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[[Tide table]]s can be used for any given locale to find the predicted times and [[amplitude]] (or "[[tidal range]]").
The predictions are influenced by many factors including the alignment of the Sun and Moon, the [[#Phase and amplitude|phase and amplitude of the tide]] (pattern of tides in the deep ocean), the [[amphidromic]] systems of the oceans, and the shape of the coastline and near-shore [[bathymetry]] (see ''[[#Timing|Timing]]''). They are however only predictions, the actual time and height of the tide is affected by wind and atmospheric pressure. Many shorelines experience [[semi-diurnal]] tides—two nearly equal high and low tides each day. Other locations have a [[diurnal cycle|diurnal]] tide—one high and low tide each day. A "mixed tide"—two uneven magnitude tides a day—is a third regular category.<ref name=Reddy>{{cite book |title=Descriptive physical oceanography: State of the Art |last1=Reddy |first1=M.P.M. |last2=Affholder |first2=M. |name-list-style=amp |url=https://books.google.com/books?id=2NC3JmKI7mYC&q=centrifugal&pg=PA436 |isbn=90-5410-706-5 |date=2002 |publisher=[[Taylor & Francis]] |page=249 |oclc=223133263 |via=[[Google Books]]}}</ref><ref name=Hubbard>{{cite book |title=Boater's Bowditch: The Small Craft American Practical Navigator |last=Hubbard |first=Richard |url=https://books.google.com/books?id=nfWSxRr8VP4C&q=centrifugal+revolution+and+rotation&pg=PA54 |isbn=0-07-136136-7 |publisher=[[McGraw-Hill]] Professional |date=1893 |page=54 |oclc=44059064 |via=[[Google Books]]}}</ref>{{efn|Coastal orientation and geometry affects the phase, direction, and amplitude of [[amphidromic system]]s, coastal [[Kelvin wave]]s as well as resonant [[seiche]]s in bays. In [[estuary|estuaries]], seasonal river outflows influence tidal flow.}}
Tides vary on timescales ranging from hours to years due to a number of factors, which determine the [[lunitidal interval]]. To make accurate records, [[tide gauge]]s at fixed stations measure water level over time. Gauges ignore variations caused by waves with periods shorter than minutes. These data are compared to the reference (or datum) level usually called [[mean sea level]].<ref>{{cite web |url=http://www.oceanservice.noaa.gov/education/kits/tides/media/supp_tide05.html |title=Tidal lunar day |publisher=[[NOAA]]}} Do not confuse with the astronomical [[lunar day]] on the Moon. A lunar zenith is the Moon's highest point in the sky.</ref>
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[[File:Tide schematic.svg|thumb|left|alt=Spring tide: the Sun, moon, and earth form a straight line. Neap tide: the Sun, moon, and earth form a right angle.|The types of tides]]
{{anchor|springtide|Spring}}The semi-diurnal range (the difference in height between high and low waters over about half a day) varies in a two-week cycle. Approximately twice a month, around [[new moon]] and [[full moon]] when the Sun, Moon, and Earth form a line (a configuration known as a [[syzygy (astronomy)|syzygy]]<ref>{{Cite book |title=Mathematical astronomy in Copernicus's De revolutionibus |volume=1 |first1=Noel M. |last1=Swerdlow |first2=Otto |last2=Neugebauer |publisher=Springer-Verlag |date=1984 |isbn=0-387-90939-7 |page=76 |url=https://books.google.com/books?id=4YDvAAAAMAAJ&q=Syzygy |via=[[Google Books]]}}</ref>), the [[tidal force]] due to the Sun reinforces that due to the Moon. The tide's range is then at its maximum; this is called the '''spring tide'''. It is not named after the [[Spring (season)|season]], but, like that word, derives from the meaning "jump, burst forth, rise", as in a natural [[Spring (hydrosphere)|spring]].
Spring tides are sometimes referred to as ''syzygy tides''.<ref name="Harris1981">{{cite book |last=Harris |first=D.L. |title=Tides and Tidal Datums in the United States |publisher=[[United States Army Corps of Engineers]], Coastal Engineering Research Center |series=Special report (Coastal Engineering Research Center (U.S.))) |year=1981 |url=https://books.google.com/books?id=kbIse3HQ74wC&pg=PA32 |access-date=2021-08-24 |page=32 |via=[[Google Books]]}}</ref>
{{anchor|Neap}}When the Moon is at [[Gibbous|first quarter]] or third quarter, the Sun and Moon are separated by 90° when viewed from the Earth, and the solar tidal force partially cancels the Moon's tidal force. At these points in the lunar cycle, the tide's range is at its minimum; this is called the '''neap tide''', or '''neaps'''. "Neap" is an Anglo-Saxon word meaning "without the power", as in ''forðganges nip'' (forth-going without-the-power).<ref>{{cite OED2|neap²}} Old English (example given from AD 469: ''forðganges nip'' – without the power of advancing). The Danish ''niptid'' is probably from the English. The English term neap-flood (from which neap tide comes) seems to have been in common use by AD 725.</ref>
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These include solar gravitational effects, the obliquity (tilt) of the Earth's Equator and rotational axis, the inclination of the plane of the lunar orbit and the elliptical shape of the Earth's orbit of the Sun.
A compound tide (or overtide) results from the shallow-water interaction of its two parent waves.<ref name="leprovost">{{cite book |last=Le Provost
=== 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=[[Philosophical Transactions of the Royal Society of London]] A |volume=290 |issue=1368 |date=November 28, 1978 |pages=235–266 |last1=Accad |first1=Y. |last2=Pekeris |first2=C.L. |name-list-style=amp |doi=10.1098/rsta.1978.0083 |bibcode=1978RSPTA.290..235A |s2cid=119526571}}</ref><ref>{{cite web |url=http://www.niwa.cri.nz/rc/prog/chaz/news/coastal#tide |title=Tide forecasts |publisher=National Institute of Water & Atmospheric Research |location=New Zealand |access-date=2008-11-07 |url-status=dead |archive-url=https://web.archive.org/web/20081014152423/http://www.niwa.cri.nz/rc/prog/chaz/news/coastal#tide |archive-date=2008-10-14}} Including animations of the M2, S2 and K1 tides for New Zealand.
</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.
For an ocean in the shape of a circular basin enclosed by a coastline, the cotidal lines point radially inward and must eventually meet at a common point, the [[amphidromic point]]. The amphidromic point is at once cotidal with high and low waters, which is satisfied by ''zero'' tidal motion. (The rare exception occurs when the tide encircles an island, as it does around New Zealand, [[Iceland]] and [[Madagascar]].) Tidal motion generally lessens moving away from continental coasts, so that crossing the cotidal lines are contours of constant ''amplitude'' (half the distance between high and low water) which decrease to zero at the amphidromic point. For a semi-diurnal tide the amphidromic point can be thought of roughly like the center of a clock face, with the hour hand pointing in the direction of the high water cotidal line, which is directly opposite the low water cotidal line. High water rotates about the amphidromic point once every 12 hours in the direction of rising cotidal lines, and away from ebbing cotidal lines. This rotation, caused by the [[Coriolis effect]], is generally clockwise in the southern hemisphere and counterclockwise in the northern hemisphere. The difference of cotidal phase from the phase of a reference tide is the ''epoch''. The reference tide is the hypothetical constituent "equilibrium tide" on a landless Earth measured at 0° longitude, the Greenwich meridian.<ref>{{cite book |last=Schureman |first=Paul |title=Manual of harmonic analysis and prediction of tides |date=1971 |publisher=U.S. Coast and geodetic survey |page=204 |url=https://www.biodiversitylibrary.org/ia/manualofharmonic00schu#page/220/mode/1up}}</ref>
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[[Seleucus of Seleucia]] theorized around 150 BC that tides were caused by the Moon. The influence of the Moon on bodies of water was also mentioned in [[Ptolemy]]'s ''[[Tetrabiblos]]''.{{efn|"The moon, too, as the heavenly body nearest the earth, bestows her effluence most abundantly upon mundane things, for most of them, animate or inanimate, are sympathetic to her and change in company with her; the rivers increase and diminish their streams with her light, the seas turn their own tides with her rising and setting, … "<ref>{{Cite book |author=[[Ptolemy]] |translator-first=Frank E. |translator-last=Robbins |title=Tetrabiblos |location=Cambridge, Massachusetts |publisher=[[Harvard University Press]] |date=1940 |volume=1 |chapter=2}}</ref>}}
In {{lang|la|De temporum ratione}} (''[[The Reckoning of Time]]'') of 725 [[Bede]] linked semidurnal tides and the phenomenon of varying tidal heights to the Moon and its phases. Bede starts by noting that the tides rise and fall 4/5 of an hour later each day, just as the Moon rises and sets 4/5 of an hour later.<ref name=Wallis>{{cite book |author=Bede |author-link=Bede |translator-last=Wallis |translator-first=Faith |title=The Reckoning of Time |year=1999 |publisher=[[Liverpool University Press]] |isbn=0-85323-693-3 |url=https://books.google.com/books?id=yFsw-Vaup6sC |access-date=1 June 2018 |page=82 |via=[[Google Books]]}}</ref> He goes on to emphasise that in two lunar months (59 days) the Moon circles the Earth 57 times and there are 114 tides.{{sfn|Bede|1999|p=83}} Bede then observes that the height of tides varies over the month. Increasing tides are called ''malinae'' and decreasing tides ''ledones'' and that the month is divided into four parts of seven or eight days with alternating ''malinae'' and ''ledones''.{{sfn|Bede|1999|p=84}} In the same passage he also notes the effect of winds to hold back tides.{{sfn|Bede|1999|p=84}} Bede also records that the time of tides varies from place to place. To the north of Bede's location ([[Monkwearmouth]]) the tides are earlier, to the south later.{{sfn|Bede|1999|p=85}} He explains that the tide "deserts these shores in order to be able all the more to be able to flood other [shores] when it arrives there" noting that "the Moon which signals the rise of tide here, signals its retreat in other regions far from this quarter of the heavens".{{sfn|Bede|1999|p=85}}
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 />
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[[Galileo Galilei]] in his 1632 ''[[Dialogue Concerning the Two Chief World Systems]]'', whose working title was ''Dialogue on the Tides'', gave an explanation of the tides. The resulting theory, however, was incorrect as he attributed the tides to the sloshing of water caused by the Earth's movement around the Sun. He hoped to provide mechanical proof of the Earth's movement. The value of his tidal theory is disputed. Galileo rejected Kepler's explanation of the tides.
[[Isaac Newton]] (1642–1727) was the first person to explain tides as the product of the gravitational attraction of astronomical masses. His explanation of the tides (and many other phenomena) was published in the ''[[Philosophiae Naturalis Principia Mathematica|Principia]]'' (1687)<ref name=slc-ch2>{{cite book |author-link=Eugenie Lisitzin |last=Lisitzin |first=E. |title=Sea-Level Changes, (Elsevier Oceanography Series) |volume=8 |date=1974 |chapter=2 "Periodical sea-level changes: Astronomical tides" |page=5}}</ref><ref>{{cite web |publisher=U.S. [[National Oceanic and Atmospheric Administration]] (NOAA) National Ocean Service (Education section) |url=http://oceanservice.noaa.gov/education/kits/tides/tides02_cause.html |title=What Causes Tides?}}</ref> and used his [[Newton's law of universal gravitation|theory of universal gravitation]] to explain the lunar and solar attractions as the origin of the tide-generating forces.{{efn|1=See for example, in the 'Principia' (Book 1) (1729 translation), [https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA251 Corollaries 19 and 20 to Proposition 66, on pages 251–254], referring back to page 234 et seq.; and in Book 3 [https://archive.org/details/bub_gb_6EqxPav3vIsC/page/n279 <!-- pg=255 --> Propositions 24, 36 and 37, starting on page 255].}}
Newton and others before [[Pierre-Simon Laplace]] worked the problem from the perspective of a static system (equilibrium theory), that provided an approximation that described the tides that would occur in a non-inertial ocean evenly covering the whole Earth.<ref name=slc-ch2 /> The tide-generating force (or its corresponding [[scalar potential|potential]]) is still relevant to tidal theory, but as an intermediate quantity (forcing function) rather than as a final result; theory must also consider the Earth's accumulated dynamic tidal response to the applied forces, which response is influenced by ocean depth, the Earth's rotation, and other factors.<ref>{{cite book |last=Wahr |first=J. |title=Earth Tides in "Global Earth Physics", American Geophysical Union Reference Shelf #1 |pages=40–46 |date=1995}}</ref>
In 1740, the [[Académie Royale des Sciences]] in Paris offered a prize for the best theoretical essay on tides. [[Daniel Bernoulli]], [[Leonhard Euler]], [[Colin Maclaurin]] and [[Antoine Cavalleri]] shared the prize.<ref name="EulerAiton1996">{{cite book |first1=Leonhard |last1=Euler |author1-link=Leonhard Euler |first2=Eric J. |last2=Aiton |title=Commentationes mechanicae et astronomicae ad physicam pertinentes |url=https://books.google.com/books?id=b1yCADlGTkgC&pg=PR19 |year=1996 |publisher=[[Springer Science+Business Media]] |isbn=978-3-7643-1459-0 |pages=19– |via=[[Google Books]]}}</ref>
Maclaurin used Newton's theory to show that a smooth sphere covered by a sufficiently deep ocean under the tidal force of a single deforming body is a [[prolate]] spheroid (essentially a three-dimensional oval) with major axis directed toward the deforming body. Maclaurin was the first to write about the Earth's [[Coriolis effect|rotational effects]] on motion. Euler realized that the tidal force's ''horizontal'' component (more than the vertical) drives the tide. In 1744 [[Jean le Rond d'Alembert]] studied tidal equations for the atmosphere which did not include rotation.
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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=[[Proceedings of the Royal Society of London]] A |volume=100 |issue=704 |pages=305–329 |bibcode=1921RSPSA.100..305D |doi=10.1098/rspa.1921.0088 |doi-access=free}}</ref> the first modern development of the tide-generating potential in harmonic form: Doodson distinguished 388 tidal frequencies.<ref>{{cite journal |title=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 |journal=[[AAS Division on Dynamical Astronomy]] |date=April 2004 |volume=36 |issue=2 |page=67 |last1=Casotto |first1=S. |last2=Biscani |first2=F. |name-list-style=amp |bibcode=2004DDA....35.0805C}}</ref> Some of his methods remain in use.<ref>{{cite book |last=Moyer
=== History of tidal observation ===
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From ancient times, tidal observation and discussion has increased in sophistication, first marking the daily recurrence, then tides' relationship to the Sun and moon. [[Pytheas]] travelled to the [[British Isles]] about 325 BC and seems to be the first to have related spring tides to the phase of the moon.
In the 2nd century BC, the [[Hellenistic astronomer]] [[Seleucus of Seleucia]] correctly described the phenomenon of tides in order to support his [[Heliocentrism|heliocentric]] theory.<ref>{{cite book |title=Flussi e riflussi |language=it |trans-title=Ebbs and flows |publisher=Feltrinelli |location=Milano |date=2003 |isbn=88-07-10349-4}}</ref> He correctly theorized that tides were caused by the [[moon]], although he believed that the interaction was mediated by the [[pneuma]]. He noted that tides varied in time and strength in different parts of the world. According to [[Strabo]] (1.1.9), Seleucus was the first to link tides to the lunar attraction, and that the height of the tides depends on the moon's position relative to the Sun.<ref>{{cite journal |last=van der Waerden |first=B.L. |author-link=Bartel Leendert van der Waerden |date=1987 |title=The Heliocentric System in Greek, Persian and Hindu Astronomy |journal=[[Annals of the New York Academy of Sciences]] |volume=500 |issue=1 |pages=525–545 [527] |doi=10.1111/j.1749-6632.1987.tb37224.x |bibcode
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 book |last=Cartwright
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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The first known sea-level record of an entire spring–neap cycle was made in 1831 on the Navy Dock in the [[Thames Estuary]]. Many large ports had automatic tide gauge stations by 1850.
[[Sir John Lubbock, 3rd Baronet|John Lubbock]] was one of the first to map co-tidal lines, for Great Britain, Ireland and adjacent coasts, in 1840.<ref>{{cite journal |last1=Lubbock |first1=J.W. |title=On the tides on the coast of Great Britain |journal=The Philosophical Magazine |date=1831 |volume=9 |issue=53 |pages=333–335 |doi=10.1080/14786443108647618 |url=https://archive.org/details/lubbock-1831-philosophical-magazine-s-2id-13416500}}</ref> [[William Whewell]] expanded this work ending with a nearly global chart in 1836.<ref>{{cite journal |last1=Whewell |first1=William |title=Researches on the tides, sixth series. On the results of an extensive system of tide observations made on the coasts of Europe and America in June 1835 |journal=[[Philosophical Transactions of the Royal Society of London]] |date=1836 |volume=126 |pages=289–341 |url=https://archive.org/details/jstor-108036}}</ref> In order to make these maps consistent, he hypothesized the existence of a region with no tidal rise or fall where co-tidal lines meet in the mid-ocean. The existence of such an [[amphidromic point]], as they are now known, was confirmed in 1840 by [[William Hewett (died 1840)|Captain William Hewett, RN]], from careful soundings in the [[North Sea]].<ref>{{cite journal |last1=Hewett |first1=William |title=Tide observations in the North Sea |journal=The Nautical Magazine |date=1841 |pages=180–183 |url=https://archive.org/details/199-1841-hewett-fairy-the-nautical-magazine-1841}}</ref><ref name="Cartwright2000">{{cite book |first1=David Edgar |last1=Cartwright |date=17 August 2000 |title=Tides: A Scientific History |publisher=[[Cambridge University Press]] |isbn=978-0-521-79746-7 |oclc=1001932580}}</ref><ref name="tidhist"/>
== Physics ==
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The tidal force produced by a massive object (Moon, hereafter) on a small particle located on or in an extensive body (Earth, hereafter) is the vector difference between the gravitational force exerted by the Moon on the particle, and the gravitational force that would be exerted on the particle if it were located at the Earth's center of mass.
Whereas the [[gravitational force]] subjected by a celestial body on Earth varies inversely as the square of its distance to the Earth, the maximal tidal force varies inversely as, approximately, the cube of this distance.<ref>{{cite book |last=Young
:<math>\text{tidal force}\propto\frac M{d^3}\propto\rho\left(\frac rd\right)^3</math>
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The shape of the shoreline and the ocean floor changes the way that tides propagate, so there is no simple, general rule that predicts the time of high water from the Moon's position in the sky. Coastal characteristics such as underwater [[bathymetry]] and coastline shape mean that individual location characteristics affect tide forecasting; actual high water time and height may differ from model predictions due to the coastal morphology's effects on tidal flow. However, for a given location the relationship between lunar [[altitude (astronomy)|altitude]] and the time of high or low tide (the [[lunitidal interval]]) is relatively constant and predictable, as is the time of high or low tide relative to other points on the same coast. For example, the high tide at [[Norfolk, Virginia]], U.S., predictably occurs approximately two and a half hours before the Moon passes directly overhead.
Land masses and ocean basins act as barriers against water moving freely around the globe, and their varied shapes and sizes affect the size of tidal frequencies. As a result, tidal patterns vary. For example, in the U.S., the East coast has predominantly semi-diurnal tides, as do Europe's Atlantic coasts, while the West coast predominantly has mixed tides.<ref name=noaa7b>{{cite web |website=U.S. [[National Oceanic and Atmospheric Administration]] (NOAA) National Ocean Service (Education section)
== Observation and prediction ==
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=== Timing ===
[[File:Diurnal tide types map.jpg|thumb|The same tidal forcing has different results depending on many factors, including coast orientation, continental shelf margin, water body dimensions.|alt=World map showing the location of diurnal, semi-diurnal, and mixed semi-diurnal tides. The European and African west coasts are exclusively semi-diurnal, and North America's West coast is mixed semi-diurnal, but elsewhere the different patterns are highly intermixed, although a given pattern may cover {{convert|200|-|2000|km|mi}}.]]
The tidal forces due to the Moon and Sun generate very long waves which travel all around the ocean following the paths shown in [[#Phase and amplitude|co-tidal charts]]. The time when the crest of the wave reaches a port then gives the time of high water at the port. The time taken for the wave to travel around the ocean also means that there is a delay between the phases of the Moon and their effect on the tide. Springs and neaps in the [[North Sea]], for example, are two days behind the new/full moon and first/third quarter moon. This is called the tide's ''age''.<ref>{{cite web |url=https://glossary.ametsoc.org/wiki/Age |title=Glossary of Meteorology |website=[[American Meteorological Society]]}}</ref><ref>{{cite book
|title=The elements of physics |first1=Thomas |last1=Webster |publisher=Printed for Scott, Webster, and Geary |date=1837 |page=[https://archive.org/details/elementsphysics00websgoog/page/n184 168] |url=https://archive.org/details/elementsphysics00websgoog}}</ref>
The ocean [[bathymetry]] greatly influences the tide's exact time and height at a particular [[coast]]al point. There are some extreme cases; the [[Bay of Fundy]], on the east coast of Canada, is often stated to have the world's highest tides because of its shape, bathymetry, and its distance from the continental shelf edge.<ref>{{cite web |url=http://www.waterlevels.gc.ca/english/FrequentlyAskedQuestions.shtml#importantes |title=FAQ |access-date=June 23, 2007 |archive-date=February 12, 2012 |archive-url=https://web.archive.org/web/20120212231548/http://www.waterlevels.gc.ca/english/FrequentlyAskedQuestions.shtml#importantes |url-status=dead}}</ref> Measurements made in November 1998 at Burntcoat Head in the Bay of Fundy recorded a maximum range of {{convert|16.3|m|ft}} and a highest predicted extreme of {{convert|17|m|ft}}.<ref name=BIO2004>{{cite journal |last1=O'Reilly |first1=C.T.R. |first2=Ron |last2=Solvason |first3=Christian |last3=Solomon |name-list-style=amp |title=Where are the World's Largest Tides |journal=BIO Annual Report "2004 in Review" |date=2005 |pages=44–46 |editor1-first=J. |editor1-last=Ryan |publisher=Biotechnol. Ind. Org. |location=Washington, D.C.}}</ref><ref name=FundyWorkshop>{{cite book |first1=Charles T. |last1=O'reilly
[[Southampton]] in the United Kingdom has a double high water caused by the interaction between the ''M''<sub>2</sub> and [[Theory of tides#Short period|''M''<sub>4</sub> tidal constituents]] (Shallow water overtides of principal lunar).<ref name=Pingree_1978>{{cite journal |last1=Pingree |first1=R.D. |first2=L. |last2=Maddock |title=Deep-Sea Research |year=1978 |volume=25 |pages=53–63}}</ref> [[Isle of Portland|Portland]] has double low waters for the same reason. The ''M''<sub>4</sub> tide is found all along the south coast of the United Kingdom, but its effect is most noticeable between the [[Isle of Wight]] and [[Isle of Portland|Portland]] because the ''M''<sub>2</sub> tide is lowest in this region.
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Remember that astronomical tides do ''not'' include weather effects. Also, changes to local conditions (sandbank movement, dredging harbour mouths, etc.) away from those prevailing at the measurement time affect the tide's actual timing and magnitude. Organisations quoting a "highest astronomical tide" for some location may exaggerate the figure as a safety factor against analytical uncertainties, distance from the nearest measurement point, changes since the last observation time, ground subsidence, etc., to avert liability should an engineering work be overtopped. Special care is needed when assessing the size of a "weather surge" by subtracting the astronomical tide from the observed tide.
Careful Fourier [[data analysis]] over a nineteen-year period (the ''National Tidal Datum Epoch'' in the U.S.) uses frequencies called the ''tidal harmonic constituents''. Nineteen years is preferred because the Earth, Moon and Sun's relative positions repeat almost exactly in the [[Metonic cycle]] of 19 years, which is long enough to include the 18.613 year [[Lunar node|lunar nodal]] [[Earth tide#Tidal constituents|tidal constituent]]. This analysis can be done using only the knowledge of the forcing ''period'', but without detailed understanding of the mathematical derivation, which means that useful tidal tables have been constructed for centuries.<ref>
In the ''M''<sub>2</sub> plot above, each cotidal line differs by one hour from its neighbors, and the thicker lines show tides in phase with equilibrium at Greenwich. The lines rotate around the [[amphidromic point]]s counterclockwise in the northern hemisphere so that from [[Baja California Peninsula]] to [[Alaska]] and from [[France]] to [[Ireland]] the ''M''<sub>2</sub> tide propagates northward. In the southern hemisphere this direction is clockwise. On the other hand, ''M''<sub>2</sub> tide propagates counterclockwise around New Zealand, but this is because the islands act as a dam and permit the tides to have different heights on the islands' opposite sides. (The tides do propagate northward on the east side and southward on the west coast, as predicted by theory.)
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Tidal flows are important for navigation, and significant errors in position occur if they are not accommodated. Tidal heights are also important; for example many rivers and harbours have a shallow "bar" at the entrance which prevents boats with significant [[Draft (hull)|draft]] from entering at low tide.
Until the advent of automated navigation, competence in calculating tidal effects was important to naval officers. The certificate of examination for lieutenants in the [[Royal Navy]] once declared that the prospective officer was able to "shift his tides".<ref>{{cite book |title=The Mariner's Mirror |url=https://books.google.com/books?id=lagPAAAAIAAJ&q=%22shift+his+tides%22 |author=Society for Nautical Research |date=1958 |access-date=2009-04-28 |via=[[Google Books]]}}</ref>
Tidal flow timings and velocities appear in ''tide charts'' or a [[tidal stream atlas]]. Tide charts come in sets. Each chart covers a single hour between one high water and another (they ignore the leftover 24 minutes) and show the average tidal flow for that hour. An arrow on the tidal chart indicates the direction and the average flow speed (usually in [[Knot (unit)|knots]]) for spring and neap tides. If a tide chart is not available, most nautical charts have "[[tidal diamond]]s" which relate specific points on the chart to a table giving tidal flow direction and speed.
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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
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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