Methods of Tide PredictionThe following is quoted from http://en.wikipedia.org/wiki/Tide.
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 forces, 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.
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
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.
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 + pa)]*cos(w*t + p)
Given that for any x and y
cos(x)*cos(y) = 1/2*cos( x + y ) + 1/2*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.
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 semi-annual.
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).
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