How Tide Predictions Are Calculated

How tides are predicted: gravitational forcing from the Moon and Sun, harmonic constituents such as M2 and S2, and why observed water levels differ from the astronomical prediction.

Tides can be predicted years in advance because they are driven by the motions of the Moon and Sun, which are known with great precision. The tide at a place is the ocean's response to that gravitational forcing, shaped by local coastline and depth. The method that turns astronomy into a tide table is called harmonic analysis.

The tide as a sum of constituents

Harmonic analysis treats the tide as the sum of many simple waves, each called a constituent. Every constituent has a frequency fixed by astronomy — how the Earth, Moon, and Sun move relative to one another — while its amplitude and timing are found by fitting the constituents to a period of real water-level observations at that location. The main constituents include:

M2 — the principal lunar semidiurnal, period 12.42 hours; usually the largest.
S2 — the principal solar semidiurnal, period 12.00 hours.
N2 — a lunar constituent from the Moon's elliptical orbit.
K1 and O1 — the main diurnal constituents, which make one high water a day larger than the other in many places.

The familiar spring-neap cycle emerges naturally from this: M2 and S2 have slightly different periods, so they drift in and out of step, reinforcing every 14.8 days to make springs and partly cancelling to make neaps.

Predicting forward

Once the amplitude and phase of each constituent are known for a location, predicting the tide is simply a matter of evaluating the sum at any future time. The astronomical frequencies never change, so the calculation is deterministic and can run decades ahead.

Why the real tide differs

A prediction is the astronomical tide only. It does not include the weather. Strong winds, low or high atmospheric pressure, heavy rainfall, and river flow can raise or lower the actual water level — a storm surge can add a metre or more above prediction, while high pressure and offshore winds can hold the water well below it. Predictions are therefore an excellent guide to the timing and size of the tide, but the observed level on the day can differ.