History
A little knowledge of history is helpful for a clear understanding of tides.
From prehistoric times people have observed the connection between the motion of the moon and sun and the rise and fall of the tide, and put this knowledge to practical use. But why did the sea follow the moon and sun, what connected them ?
Australia
The oldest oral traditions are probably those of the Aboriginal and Torres Strait Islander communities. Stories tell how the tide varies during the lunar cycle; water fills as the moon rises, and drains as the moon sets. The moon wanes, disappearing at new moon, grows until full, then the cycle begins again.
The stories combine practical knowledge and mythical explanations, attributing the mysterious connection between moon and tides to the actions of supernatural figures and forces.
Polynesia
The ancestors of the Polynesians migrated from southeast Asia and settled the Pacific, starting about 1500BC. As they settled different island groups their languages and oral traditions diverged, but not completely, common elements remained.
In Polynesian tradition Tangaroa is god of the seas, and controls the tides. In some stories Marama (the moon) is his wife. In other stories the relationship between the seas and the moon is more complicated, but still intimate.
In some stories the moon has different names for the dark and light phases. In one Maori myth Rona (the moon) is one of two controllers of the tides. In the Takitimu (Hawaiian) tradition Tane instructed Te Marama (the moon) to influence the ocean waters in causing tides.
Recorded history
The earliest texts linking the moon and tides are in Tamil and Sanskrit, written about the 4th century BC, or perhaps later. Verses describe how waters swell with the increase of the moon, and contract as the moon wanes, in the light and dark fortnights.
The ancient Greeks wrote about tides, probably based on Arab or Indian sources. The Mediterranean Sea experiences only small tides, of little importance to the people of it's shores. For the Greeks tides were an exotic phenomena.
In their writings they posed the question—what causes the tide—but offered no useful answer. Nor did anyone else; another two thousand years elapses before the question is answered satisfactorily.
Tide tables
Over the following centuries the written record improves in detail; how the tides respond to the motion of the moon and sun, in daily and lunar-monthly cycles, and how the time of high tide lags the transit of the moon, by an interval that varies from place to place.
The oldest surviving tide tables were carved on a slab of stone at Yanguan in Hangzhou Bay, some time before the 9th century. Tide tables in Europe and the Middle East appear about two hundred years later.
Scholars and philosophers documented the rule-of-thumb knowledge, gradually obtained from sailors, fisherman, and farmers. But nobody explained the mechanism satisfactorily. Up to the 17th century the written record is empirical; a good description of what happens, but no useful explanation why.
Newton
Isaac Newton published his Principia in 1687 explaining how the gravitational attraction of the moon and sun acts on the earth, and causes the tidal force. He built on the work of Johannes Kepler. His explanation is succinct yet gives us everything we need to predict the gravitational field at earth's surface, assuming we know where the earth, moon and sun are, in relation to each other.
See the astronomic tide for a non-mathematical summary of the effects of gravity, as relevant to tides.
Newton's theory was a huge step forward in our understanding, but left two important questions unanswered: how does gravity act on the oceans, to cause the tides we see ? And what is the nature of gravity ?
Reasoning about tidal flow
With the discovery of the law of gravity, the problem was now to find a formula to connect the gravitational forces with the rise and fall of the seas.
Plainly the dimensions of the oceans and inertia of the water played an important part. In the early 18th century Daniel Bernoulli and Leonhard Euler advanced their theories though both fell a little short of a correct treatment of the fluid motion. Meanwhile astronomers analysed the motion of heavenly bodies with ever greater precision.
Laplace
In the late 18th century Pierre Simon Laplace introduced his dynamic theory, and described physical wave phenomena; tide resonance, standing waves, propagating waves, node points, reflection and refraction. His equations were of great conceptual importance but did not lead to a practical solution, owing to the intractably complex conditions found in the world's oceans.
More influential was his observation was that the physical traits of water and waves could, for the most part, be disregarded. Rather, the tide may be conceived as made up of a number of simple harmonic waves. The cyclical components of tides matches the major astronomic cycles, the motions of the moon and sun.
Laplace put this age-old observation into the language of contemporary mathematics. Subsequent investigators have favoured the same approach.
Fluid dynamics
In the 19th century explorers investigated the geographical distribution of tides. William Whewell attempted to describe a single world-wide tidal system, while Rollin Harris had more success describing tides as regional phenomena, caused by standing waves in the various oceans.
George Airy described propagation of waves for non-viscous fluid, and later Gabriel Stokes extended the analysis to viscous fluids, laying the groundwork for modern fluid dynamics.
Tide stations were established and readings collected, using tide staffs then automatic tide gauges. The improved knowledge showed that a simple, elegant solution was not to be found; no single equation could predict the tide everywhere.
Harmonic analysis
Harmonic analysis can be applied to tides, and does not require physical modelling of fluids or tidal flows. The method assumes the tidal signal contains waves of the same frequency as the astronomic cycles; for each frequency, the method finds the amplitude and phase of the constituent sine-wave.
In the 1860s William Thomson (Lord Kelvin) devised a mathematical process to apply Fourier analysis to the tidal readings, to extract the tidal constituents. His harmonic analysis technique was later improved by George Darwin and Arthur Doodson.
Kelvin designed the tide predicting machine, a kind of analog mechanical computer. With this machine the tidal constituents for a location can be determined. Once the constituents are known, the same machine can be used to predict the tide at a future time.
In the mid 20th century the analog machines were retired and replaced with digital computers, using the same harmonic analysis method, implemented as software.
Unresolved problems
Our present-day understanding of tides and tide prediction is not perfect, a few problems remain.
Nature of gravity
Why is gravity the way it is ? Newton said that gravitational force acts instantaneously, and at a distance across the vacuum of space, not via any substance or medium. If gravity acted any speed slower than infinite then angular momentum would not be conserved; planets would not orbit the sun, the moon would not orbit earth.
A force of infinite speed was a troublesome idea for his contemporaries. It remained troublesome until Einstein proposed that space-time is curved, and the apparently infinite speed of gravity is illusory, an observer effect.
The story is not finished, the nature of gravity remains at the heart of unresolved questions in physics; but as far as tides are concerned our current knowledge is sufficient.
Perfect harmony
To predict tides at a location using the harmonic analysis method you need to know the tidal constituents. To determine these constituents you need a series of height measurements recorded at that location, at closely spaced intervals, for at least several months, preferably a year or more.
The method is sensitive to noise, raw readings need to be filtered and smoothed. Asymmetrical tidal flow (as found in shallow waters or up-river) requires special handling.
Tidal constituents have a limited lifetime, predictions created using harmonic analysis will start to drift from reality after a year or so. You need to periodically re-calculate the coefficients, either from fresh readings or by corrections.
The method only allows you to predict the tide at that one specific location, for which you have measurements. Tides at nearby locations may be similar, or may be substantially different. This limitation arises because harmonic analysis ignores fluid dynamics.
Fluid dynamics is hard
If we want an alternative to harmonic analysis the obvious place to look is fluid dynamics. With the Navier-Stokes equations and modern computers we can simulate the behaviour of fluids. For modest-scale scenarios this is not too difficult. But the oceans slop around the undersea topography of the entire world.
To model the oceans we would need a detailed chart of the world's coastlines and sea-bed contours, and a computational model of the physics and fluid dynamics, with sufficiently small grid-cells and time-steps.
A fluid-dynamics model is always an approximation, you must simplify. Building a simulation that is practical yet accurate enough for some particular purpose is not easy. For tides, the challenge is roughly similar to that of creating an atmospheric model for predicting the weather.
Sources
Philip Clarke (1989) An Overview of Australian Aboriginal Ethnoastronomy
Elsdon Best (1982) Maori Religion and Mythology
Horace Wilson (1840) The Vishnu Purana (English translation)
Yang Zuosheng (1989) Historical development and use of thousand-year-old tide-prediction tables
George Smith (2007) Newton's Philosophiae Naturalis Principia Mathematica
David Cartwright (1999) Understanding Tides—From Ancient Beliefs to Present-day Solutions to the Laplace Equations (review by James Case)
Harry Marmer (1922) The Problems of the Tide
history last updated 2023-07-31 visit oceantide.io