Rule of twelfths

The rule of twelfths is an approximation to a sine curve. It can be used as a rule of thumb for estimating the height of the tide at any time, given only the time and height of high and low water. This is important when navigating a boat or a ship in shallow water, and when launching and retrieving boats on slipways on a tidal shore. The rule is also useful for estimating the monthly change in sunrise/set and day length.

The rule assumes that the rate of flow of a tide increases smoothly to a maximum halfway between high and low tide before smoothly decreasing to zero again and that the interval between low and high tides is approximately six hours. For the six hours, the rule says that in the first hour after low tide the water level rises by one twelfth of the range, in the second hour two twelfths, and so on according to the sequence - 1:2:3:3:2:1.

If a tide table gave us the information that tomorrow's low water would be at noon and that the water level at this time would be two metres above chart datum and further, that at the following high tide the water level would be 14 metres. We could work out the height of water at 3:00 p.m. as follows:

This represents only the increase - the total depth of the water (relative to chart datum) will include the 2 m depth at low tide: 6 m + 2 m = 8 metres.

Obviously the calculation can be simplified by adding twelfths together and reducing the fraction beforehand i.e.

Rise of tide in three hours ${\displaystyle =\left({1 \over 12}+{2 \over 12}+{3 \over 12}\right)\times 12\ \mathrm {m} =\left({6 \over 12}\right)\times 12\ \mathrm {m} =\left({1 \over 2}\right)\times 12\ \mathrm {m} =6\ \mathrm {m} }$

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