Some shorelines experience two almost equal high tides and two low tides each day, called a semi-diurnal tide. Some locations experience only one high and one low tide each day, called a diurnal tide. Some locations experience two uneven tides a day, or sometimes one high and one low each day; this is called a mixed tide. The times and amplitude of the tides at a locale are influenced by the alignment of the Sun and Moon, by the pattern of tides in the deep ocean, by the amphidromic systems of the oceans, and by the shape of the coastline and near-shore bathymetry (see Timing).
Tides vary on timescales ranging from hours to years due to numerous influences. To make accurate records, tide gauges at fixed stations measure the 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.
While tides are usually the largest source of short-term sea-level fluctuations, sea levels are also subject to forces such as wind and barometric pressure changes, resulting in storm surges, 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 solid part of the Earth is affected by tides, though this is not as easily seen as the water tidal movements.
Tide changes proceed via the following stages:
- Sea level rises over several hours, covering the intertidal zone; flood tide.
- The water rises to its highest level, reaching high tide.
- Sea level falls over several hours, revealing the intertidal zone; ebb tide.
- The water stops falling, reaching low tide.
Tides produce oscillating currents known as tidal streams. 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.
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 tables. 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 equator.
Observation and predictionEdit
From ancient times, tidal observation and discussion has increased in sophistication, first marking the daily recurrence, then tides' relationship to the sun and moon. Pytheas travelled to the British Isles about 325 BC and mentions spring tides.
In the 2nd century BC, the Babylonian astronomer, Seleucus of Seleucia, correctly described the phenomenon of tides in order to support his heliocentric theory. He correctly theorized that tides were caused by the moon, although he believed that the interaction was mediated by the pneuma. He noted that tides varied in time and strength in different parts of the world. According to Strabo (1.1.9), Seleucus was the first to link tides to the lunar attraction, and that the height of the tides depends on the moon's position relative to the sun.
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 Euripus Strait and the Strait of Messina puzzled Aristotle.) Philostratus discussed tides in Book Five of The Life of Apollonius of Tyana. Philostratus mentions the moon, but attributes tides to "spirits". In Europe around 730 AD, the Venerable Bede described how the rising tide on one coast of the British Isles coincided with the fall on the other and described the time progression of high water along the Northumbrian coast.
In the 9th century, the Arabian earth-scientist, Al-Kindi (Alkindus), wrote a treatise entitled Risala fi l-Illa al-Failali l-Madd wa l-Fazr (Treatise on the Efficient Cause of the Flow and Ebb), in which he presents an argument on tides which "depends on the changes which take place in bodies owing to the rise and fall of temperature." He describes a clear and precise laboratory experiment in order to prove his argument. Al-Kindi explained the influence of Sun and Moon on the tide phenomenon, according to the Islamic Encyclopedia, as follows: "The sun and the moon warm the water and hence cause it to expand. It is this expansion that makes the water spring out of the center of the earth, and depending on the yearly, monthly and daily movement of the sun and the moon, it also causes the ebb and flow of the sea water known as tides."
The first tide table in China was recorded in 1056 AD primarily for visitors wishing to see the famous tidal bore in the Qiantang River. The first known British tide table is thought to be that of John Wallingford, who died Abbot of St. Albans in 1213, based on high water occurring 48 minutes later each day, and three hours earlier at the Thames mouth than upriver at London.
William Thomson (Lord Kelvin) led the first systematic harmonic analysis of tidal records starting in 1867. The main result was the building of a 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.
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 gage stations by 1850.
William Whewell first mapped co-tidal lines ending with a nearly global chart in 1836. In order to make these maps consistent, he hypothesized the existence of amphidromes where co-tidal lines meet in the mid-ocean. These points of no tide were confirmed by measurement in 1840 by Captain Hewett, RN, from careful soundings in the North Sea.
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.
The ocean bathymetry greatly influences the tide's exact time and height at a particular coastal 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. Measurements made in November 1998 at Burntcoat Head in the Bay of Fundy recorded a maximum range of 16.3 meters (53 ft) and a highest predicted extreme of 17 meters (56 ft).   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 16.2 meters (53 ft) and a highest predicted extreme of 16.8 meters (55 ft). Ungava Bay and the Bay of Fundy lie similar distances from the continental shelf edge, but Ungava Bay is free of pack ice for only 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 region's different tidal harmonics, caused primarily by the east/west orientation of the English Channel and the fact that when it is high water at Dover it is low water at Land's End (some 300 nautical miles distant) and vice versa. This is contrary to the popular belief that the flow of water around the Isle of Wight creates two high waters. The Isle of Wight is important, however, since it is responsible for the 'Young Flood Stand', which describes the pause of the incoming tide about three hours after low water.
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 amphidrome.
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 detailed understanding. Although it may seem that tides could be predicted via a sufficiently detailed knowledge of the instantaneous astronomical forcings, the actual tide at a given location is determined by astronomical forces accumulated over many days. Precise results 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 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 the 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 sinusoidal 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) 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. 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
- A·cos(w·t + p)
where A is the amplitude, w is the angular frequency usually given in degrees per hour corresponding to t measured in hours, and p 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 p of the first harmonic for the moon term is called the lunitidal interval or high water interval. The next step is to accommodate the harmonic terms due to the elliptical shape of the orbits. Accordingly, the value of A is not a constant but also varying with time, slightly, about some average figure. Replace it then by A(t) where A is another sinusoid, similar to the cycles and epicycles of Ptolemaic theory. Accordingly,
- A(t) = A·(1 + Aa·cos(wa·t + pa)) ,
which is to say an average value A with a sinusoidal variation about it of magnitude Aa , with frequency wa and phase pa . Thus the simple term is now the product of two cosine factors:
- A·[1 + Aa·cos(wa ·t + pa)]·cos(w·t + p)
Given that for any x and y
- cos(x)·cos(y) = ½·cos( x + y ) + ½·cos( x–y ) ,
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 (1 + cos(x))·cos(y) .) 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.
Remember that astronomical tides do not include weather effects. Also, changes to local conditions (sandbank movement, dredging harbour mouths, etc.) away from those prevailing at the measurement time affect the tide's actual timing and magnitude. Organisations quoting a "highest astronomical tide" for some location may exaggerate the figure as a safety factor against analytical uncertainties, distance from the nearest measurement point, changes since the last observation time, ground subsidence, etc., to avert liability should an engineering work be overtopped. Special care is needed when assessing the size of a "weather surge" by subtracting the astronomical tide from the observed tide.
Careful Fourier data analysis over a nineteen-year period (the National Tidal Datum Epoch in the U.S.) uses frequencies called the tidal harmonic constituents. Nineteen years is preferred because the Earth, moon and sun's relative positions repeat almost exactly in the Metonic cycle of 19 years, which is long enough to include the 18.613 year lunar nodal 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. The resulting amplitudes and phases can then be used to predict the expected tides. These are usually dominated by the constituents near 12 hours (the semi-diurnal constituents), but there are major constituents near 24 hours (diurnal) as well. Longer term constituents are 14 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 M2 (lunar) and S2 (solar) periods differ slightly, so that the relative phases, and thus the amplitude of the combined tide, change fortnightly (14 day period).
In the M2 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 points counterclockwise in the northern hemisphere so that from Baja California Peninsula to Alaska and from France to Ireland the M2 tide propagates northward. In the southern hemisphere this direction is clockwise. On the other hand M2 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 M2 patterns cannot be used for other tide components.
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 semidiurnal 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.
The tides' influence on current flow is much more difficult to analyse, and data is much more difficult to collect. A tidal height is a simple number which applies to a wide region simultaneously. A flow has both a magnitude and a 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 is the same flow, 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, eddies may form in only one flow direction.
Nevertheless, current analysis is similar to tidal 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 charts, 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 north side (e.g. at Nelson) follows the common bi-weekly spring–neap tide cycle (as found along the west side of the country), but the south side's tidal pattern has only one cycle per month, as on the east side: Wellington, and 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 end (Nelson) has a counterpart spring tide at the south end (Wellington), so the resulting behaviour follows neither reference harbour.
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, 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.
History of tidal physicsEdit
Investigation into tidal physics was important in the early development of heliocentrism and 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.
Galileo Galilei in his 1632 Dialogue Concerning the Two Chief World Systems, whose working title was Dialogue on the Tides, gave an explanation of the tides. The resulting theory, however, was incorrect as he attributed the tides to the sloshing of water caused by the Earth's movement around the sun. He hoped to provide mechanical proof of the Earth's movement – the value of his tidal theory is disputed. At the same time Johannes Kepler correctly suggested that the Moon caused the tides, which he based upon ancient observations and correlations, an explanation which was rejected by Galileo. It was originally mentioned in Ptolemy's Tetrabiblos as having derived from ancient observation.
Isaac Newton (1642–1727) explained 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) and used his theory of universal gravitation to explain the lunar and solar attractions as the origin of the tide-generating forces. Newton and others before Pierre-Simon Laplace worked the problem from the perspective of a static system (equilibrium theory), that provided an approximation that described the tides that would occur in a non-inertial ocean evenly covering the whole Earth. The tide-generating force (or its corresponding 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 bathymetry, Earth's rotation, and other factors.
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 wrote 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 Jean le Rond d'Alembert studied tidal equations for the atmosphere which did not include rotation.
Pierre-Simon Laplace formulated a system of partial differential equations relating the ocean's horizontal flow to its surface height, a major dynamic theory for water tides. The Laplace tidal equations are still in use today. William Thomson, 1st Baron Kelvin, rewrote Laplace's equations in terms of vorticity which allowed for solutions describing tidally driven coastally trapped waves, known as Kelvin waves.
Others including Kelvin and Henri Poincaré further developed Laplace's theory. Based on these developments and the lunar theory of E W Brown describing the motions of the Moon, Arthur Thomas Doodson developed and published in 1921 the first modern development of the tide-generating potential in harmonic form: Doodson distinguished 388 tidal frequencies. Some of his methods remain in use.
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. 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 solar tidal force is 46% as large as the lunar. More precisely, the lunar tidal acceleration (along the Moon–Earth axis, at the Earth's surface) is about 1.1 × 10−7 g, while the solar tidal acceleration (along the Sun–Earth axis, at the Earth's surface) is about 0.52 × 10−7 g, where g is the gravitational acceleration at the Earth's surface. Venus has the largest effect of the other planets, at 0.000113 times the solar effect.
The ocean's surface is closely approximated by an equipotential surface, (ignoring ocean currents) commonly referred to as the geoid. Since the gravitational force is equal to the potential's gradient, there are no tangential forces on such a surface, and the ocean surface is thus in gravitational equilibrium. Now consider the effect of massive external bodies such as the Moon and Sun. These bodies have strong gravitational fields that diminish with distance in space and which act to alter the shape of an equipotential surface on the Earth. This deformation has a fixed spatial orientation relative to the influencing body. The Earth's rotation relative to this shape causes the daily tidal cycle. Gravitational forces follow an inverse-square law (force is inversely proportional to the square of the distance), but tidal forces are inversely proportional to the cube of the distance. The ocean surface moves because of the changing tidal equipotential, rising when the tidal potential is high, which occurs on the parts of the Earth nearest to and furthest from the Moon. When the tidal equipotential changes, the ocean surface is no longer aligned with it, so the apparent direction of the vertical shifts. The surface then experiences a down slope, in the direction that the equipotential has risen.
Laplace's tidal equationsEdit
- The vertical (or radial) velocity is negligible, and there is no vertical shear—this is a sheet flow.
- The forcing is only horizontal (tangential).
- 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 currents moving towards the equator to the west and toward the east for flows moving away from the equator, allowing coastally trapped waves. Finally, a dissipation term can be added which is an analog to viscosity.
Tsunamis, 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 actual link to the tide. Other phenomena unrelated to tides but using the word tide are rip tide, storm tide, hurricane tide, and black or red tides.
- ↑ Reddy, M.P.M. & Affholder, M. (2002). Descriptive physical oceanography: State of the Art. Taylor and Francis. p. 249. ISBN 90-5410-706-5. OCLC 223133263 47801346. http://books.google.com/?id=2NC3JmKI7mYC&pg=PA436&dq=tides+centrifugal+%22equilibrium+theory%22+date:2000-2010.
- ↑ Hubbard, Richard (1893). Boater's Bowditch: The Small Craft American Practical Navigator. McGraw-Hill Professional. p. 54. ISBN 0-07-136136-7. OCLC 44059064. http://books.google.com/?id=nfWSxRr8VP4C&pg=PA54&dq=centrifugal+revolution+and+rotation+date:1970-2009.
- ↑ Coastal orientation and geometry affects the phase, direction, and amplitude of amphidromic systems, coastal Kelvin waves as well as resonant seiches in bays. In estuaries seasonal river outflows influence tidal flow.
- ↑ "Tidal lunar day". NOAA. http://www.oceanservice.noaa.gov/education/kits/tides/media/supp_tide05.html. Do not confuse with the astronomical lunar day on the Moon. A lunar zenith is the Moon's highest point in the sky.
- ↑ Mellor, George L. (1996). Introduction to physical oceanography. Springer. p. 169. ISBN 1-56396-210-1.
- ↑ 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."Glossary of Coastal Terminology: H–M". Washington Department of Ecology, State of Washington. http://www.ecy.wa.gov/programs/sea/swces/products/publications/glossary/words/H_M.htm. Retrieved on 5 April 2007.
- ↑ Flussi e riflussi. Milano: Feltrinelli. 2003. ISBN 88-07-10349-4.
- ↑ van der Waerden, B.L. (1987). "The Heliocentric System in Greek, Persian and Hindu Astronomy". Annals of the New York Academy of Sciences 500 (1): 525–545 . doi:10.1111/j.1749-6632.1987.tb37224.x. Bibcode: 1987NYASA.500..525V.
- ↑ "Baike.baidu.com". Baike.baidu.com. 2012-05-02. http://baike.baidu.com/view/135336.htm. Retrieved on 2012-08-28.
- ↑ Al-Kindi, FSTC
- ↑ Plinio Prioreschi, "Al-Kindi, A Precursor Of The Scientific Revolution", Journal of the International Society for the History of Islamic Medicine, 2002 (2): 17-19 
- ↑ The Times of al-Kindi, Islamic Encyclopedia
- ↑ Cartwright, D.E. (1999). Tides, A Scientific History: 11, 18
- ↑ "The Doodson–Légé Tide Predicting Machine". Proudman Oceanographic Laboratory. http://www.pol.ac.uk/home/insight/doodsonmachine.html. Retrieved on 2008-10-03.
- ↑ 15.0 15.1 Zuosheng, Y.; Emery, K.O. & Yui, X. (July 1989). "Historical Development and Use of Thousand-Year-Old Tide-Prediction Tables". Limnology and Oceanography 34 (5): 953–957. doi:10.4319/lo.1989.34.5.0953.
- ↑ Glossary of Meteorology American Meteorological Society.
- ↑ Webster, Thomas (1837). The elements of physics. Printed for Scott, Webster, and Geary. p. 168. http://books.google.com/books?id=dUwEAAAAQAAJ.
- ↑ "FAQ". http://www.waterlevels.gc.ca/english/FrequentlyAskedQuestions.shtml#importantes. Retrieved on June 23, 2007.
- ↑ 19.0 19.1 O'Reilly, C.T.R.; Ron Solvason and Christian Solomon (2005). "Where are the World's Largest Tides". BIO Annual Report "2004 in Review": 44–46. Washington, D.C.: Biotechnol. Ind. Org..
- ↑ 20.0 20.1 Charles T. O'reilly, Ron Solvason, and Christian Solomon. "Resolving the World's largest tides", in J.A Percy, A.J. Evans, P.G. Wells, and S.J. Rolston (Editors) 2005: 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.
- ↑ "English Channel double tides". Bristolnomads.org.uk. http://www.bristolnomads.org.uk/stuff/double_tides.htm. Retrieved on 2012-08-28.
- ↑ To demonstrate this 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.
- ↑ Center for Operational Oceanographic Products and Services, National Ocean Service, National Oceanic and Atmospheric Administration (January 2000). "Tide and Current Glossary". Silver Spring, MD. http://tidesandcurrents.noaa.gov/publications/glossary2.pdf.
- ↑ Harmonic Constituents, NOAA.
- ↑ 25.0 25.1 Lisitzin, E. (1974). "2 "Periodical sea-level changes: Astronomical tides"". Sea-Level Changes, (Elsevier Oceanography Series). 8. p. 5.
- ↑ "What Causes Tides?". U.S. National Oceanic and Atmospheric Administration (NOAA) National Ocean Service (Education section). http://oceanservice.noaa.gov/education/kits/tides/tides02_cause.html.
- ↑ See for example, in the 'Principia' (Book 1) (1729 translation), Corollaries 19 and 20 to Proposition 66, on pages 251–254, referring back to page 234 et seq.; and in Book 3 Propositions 24, 36 and 37, starting on page 255.
- ↑ Wahr, J. (1995). Earth Tides in "Global Earth Physics", American Geophysical Union Reference Shelf #1,. pp. 40–46.
- ↑ Cartwright, David E. (1999). Tides: A Scientific History. Cambridge, UK: Cambridge University Press.
- ↑ Case, James (March 2000). "Understanding Tides—From Ancient Beliefs to Present-day Solutions to the Laplace Equations". SIAM News 33 (2).
- ↑ Doodson, A.T. (December 1921). "The Harmonic Development of the Tide-Generating Potential". Proceedings of the Royal Society of London. Series A 100 (704): 305–329. doi:10.1098/rspa.1921.0088. Bibcode: 1921RSPSA.100..305D.
- ↑ Casotto, S. & Biscani, F. (April 2004). "A fully analytical approach to the harmonic development of the tide-generating potential accounting for precession, nutation, and perturbations due to figure and planetary terms". AAS Division on Dynamical Astronomy 36 (2).
- ↑ Moyer, T.D. (2003) "Formulation for observed and computed values of Deep Space Network data types for navigation", vol. 3 in Deep-space communications and navigation series, Wiley, pp. 126–8, ISBN 0-471-44535-5.
- ↑ According to NASA the lunar tidal force is 2.21 times larger than the solar.
- ↑ See Tidal force – Mathematical treatment and sources cited there.
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