View source for Astronomy:Tide

{{short description|Rise and fall of the sea level under astronomical gravitational influences}}
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[[File:tide overview.svg|300px|thumb|upright=1|Simplified schematic of only the lunar portion of Earth's tides, showing (exaggerated) high tides at the sublunar point and its [[Antipodal point|antipode]] for the hypothetical case of an ocean of constant depth without land, and on the assumption that Earth is not rotating; otherwise there is a lag angle. Solar tides not shown.|alt=]]
[[File:Tidal circularization figure1.svg|right|thumb|Earth's rotation drags the position of the tidal bulge ahead of the position directly under the Moon showing the lag angle.]]
[[File:Tide and Moon.jpg|thumb|In Maine (U.S.), low tide occurs roughly at moonrise and high tide with a high Moon, corresponding to the simple gravity model of two tidal bulges; at most places however, the Moon and tides have a phase shift.]]
[[File:Tide St. Simons, GA 2018.webm|right|thumb|Tide coming in, video stops about {{frac|1|1|2}} hours before high tide]]

'''Tides''' are the rise and fall of [[Sea level|sea level]]s caused by the combined effects of the [[Physics:Gravity|gravitational]] forces exerted by the [[Astronomy:Moon|Moon]] (and to a much lesser extent, the [[Astronomy:Sun|Sun]]) and are also caused by the [[Astronomy:Earth|Earth]] and [[Astronomy:Moon|Moon]] orbiting one another.

[[Tide table]]s can be used for any given locale to find the predicted times and [[Physics:Amplitude|amplitude]] (or "[[Earth:Tidal range|tidal range]]"). 
The predictions are influenced by many factors including the alignment of the Sun and Moon, the 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 [[Earth:Bathymetry|bathymetry]] (see ''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 [[Earth: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]] |access-date=2022-01-05 |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153028/https://books.google.com/books?id=2NC3JmKI7mYC&q=centrifugal&pg=PA436 |url-status=live }}</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]] |access-date=2022-01-05 |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153028/https://books.google.com/books?id=nfWSxRr8VP4C&q=centrifugal+revolution+and+rotation&pg=PA54 |url-status=live }}</ref>{{efn|Coastal orientation and geometry affects the phase, direction, and amplitude of amphidromic systems, coastal [[Physics:Kelvin wave|Kelvin wave]]s as well as resonant [[Physics:Seiche|seiche]]s in bays. In [[Earth: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 [[Earth:Lunitidal interval|lunitidal interval]]. To make accurate records, [[Earth:Tide gauge|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 |access-date=2007-04-07 |archive-date=2018-08-17 |archive-url=https://web.archive.org/web/20180817075116/https://oceanservice.noaa.gov/education/kits/tides/media/supp_tide05.html |url-status=live }} Do not confuse with the astronomical [[Astronomy:Lunar day|lunar day]] on the Moon. A lunar zenith is the Moon's highest point in the sky.</ref>

While tides are usually the largest source of short-term sea-level fluctuations, sea levels are also subject to change from [[Physics:Thermal expansion|thermal expansion]], wind, and barometric pressure changes, resulting in [[Physics:Storm surge|storm surge]]s, especially in shallow seas and near coasts.

Tidal phenomena are not limited to the oceans, but can occur in other systems whenever a gravitational field that varies in time and space is present. For example, the shape of the solid part of the Earth is affected slightly by [[Earth:Earth tide|Earth tide]], though this is not as easily seen as the water tidal movements.

== Characteristics ==
[[File:Tide type.svg|thumb|Types of tides (See ''Timing'' (below) for coastal map)|alt=Three graphs. The first shows the twice-daily rising and falling tide pattern with nearly regular high and low elevations. The second shows the much more variable high and low tides that form a "mixed tide". The third shows the day-long period of a diurnal tide.]]

Four stages in the tidal cycle are named:
* {{anchor|Low|Low tide}}The water stops falling, reaching a local minimum called '''low tide'''.
* {{anchor|Flood|Flood tide}}Sea level rises over several hours, covering the [[Earth:Intertidal zone|intertidal zone]]; '''flood tide'''.
* {{anchor|High|High tide}}The water stops rising, reaching a local maximum called '''high tide'''.
* {{anchor|Ebb|Ebb tide}}Sea level falls over several hours, revealing the intertidal zone; '''ebb tide'''.

Oscillating [[Physics:Current (fluid)|currents]] produced by tides are known as '''tidal streams''' or '''tidal currents'''. The moment that the tidal current ceases is called ''slack water'' or ''slack tide''. The tide then reverses direction and is said to be turning. Slack water usually occurs near high water and low water, but there are locations where the moments of slack tide differ significantly from those of high and low water.<ref>{{Cite book |first=George L. |last=Mellor |title=Introduction to physical oceanography |publisher=Springer |date=1996 |isbn=1-56396-210-1 |page=169}}</ref>

Tides are commonly ''semi-diurnal'' (two high waters and two low waters each day), or ''diurnal'' (one tidal cycle per day). The two high waters on a given day are typically not the same height (the daily inequality); these are the ''higher high water'' and the ''lower high water'' in [[Tide table|tide table]]s. Similarly, the two low waters each day are the ''higher low water'' and the ''lower low water''. The daily inequality is not consistent and is generally small when the Moon is over the [[Astronomy:Equator|Equator]].{{efn|Tide tables usually list ''mean lower low water'' (mllw, the 19 year average of mean lower low waters), ''mean higher low water'' (mhlw), ''mean lower high water'' (mlhw), ''mean higher high water'' (mhhw), as well as ''perigean tides''. These are ''mean'' values in the sense that they derive from mean data.<ref>{{cite web |url=http://www.ecy.wa.gov/programs/sea/swces/products/publications/glossary/words/H_M.htm |title=Glossary of Coastal Terminology: H–M |publisher=Washington Department of Ecology, State of Washington |access-date=5 April 2007 |archive-date=21 November 2017 |archive-url=https://web.archive.org/web/20171121042259/http://www.ecy.wa.gov/programs/sea/swces/products/publications/glossary/words/H_M.htm |url-status=live }}</ref>}}

=== Reference levels ===

The following reference tide levels can be defined, from the highest level to the lowest:
* ''Highest astronomical tide'' (HAT) – The highest tide which can be predicted to occur. Note that meteorological conditions may add extra height to the HAT.
* ''[[Earth:Mean high water springs|Mean high water springs]]'' (MHWS) – The average of the two high tides on the days of spring tides.
* ''Mean high water neaps'' (MHWN) – The average of the two high tides on the days of neap tides.
* ''Mean sea level'' (MSL) – This is the average sea level. The MSL is constant for any location over a long period.
* ''Mean low water neaps'' (MLWN) – The average of the two low tides on the days of neap tides.
* ''Mean low water springs'' (MLWS) – The average of the two low tides on the days of spring tides.
* ''Lowest astronomical tide'' (LAT) – The lowest tide which can be predicted to occur.<ref>{{cite web |title=Definitions of tidal terms |url=http://www.linz.govt.nz/hydro/tidal-info/tidal-intro/definitions |website=Land Information New Zealand |access-date=20 February 2017 |archive-date=30 August 2014 |archive-url=https://web.archive.org/web/20140830114240/http://www.linz.govt.nz/hydro/tidal-info/tidal-intro/definitions |url-status=live }}</ref>


[[File:Tide terms.png|thumb|upright=3.65|center|Illustration by the course of half a month]]

== Tidal constituents{{anchor|Constituents}} == <!-- tidal constituent and tidal constituents redirect here -->

''Tidal constituents'' are the net result of multiple influences impacting tidal changes over certain periods of time. Primary constituents include the Earth's rotation, the position of the Moon and Sun relative to the Earth, the Moon's altitude (elevation) above the Earth's Equator, and [[Earth:Bathymetry|bathymetry]]. Variations with periods of less than half a day are called ''harmonic constituents''. Conversely, cycles of days, months, or years are referred to as ''long period'' constituents.

Tidal forces [[Earth:Earth tide|affect the entire earth]], but the movement of solid Earth occurs by mere centimeters. In contrast, the atmosphere is much more fluid and compressible so its surface moves by kilometers, in the sense of the contour level of a particular low pressure in the outer atmosphere.

=== Principal lunar semi-diurnal constituent ===
[[File:Global surface elevation of M2 ocean tide.webm|thumb|upright=1.7|Global surface elevation of M2 ocean tide (NASA)<ref name=NASA2016>{{Cite web |url=https://svs.gsfc.nasa.gov/4541 |title=Ocean Tides and Magnetic Fields |website=NASA Visualization Studio |publisher=[[Organization:NASA|NASA]] |date=30 December 2016 |access-date=20 November 2020 |archive-date=27 November 2020 |archive-url=https://web.archive.org/web/20201127195922/https://svs.gsfc.nasa.gov/4541 |url-status=live }}</ref>]]

In most locations, the largest constituent is the ''principal lunar semi-diurnal'', also known as the ''M2 tidal constituent'' or ''M<sub>2</sub> tidal constituent''. Its period is about 12 hours and 25.2 minutes, exactly half a ''tidal lunar day'', which is the average time separating one lunar [[Zenith|zenith]] from the next, and thus is the time required for the Earth to rotate once relative to the Moon. Simple tide clocks track this constituent. The lunar day is longer than the Earth day because the Moon orbits in the same direction the Earth spins. This is analogous to the minute hand on a watch crossing the hour hand at 12:00 and then again at about 1:05½ (not at 1:00).

The Moon orbits the Earth in the same direction as the Earth rotates on its axis, so it takes slightly more than a day—about 24 hours and 50 minutes—for the Moon to return to the same location in the sky. During this time, it has passed overhead ([[Astronomy:Culmination|culmination]]) once and underfoot once (at an [[Hour angle|hour angle]] of 00:00 and 12:00 respectively), so in many places the period of strongest tidal forcing is the above-mentioned, about 12 hours and 25 minutes. The moment of highest tide is not necessarily when the Moon is nearest to [[Zenith|zenith]] or [[Nadir|nadir]], but the period of the forcing still determines the time between high tides.

Because the gravitational field created by the Moon weakens with distance from the Moon, it exerts a slightly stronger than average force on the side of the Earth facing the Moon, and a slightly weaker force on the opposite side. The Moon thus tends to "stretch" the Earth slightly along the line connecting the two bodies. The solid Earth deforms a bit, but ocean water, being fluid, is free to move much more in response to the tidal force, particularly horizontally (see equilibrium tide).

As the Earth rotates, the magnitude and direction of the tidal force at any particular point on the Earth's surface change constantly; although the ocean never reaches equilibrium—there is never time for the fluid to "catch up" to the state it would eventually reach if the tidal force were constant—the changing tidal force nonetheless causes rhythmic changes in sea surface height.

When there are two high tides each day with different heights (and two low tides also of different heights), the pattern is called a ''mixed semi-diurnal tide''.<ref name=noaa7>{{cite web |publisher=[[Organization:National Oceanic and Atmospheric Administration|U.S. National Oceanic and Atmospheric Administration]] (NOAA) National Ocean Service (Education section) |url=http://oceanservice.noaa.gov/education/kits/tides/tides07_cycles.html |title=Types and causes of tidal cycles |archive-url=https://web.archive.org/web/20120201145550/http://oceanservice.noaa.gov/education/kits/tides/tides07_cycles.html |archive-date=February 1, 2012}}</ref>

=== Range variation: springs and neaps ===
{{further|Earth:Tidal range}}
[[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 [[Astronomy:New moon|new moon]] and [[Astronomy:Full moon|full moon]] when the Sun, Moon, and Earth form a line (a configuration known as a [[Astronomy:Syzygy|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]] |access-date=2020-11-22 |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153030/https://books.google.com/books?id=4YDvAAAAMAAJ&q=Syzygy |url-status=live }}</ref>), the [[Physics:Tidal force|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 season, but, like that word, derives from the meaning "jump, burst forth, rise", as in a natural 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]] |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153028/https://books.google.com/books?id=kbIse3HQ74wC&pg=PA32 |url-status=live }}</ref>

{{anchor|Neap}}When the Moon is at 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>
Neap tides are sometimes referred to as ''quadrature tides''.<ref name="Harris1981"/>

Spring tides result in high waters that are higher than average, low waters that are lower than average, "slack water" time that is shorter than average, and stronger tidal currents than average. Neaps result in less extreme tidal conditions. There is about a seven-day interval between springs and neaps.



{{clear}}
<gallery class="center">
File:High tide sun moon same side beginning.png|''Spring tide:'' Sun and Moon on the same side (0°)
File:Low tide sun moon 90 degrees.png|''Neap tide:'' Sun and Moon at 90°
File:High tide sun moon opposite side.png|''Spring tide:'' Sun and Moon at opposite sides (180°)
File:Low tide sun moon 270 degrees.png|''Neap tide:'' Sun and Moon at 270°
File:High tide sun moon same side end.png|''Spring tide:'' Sun and Moon at the same side (cycle restarts)
</gallery>
{{clear}}

=== Lunar distance ===
[[File:Bangchuidao Island.JPG|thumb|Low tide at Bangchuidao scenic area, Dalian, Liaoning Province, {{wipe|China}}]]
[[File:Negative low tide at Ocean Beach 1.jpg|thumb|Low tide at Ocean Beach in {{wipe|San Francisco}}, {{wipe|California}}, U.S.]]
[[File:Atlantic coast at low tide, Bar Harbor IMG 2262.JPG|thumb|Low tide at Bar Harbor, Maine, U.S. (2014)]]
 
The changing distance separating the Moon and Earth also affects tide heights. When the Moon is closest, at perigee, the range increases, and when it is at apogee, the range shrinks. Six or eight times a year perigee coincides with either a new or full moon causing perigean spring tides with the largest ''[[Earth:Tidal range|tidal range]]''. The difference between the height of a tide at perigean spring tide and the spring tide when the moon is at apogee depends on location but can be large as a foot higher.<ref>{{cite web |title=What is a perigean spring tide? |url=https://oceanservice.noaa.gov/facts/perigean-spring-tide.html |publisher=National Oceanic and Atmospheric Administration |date=26 February 2021 |access-date=16 July 2021 |archive-date=30 July 2021 |archive-url=https://web.archive.org/web/20210730210313/https://oceanservice.noaa.gov/facts/perigean-spring-tide.html |url-status=live }}</ref>

=== Other constituents ===
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 |first=Christian |date=1991 |chapter=Generation of Overtides and compound tides (review) |editor1-last=Parker |editor1-first=Bruce B. |title=Tidal Hydrodynamics |publisher={{wipe|John Wiley & Sons}} |isbn=978-0-471-51498-5}}</ref>

=== 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. Around these convergences, called [[Physics:Amphidromic point|amphidromic point]]s, curved arrows 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 |publisher=Springer |via=[[Google Books]] |access-date=22 November 2020 |archive-date=16 September 2023 |archive-url=https://web.archive.org/web/20230916153029/https://books.google.com/books?id=E3uhBQAAQBAJ&q=tidal+map&pg=PT28 |url-status=live }}</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 [[Physics:Amphidromic point|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 {{wipe|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 |access-date=2018-01-14 |archive-date=2017-08-08 |archive-url=https://web.archive.org/web/20170808200945/http://www.biodiversitylibrary.org/ia/manualofharmonic00schu#page/220/mode/1up |url-status=live }}</ref>

In the North Atlantic, because the cotidal lines circulate counterclockwise around the amphidromic point, the high tide passes New York Harbor approximately an hour ahead of Norfolk Harbor. South of Cape Hatteras the tidal forces are more complex, and cannot be predicted reliably based on the North Atlantic cotidal lines.

== History ==

=== History of tidal theory ===

Investigation into tidal physics was important in the early development of [[Physics:Celestial mechanics|celestial mechanics]], with the existence of two daily tides being explained by the Moon's gravity. Later the daily tides were explained more precisely by the interaction of the Moon's and the Sun's gravity.

[[Biography:Seleucus of Seleucia|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 [[Biography:Ptolemy|Ptolemy]]'s ''[[Philosophy:Tetrabiblos|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=[[Biography:Ptolemy|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 [[Biography:Bede|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 |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]] |archive-date=9 April 2023 |archive-url=https://web.archive.org/web/20230409160418/https://books.google.com/books?id=yFsw-Vaup6sC |url-status=live }}</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}}

Later medieval understanding of the tides was primarily based on works of Muslim astronomers, which became available through Latin translation starting from the 12th century.<ref name=Tolmacheva>{{Cite book |publisher=[[Company:Routledge|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 [[Astronomy:Neap tide|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 |access-date=2014-06-01 |archive-date=2014-08-05 |archive-url=https://web.archive.org/web/20140805054735/http://www.vliz.be/imisdocs/publications/224466.pdf |url-status=live }}</ref><ref>{{Cite book |last1=Palmerino |first1=Carla Rita |first2=J.M.M.H. |last2=Thijssen |url=https://books.google.com/books?id=a5lkdlMPi1AC&dq=%22johannes+kepler%22+%22simon+stevin%22+ebb&pg=PA200 |title=The Reception of the Galilean Science of Motion in Seventeenth-Century Europe |date=31 August 2004 |publisher=[[Company:Springer Science+Business Media|Springer Science+Business Media]] |isbn=978-1-4020-2455-9 |page=200 |via=[[Google Books]] |access-date=29 November 2022 |archive-date=12 April 2022 |archive-url=https://web.archive.org/web/20220412060701/https://books.google.com/books?id=a5lkdlMPi1AC&pg=PA200&dq=%22johannes+kepler%22+%22simon+stevin%22+ebb |url-status=live }}</ref>

In 1609 [[Biography:Johannes Kepler|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.

[[Biography:Galileo Galilei|Galileo Galilei]] in his 1632 ''[[Physics:Dialogue Concerning the Two Chief World Systems|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.

[[Biography:Isaac Newton|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 ''Principia'' (1687)<ref name=slc-ch2>{{cite book |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. [[Organization:National Oceanic and Atmospheric Administration|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? |access-date=2009-09-06 |archive-date=2016-08-20 |archive-url=https://web.archive.org/web/20160820055655/http://oceanservice.noaa.gov/education/kits/tides/tides02_cause.html |url-status=live }}</ref> and used his [[Physics: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 [[Biography:Pierre-Simon Laplace|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. [[Biography:Daniel Bernoulli|Daniel Bernoulli]], [[Biography:Leonhard Euler|Leonhard Euler]], Colin Maclaurin and Antoine Cavalleri shared the prize.<ref name="EulerAiton1996">{{cite book |first1=Leonhard |last1=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=[[Company:Springer Science+Business Media|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 rotational effects on motion. Euler realized that the tidal force's ''horizontal'' component (more than the vertical) drives the tide. In 1744 [[Biography:Jean le Rond d'Alembert|Jean le Rond d'Alembert]] studied tidal equations for the atmosphere which did not include rotation.

In 1770 James Cook's [[Engineering:Barque|barque]] [[Engineering:HMS Endeavour|HMS ''Endeavour'']] grounded on the Great Barrier Reef. Attempts were made to refloat her on the following tide which failed, but the tide after that lifted her clear with ease. Whilst she was being repaired in the mouth of the Endeavour River Cook observed the tides over a period of seven weeks. At neap tides both tides in a day were similar, but at springs the tides rose {{convert|7|feet}} in the morning but {{convert|9|feet}} in the evening.<ref name=Cook>{{cite journal |editor-last=Thomson |editor-first=Thomas |title=On Capt. Cook's Account of the Tides |date=March 1819 |publisher=Baldwin, Cradock and Joy |place=London |journal=[[Philosophy:Annals of Philosophy|Annals of Philosophy]] |volume=XIII |page=204 |url=https://www.biodiversitylibrary.org/page/15877750 |access-date=25 July 2015 |archive-date=26 August 2016 |archive-url=https://web.archive.org/web/20160826094842/http://biodiversitylibrary.org/page/15877750 |url-status=live }}</ref>

Pierre-Simon Laplace formulated a system of [[Partial differential equation|partial differential equation]]s relating the ocean's horizontal flow to its surface height, the first major dynamic theory for water tides. The Laplace tidal equations are still in use today. [[Biography:William Thomson, 1st Baron Kelvin|William Thomson, 1st Baron Kelvin]], rewrote Laplace's equations in terms of [[Vorticity|vorticity]] which allowed for solutions describing tidally driven coastally trapped waves, known as [[Physics:Kelvin wave|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={{wipe|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 [[Biography:Henri Poincaré|Henri Poincaré]] further developed Laplace's theory. Based on these developments and the [[Physics:Lunar theory|lunar theory]] of 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 |first=T.D. |date=2003 |url=http://descanso.jpl.nasa.gov/Monograph/series2/Descanso2_all.pdf |title=Formulation for observed and computed values of Deep Space Network data types for navigation |archive-url=https://web.archive.org/web/20041016204145/http://descanso.jpl.nasa.gov/Monograph/series2/Descanso2_all.pdf |archive-date=2004-10-16 |volume=3 |series=Deep-space communications and navigation |publisher=[[Company:Wiley (publisher)|Wiley]] |pages=126–128 |isbn=0-471-44535-5}}</ref>

=== History of tidal observation ===
[[File:Brouscon Almanach 1546 Compass bearing of high waters in the Bay of Biscay left Brittany to Dover right.jpg|thumb|Brouscon's Almanach of 1546: Compass bearings of high waters in the Bay of Biscay (left) and the coast from Brittany to Dover (right).]]
[[File:Brouscon Almanach 1546 Tidal diagrams according to the age of the Moon.jpg|thumb|Brouscon's Almanach of 1546: Tidal diagrams "according to the age of the moon".]]
From ancient times, tidal observation and discussion has increased in sophistication, first marking the daily recurrence, then tides' relationship to the Sun and moon. [[Biography:Pytheas|Pytheas]] travelled to the [[Place:British Isles|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 [[Biography:Seleucus of Seleucia|Seleucus of Seleucia]] correctly described the phenomenon of tides in order to support his [[Astronomy: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 [[Astronomy:Moon|moon]], although he believed that the interaction was mediated by the [[Philosophy:Pneuma|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. |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=1987NYASA.500..525V |s2cid=222087224}}</ref>

The ''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 [[Biology:Euripus Strait|Euripus Strait]] and the Strait of Messina puzzled [[Biography:Aristotle|Aristotle]].) Philostratus discussed tides in Book Five of ''The Life of [[Biography:Apollonius of Tyana|Apollonius of Tyana]]''. Philostratus mentions the moon, but attributes tides to "spirits". In Europe around 730 AD, the Venerable [[Biography:Bede|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|tide table]] in {{wipe|China}} was recorded in 1056 AD primarily for visitors wishing to see the famous [[Physics:Tidal bore|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 [[Place:London|London]].<ref>{{cite journal |last=Cartwright |first=D.E. |date=1999 |title=Tides, A Scientific History |journal=Eos Transactions |volume=80 |issue=36 |pages=11, 18|doi=10.1029/99EO00304 |bibcode=1999EOSTr..80..408A |doi-access=free }}</ref>

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>

[[Biography:William Thomson, 1st Baron Kelvin|William Thomson (Lord Kelvin)]] led the first systematic [[Physics:Harmonic analysis|harmonic analysis]] of tidal records starting in 1867. The main result was the building of a [[Tide-predicting machine|tide-predicting machine]] using a system of pulleys to add together six harmonic time functions. It was "programmed" by resetting gears and chains to adjust phasing and amplitudes. Similar machines were used until the 1960s.<ref>{{cite web |url=http://www.pol.ac.uk/home/insight/doodsonmachine.html |title=The Doodson–Légé Tide Predicting Machine |access-date=2008-10-03 |publisher=Proudman Oceanographic Laboratory |url-status=dead |archive-url=https://web.archive.org/web/20090320184406/http://www.pol.ac.uk/home/insight/doodsonmachine.html |archive-date=2009-03-20}}</ref>

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.

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> [[Biography:William Whewell|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 [[Physics:Amphidromic point|amphidromic point]], as they are now known, was confirmed in 1840 by 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={{wipe|Cambridge University Press}} |isbn=978-0-521-79746-7 |oclc=1001932580}}</ref><ref name="tidhist"/>

== Physics ==
{{main|Astronomy:Theory of tides}}

=== Forces ===
{{main|Physics:Tidal force}}
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 |first=C. A. |date=1889 |title=A Textbook of General Astronomy |url=https://www.gutenberg.org/files/37275/37275-pdf.pdf |page=288 |access-date=2018-08-13 |archive-date=2019-10-05 |archive-url=https://web.archive.org/web/20191005003655/http://www.gutenberg.org/files/37275/37275-pdf.pdf |url-status=live }}</ref> If the tidal force caused by each body were instead equal to its full gravitational force (which is not the case due to the [[Physics:Free fall|free fall]] of the whole Earth, not only the oceans, towards these bodies) a different pattern of tidal forces would be observed, e.g. with a much stronger influence from the Sun than from the Moon: The solar gravitational force on the Earth is on average 179 times stronger than the lunar, but because the Sun is on average 389 times farther from the Earth, its field gradient is weaker. The overall proportionality is

: <math>\text{tidal force} \propto \frac{M}{d^3} \propto \rho\left(\frac{r}{d}\right)^3,</math>

where {{mvar|M}} is the mass of the heavenly body, {{mvar|d}} is its distance, {{mvar|ρ}} is its average density, and {{mvar|r}} is its radius. The ratio {{math|''r''/''d''}} is related to the angle subtended by the object in the sky. Since the Sun and the Moon have practically the same diameter in the sky, the tidal force of the Sun is less than that of the Moon because its average density is much less, and it is only 46% as large as the lunar,<!-- numbers double-checked by User:JEBrown87544, 03 Jan 2007 -->{{efn|According to [https://web.archive.org/web/19980224174548/http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/961029b.html NASA] the lunar tidal force is 2.21 times larger than the solar.}} thus during a spring tide, the Moon contributes 69% while the Sun contributes 31%. More precisely, the lunar tidal acceleration (along the Moon–Earth axis, at the Earth's surface) is about 1.1{{e|−7}}&nbsp;''g'', while the solar tidal acceleration (along the Sun–Earth axis, at the Earth's surface) is about 0.52{{e|−7}}&nbsp;''g'', where ''g'' is the [[Physics:Standard gravity|gravitational acceleration]] at the Earth's surface.{{efn|See [[Physics:Tidal force#Formulation|Tidal force – Mathematical treatment]] and sources cited there.}} The effects of the other planets vary as their distances from Earth vary. When Venus is closest to Earth, its effect is 0.000113 times the solar effect.<ref name="Science Mission Directorate 2000">{{cite web | title=Interplanetary Low Tide | website=Science Mission Directorate | date=2000-05-03 | url=https://science.nasa.gov/science-news/science-at-nasa/2000/ast04may_1m | access-date=2023-06-25 | archive-date=2023-06-04 | archive-url=https://web.archive.org/web/20230604014510/https://science.nasa.gov/science-news/science-at-nasa/2000/ast04may_1m | url-status=live }}</ref> At other times, Jupiter or Mars may have the most effect.

[[File:Tidal field and gravity field.svg|thumb|The lunar [[Physics:Gravity|gravity]] residual [[Vector field|field]] at the Earth's surface is known as the ''tide-generating force''. This is the primary mechanism that drives tidal action and explains two simultaneous tidal bulges; Earth's rotation further accounts for two daily high waters at any location. The figure shows both the tidal field (thick red arrows) and the gravity field (thin blue arrows) exerted on Earth's surface and center (label O) by the Moon (label S).|alt=Diagram showing a circle with closely spaced arrows pointing away from the reader on the left and right sides, while pointing towards the user on the top and bottom.]]

The ocean's surface is approximated by a surface referred to as the [[Geoid|geoid]], which takes into consideration the gravitational force exerted by the earth as well as [[Physics:Centrifugal force|centrifugal force]] due to rotation. Now consider the effect of massive external bodies such as the Moon and Sun. These bodies have strong gravitational fields that diminish with distance and cause the ocean's surface to deviate from the geoid. They establish a new equilibrium ocean surface which bulges toward the moon on one side and away from the moon on the other side. The earth's rotation relative to this shape causes the daily tidal cycle. The ocean surface tends toward this equilibrium shape, which is constantly changing, and never quite attains it. When the ocean surface is not aligned with it, it's as though the surface is sloping, and water accelerates in the down-slope direction.

=== Equilibrium ===
The '''equilibrium tide''' is the idealized tide assuming a landless Earth.<ref name="AMS Glossary 2020">{{cite web |title=Equilibrium tide |website=AMS Glossary |date=2020-09-02 |url=http://glossary.ametsoc.org/wiki/Equilibrium_tide |access-date=2020-09-02 |archive-date=2020-08-01 |archive-url=https://web.archive.org/web/20200801165541/http://glossary.ametsoc.org/wiki/Equilibrium_tide |url-status=live }}</ref>
It would produce a tidal bulge in the ocean, elongated towards the attracting body (Moon or Sun).
It is ''not'' caused by the vertical pull nearest or farthest from the body, which is very weak; rather, it is caused by the tangential or tractive tidal force, which is strongest at about 45 degrees from the body, resulting in a horizontal tidal current.{{efn|"The ocean does not produce tides as a direct response to the vertical forces at the bulges. The tidal force is only about 1 ten millionth the size of the gravitational force owing to the Earth's gravity. It is the horizontal component of the tidal force that produces the tidal bulge, causing fluid to converge at the sublunar and antipodal points and move away from the poles, causing a contraction there." (...) "The projection of the tidal force onto the horizontal direction is called the tractive force (see Knauss, Fig. 10.11). This force causes an acceleration of water towards the sublunar and antipodal points, building up water until the pressure gradient force from the bulging sea surface exactly balances the tractive force field."<ref>{{Cite web |first=LuAnne |last=Thompson |year=2006 |url=http://faculty.washington.edu/luanne/pages/ocean420/notes/TidesIntro.pdf |title=Physical Processes in the Ocean |access-date=2020-06-27 |archive-date=2020-09-28 |archive-url=https://web.archive.org/web/20200928191813/http://faculty.washington.edu/luanne/pages/ocean420/notes/TidesIntro.pdf |url-status=live }}</ref>}}
{{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 |archive-date=2022-01-20 |archive-url=https://web.archive.org/web/20220120232639/http://www.tidesandcurrents.noaa.gov/publications/Understanding_Tides_by_Steacy_finalFINAL11_30.pdf |url-status=live }}</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 |publisher=World Scientific |isbn=9789814338189 |url=https://books.google.com/books?id=aKLICgAAQBAJ&q=tractal |via=[[Google Books]] |access-date=2022-01-05 |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153030/https://books.google.com/books?id=aKLICgAAQBAJ&q=tractal |url-status=live }}</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 |archive-date=2020-10-21 |archive-url=https://web.archive.org/web/20201021020010/https://www.pbslearningmedia.org/resource/what-physics-teachers-pbs-space-time/what-physics-teachers-pbs-space-time/ |url-status=live }}</ref>

=== Laplace's tidal equations ===
Ocean depths are much smaller than their horizontal extent. Thus, the response to tidal forcing can be modelled using the Laplace tidal equations which incorporate the following features:

* The vertical (or radial) velocity is negligible, and there is no vertical [[Physics:Wind shear|shear]]—this is a sheet flow.
* The forcing is only horizontal ([[Tangent|tangent]]ial).
* The Coriolis effect appears as an inertial force (fictitious) acting laterally to the direction of flow and proportional to velocity.
* The surface height's rate of change is proportional to the negative divergence of velocity multiplied by the depth. As the horizontal velocity stretches or compresses the ocean as a sheet, the volume thins or thickens, respectively.

The boundary conditions dictate no flow across the coastline and free slip at the bottom.

The Coriolis effect (inertial force) steers flows moving towards the Equator to the west and flows moving away from the Equator toward the east, allowing coastally trapped waves. Finally, a dissipation term can be added which is an analog to viscosity.

=== Amplitude and cycle time ===
The theoretical [[Physics:Amplitude|amplitude]] of oceanic tides caused by the Moon is about {{convert|54|cm|in}} at the highest point, which corresponds to the amplitude that would be reached if the ocean possessed a uniform depth, there were no landmasses, and the Earth were rotating in step with the Moon's orbit. The Sun similarly causes tides, of which the theoretical amplitude is about {{convert|25|cm|in}} (46% of that of the Moon) with a cycle time of 12 hours. At spring tide the two effects add to each other to a theoretical level of {{convert|79|cm|in}}, while at neap tide the theoretical level is reduced to {{convert|29|cm|in}}. Since the orbits of the Earth about the Sun, and the Moon about the Earth, are elliptical, tidal amplitudes change somewhat as a result of the varying Earth–Sun and Earth–Moon distances. This causes a variation in the tidal force and theoretical amplitude of about ±18% for the Moon and ±5% for the Sun. If both the Sun and Moon were at their closest positions and aligned at new moon, the theoretical amplitude would reach {{convert|93|cm|in}}.

Real amplitudes differ considerably, not only because of depth variations and continental obstacles, but also because wave propagation across the ocean has a natural period of the same order of magnitude as the rotation period: if there were no land masses, it would take about 30 hours for a long wavelength surface wave to propagate along the Equator halfway around the Earth (by comparison, the Earth's [[Earth:Lithosphere|lithosphere]] has a natural period of about 57 minutes). [[Earth:Earth tide|Earth tide]]s, which raise and lower the bottom of the ocean, and the tide's own gravitational self attraction are both significant and further complicate the ocean's response to tidal forces.

=== Dissipation ===
<!--this seems misleading in terms of the oceanography. Is this global average? Cites please!: Because the lunar tidal forces drive the oceans with a period of about 12.42 hours, which is considerably less than the natural period of the oceans, complex resonance phenomena take place. This, as well as the effects of friction, gives rise to an average lag time of 11 minutes between the occurrence of high water and lunar zenith. This tidal lag time corresponds to an angle of about 3 degrees between the position of the Moon, the center of the Earth, and the location of the global average high water.

Regarding the Earth–Moon system by itself (excluding the sun for the moment) unless both bodies' spin axes align perpendicularly to the orbital plane, oscillations result. Such oscillations contribute to tidal dissipation.

Dissipation by internal fluctuating deformations of the Earth due to the lunar tidal force is small compared with the dissipation in the Earth's oceans and seas, which account for 98% of the reduction of the Earth's rotational energy.<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 |volume=381 |issue=6583 |pages=595 |doi=10.1038/381595a0 |last2=Eanes |first2=R.J. |last3=Chao |first3=B.F. |bibcode=1996Natur.381..595R}}</ref> This lack of alignment is the case for the Earth–Moon system. Thus, besides tidal bulges, opposite to each other and comparable in size, that are associated with the so-called equilibrium tide,<ref name=Boon>{{cite book |title=Secrets of the Tide: Tide and Tidal Current Analysis and Applications, Storm Surges and Sea Level Trends |last=Boon |first=John D. |url=https://books.google.com/books?id=l75xhGEZ550C&pg=PA13&dq=%22equilibrium+tide%22 |isbn=1-904275-17-6 |publisher=Hollywood Publishing |year=2004 |pages=Chapter 2 pp. 13–end |oclc=57495983 |no-pp=true |via=[[Google Books]]}}</ref> additionally, surface oscillations commonly known as the dynamical tide, characterized by a wide variety of harmonic frequencies, is established.<ref name=Toledano>{{cite journal |url=https://arxiv.org/abs/astro-ph/0610563v1 |first1=Oswaldo |last1=Toledano |first2=Edmundo |last2=Moreno |first3=Gloria |last3=Koenigsberger |first4=R. |last4=Detmers |first5=Norbert |last5=Langer |date=18 October 2006 |title=Tides in asynchronous binary systems |journal=Astrophysics</ref><ref name=Lamb>{{cite book |title=Hydrodynamics |last=Lamb |first=Horace |url=https://archive.org/details/hydrodynamics02lambgoog |quote=dynamical tide. |year=1916 |publisher={{wipe|Cambridge University Press}} |edition=4th |page=[https://archive.org/details/hydrodynamics02lambgoog/page/n359 339] |isbn=0-521-45868-4 |oclc=30070401 31079426 33629948}} </ref><ref name=Americana>{{cite book |title=The Encyclopedia Americana: A Library of Universal Knowledge |last=Harris |first=Rollin A. |publisher=Encyclopedia Americana |year=1918 |url=https://books.google.com/books?d=CF4fijqC9GgC&pg=RA1-PA613&dq=%22equilibrium+tide%22 |pages=Article on Tides, pp. 613–614 |no-pp=true |via=[[Google Books]]}}</ref>
-->

Earth's tidal oscillations introduce dissipation at an [[Average|average]] rate of about 3.75 [[Terawatt|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 |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, [[Organization:Woods Hole Oceanographic Institution|WHOI]] Notes by Yaron Toledo and Marshall Ward.</ref>}}

[[Astronomy: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.

=== Bathymetry ===
[[File:Gorey Harbour at low tide.JPG|thumb|The harbour of Gorey, Jersey falls dry at low tide.]]
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 [[Earth:Bathymetry|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 and the time of high or low tide (the [[Earth:Lunitidal interval|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. [[Organization:National Oceanic and Atmospheric Administration|National Oceanic and Atmospheric Administration]] (NOAA) National Ocean Service (Education section) |url=http://oceanservice.noaa.gov/education/kits/tides/media/supp_tide07b.html |title=map showing world distribution of tide patterns, semi-diurnal, diurnal and mixed semi-diurnal |access-date=2009-09-05 |archive-date=2018-08-27 |archive-url=https://web.archive.org/web/20180827211303/http://oceanservice.noaa.gov/education/kits/tides/media/supp_tide07b.html |url-status=live }}</ref><ref>{{cite book |first=H.V. |last=Thurman |date=1994 |title=Introductory Oceanography |edition=7th |location=New York |publisher=Macmillan Publishers |pages=252–276}}ref</ref><ref>{{cite book |first=D.A. |last=Ross |date=1995 |title=Introduction to Oceanography |location=New York |publisher=HarperCollins |pages=236–242}}</ref> Human changes to the landscape can also significantly alter local tides.<ref>{{cite news |last1=Witze |first1=Alexandra |title=How humans are altering the tides of the oceans |url=https://www.bbc.com/future/article/20200703-how-humans-are-altering-the-tides-of-the-oceans |access-date=8 July 2020 |work=BBC Future |publisher=BBC |date=5 July 2020 |archive-date=6 July 2020 |archive-url=https://web.archive.org/web/20200706134113/https://www.bbc.com/future/article/20200703-how-humans-are-altering-the-tides-of-the-oceans |url-status=live }}</ref>

== Observation and prediction ==

=== 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 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 |access-date=2021-04-02 |archive-date=2021-05-08 |archive-url=https://web.archive.org/web/20210508194219/https://glossary.ametsoc.org/wiki/Age |url-status=live }}</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 [[Earth:Bathymetry|bathymetry]] greatly influences the tide's exact time and height at a particular [[Earth:Coast|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 |first2=Ron |last2=Solvason |first3=Christian |last3=Solomon |chapter-url=http://www.bofep.org/PDFfiles/BoFEP6thProceedings.pdf |chapter=Resolving the World's largest tides |editor1-first=J.A |editor1-last=Percy |editor2-first=A.J. |editor2-last=Evans |editor3-first=P.G. |editor3-last=Wells |editor4-first=S.J. |editor4-last=Rolston |date=2005 |title=The Changing Bay of Fundy-Beyond 400 years, Proceedings of the 6th Bay of Fundy Workshop, Cornwallis, Nova Scotia, Sept. 29, 2004 to October 2, 2004. Environment Canada-Atlantic Region, Occasional Report no. 23. Dartmouth, N.S. and Sackville, N.B. |access-date=April 1, 2013 |archive-date=August 27, 2016 |archive-url=https://web.archive.org/web/20160827202033/http://www.bofep.org/PDFfiles/BoFEP6thProceedings.pdf |url-status=live }}</ref> Similar measurements made in March 2002 at Leaf Basin, Ungava Bay in northern Quebec gave similar values (allowing for measurement errors), a maximum range of {{convert|16.2|m|ft}} and a highest predicted extreme of {{convert|16.8|m|ft}}.<ref name=BIO2004 /><ref name=FundyWorkshop /> Ungava Bay and the Bay of Fundy lie similar distances from the continental shelf edge, but Ungava Bay is only free of pack ice for about four months every year while the Bay of Fundy rarely freezes.

Southampton in the United Kingdom has a double high water caused by the interaction between the ''M''<sub>2</sub> and [[Astronomy: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> 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 Portland because the ''M''<sub>2</sub> tide is lowest in this region.

Because the oscillation modes of the [[Place:Mediterranean Sea|Mediterranean Sea]] and the [[Place:Baltic Sea|Baltic Sea]] do not coincide with any significant astronomical forcing period, the largest tides are close to their narrow connections with the Atlantic Ocean. Extremely small tides also occur for the same reason in the Gulf of Mexico and Sea of Japan. Elsewhere, as along the southern coast of {{wipe|Australia}}, low tides can be due to the presence of a nearby [[Physics:Amphidromic point|amphidrome]].

=== Analysis ===
[[File:Water surface level changes with tides.svg|thumb|upright=2|A regular water level chart]]
[[Biography:Isaac Newton|Isaac Newton]]'s theory of gravitation first enabled an explanation of why there were generally two tides a day, not one, and offered hope for a detailed understanding of tidal forces and behavior. Although it may seem that tides could be predicted via a sufficiently detailed knowledge of instantaneous astronomical forcings, the actual tide at a given location is determined by astronomical forces accumulated by the body of water over many days. In addition, accurate results would require detailed knowledge of the shape of all the ocean basins—their [[Earth:Bathymetry|bathymetry]], and coastline shape.

Current procedure for analysing tides follows the method of harmonic analysis introduced in the 1860s by [[Biography:Lord Kelvin|William Thomson]]. It is based on the principle that the astronomical theories of the motions of Sun and Moon determine a large number of component frequencies, and at each frequency there is a component of force tending to produce tidal motion, but that at each place of interest on the Earth, the tides respond at each frequency with an amplitude and phase peculiar to that locality. At each place of interest, the tide heights are therefore measured for a period of time sufficiently long (usually more than a year in the case of a new port not previously studied) to enable the response at each significant tide-generating frequency to be distinguished by analysis, and to extract the tidal constants for a sufficient number of the strongest known components of the astronomical tidal forces to enable practical tide prediction. The tide heights are expected to follow the tidal force, with a constant amplitude and phase delay for each component. Because astronomical frequencies and phases can be calculated with certainty, the tide height at other times can then be predicted once the response to the harmonic components of the astronomical tide-generating forces has been found.

The main patterns in the tides are
* the twice-daily variation
* the difference between the first and second tide of a day
* the spring–neap cycle
* the annual variation
The ''Highest Astronomical Tide'' is the perigean spring tide when both the Sun and Moon are closest to the Earth.

When confronted by a periodically varying function, the standard approach is to employ [[Fourier series]], a form of analysis that uses [[Physics:Sine wave|sinusoid]]al functions as a ''basis'' set, having frequencies that are zero, one, two, three, etc. times the frequency of a particular fundamental cycle. These multiples are called ''harmonics'' of the fundamental frequency, and the process is termed [[Physics:Harmonic analysis|harmonic analysis]]. If the basis set of sinusoidal functions suit the behaviour being modelled, relatively few harmonic terms need to be added. Orbital paths are very nearly circular, so sinusoidal variations are suitable for tides.

For the analysis of tide heights, the Fourier series approach has in practice to be made more elaborate than the use of a single frequency and its harmonics. The tidal patterns are decomposed into many sinusoids having many fundamental frequencies, corresponding (as in the [[Physics:Lunar theory#Results of the theories|lunar theory]]) to many different combinations of the motions of the Earth, the Moon, and the angles that define the shape and location of their orbits.

For tides, then, ''harmonic analysis'' is not limited to harmonics of a single frequency.{{efn|To demonstrate this [http://www.arachnoid.com/tides/index.html Tides Home Page] offers a tidal height pattern converted into an ''.mp3'' sound file, and the rich sound is quite different from a pure tone.}} In other words, the harmonies are multiples of many fundamental frequencies, not just of the fundamental frequency of the simpler Fourier series approach. Their representation as a Fourier series having only one fundamental frequency and its (integer) multiples would require many terms, and would be severely limited in the time-range for which it would be valid.

The study of tide height by harmonic analysis was begun by Laplace, William Thomson (Lord Kelvin), and George Darwin. A.T. Doodson extended their work, introducing the ''Doodson Number'' notation to organise the hundreds of resulting terms. This approach has been the international standard ever since, and the complications arise as follows: the tide-raising force is notionally given by sums of several terms. Each term is of the form
: <math> A_o \cos(\omega t + p),</math>
where
: {{math|<var>A<sub>o</sub></var>}} is the amplitude,
: {{math|<var>&omega;</var>}} is the angular frequency, usually given in degrees per hour, corresponding to {{math|<var>t</var>}} measured in hours,
: {{math|<var>p</var>}} is the phase offset with regard to the astronomical state at time ''t'' = 0. 

There is one term for the Moon and a second term for the Sun. The phase {{math|<var>p</var>}} of the first harmonic for the Moon term is called the [[Earth:Lunitidal interval|lunitidal interval]] or high water interval. 

The next refinement is to accommodate the harmonic terms due to the elliptical shape of the orbits. To do so, the value of the amplitude is taken to be not a constant, but varying with time, about the average amplitude {{math|<var>A<sub>o</sub></var>}}. To do so, replace {{math|<var>A<sub>o</sub></var>}} in the above equation with {{math|<var>A</var>(<var>t</var>)}} where {{math|A}} is another sinusoid, similar to the cycles and epicycles of [[Biography:Ptolemy|Ptolemaic theory]]. This gives

: <math> A(t) = A_o\bigl(1 + A_a \cos(\omega_a t + p_a)\bigr),</math>

which is to say an average value {{math|<var>A<sub>o</sub></var>}} with a sinusoidal variation about it of magnitude {{math|<var>A<sub>a</sub></var>}}, with frequency {{math|<var>&omega;<sub>a</sub></var>}} and phase {{math|<var>p<sub>a</sub></var>}}. Substituting this for {{math|<var>A<sub>o</sub></var>}} in the original equation gives a product of two cosine factors:

: <math> A_o \bigl( 1 + A_a \cos(\omega_a t + p_a)\bigr) \cos(\omega t + p).</math>

Given that for any {{math|<var>x</var>}} and {{math|<var>y</var>}}
: <math>\cos x \cos y = \tfrac{1}{2} \cos(x + y) + \tfrac{1}{2} \cos(x - y),</math>
it is clear that a compound term involving the product of two cosine terms each with their own frequency is the same as ''three'' simple cosine terms that are to be added at the original frequency and also at frequencies which are the sum and difference of the two frequencies of the product term. (Three, not two terms, since the whole expression is <math>(1 + \cos x) \cos y</math>.) Consider further that the tidal force on a location depends also on whether the Moon (or the Sun) is above or below the plane of the Equator, and that these attributes have their own periods also incommensurable with a day and a month, and it is clear that many combinations result. With a careful choice of the basic astronomical frequencies, the Doodson Number annotates the particular additions and differences to form the frequency of each simple cosine term.

[[File:Tidal constituent sum.gif|thumb|left|Tidal prediction summing constituent parts. The tidal coefficients are defined on the page theory of tides.|alt=Graph showing one line each for M&nbsp;<sub>2</sub>, S&nbsp;<sub>2</sub>, N&nbsp;<sub>2</sub>, K&nbsp;<sub>1</sub>, O&nbsp;<sub>1</sub>, P&nbsp;<sub>1</sub>, and one for their summation, with the X axis spanning slightly more than a single day]]

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|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 [[Astronomy:Metonic cycle|Metonic cycle]] of 19&nbsp;years, which is long enough to include the 18.613 year [[Astronomy:Lunar node|lunar nodal]] [[Earth: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>{{cite web |url=http://tidesandcurrents.noaa.gov/publications/glossary2.pdf |title=Tide and Current Glossary |author=Center for Operational Oceanographic Products and Services, National Ocean Service, [[Organization:National Oceanic and Atmospheric Administration|National Oceanic and Atmospheric Administration]] |location=Silver Spring, MD |date=January 2000 |access-date=2007-04-05 |archive-date=2007-01-28 |archive-url=https://web.archive.org/web/20070128171521/http://tidesandcurrents.noaa.gov/publications/glossary2.pdf |url-status=live }}</ref> The resulting amplitudes and phases can then be used to predict the expected tides. These are usually dominated by the constituents near 12&nbsp;hours (the ''semi-diurnal'' constituents), but there are major constituents near 24&nbsp;hours (''diurnal'') as well. Longer term constituents are 14&nbsp;day or ''fortnightly'', monthly, and semiannual. Semi-diurnal tides dominated coastline, but some areas such as the South China Sea and the Gulf of Mexico are primarily diurnal. In the semi-diurnal areas, the primary constituents ''M''<sub>2</sub>&nbsp;(lunar) and ''S''<sub>2</sub>&nbsp;(solar) periods differ slightly, so that the relative phases, and thus the amplitude of the combined tide, change fortnightly (14&nbsp;day period).<ref>{{Cite web |url=http://tidesandcurrents.noaa.gov/harmonic_cons_defs.html |title=Harmonic Constituents |website=NOAA |access-date=2007-04-05 |archive-date=2007-03-17 |archive-url=https://web.archive.org/web/20070317043911/http://tidesandcurrents.noaa.gov/harmonic_cons_defs.html |url-status=live }}</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 [[Physics:Amphidromic point|amphidromic point]]s counterclockwise in the northern hemisphere so that from Baja California Peninsula to [[Earth:Alaska|Alaska]] and from {{wipe|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.)

The exception is at Cook Strait where the tidal currents periodically link high to low water. This is because cotidal lines 180° around the amphidromes are in opposite phase, for example high water across from low water at each end of Cook Strait. Each tidal constituent has a different pattern of amplitudes, phases, and amphidromic points, so the ''M''<sub>2</sub> patterns cannot be used for other tide components.

=== Example calculation ===
[[File:Tide.Bridgeport.50h.svg|thumb|Tides at Bridgeport, Connecticut, U.S. during a 50-hour period.|alt=Graph with a single line rising and falling between 4 peaks around 3 and four valleys around −3]]
[[File:Tide.Bridgeport.30d.svg|thumb|Tides at Bridgeport, Connecticut, U.S. during a 30-day period.|alt=Graph with a single line showing tidal peaks and valleys gradually cycling between higher highs and lower highs over a 14-day period]]
[[File:Tide.Bridgeport.400d.svg|thumb|Tides at Bridgeport, Connecticut, U.S. during a 400-day period.|alt=Graph showing with a single line showing only a minimal annual tidal fluctuation]]
[[File:Tide.NZ.November.png|thumb|Tidal patterns in Cook Strait. The south part (Nelson) has two spring tides per month, versus only one on the north side (Wellington and Napier).|alt=Graph showing 6 lines with two lines for each of three cities. Nelson has two monthly spring tides, while Napier and Wellington each have one.]]

Because the Moon is moving in its orbit around the Earth and in the same sense as the Earth's rotation, a point on the Earth must rotate slightly further to catch up so that the time between semi-diurnal tides is not twelve but 12.4206 hours—a bit over twenty-five minutes extra. The two peaks are not equal. The two high tides a day alternate in maximum heights: lower high (just under three feet), higher high (just over three feet), and again lower high. Likewise for the low tides.

When the Earth, Moon, and Sun are in line (Sun–Earth–Moon, or Sun–Moon–Earth) the two main influences combine to produce spring tides; when the two forces are opposing each other as when the angle Moon–Earth–Sun is close to ninety degrees, neap tides result. As the Moon moves around its orbit it changes from north of the Equator to south of the Equator. The alternation in high tide heights becomes smaller, until they are the same (at the lunar equinox, the Moon is above the Equator), then redevelop but with the other polarity, waxing to a maximum difference and then waning again.

=== Current ===
The tides' influence on [[Earth:Ocean current|current]] or flow is much more difficult to analyze, and data is much more difficult to collect. A tidal height is a scalar quantity and varies smoothly over a wide region. A flow is a vector quantity, with magnitude and direction, both of which can vary substantially with depth and over short distances due to local bathymetry. Also, although a water channel's center is the most useful measuring site, mariners object when current-measuring equipment obstructs waterways. A flow proceeding up a curved channel may have similar magnitude, even though its direction varies continuously along the channel. Surprisingly, flood and ebb flows are often not in opposite directions. Flow direction is determined by the upstream channel's shape, not the downstream channel's shape. Likewise, [[Physics:Eddy (fluid dynamics)|eddies]] may form in only one flow direction.

Nevertheless, tidal current analysis is similar to tidal heights analysis: in the simple case, at a given location the flood flow is in mostly one direction, and the ebb flow in another direction. Flood velocities are given positive sign, and ebb velocities negative sign. Analysis proceeds as though these are tide heights.

In more complex situations, the main ebb and flood flows do not dominate. Instead, the flow direction and magnitude trace an ellipse over a tidal cycle (on a polar plot) instead of along the ebb and flood lines. In this case, analysis might proceed along pairs of directions, with the primary and secondary directions at right angles. An alternative is to treat the tidal flows as complex numbers, as each value has both a magnitude and a direction.

Tide flow information is most commonly seen on [[Nautical chart|nautical chart]]s, presented as a table of flow speeds and bearings at hourly intervals, with separate tables for spring and neap tides. The timing is relative to high water at some harbour where the tidal behaviour is similar in pattern, though it may be far away.

As with tide height predictions, tide flow predictions based only on astronomical factors do not incorporate weather conditions, which can ''completely'' change the outcome.

The tidal flow through Cook Strait between the two main islands of New Zealand is particularly interesting, as the tides on each side of the strait are almost exactly out of phase, so that one side's high water is simultaneous with the other's low water. Strong currents result, with almost zero tidal height change in the strait's center. Yet, although the tidal surge normally flows in one direction for six hours and in the reverse direction for six hours, a particular surge might last eight or ten hours with the reverse surge enfeebled. In especially boisterous weather conditions, the reverse surge might be entirely overcome so that the flow continues in the same direction through three or more surge periods.

A further complication for Cook Strait's flow pattern is that the tide at the south side (e.g. at [[Place:Nelson, New Zealand|Nelson]]) follows the common bi-weekly spring–neap tide cycle (as found along the west side of the country), but the north side's tidal pattern has only ''one'' cycle per month, as on the east side: [[Place:Wellington|Wellington]], and [[Place:Napier, New Zealand|Napier]].

The graph of Cook Strait's tides shows separately the high water and low water height and time, through November 2007; these are ''not'' measured values but instead are calculated from tidal parameters derived from years-old measurements. Cook Strait's nautical chart offers tidal current information. For instance the January 1979 edition for 41°13·9’S 174°29·6’E (north west of Cape Terawhiti) refers timings to Westport while the January 2004 issue refers to Wellington. Near Cape Terawhiti in the middle of Cook Strait the tidal height variation is almost nil while the tidal current reaches its maximum, especially near the notorious Karori Rip. Aside from weather effects, the actual currents through Cook Strait are influenced by the tidal height differences between the two ends of the strait and as can be seen, only one of the two spring tides at the north west end of the strait near Nelson has a counterpart spring tide at the south east end (Wellington), so the resulting behaviour follows neither reference harbour.{{Citation needed|date=September 2009}}

=== Power generation ===
{{Main|Engineering:Tidal power}}
Tidal energy can be extracted by two means: inserting a water [[Physics:Turbine|turbine]] into a tidal current, or building ponds that release/admit water through a turbine. In the first case, the energy amount is entirely determined by the timing and tidal current magnitude. However, the best currents may be unavailable because the turbines would obstruct ships. In the second, the impoundment dams are expensive to construct, natural water cycles are completely disrupted, ship navigation is disrupted. However, with multiple ponds, power can be generated at chosen times. So far, there are few installed systems for tidal power generation (most famously, La Rance at Saint Malo, France) which face many difficulties. Aside from environmental issues, simply withstanding corrosion and biological fouling pose engineering challenges.

Tidal power proponents point out that, unlike wind power systems, generation levels can be reliably predicted, save for weather effects. While some generation is possible for most of the tidal cycle, in practice turbines lose efficiency at lower operating rates. Since the power available from a flow is proportional to the cube of the flow speed, the times during which high power generation is possible are brief.

== Navigation ==
[[File:Tide legal use.gif|thumb|left|US civil and maritime uses of tidal data|alt=Chart illustrating that tidal heights enter in calculations of legally significant data such as ''boundary lines'' between the high seas and territorial waters. Chart shows an exemplar coastline, identifying bottom features such as longshore bar and berms, tidal heights such as mean higher high water, and distances from shore such as the 12 mile limit.]]

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 [[Engineering: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]] |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153531/https://books.google.com/books?id=lagPAAAAIAAJ&q=%22shift+his+tides%22 |url-status=live }}</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 [[Physics:Knot (unit)|knots]]) for spring and neap tides. If a tide chart is not available, most nautical charts have "[[Earth:Tidal diamond|tidal diamond]]s" which relate specific points on the chart to a table giving tidal flow direction and speed.

The standard procedure to counteract tidal effects on navigation is to (1) calculate a "[[Earth:Dead reckoning|dead reckoning]]" position (or DR) from travel distance and direction, (2) mark the chart (with a vertical cross like a plus sign) and (3) draw a line from the DR in the tide's direction. The distance the tide moves the boat along this line is computed by the tidal speed, and this gives an "estimated position" or EP (traditionally marked with a dot in a triangle).

[[File:Tidal Indicator Delaware River ca1897.jpg|thumb|Tidal Indicator, Delaware River, Delaware c. 1897. At the time shown in the figure, the tide is {{frac|1|1|4}} feet above mean low water and is still falling, as indicated by pointing of the arrow. Indicator is powered by system of pulleys, cables and a float. (Report Of The Superintendent Of The Coast & Geodetic Survey Showing The Progress Of The Work During The Fiscal Year Ending With June 1897 (p. 483))]]

[[Nautical chart]]s display the water's "charted depth" at specific locations with "[[Earth:Echo sounding|sounding]]s" and the use of bathymetric [[Contour line|contour line]]s to depict the submerged surface's shape. These depths are relative to a "[[Chart datum|chart datum]]", which is typically the water level at the lowest possible astronomical tide (although other datums are commonly used, especially historically, and tides may be lower or higher for meteorological reasons) and are therefore the minimum possible water depth during the tidal cycle. "Drying heights" may also be shown on the chart, which are the heights of the exposed [[Earth:Seabed|seabed]] at the lowest astronomical tide.

Tide tables list each day's high and low water heights and times. To calculate the actual water depth, add the charted depth to the published tide height. Depth for other times can be derived from tidal curves published for major ports. The [[Rule of twelfths|rule of twelfths]] can suffice if an accurate curve is not available. This approximation presumes that the increase in depth in the six hours between low and high water is: first hour — 1/12, second — 2/12, third — 3/12, fourth — 3/12, fifth — 2/12, sixth — 1/12.

== Biological aspects ==

=== Intertidal ecology ===
[[File:Intertide zonation at Kalaloch.jpg|thumb|left|upright|A rock, seen at low water, exhibiting typical intertidal zonation.|alt=Photo of partially submerged rock showing horizontal bands of different color and texture, where each band represents a different fraction of time spent submerged.]]
{{Main|Biology:Intertidal ecology}}
{{Further|Earth:Intertidal zone}}

Intertidal ecology is the study of [[Earth:Ecosystem|ecosystem]]s between the low- and high-water lines along a shore. At low water, the intertidal zone is exposed (or ''emersed''), whereas at high water, it is underwater (or ''immersed''). Intertidal ecologists therefore study the interactions between intertidal organisms and their environment, as well as among the [[Biology:Biological interaction|different species]]. The most important interactions may vary according to the type of intertidal community. The broadest classifications are based on substrates — [[Earth:Rocky shore|rocky shore]] or soft bottom.

Intertidal organisms experience a highly variable and often hostile environment, and have [[Biology:Adaptation|adapted]] to cope with and even exploit these conditions. One easily visible feature is vertical zonation, in which the community divides into distinct horizontal bands of specific species at each elevation above low water. A species' ability to cope with [[Earth:Desiccation|desiccation]] determines its upper limit, while [[Biology:Competition|competition]] with other species sets its lower limit.

Humans use [[Biology:Intertidal ecology|intertidal regions]] for food and recreation. [[Earth:Overexploitation|Overexploitation]] can damage intertidals directly. Other anthropogenic actions such as introducing [[Earth:Invasive species|invasive species]] and [[Earth:Global warming|climate change]] have large negative effects. Marine Protected Areas are one option communities can apply to protect these areas and aid scientific [[Research|research]].

=== Biological rhythms ===
The approximately 12-hour and fortnightly tidal cycle has large effects on intertidal<ref>{{cite journal |last1=Bos |first1=A.R. |last2=Gumanao |first2=G.S. |last3=van Katwijk |first3=M.M. |last4=Mueller |first4=B. |last5=Saceda |first5=M.M. |last6=Tejada |first6=R.P. |name-list-style=amp |date=2011 |title=Ontogenetic habitat shift, population growth, and burrowing behavior of the Indo-Pacific beach star ''Archaster typicus'' (Echinodermata: Asteroidea) |journal=Marine Biology |volume=158 |issue=3 |pages=639–648 |doi=10.1007/s00227-010-1588-0 |pmid=24391259 |pmc=3873073}}</ref> and marine organisms.<ref>{{cite journal |last1=Bos |first1=A.R. |last2=Gumanao |first2=G.S. |name-list-style=amp |date=2012 |title=The lunar cycle determines availability of coral reef fishes on fish markets |journal=Journal of Fish Biology |volume=81 |issue=6 |pages=2074–2079 |doi=10.1111/j.1095-8649.2012.03454.x |pmid=23130702}}</ref> Hence their [[Biology:Biological rhythm|biological rhythm]]s tend to occur in rough multiples of these periods.<ref>{{Cite book |last=Naylor |first=Ernest |url=https://books.google.com/books?id=7zLqjQlzM60C |title=Chronobiology of Marine Organisms |date=2010-02-04 |publisher={{wipe|Cambridge University Press}} |isbn=978-1-139-48494-7 |language=en |chapter=Chapter 5: Lunar and semilunar biorhythms |via=[[Google Books]] |access-date=2022-01-03 |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153532/https://books.google.com/books?id=7zLqjQlzM60C |url-status=live }}</ref> Many other animals such as the [[Biology:Vertebrate|vertebrate]]s, display similar circatidal rhythms.<ref>{{Cite journal |last1=Zhu |first1=Bokai |last2=Dacso |first2=Clifford C. |last3=O’Malley |first3=Bert W. |date=2018-07-01 |title=Unveiling "Musica Universalis" of the Cell: A Brief History of Biological 12-Hour Rhythms |url=https://doi.org/10.1210/js.2018-00113 |journal=Journal of the Endocrine Society |volume=2 |issue=7 |pages=727–752 |doi=10.1210/js.2018-00113 |pmid=29978151 |pmc=6025213 |issn=2472-1972 |access-date=2022-01-03 |archive-date=2023-09-16 |archive-url=https://web.archive.org/web/20230916153533/https://academic.oup.com/jes/article/2/7/727/5033317 |url-status=live }}</ref> Examples include gestation and egg hatching. In humans, the [[Medicine:Menstrual cycle|menstrual cycle]] lasts roughly a [[Astronomy:Lunar month|lunar month]], an even multiple of the tidal period. Such parallels at least hint at the [[Biology:Common descent|common descent]] of all animals from a marine ancestor.<ref name=Descent>{{cite book |last=Darwin |first=Charles |date=1871 |title=The Descent of Man, and Selection in Relation to Sex |publisher=John Murray |location=London}}</ref>

== Other tides ==
When oscillating tidal currents in the stratified ocean flow over uneven bottom topography, they generate [[Internal wave|internal wave]]s with tidal frequencies. Such waves are called ''[[Earth:Internal tide|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=[[Earth:Geographical Review|Geographical Review]] |date=April 1924 |volume=14 |issue=2 |pages=282–286 |doi=10.2307/208104 |jstor=208104 |access-date=4 February 2012 |archive-date=16 September 2023 |archive-url=https://web.archive.org/web/20230916153559/https://www.jstor.org/stable/208104 |url-status=live }}</ref>

In addition to oceanic tides, large lakes can experience small tides and even planets can experience ''[[Earth:Atmospheric tide|atmospheric tide]]s'' and ''[[Earth:Earth tide|Earth tide]]s''. These are [[Physics:Continuum mechanics|continuum mechanical]] phenomena. The first two take place in [[Physics:Fluid mechanics|fluid]]s. The third affects the Earth's thin [[Physics:Solid mechanics|solid]] crust surrounding its semi-liquid interior (with various modifications).

=== Lake tides ===
Large lakes such as Superior and Erie can experience tides of {{convert|1|to|4|cm|in|abbr=on|sigfig=2}}, but these can be masked by meteorologically induced phenomena such as [[Physics:Seiche|seiche]].<ref>{{cite web |title=Do the Great Lakes have tides? |date=October 1, 2000 |url=http://www.great-lakes.net/teach/chat/answers/100100_tides.html |publisher=Great Lakes Information Network |access-date=2010-02-10 |archive-date=2017-12-30 |archive-url=https://web.archive.org/web/20171230003511/http://www.great-lakes.net/teach/chat/answers/100100_tides.html |url-status=dead}}</ref> The tide in Lake Michigan is described as {{convert|0.5|to|1.5|in|cm|abbr=on|order=flip}}<ref>{{cite web |last=Calder |first=Vince |title=Tides on Lake Michigan |url=https://wat.lewiscollard.com/archive/www.newton.dep.anl.gov/askasci/phy00/phy00330.htm |publisher=Argonne National Laboratory |access-date=2019-08-14 |archive-date=2019-08-15 |archive-url=https://web.archive.org/web/20190815033734/https://wat.lewiscollard.com/archive/www.newton.dep.anl.gov/askasci/phy00/phy00330.htm |url-status=live }}</ref> or {{convert|1+3/4|in|cm|order=flip|abbr=on}}.<ref>{{cite web |title=moon and Tides |url=http://www.thespaceguy.com/moontides.htm |publisher=Astronomy Briefly |last=Dunkerson |first=Duane |access-date=2010-02-10 |archive-date=2010-01-15 |archive-url=https://web.archive.org/web/20100115000607/http://www.thespaceguy.com/moontides.htm |url-status=live }}</ref> This is so small that other larger effects completely mask any tide, and as such these lakes are considered non-tidal.<ref>{{cite web |title=Do the Great Lakes have tides? |url=http://oceanservice.noaa.gov/facts/gltides.html |website=National Ocean Service |publisher=NOAA |access-date=2016-04-26 |archive-date=2016-04-23 |archive-url=https://web.archive.org/web/20160423224702/http://oceanservice.noaa.gov/facts/gltides.html |url-status=live }}</ref>

=== Atmospheric tides ===
{{main|Earth:Atmospheric tide}}
Atmospheric tides are negligible at ground level and aviation altitudes, masked by [[Earth:Weather|weather]]'s much more important effects.<!-- and solar thermal tides. However, there is no strict upper limit to the Earth's atmosphere, and the tidal pull increases with the distance from the Earth's center.--> Atmospheric tides are both gravitational and thermal in origin and are the dominant dynamics from about {{convert|80| to |120|km|mi}}, above which the molecular density becomes too low to support fluid behavior.

=== Earth tides ===
{{Main|Earth:Earth tide}}

Earth tides or terrestrial tides affect the entire Earth's mass, which acts similarly to a liquid [[Engineering:Gyroscope|gyroscope]] with a very thin crust. The Earth's crust shifts (in/out, east/west, north/south) in response to lunar and solar gravitation, ocean tides, and atmospheric loading. While negligible for most human activities, terrestrial tides' semi-diurnal amplitude can reach about {{convert|55|cm|in}} at the Equator—{{convert|15|cm|in}} due to the Sun—which is important in [[Engineering:GPS|GPS]] calibration and VLBI measurements. Precise astronomical angular measurements require knowledge of the Earth's rotation rate and [[Astronomy:Polar motion|polar motion]], both of which are influenced by Earth tides. The semi-diurnal ''M''<sub>2</sub> Earth tides are nearly in phase with the Moon with a lag of about two hours.{{Citation needed|date=August 2008}}

=== Galactic tides ===
''[[Astronomy:Galactic tide|Galactic tide]]s'' are the tidal forces exerted by galaxies on stars within them and [[Astronomy:Satellite galaxy|satellite galaxies]] orbiting them. The galactic tide's effects on the [[Astronomy:Solar System|Solar System]]'s [[Astronomy:Oort cloud|Oort cloud]] are believed to cause 90 percent of long-period comets.<ref>{{cite journal |title=Periodic variation of Oort Cloud flux and cometary impacts on the Earth and Jupiter
|last1=Nurmi |first1=P. |last2=Valtonen |first2=M.J. |last3=Zheng |first3=J.Q. |name-list-style=amp |journal=[[Organization:Monthly Notices of the Royal Astronomical Society|Monthly Notices of the Royal Astronomical Society]] |volume=327 |issue=4 |pages=1367–1376 |date=2001 |bibcode=2001MNRAS.327.1367N |doi=10.1046/j.1365-8711.2001.04854.x |doi-access=free}}</ref>

== Misnomers ==
[[Tsunami]]s, the large waves that occur after earthquakes, are sometimes called ''tidal waves'', but this name is given by their ''resemblance'' to the tide, rather than any causal link to the tide. Other phenomena unrelated to tides but using the word ''tide'' are [[Earth:Rip current|rip tide]], storm tide, hurricane tide, and [[Earth:Oil spill|black]] or [[Biology:Red tide|red tide]]s. Many of these usages are historic and refer to the earlier meaning of tide as "a portion of time, a season" and "a stream, current or flood".<ref>{{cite OED2|tide|volume=XVIII|page=64}}</ref>

== See also ==
{{colbegin}}
* {{annotated link|Physics:Clairaut's theorem}}
* {{annotated link|Earth:Coastal erosion}}
* {{annotated link|Astronomy:Establishment of a port}}
* {{annotated link|Hough function}}
* {{annotated link|Astronomy:Lunar Laser Ranging experiment}}
* {{annotated link|Astronomy:Lunar phase}}
* {{annotated link|Earth:Raised beach|aka=Marine terrace}}
* {{annotated link|Earth:Mean high water spring}}
* {{annotated link|Earth:Mean low water spring}}
* {{annotated link|Astronomy:Orbit of the Moon}}
* {{annotated link|Earth:Primitive equations}}
* {{annotated link|Earth:Tidal island}}
* {{annotated link|Astronomy:Tidal locking}}
* {{annotated link|Earth:Tidal resonance}}
* {{annotated link|Earth:Tidal river}}
* {{annotated link|Earth:Tidal triggering of earthquakes}}
* {{annotated link|Earth:Tide pool}}
* {{annotated link|Earth:Tideline}}
* {{annotated link|Astronomy:Tides in marginal seas}}
{{colend}}

== Notes ==
{{notelist}}

== References ==
{{Reflist|30em}}

== Further reading ==
{{refbegin}}
* [http://tidesandcurrents.noaa.gov/publications/150_years_of_tides.pdf 150 Years of Tides on the Western Coast: The Longest Series of Tidal Observations in the Americas] {{Webarchive|url=https://web.archive.org/web/20110505153935/http://tidesandcurrents.noaa.gov/publications/150_years_of_tides.pdf |date=2011-05-05 }} NOAA (2004).
* [http://faculty.ifmo.ru/butikov/Projects/tides1.pdf Eugene I. Butikov: ''A dynamical picture of the ocean tides''] {{Webarchive|url=https://web.archive.org/web/20080911141308/http://faculty.ifmo.ru/butikov/Projects/tides1.pdf |date=2008-09-11 }}
* [http://www.vialattea.net/maree/eng/index.htm Tides and centrifugal force] {{Webarchive|url=https://web.archive.org/web/20070512004000/http://www.vialattea.net/maree/eng/index.htm |date=2007-05-12 }}: Why the centrifugal force does not explain the tide's opposite lobe (with nice animations).
* [https://arxiv.org/abs/astro-ph/0610563v1 O. Toledano ''et al.'' (2008): ''Tides in asynchronous binary systems''] {{Webarchive|url=https://web.archive.org/web/20170809040314/https://arxiv.org/abs/astro-ph/0610563v1 |date=2017-08-09 }}
* Gaylord Johnson [https://books.google.com/books?id=uSgDAAAAMBAJ&pg=PA50 "How Moon and Sun Generate the Tides"] {{Webarchive|url=https://web.archive.org/web/20230916153532/https://books.google.com/books?id=uSgDAAAAMBAJ&pg=PA50 |date=2023-09-16 }} ''Popular Science'', April 1934
{{refend}}
* {{cite book |last1=Simon |first1=Bernard |translator-last=Manley |translator-first=David |year=2013 |orig-date=2007 |title=Coastal Tides |url=https://diffusion.shom.fr/pro/coastal-tides-version-anglaise-de-la-maree-oceanique-cotiere.html |publisher=Institut océanographique, Fondation Albert Ier, Prince de Monaco |isbn=978-2-903581-83-1 |access-date=2021-10-18 |archive-date=2022-11-13 |archive-url=https://web.archive.org/web/20221113030309/https://diffusion.shom.fr/pro/coastal-tides-version-anglaise-de-la-maree-oceanique-cotiere.html |url-status=dead }}

== External links ==
* [http://tidesandcurrents.noaa.gov/ NOAA Tides and Currents information and data]
* [http://www.co-ops.nos.noaa.gov/predhist.html History of tide prediction] {{Webarchive|url=https://web.archive.org/web/20150509125035/http://co-ops.nos.noaa.gov/predhist.html |date=2015-05-09 }}
* [http://oceanworld.tamu.edu/resources/ocng_textbook/chapter17/chapter17_04.htm Department of Oceanography, Texas A&M University] {{Webarchive|url=https://web.archive.org/web/20160304000526/http://oceanworld.tamu.edu/resources/ocng_textbook/chapter17/chapter17_04.htm |date=2016-03-04 }}
* [http://easytide.ukho.gov.uk/EasyTide/EasyTide/index.aspx UK Admiralty Easytide]
* [https://web.archive.org/web/20050406202200/http://www.pol.ac.uk/ntslf/tidalp.html UK, South Atlantic, British Overseas Territories and Gibraltar tide times from the UK National Tidal and Sea Level Facility]
* [http://www.bom.gov.au/oceanography/tides/index.shtml Tide Predictions for Australia, South Pacific & Antarctica]
* [http://tbone.biol.sc.edu/tide/index.html Tide and Current Predictor, for stations around the world]

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