Rule of twelfths tidal calculator
The rule of twelfths approximates a tide as a sine wave: it rises one twelfth of the range in the first hour, two in the second, three in the third and so on.
hours
Tidal height
3.668m
- Risen since low water
- 2.468
- Depth of water
- 6.168
- Clearance under the keel
- 4.368
- More than half a metre of clearance
- 1
- Rate of rise
- 1.3158 m/h
- Tidal range
- 5.20
- Twelfths elapsed
- 5.81
Show the working
- Low water height = 1.2
- High water height = 6.4
- Hours after low water = 3
- Length of the tide = 6.2
- Charted depth = 2.5
- Boat draft = 1.8
- Tidal range = 6.4 - 1.2 = 5.2
- fraction = 6.2 > 0 ? 3 / 6.2 : 0 = 0.483 871
- sine_rise = (1 - cos(pi * clamp(0.483 871, 0, 1))) / 2 = 0.474 675
- Tidal height = 1.2 + 5.2 * 0.474 675 = 3.668 31
- Risen since low water = 5.2 * 0.474 675 = 2.468 31
- Depth of water = 2.5 + 3.668 31 = 6.168 31
- Clearance under the keel = 6.168 31 - 1.8 = 4.368 31
- More than half a metre of clearance = 4.368 31 > 0.5 ? 1 : 0 = 1
- Rate of rise = 6.2 > 0 ? 5.2 * pi * sin(pi * clamp(0.483 871, 0, 1)) / (2 * 6.2) : 0 = 1.315 75
- Twelfths elapsed = 0.483 871 * 12 = 5.806 45
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- height = low water + range x (1 - cos(pi t / T)) / 2
- the rule of twelfths is this sine curve rounded to 1, 2, 3, 3, 2, 1 twelfths per hour
The sine approximation holds well in open coastal water and badly in estuaries, where the flood and ebb are not symmetric.
Always keep a safety margin under the keel. Charted depths are minima, not guarantees, and swell adds motion.