Eight cosines

Every curve on this site is the same arithmetic run at a different station. There is no model of the sea in it, no simulation and no smoothing.

The tide-raising forces of the moon and the sun can be separated into a set of components, each of which has a period fixed by the geometry of the orbits. Those periods are the same at Halberry Roads as they are anywhere else on earth. What a tidal analysis of a real gauge record produces is not the periods — it is, for each of them, an amplitude in metres and a phase lag in degrees. Two numbers per constituent. That is the whole station.

h(t) = Z0 + Σ A_i · cos( 2π t / T_i − g_i )

Z0 is mean level above chart datum, A is amplitude, T is period, g is phase lag. Run it forward and you have a prediction; run it for a fortnight and springs, neaps, unequal highs and, in shallow water, a second high water all appear without any of them being put there.

The eight we carry

M2 — Principal lunar semidiurnal

The moon, twice a day. On most coasts it is the tide and everything else is a correction to it.

S2 — Principal solar semidiurnal

The sun, twice a day. Because it runs slightly faster than M2 the two drift in and out of step — that beat is springs and neaps, and nobody draws it.

N2 — Larger lunar elliptic semidiurnal

The moon is not on a circle. N2 is the part of M2 that the varying earth–moon distance will not sit still for.

K2 — Lunisolar semidiurnal

The declination of both bodies at once. Small, and the reason a spring tide is never quite the height of the last one.

K1 — Lunisolar diurnal

Once a day. Where the diurnals are strong the two highs of a day stop matching.

O1 — Principal lunar diurnal

The lunar half of the once-a-day term. K1 and O1 together are what the form factor measures.

P1 — Principal solar diurnal

The solar half. Small enough to leave out of a rough prediction and large enough to be missed in one.

M4 — Shallow water overtide of M2

Not astronomy — geometry. In a shoaling estuary the crest of M2 travels faster than its trough and the wave loses its symmetry; M4 is the arithmetic of that. It is what puts a second high water on a port that should only have one.

Why the periods are carried to seven places

M2 is 12.4206012 hours. Rounded to 12.42 it is 2.2 seconds short, which is nothing, and after a fortnight it is 6 minutes of high water, which is not. Rounding a physical constant is the one shortcut in a predictor that gets worse the longer you leave it running, and it is the reason the tables in every station page print all seven places.

Springs and neaps are not a feature

S2 runs at exactly 12.0000000 hours and M2 at 12.4206012. They therefore go in and out of step with a beat period of 14.7653 days — half a synodic month. Nobody wrote a spring–neap cycle into this software. It is a consequence of two numbers we did not choose, and taking S2 out of the sum on the front page is what shows it.

And what the eight cannot do

Eight constituents is a coastal prediction, not a port operations one — a full analysis carries sixty or more, and the shallow-water ports need most of them. Where we know the eight are not enough we say so on the station page rather than in a table of accuracies nobody reads. The residuals we publish.

Get Wrackline

All 8 constituent tables ship inside the app, so the arithmetic on this page can be run against ours. Wrackline is an aid to planning and is not a substitute for an official tide table. Do not navigate on it.