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=== Phase and amplitude ===
[[File:M2 tidal constituent.jpg|thumb|''M''<sub>2</sub> tidal constituent. Red is most extreme (highest highs, lowest lows), with blues being least extreme. White cotidal lines converge in blue areas indicating little or no tide. The curved arcs around these convergent areas are [[amphidromic point]]s. They show the direction of the tides, each indicating a synchronized 6-hour period. Tidal ranges generally increase with increasing distance from amphidromic points. Tide waves move around these points, generally counterclockwise in the N. Hemisphere and clockwise in the S. Hemisphere <ref>{{cite journal |title=Solution of the Tidal Equations for the M<sub>2</sub> and S<sub>2</sub> Tides in the World Oceans from a Knowledge of the Tidal Potential Alone |journal=
</ref>|alt=Map showing relative tidal magnitudes of different ocean areas]]
Because the ''M''<sub>2</sub> tidal constituent dominates in most locations, the stage or ''phase'' of a tide, denoted by the time in hours after high water, is a useful concept. Tidal stage is also measured in degrees, with 360° per tidal cycle. Lines of constant tidal phase are called ''cotidal lines'', which are analogous to [[contour lines]] of constant altitude on [[topographical maps]], and when plotted form a ''cotidal map'' or ''cotidal chart''.<ref>{{Cite book |url=https://books.google.com/books?id=E3uhBQAAQBAJ&q=tidal+map&pg=PT28 |title=Dynamics of Ocean Tides |isbn=9789400925717 |last1=Marchuk |first1=Guri I. |last2=Kagan |first2=B. A. |date=6 December 2012 |via=[[Google Books]]}}</ref> High water is reached simultaneously along the cotidal lines extending from the coast out into the ocean, and cotidal lines (and hence tidal phases) advance along the coast. Semi-diurnal and long phase constituents are measured from high water, diurnal from maximum flood tide. This and the discussion that follows is precisely true only for a single tidal constituent.
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Medieval understanding of the tides was primarily based on works of [[Muslim astronomers]], which became available through [[Latin translations of the 12th century|Latin translation]] starting from the 12th century.<ref name=Tolmacheva>{{Cite book |publisher=[[Routledge]] |isbn=978-1135459321 |editor1-last=Glick |editor1-first=Thomas F. |work=Medieval Science, Technology, and Medicine: An Encyclopedia |year=2014 |title=Geography, Chorography |page=188 |first=Marina |last=Tolmacheva}}</ref> [[Abu Ma'shar al-Balkhi]] (d. circa 886), in his {{lang|la|Introductorium in astronomiam}}, taught that ebb and flood tides were caused by the Moon.<ref name=Tolmacheva /> Abu Ma'shar discussed the effects of wind and Moon's phases relative to the Sun on the tides.<ref name=Tolmacheva /> In the 12th century, [[al-Bitruji]] (d. circa 1204) contributed the notion that the tides were caused by the general circulation of the heavens.<ref name=Tolmacheva />
[[Simon Stevin]], in his 1608 {{lang|nl|De spiegheling der Ebbenvloet}} (''The theory of ebb and flood''), dismissed a large number of misconceptions that still existed about ebb and flood. Stevin pleaded for the idea that the attraction of the Moon was responsible for the tides and spoke in clear terms about ebb, flood, [[spring tide]] and [[neap tide]], stressing that further research needed to be made.<ref>{{cite web |url=http://www.vliz.be/imisdocs/publications/224466.pdf |title=Simon Stevin |publisher=Flanders Marine Institute |type=pdf |language=nl}}</ref><ref>{{Cite book |last1=Palmerino |first1=Carla Rita |first2=J.M.M.H. |last2=Thijssen |url=https://books.google.com/books?id=a5lkdlMPi1AC
In 1609 [[Johannes Kepler]] also correctly suggested that the gravitation of the Moon caused the tides,{{efn|''"Orbis virtutis tractoriæ, quæ est in Luna, porrigitur utque ad Terras, & prolectat aquas sub Zonam Torridam, … Celeriter vero Luna verticem transvolante, cum aquæ tam celeriter sequi non possint, fluxus quidem fit Oceani sub Torrida in Occidentem, … "'' (The sphere of the lifting power, which is [centered] in the moon, is extended as far as to the earth and attracts the waters under the torrid zone, … However the moon flies swiftly across the zenith ; because the waters cannot follow so quickly, the tide of the ocean under the torrid [zone] is indeed made to the west, …"<ref>Johannes Kepler, ''Astronomia nova'' … (1609), p. 5 of the ''Introductio in hoc opus'' (Introduction to this work). [https://archive.org/stream/Astronomianovaa00Kepl#page/n24/mode/1up From page 5:]</ref>}} which he based upon ancient observations and correlations.
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Pierre-Simon Laplace formulated a system of [[partial differential equation]]s relating the ocean's horizontal flow to its surface height, the first major dynamic theory for water tides. The [[Laplace's tidal equations|Laplace tidal equations]] are still in use today. [[William Thomson, 1st Baron Kelvin]], rewrote Laplace's equations in terms of [[vorticity]] which allowed for solutions describing tidally driven coastally trapped waves, known as [[Kelvin wave]]s.<ref name="tidhist">{{cite journal |title=Historical Development and Use of Thousand-Year-Old Tide-Prediction Tables |journal=Limnology and Oceanography |volume=34 |issue=5 |date=July 1989 |pages=953–957 |last1=Zuosheng |first1=Y. |last2=Emery |first2=K.O. |last3=Yui |first3=X. |name-list-style=amp |doi=10.4319/lo.1989.34.5.0953 |bibcode=1989LimOc..34..953Z |doi-access=free}}</ref><ref>{{cite book |title=Tides: A Scientific History |url=https://archive.org/details/tidesscientifich0000cart |url-access=registration |last=Cartwright |first=David E. |publisher=[[Cambridge University Press]] |location=Cambridge, UK |date=1999 |isbn=9780521621458}}</ref><ref>{{cite journal |title=Understanding Tides – From Ancient Beliefs to Present-day Solutions to the Laplace Equations |first=James |last=Case |journal=SIAM News |volume=33 |issue=2 |date=March 2000}}</ref>
Others including Kelvin and [[Henri Poincaré]] further developed Laplace's theory. Based on these developments and the [[lunar theory]] of [[Ernest William Brown|E W Brown]] describing the motions of the Moon, [[Arthur Thomas Doodson]] developed and published in 1921<ref>{{cite journal |last=Doodson |first=A.T. |date=December 1921 |title=The Harmonic Development of the Tide-Generating Potential |journal=
=== History of tidal observation ===
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The [[Natural History (Pliny)|''Naturalis Historia'']] of [[Pliny the Elder]] collates many tidal observations, e.g., the spring tides are a few days after (or before) new and full moon and are highest around the equinoxes, though Pliny noted many relationships now regarded as fanciful. In his ''Geography'', Strabo described tides in the [[Persian Gulf]] having their greatest range when the moon was furthest from the plane of the Equator. All this despite the relatively small amplitude of [[Mediterranean]] basin tides. (The strong currents through the [[Euripus Strait]] and the [[Strait of Messina]] puzzled [[Aristotle]].) [[Philostratus]] discussed tides in Book Five of ''The Life of [[Apollonius of Tyana]]''. Philostratus mentions the moon, but attributes tides to "spirits". In Europe around 730 AD, the Venerable [[Bede]] described how the rising tide on one coast of the British Isles coincided with the fall on the other and described the time progression of high water along the Northumbrian coast.
The first [[tide table]] in [[China]] was recorded in 1056 AD primarily for visitors wishing to see the famous [[tidal bore]] in the [[Qiantang River]]. The first known British tide table is thought to be that of John Wallingford, who died Abbot of St. Albans in 1213, based on high water occurring 48 minutes later each day, and three hours earlier at the [[Thames]] mouth than upriver at [[London]].<ref>{{cite
In 1614 [[Claude d'Abbeville]] published the work “{{lang|fr|Histoire de la mission de pères capucins en l’Isle de Maragnan et terres circonvoisines}}”, where he exposed that the [[Tupinambá people]] already had an understanding of the relation between the Moon and the tides before Europe.<ref>{{Cite web |url=https://mundogeo.com/2009/06/19/astronomia-indigena-preve-influencia-da-lua-sobre-as-mares-antes-de-galileu-e-newton/ |title=Astronomia indígena prevê influência da lua sobre as marés antes de Galileu e Newton |trans-title=Indigenous astronomy predicts moon's influence on tides before Galileo and Newton |date=2009-06-19 |access-date=2021-12-11 |lang=pt-br}}</ref>
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{{efn|"While the solar and lunar envelopes are thought of as representing the actual ocean waters, another very important factor must be recognized. The components of the tide-generating forces acting tangentially along the water surface turn out to be the most important. Just as it is easier to slide a bucket of water across a floor rather than to lift it, the horizontal tractive components move the waters toward the points directly beneath and away from the sun or moon far more effectively than the vertical components can lift them. These tractive forces are most responsible for trying to form
the ocean into the symmetrical egg-shaped distensions (the tide potential, the equilibrium tide). They reach their maximums in rings 45° from the points directly beneath and away from the sun or moon."<ref name="Hicks2006">{{cite report |last=Hicks |first=S.D. |title=Understanding Tides |publisher=[[NOAA]] |year=2006 |url=https://tidesandcurrents.noaa.gov/publications/Understanding_Tides_by_Steacy_finalFINAL11_30.pdf |language=en |access-date=2020-09-02}}</ref>}}
{{efn|"... the gravitational effect that causes the tides is much too weak to lift the oceans 12 inches vertically away from the earth. It is possible, however, to move the oceans horizontally within the earth's gravitational field. This gathers the oceans toward two points where the height of the water becomes elevated by the converging volume of water."<ref>{{cite book |first=James Greig |last=Mccully |date=2006 |title=Beyond The Moon: A Conversational, Common Sense Guide To Understanding The Tides, World Scientific |isbn=9789814338189 |url=https://books.google.com/books?id=aKLICgAAQBAJ&q=tractal |via=[[Google Books]]}}</ref>}}<ref name="PBS LearningMedia 2020">{{cite web |title=What Physics Teachers Get Wrong about Tides! - PBS Space Time |website=[[PBS]] LearningMedia |date=2020-06-17 |url=https://www.pbslearningmedia.org/resource/what-physics-teachers-pbs-space-time/what-physics-teachers-pbs-space-time/ |access-date=2020-06-27}}</ref>
=== Laplace's tidal equations ===
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Earth's tidal oscillations introduce dissipation at an [[average]] rate of about 3.75 [[terawatt]]s.<ref name=Munk1998>{{Cite journal |last1=Munk |first1=W. |date=1998 |title=Abyssal recipes II: energetics of tidal and wind mixing |journal=Deep-Sea Research Part I |volume=45 |issue=12 |page=1977 |doi=10.1016/S0967-0637(98)00070-3 |last2=Wunsch |first2=C. |bibcode=1998DSRI...45.1977M}}</ref> About 98% of this dissipation is by marine tidal movement.<ref name="Ray1996">{{Cite journal |last1=Ray |first1=R.D. |year=1996 |title=Detection of tidal dissipation in the solid Earth by satellite tracking and altimetry |journal=[[Nature (journal)|Nature]] |volume=381 |issue=6583 |pages=595 |doi=10.1038/381595a0 |last2=Eanes |first2=R.J. |last3=Chao |first3=B.F. |bibcode=1996Natur.381..595R|s2cid=4367240 }}</ref> Dissipation arises as basin-scale tidal flows drive smaller-scale flows which experience turbulent dissipation. This tidal drag creates torque on the moon that gradually transfers angular momentum to its orbit, and a gradual increase in Earth–moon separation. The equal and opposite torque on the Earth correspondingly decreases its rotational velocity. Thus, over geologic time, the moon recedes from the Earth, at about {{convert|3.8|cm|in}}/year, lengthening the terrestrial day.{{efn|The day is currently lengthening at a rate of about 0.002 seconds per century.<ref>Lecture 2: The Role of Tidal Dissipation and the Laplace Tidal Equations by Myrl Hendershott. GFD Proceedings Volume, 2004, [[Woods Hole Oceanographic Institution|WHOI]] Notes by Yaron Toledo and Marshall Ward.</ref>}}
[[Tidal acceleration|Day length has increased]] by about 2 hours in the last 600 million years. Assuming (as a crude approximation) that the deceleration rate has been constant, this would imply that 70 million years ago, day length was on the order of 1% shorter with about 4 more days per year.
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When oscillating tidal currents in the stratified ocean flow over uneven bottom topography, they generate [[internal wave]]s with tidal frequencies. Such waves are called ''[[internal tide]]s''.
Shallow areas in otherwise open water can experience rotary tidal currents, flowing in directions that continually change and thus the flow direction (not the flow) completes a full rotation in {{frac|12|1|2}} hours (for example, the [[Nantucket Shoals]]).<ref name=rocur>{{cite journal |last=Le Lacheur |first=Embert A. |url=https://www.jstor.org/pss/208104 |title=Tidal currents in the open sea: Subsurface tidal currents at Nantucket Shoals Light Vessel |journal=[[Geographical Review]] |date=April 1924 |volume=14 |issue=2 |pages=282–286 |doi=10.2307/208104 |jstor=208104 |access-date=4 February 2012}}</ref>
In addition to oceanic tides, large lakes can experience small tides and even planets can experience ''[[atmospheric tide]]s'' and ''[[Earth tide]]s''. These are [[continuum mechanics|continuum mechanical]] phenomena. The first two take place in [[fluid mechanics|fluid]]s. The third affects the Earth's thin [[Solid mechanics|solid]] crust surrounding its semi-liquid interior (with various modifications).
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