Theory of tides
[3] Crates of Mallus attributed the tides to "the counter-movement (ἀντισπασμός) of the sea” and Apollodorus of Corcyra to "the refluxes from the Ocean".
[7] An ancient Indian Purana text dated to 400-300 BC refers to the ocean rising and falling because of heat expansion from the light of the Moon.
[a][8] Ultimately the link between the Moon (and Sun) and the tides became known to the Greeks, although the exact date of discovery is unclear; references to it are made in sources such as Pytheas of Massilia in 325 BC and Pliny the Elder's Natural History in 77 AD.
[4] Classicists Thomas Little Heath claimed that both Pytheas and Posidonius connected the tides with the moon, "the former directly, the latter through the setting up of winds".
[3][10] Seleucus of Seleucia is thought to have theorized around 150 BC that tides were caused by the Moon as part of his heliocentric model.
An apocryphal legend claims that he committed suicide in frustration with his failure to fully understand the tides.
[7] Philostratus discusses tides in Book Five of Life of Apollonius of Tyana (circa 217-238 AD); he was vaguely aware of a correlation of the tides with the phases of the Moon but attributed them to spirits moving water in and out of caverns, which he connected with the legend that spirits of the dead cannot move on at certain phases of the Moon.
[15][3] Medieval European understanding of the tides was often based on works of Muslim astronomers that became available through Latin translation starting from the 12th century.
[16] Abu Ma'shar al-Balkhi, in his Introductorium in astronomiam, taught that ebb and flood tides were caused by the Moon.
[16] Abu Ma'shar discussed the effects of wind and Moon's phases relative to the Sun on the tides.
[17][18] In 1609, Johannes Kepler correctly suggested that the gravitation of the Moon causes the tides,[c] which he compared to magnetic attraction[20][4][21][22] basing his argument upon ancient observations and correlations.
[31] While Newton explained the tides by describing the tide-generating forces and Daniel Bernoulli gave a description of the static reaction of the waters on Earth to the tidal potential, the dynamic theory of tides, developed by Pierre-Simon Laplace in 1775,[32] describes the ocean's real reaction to tidal forces.
[34] The equilibrium theory—based on the gravitational gradient from the Sun and Moon but ignoring the Earth's rotation, the effects of continents, and other important effects—could not explain the real ocean tides.
Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within a few centimeters.
For a fluid sheet of average thickness D, the vertical tidal elevation ζ, as well as the horizontal velocity components u and v (in the latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations:[44] where Ω is the angular frequency of the planet's rotation, g is the planet's gravitational acceleration at the mean ocean surface, a is the planetary radius, and U is the external gravitational tidal-forcing potential.
William Thomson (Lord Kelvin) rewrote Laplace's momentum terms using the curl to find an equation for vorticity.
Thomson's work in this field was further developed and extended by George Darwin, applying the lunar theory current in his time.
Darwin's harmonic developments of the tide-generating forces were later improved when A.T. Doodson, applying the lunar theory of E.W.
Amplitudes (half of peak-to-peak amplitude) of tidal constituents are given below for six example locations: Eastport, Maine (ME),[48] Biloxi, Mississippi (MS), San Juan, Puerto Rico (PR), Kodiak, Alaska (AK), San Francisco, California (CA), and Hilo, Hawaii (HI).
refer to an alternative set of fundamental angular arguments (usually preferred for use in modern lunar theory), in which:- It is possible to define several auxiliary variables on the basis of combinations of these.
