NATARAJATides — the Moon's pull, the rule of twelfths & the keel
Learning outcomes & permitted supports
- Read a tide curve the way the table means it: heights above CHART DATUM, so charted depth and height of tide simply add
- Explain springs and neaps as the beat of the lunar M2 against the solar S2 — big ranges at BOTH new and full moon
- Work the rule of twelfths (1,2,3,3,2,1) and state its honest error against the real sinusoid — exact mid-rise, 1.63% out at hours 1 and 5
- Find when there is water enough: height at a time, and time for a height, both ways
- Keep the keel clear: UKC = charted + tide − draft − squat, with Barrass' Cb·V²/100 and why halving speed quarters the squat
Supports in this lab: Calculator. Two-constituent harmonic port (model stated — the Admiralty method stacks dozens; two make every examined idea real); Barrass squat as the stated teaching rule. Golden 961/961.
Glossary: chart datum · springs & neaps · rule of twelfths · UKC & squat
The rhythm — and where it comes from
The Moon drags two bulges of ocean around the Earth: high water roughly every 12 h 25 min — the lunar semidiurnal beat, M2. The Sun plays the same tune at exactly 12 h — S2, weaker. When the two crests coincide (full and new Moon) their amplitudes ADD: springs, range 4.8 m here. A week later they oppose: neaps, 2.4 m. Slide the clock and watch the curve breathe through the cycle — the beat of two frequencies, nothing more mystical.
Model stated: two harmonic constituents, h(t) = Z₀ + aM2cos + aS2cos. The Admiralty method stacks dozens; two are enough to make every examined idea real. Heights are metres above chart datum — the same zero the chart's soundings hang from, which is the whole reason tide and charted depth simply ADD.
Metres under the keel
The cross-section on the bench keeps the whole sum in one picture: charted depth from the datum DOWN, height of tide from the datum UP, your draft, and squat — the ship sitting deeper because she is moving (Barrass: Cb·V²/100 open water, twice that confined). What is left is the under-keel clearance, and it goes red before it goes aground.
Passage planning, live: for 0.5 m clearance at 10.0 kn you need the tide at 2.90 m. There is water RIGHT NOW — the tide already stands above what you need. The next question is how long it lasts.
The rule of twelfths — a seaman's approximation, priced
Between LW and HW the tide does not rise evenly: slow, fast, slow — a sinusoid. The mental shortcut: in the six hours of rise she makes 1, 2, 3, 3, 2, 1 twelfths per hour. Against the exact curve the rule is PERFECT at hours 2, 3, 4 and 6, and out by 1.63% of the range at hours 1 and 5 — the golden suite pins that gap as a fact, not a secret.
| after LW | rule (cum.) | exact sinusoid | gap |
|---|---|---|---|
| 1 h | 1/12 = 8.3% | 6.7% | 1.63% |
| 2 h | 3/12 = 25.0% | 25.0% | exact |
| 3 h | 6/12 = 50.0% | 50.0% | exact |
| 4 h | 9/12 = 75.0% | 75.0% | exact |
| 5 h | 11/12 = 91.7% | 93.3% | 1.63% |
| 6 h | 12/12 = 100.0% | 100.0% | exact |
Worked the exam's way: LW 1.2 m rising to HW 5.0 m; 2½ h after LW the rule gives 2.63 m — cumulative 3/12 at 2 h, halfway to 6/12 by 2½.
| Height of tide now | 5.69 m above datum |
|---|---|
| Next events | LW 1.1 m at d1 0855 · HW 5.7 m at d1 1504 · LW 1.1 m at d1 2114 |
| Springs / neaps range | 4.8 m / 2.4 m |
| Depth available | 13.69 m = charted 8.0 + tide 5.69 |
| Squat at 10.0 kn (open) | 0.80 m — Cb·V²/100 |
| Under-keel clearance | 3.29 m |