Tide Height Between High and Low Water (Cosine Curve)

A tide table gives a high water and a low water with their times, but you often want the height at the minute you cross a bar, or the time at which a stated depth is reached. This formula bridges the two by treating the rise or fall of a semi-diurnal tide between consecutive extremes as half of a cosine wave. It is a smooth interpolation, not a prediction from the physics of the tide: it knows only the two heights and the time between them, and it is the smooth version of the rule of twelfths.

Here $h_1$ is the height at the start (the high or low water just before the time), $h_2$ the height at the end (the next low or high water), $x$ the time since the start and $\tau$ the time between the two extremes, about 6 h 12 min. The result $h$ is the height above chart datum. Between $x = 0$ and $x = \tau$ the bracket sweeps smoothly from its smallest to its largest value, so $h$ leaves $h_1$ and arrives exactly at $h_2$. Because a cosine is flat at its turning points, the tide changes slowly just after an extreme, fastest in the middle, and slowly again as the next extreme approaches.

The rule of twelfths says a six-hour rise of range $R$ completes 1/12, 3/12, 6/12, 9/12, 11/12 and 12/12 of $R$ at hours 1 to 6. The rule steps hour by hour; the cosine curve agrees with it exactly at hours 2, 3 and 4. In the worked example the tide rises from 0.4 m to 4.0 m over 6 h, so $R = 3.6$ m. At hour 2 the page gives 1.3 m (3/12 of the range above 0.4 m), at hour 3 it gives 2.2 m (halfway, 6/12) and at hour 4 it gives 3.1 m (9/12).

The formula runs the other way too. Given a depth, the page solves for the time: in this example 2.2 m is reached 3 h after low water. The cosine is a shape, not a measurement: it uses one high water and one low water and sees nothing else that shapes a real tide, and the time since the start must lie between 0 and $\tau$. A full harmonic prediction is built from the angular frequencies of tidal constituents, and an air-pressure correction sits on top of the predicted tide at a gauge.