Editing Tide (section)

== 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 [[#Phase and amplitude|co-tidal charts]]. The time when the crest of the wave reaches a port then gives the time of high water at the port. The time taken for the wave to travel around the ocean also means that there is a delay between the phases of the Moon and their effect on the tide. Springs and neaps in the [[North Sea]], for example, are two days behind the new/full moon and first/third quarter moon. This is called the tide's ''age''.<ref>{{cite web |url=https://glossary.ametsoc.org/wiki/Age |title=Glossary of Meteorology |website=[[American Meteorological Society]] |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 [[bathymetry]] greatly influences the tide's exact time and height at a particular [[coast]]al point. There are some extreme cases; the [[Bay of Fundy]], on the east coast of Canada, is often stated to have the world's highest tides because of its shape, bathymetry, and its distance from the continental shelf edge.<ref>{{cite web |url=http://www.waterlevels.gc.ca/english/FrequentlyAskedQuestions.shtml#importantes |title=FAQ |access-date=June 23, 2007 |archive-date=February 12, 2012 |archive-url=https://web.archive.org/web/20120212231548/http://www.waterlevels.gc.ca/english/FrequentlyAskedQuestions.shtml#importantes |url-status=dead}}</ref> Measurements made in November 1998 at Burntcoat Head in the Bay of Fundy recorded a maximum range of {{convert|16.3|m|ft}} and a highest predicted extreme of {{convert|17|m|ft}}.<ref name=BIO2004>{{cite journal |last1=O'Reilly |first1=C.T.R. |first2=Ron |last2=Solvason |first3=Christian |last3=Solomon |name-list-style=amp |title=Where are the World's Largest Tides |journal=BIO Annual Report "2004 in Review" |date=2005 |pages=44–46 |editor1-first=J. |editor1-last=Ryan |publisher=Biotechnol. Ind. Org. |location=Washington, D.C.}}</ref><ref name=FundyWorkshop>{{cite book |first1=Charles T. |last1=O'reilly |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 [[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. |last2=Maddock |first2=L. |year=1978 |title=The M4 tide in the English Channel derived from a non-linear numerical model of the M2 tide |journal=Deep-Sea Research |volume=25 |pages=53–63}}</ref> [[Isle of Portland|Portland]] has double low waters for the same reason. The ''M''<sub>4</sub> tide is found all along the south coast of the United Kingdom, but its effect is most noticeable between the [[Isle of Wight]] and [[Isle of Portland|Portland]] because the ''M''<sub>2</sub> tide is lowest in this region.

Because the oscillation modes of the [[Mediterranean Sea]] and the [[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 [[Australia]], low tides can be due to the presence of a nearby [[Amphidromic point|amphidrome]].

=== Analysis ===
[[File:Water surface level changes with tides.svg|thumb|upright=2|A regular water level chart]]
[[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 [[bathymetry]], and coastline shape.

Current procedure for analysing tides follows the method of harmonic analysis introduced in the 1860s by [[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 [[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 [[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 [[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]]. [[Arthur Thomas Doodson|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 [[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 [[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#Tidal constituents|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]] over a nineteen-year period (the ''National Tidal Datum Epoch'' in the U.S.) uses frequencies called the ''tidal harmonic constituents''. Nineteen years is preferred because the Earth, Moon and Sun's relative positions repeat almost exactly in the [[Metonic cycle]] of 19&nbsp;years, which is long enough to include the 18.613 year [[Lunar node|lunar nodal]] [[Earth tide#Tidal constituents|tidal constituent]]. This analysis can be done using only the knowledge of the forcing ''period'', but without detailed understanding of the mathematical derivation, which means that useful tidal tables have been constructed for centuries.<ref>{{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, [[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 [[amphidromic point]]s counterclockwise in the northern hemisphere so that from [[Baja California Peninsula]] to [[Alaska]] and from [[France]] to [[Ireland]] the ''M''<sub>2</sub> tide propagates northward. In the southern hemisphere this direction is clockwise. On the other hand, ''M''<sub>2</sub> tide propagates counterclockwise around New Zealand, but this is because the islands act as a dam and permit the tides to have different heights on the islands' opposite sides. (The tides do propagate northward on the east side and southward on the west coast, as predicted by theory.)

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 [[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 (physics)|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, [[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]]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 [[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: [[Wellington]], and [[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 {{coord|41|13.9|S|174|29.6|E}} (northwest of [[Cape Terawhiti]]) refers timings to [[Westport, New Zealand|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|Tidal power}}
Tidal energy can be extracted by two means: inserting a water [[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, [[Rance tidal power plant|La Rance]] at [[St Malo|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.