terms. Each term is of the form where is the amplitude. is the angular frequency, usually given in degrees per hour corresponding to measured in hours. 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 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 . To do so, replace in the above equation with where is another sinusoid, similar to the cycles and epicycles of Ptolemaic theory. This gives: which is to say an average value with a sinusoidal variation about it of magnitude , with frequency and phase . Substituting this for in the original equation gives a product of two cosine factors: Given that for any and 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 .) 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