ekman spiral</a></li> <li><a href="/wiki/Ekman_transport" title="Ekman transport">Ekman transport</a></li> <li><a href="/wiki/El_Ni%C3%B1o%E2%80%93Southern_Oscillation" title="El Niño–Southern Oscillation">El Niño–Southern Oscillation</a></li> <li><a href="/wiki/General_circulation_model" title="General circulation model">General circulation model</a></li> <li><a href="/wiki/Geochemical_Ocean_Sections_Study" title="Geochemical Ocean Sections Study">Geochemical Ocean Sections Study</a></li> <li><a href="/wiki/Geostrophic_current" title="Geostrophic current">Geostrophic current</a></li> <li><a href="/wiki/Global_Ocean_Data_Analysis_Project" title="Global Ocean Data Analysis Project">Global Ocean Data Analysis Project</a></li> <li><a href="/wiki/Gulf_Stream" title="Gulf Stream">Gulf Stream</a></li> <li><a href="/wiki/Humboldt_Current" title="Humboldt Current">Humboldt Current</a></li> <li><a href="/wiki/Hydrothermal_circulation" title="Hydrothermal circulation">Hydrothermal circulation</a></li> <li><a href="/wiki/Langmuir_circulation" title="Langmuir circulation">Langmuir circulation</a></li> <li><a href="/wiki/Longshore_drift" title="Longshore drift">Longshore drift</a></li> <li><a href="/wiki/Loop_Current" title="Loop Current">Loop Current</a></li> <li><a href="/wiki/Modular_Ocean_Model" title="Modular Ocean Model">Modular Ocean Model</a></li> <li><a href="/wiki/Ocean_current" title="Ocean current">Ocean current</a></li> <li><a href="/wiki/Ocean_dynamical_thermostat" title="Ocean dynamical thermostat">Ocean dynamical thermostat</a></li> <li><a href="/wiki/Ocean_dynamics" title="Ocean dynamics">Ocean dynamics</a></li> <li><a href="/wiki/Ocean_gyre" title="Ocean gyre">Ocean gyre</a></li> <li><a href="/wiki/Overflow_(oceanography)" title="Overflow (oceanography)">Overflow</a></li> <li><a href="/wiki/Princeton_Ocean_Model" title="Princeton Ocean Model">Princeton Ocean Model</a></li> <li><a href="/wiki/Rip_current" title="Rip current">Rip current</a></li> <li><a href="/wiki/Subsurface_ocean_current" title="Subsurface ocean current">Subsurface ocean current</a></li> <li><a href="/wiki/Sverdrup_balance" title="Sverdrup balance">Sverdrup balance</a></li> <li><a href="/wiki/Thermohaline_circulation" title="Thermohaline circulation">Thermohaline circulation</a> <ul><li><a href="/wiki/Shutdown_of_thermohaline_circulation" class="mw-redirect" title="Shutdown of thermohaline circulation">shutdown</a></li></ul></li> <li><a href="/wiki/Upwelling" title="Upwelling">Upwelling</a></li> <li><a href="/wiki/Whirlpool" title="Whirlpool">Whirlpool</a></li> <li><a href="/wiki/Wind_generated_current" title="Wind generated current">Wind generated current</a></li> <li><a href="/wiki/World_Ocean_Circulation_Experiment" title="World Ocean Circulation Experiment">World Ocean Circulation Experiment</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tide" title="Tide">Tides</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amphidromic_point" title="Amphidromic point">Amphidromic point</a></li> <li><a href="/wiki/Earth_tide" title="Earth tide">Earth tide</a></li> <li><a href="/wiki/Head_of_tide" title="Head of tide">Head of tide</a></li> <li><a href="/wiki/Internal_tide" title="Internal tide">Internal tide</a></li> <li><a href="/wiki/Lunitidal_interval" title="Lunitidal interval">Lunitidal interval</a></li> <li><a href="/wiki/Perigean_spring_tide" title="Perigean spring tide">Perigean spring tide</a></li> <li><a href="/wiki/Rip_tide" title="Rip tide">Rip tide</a></li> <li><a href="/wiki/Rule_of_twelfths" title="Rule of twelfths">Rule of twelfths</a></li> <li><a href="/wiki/Slack_tide" title="Slack tide">Slack tide</a></li> <li><a class="mw-selflink selflink">Theory of tides</a></li> <li><a href="/wiki/Tidal_bore" title="Tidal bore">Tidal bore</a></li> <li><a href="/wiki/Tidal_force" title="Tidal force">Tidal force</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal power</a></li> <li><a href="/wiki/Tidal_race" title="Tidal race">Tidal race</a></li> <li><a href="/wiki/Tidal_range" title="Tidal range">Tidal range</a></li> <li><a href="/wiki/Tidal_resonance" title="Tidal resonance">Tidal resonance</a></li> <li><a href="/wiki/Tide_gauge" title="Tide gauge">Tide gauge</a></li> <li><a href="/wiki/Tideline" title="Tideline">Tideline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Landform" title="Landform">Landforms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abyssal_fan" title="Abyssal fan">Abyssal fan</a></li> <li><a href="/wiki/Abyssal_plain" title="Abyssal plain">Abyssal plain</a></li> <li><a href="/wiki/Atoll" title="Atoll">Atoll</a></li> <li><a href="/wiki/Bathymetric_chart" title="Bathymetric chart">Bathymetric chart</a></li> <li><a href="/wiki/Carbonate_platform" title="Carbonate platform">Carbonate platform</a></li> <li><a href="/wiki/Coastal_geography" title="Coastal geography">Coastal geography</a></li> <li><a href="/wiki/Cold_seep" title="Cold seep">Cold seep</a></li> <li><a href="/wiki/Continental_margin" title="Continental margin">Continental margin</a></li> <li><a href="/wiki/Continental_rise" title="Continental rise">Continental rise</a></li> <li><a href="/wiki/Continental_shelf" title="Continental shelf">Continental shelf</a></li> <li><a href="/wiki/Contourite" title="Contourite">Contourite</a></li> <li><a href="/wiki/Guyot" title="Guyot">Guyot</a></li> <li><a href="/wiki/Hydrography" title="Hydrography">Hydrography</a></li> <li><a href="/wiki/Knoll_(oceanography)" title="Knoll (oceanography)">Knoll</a></li> <li><a href="/wiki/Ocean_bank" title="Ocean bank">Ocean bank</a></li> <li><a href="/wiki/Oceanic_basin" title="Oceanic basin">Oceanic basin</a></li> <li><a href="/wiki/Oceanic_plateau" title="Oceanic plateau">Oceanic plateau</a></li> <li><a href="/wiki/Oceanic_trench" title="Oceanic trench">Oceanic trench</a></li> <li><a href="/wiki/Passive_margin" title="Passive margin">Passive margin</a></li> <li><a href="/wiki/Seabed" title="Seabed">Seabed</a></li> <li><a href="/wiki/Seamount" title="Seamount">Seamount</a></li> <li><a href="/wiki/Submarine_canyon" title="Submarine canyon">Submarine canyon</a></li> <li><a href="/wiki/Submarine_volcano" title="Submarine volcano">Submarine volcano</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plate_tectonics" title="Plate tectonics">Plate<br />tectonics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convergent_boundary" title="Convergent boundary">Convergent boundary</a></li> <li><a href="/wiki/Divergent_boundary" title="Divergent boundary">Divergent boundary</a></li> <li><a href="/wiki/Fracture_zone" title="Fracture zone">Fracture zone</a></li> <li><a href="/wiki/Hydrothermal_vent" title="Hydrothermal vent">Hydrothermal vent</a></li> <li><a href="/wiki/Marine_geology" title="Marine geology">Marine geology</a></li> <li><a href="/wiki/Mid-ocean_ridge" title="Mid-ocean ridge">Mid-ocean ridge</a></li> <li><a href="/wiki/Mohorovi%C4%8Di%C4%87_discontinuity" title="Mohorovičić discontinuity">Mohorovičić discontinuity</a></li> <li><a href="/wiki/Oceanic_crust" title="Oceanic crust">Oceanic crust</a></li> <li><a href="/wiki/Outer_trench_swell" title="Outer trench swell">Outer trench swell</a></li> <li><a href="/wiki/Ridge_push" title="Ridge push">Ridge push</a></li> <li><a href="/wiki/Seafloor_spreading" title="Seafloor spreading">Seafloor spreading</a></li> <li><a href="/wiki/Slab_pull" title="Slab pull">Slab pull</a></li> <li><a href="/wiki/Slab_suction" title="Slab suction">Slab suction</a></li> <li><a href="/wiki/Slab_window" title="Slab window">Slab window</a></li> <li><a href="/wiki/Subduction" title="Subduction">Subduction</a></li> <li><a href="/wiki/Transform_fault" title="Transform fault">Transform fault</a></li> <li><a href="/wiki/Vine%E2%80%93Matthews%E2%80%93Morley_hypothesis" title="Vine–Matthews–Morley hypothesis">Vine–Matthews–Morley hypothesis</a></li> <li><a href="/wiki/Volcanic_arc" title="Volcanic arc">Volcanic arc</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ocean zones</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benthic_zone" title="Benthic zone">Benthic</a></li> <li><a href="/wiki/Deep_ocean_water" title="Deep ocean water">Deep ocean water</a></li> <li><a href="/wiki/Deep_sea" title="Deep sea">Deep sea</a></li> <li><a href="/wiki/Littoral_zone" title="Littoral zone">Littoral</a></li> <li><a href="/wiki/Mesopelagic_zone" title="Mesopelagic zone">Mesopelagic</a></li> <li><a href="/wiki/Oceanic_zone" title="Oceanic zone">Oceanic</a></li> <li><a href="/wiki/Pelagic_zone" title="Pelagic zone">Pelagic</a></li> <li><a href="/wiki/Photic_zone" title="Photic zone">Photic</a></li> <li><a href="/wiki/Surf_zone" title="Surf zone">Surf</a></li> <li><a href="/wiki/Swash" title="Swash">Swash</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sea_level" title="Sea level">Sea level</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deep-ocean_Assessment_and_Reporting_of_Tsunamis" title="Deep-ocean Assessment and Reporting of Tsunamis">Deep-ocean Assessment and Reporting of Tsunamis</a></li> <li><a href="/wiki/Global_Sea_Level_Observing_System" title="Global Sea Level Observing System">Global Sea Level Observing System</a></li> <li><a href="/wiki/North_West_Shelf_Operational_Oceanographic_System" title="North West Shelf Operational Oceanographic System">North West Shelf Operational Oceanographic System</a></li> <li><a href="/wiki/Sea-level_curve" title="Sea-level curve">Sea-level curve</a></li> <li><a href="/wiki/Sea_level_drop" title="Sea level drop">Sea level drop</a></li> <li><a href="/wiki/Sea_level_rise" title="Sea level rise">Sea level rise</a></li> <li><a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Acoustical_oceanography" class="mw-redirect" title="Acoustical oceanography">Acoustics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deep_scattering_layer" title="Deep scattering layer">Deep scattering layer</a></li> <li><a href="/wiki/Ocean_acoustic_tomography" title="Ocean acoustic tomography">Ocean acoustic tomography</a></li> <li><a href="/wiki/Sofar_bomb" title="Sofar bomb">Sofar bomb</a></li> <li><a href="/wiki/SOFAR_channel" title="SOFAR channel">SOFAR channel</a></li> <li><a href="/wiki/Underwater_acoustics" title="Underwater acoustics">Underwater acoustics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satellites</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jason-1" title="Jason-1">Jason-1</a></li> <li><a href="/wiki/OSTM/Jason-2" title="OSTM/Jason-2">OSTM/Jason-2</a></li> <li><a href="/wiki/Jason-3" title="Jason-3">Jason-3</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ocean_acidification" title="Ocean acidification">Acidification</a></li> <li><a href="/wiki/Argo_(oceanography)" title="Argo (oceanography)">Argo</a></li> <li><a href="/wiki/Benthic_lander" title="Benthic lander">Benthic lander</a></li> <li><a href="/wiki/Color_of_water" title="Color of water">Color of water</a></li> <li><a href="/wiki/DSV_Alvin" title="DSV Alvin">DSV <i>Alvin</i></a></li> <li><a href="/wiki/Marginal_sea" class="mw-redirect" title="Marginal sea">Marginal sea</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine energy</a></li> <li><a href="/wiki/Marine_pollution" title="Marine pollution">Marine pollution</a></li> <li><a href="/wiki/Mooring_(oceanography)" title="Mooring (oceanography)">Mooring</a></li> <li><a href="/wiki/National_Oceanographic_Data_Center" title="National Oceanographic Data Center">National Oceanographic Data Center</a></li> <li><a href="/wiki/Ocean" title="Ocean">Ocean</a></li> <li><a href="/wiki/Ocean_exploration" title="Ocean exploration">Explorations</a></li> <li><a href="/wiki/Ocean_observations" title="Ocean observations">Observations</a></li> <li><a href="/wiki/Ocean_reanalysis" title="Ocean reanalysis">Reanalysis</a></li> <li><a href="/wiki/Ocean_surface_topography" title="Ocean surface topography">Ocean surface topography</a></li> <li><a href="/wiki/Ocean_temperature" title="Ocean temperature">Ocean temperature</a></li> <li><a href="/wiki/Ocean_thermal_energy_conversion" title="Ocean thermal energy conversion">Ocean thermal energy conversion</a></li> <li><a href="/wiki/Oceanography" title="Oceanography">Oceanography</a> <ul><li><a href="/wiki/Outline_of_oceanography" title="Outline of oceanography">Outline of oceanography</a></li></ul></li> <li><a href="/wiki/Pelagic_sediment" title="Pelagic sediment">Pelagic sediment</a></li> <li><a href="/wiki/Sea_surface_microlayer" title="Sea surface microlayer">Sea surface microlayer</a></li> <li><a href="/wiki/Sea_surface_temperature" title="Sea surface temperature">Sea surface temperature</a></li> <li><a href="/wiki/Seawater" title="Seawater">Seawater</a></li> <li><a href="/wiki/Science_On_a_Sphere" title="Science On a Sphere">Science On a Sphere</a></li> <li><a href="/wiki/Ocean_stratification" title="Ocean stratification">Stratification</a></li> <li><a href="/wiki/Thermocline" title="Thermocline">Thermocline</a></li> <li><a href="/wiki/Underwater_glider" title="Underwater glider">Underwater glider</a></li> <li><a href="/wiki/Water_column" title="Water column">Water column</a></li> <li><a href="/wiki/World_Ocean_Atlas" title="World Ocean Atlas">World Ocean Atlas</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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