# Rule of twelfths

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.[1][2] 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.[3]

## Example calculations

### Tides

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:

• The total increase in water level between low and high tide would be: 14 - 2 = 12 metres.
• In the first hour the water level would rise by 1 twelfth of the total (12 m) or: 1 m
• In the second hour the water level would rise by another 2 twelfths of the total (12 m) or: 2 m
• In the third hour the water level would rise by another 3 twelfths of the total (12 m) or: 3 m
• This gives us the increase in the water level by 3:00 p.m. as 6 metres.

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 ${\displaystyle =\left({1 \over 12}+{2 \over 12}+{3 \over 12}\right)\times 12\ \mathrm {m} =\left({6 \over 12}\right)\times 12\ \mathrm {m} =\left({1 \over 2}\right)\times 12\ \mathrm {m} =6\ \mathrm {m} }$

### Daylength

If midwinter sunrise and set are at 09:00 and 15:00, and midsummer at 03:00 and 21:00, the daylight duration will shift by 0:30, 1:00, 1:30, 1:30, 1:00 and 00:30 over the six months from one solstice to the other. Likewise the day length changes by 0:30, 1:00, 1:30, 1:30, 1:00 and 00:30 each month. More equatorial latitudes change by less, but still in the same proportions; more polar by more.

## Caveats

The rule is a rough approximation only and should be applied with great caution when used for navigational purposes. Officially produced tide tables should be used in preference whenever possible.

The rule assumes that all tides behave in a regular manner, this is not true of some geographical locations, such as Poole Harbour[4] or the Solent[5] where there are "double" high waters or Weymouth Bay[6] where there is a double low water.

The rule assumes that the period between high and low tides is six hours but this is an underestimate and can vary anyway.

## References

1. ^ The Outboard Boater's Handbook: Advanced Seamanship and Practical Skills, written by David R. Getchell, published by International Marine, page 195, ISBN 978-0-07-023053-8
2. ^ Rule of Twelfths for quick tidal estimates
3. ^ The weekend navigator: simple boat navigation with GPS and electronics, written by Robert J. Sweet, page 162, ISBN 978-0-07-143035-7
4. ^ http://www.shrimperowners.org/sitefiles/Poole%20Tides.pdf
5. ^ English Channel Double Tides