How Harmonic Analysis Predicts Tides: From Kelvin to Doodson

Every tide-predicting machine is frozen mathematics. Before the first gear was ever cut, three generations of scientists had to prove that the ocean’s chaos could be written as a sum of clockwork waves — and that those waves could be measured from a finite stretch of observations. This is that story, in plain English, with the equations kept to the minimum you need.

The tide as a sum of cosines

The modern prediction formula fits on one line. The height $h$ at time $t$ is:

$h(t) = Z_0 + \sum A_i \cos(\omega_i t - \phi_i)$

In words: start from mean sea level $Z_0$, then add one cosine wave per constituent $i$. Each wave has an amplitude $A$ (how big, in metres), a speed $\omega$ (how fast, in degrees per hour — fixed by astronomy for all time), and a phase $\phi$ (where its peaks fall at your port — measured locally).

The speeds are universal. The lunar semidiurnal M2 always runs at about 28.98° per hour, completing a cycle every 12.42 hours. The solar S2 runs at exactly 30° per hour. The amplitudes and phases are local fingerprints: Liverpool and Hamburg share the same speeds but need completely different $A$ and $\phi$ values. A machine “set up for a port” is simply a machine whose crank pins encode that port’s fingerprints — see How Operators Set Up a Prediction Run for the spanner work.

My animated sketch at the top of this article shows the two biggest constituents, M2 and S2, adding live: when they align you get spring tides, when they oppose you get neaps. The bar chart in Why 37? The Harmonic Constituents shows the relative sizes of the big four.

Kelvin’s leap (1867–1872)

Lord Kelvin’s contribution was not the formula — Fourier-style thinking was in the air — but the operational programme: analyse a year of hourly gauge readings to extract the local constants, then build a machine that re-sums them for any future date. His 1872 British Association report proposed ten constituents as enough for practical purposes, and his first machine (built by Légé) proved it by reproducing known tides. The full mechanical story is in How Kelvin’s First Predictor Actually Worked.

Darwin’s analysis engine (1883)

George Darwin, Kelvin’s collaborator, turned analysis from an art into a procedure. His method of “harmonic analysis” used cleverly chosen combinations of hourly readings to isolate each constituent while cancelling the others — essentially a hand-computed Fourier transform before the term existed. With Darwin’s schedules, a skilled computer could reduce a year of observations in weeks. Ports with Darwin-analysed constants — India first among them — got dramatically better predictions, which is why the India Office funded its own machine.

The shallow-water problem

Real harbours are not ideal oceans. Friction and funnelling spawn overtides — waves at double or triple speed generated by the estuary itself, not the sky. The M4 overtide (M2’s faster child) can tilt high waters visibly in places like Southampton. Machine builders answered by adding cranks for these locally-generated waves; the American 37-component machine carried a whole family of them. If you want the cast list, Why 37? names every one.

Doodson’s rigorous overhaul (1920s)

Arthur Thomas Doodson at Bidston Observatory found Darwin’s shortcuts leaking accuracy and rebuilt the entire edifice. His 1921 analysis of Liverpool tides used the full least-squares machinery, longer data spans, and 60+ constituents where earlier workers used a dozen. The famous Doodson numbers — the six-digit codes still printed in tide software — come from this era. Doodson’s machines at Bidston embodied the new rigour; I profile them in The Bidston Observatory Machines.

From constants to paper: the prediction pipeline

The complete workflow, unchanged for ninety years, ran like this:

1. Observe. A float gauge records a year (ideally several) of hourly heights — see Tide Staff vs. Gauge vs. Predictor for the instruments.
2. Analyse. By Darwin’s schedules or Doodson’s least squares, extract $A$ and $\phi$ per constituent.
3. Set up. Adjust every crank pin on the machine to the port’s constants (a day’s work).
4. Run. Crank (or motor) through the desired year; the pen draws hourly heights on a graduated roll.
5. Read off. Clerks tabulate high and low waters from the roll into the printed tide tables — the decoding skill is covered in Reading an Old Prediction Roll.

Worked miniature: one hour, three waves

Formulas stay abstract until you push numbers through one. Take a fictitious port at one instant, mean level 2.00 m, with just three constituents:

Predicted height: 2.00 + 1.50 + 0.433 + 0.00 = 3.93 m. That is the entire machine in miniature — each crank contributes one row, the wire adds the column, the pen writes the total. Real ports use thirty such rows (see Why 37?); the arithmetic is identical, only longer. Or skip the arithmetic and move the sliders yourself in the harmonic tide synthesizer.

Objections and answers

How many constituents are really needed? Ten gives a recognisable tide; 20–40 gives harbour-grade accuracy; modern software uses 100+. The machines topped out at 37 moving cranks (America’s No. 2) because each crank cost brass, weight, and friction.

Why are speeds known but amplitudes measured? Speeds come from celestial mechanics — the moon’s orbit is the same everywhere. Amplitudes come from ocean basins, shelves, and estuaries resonating differently, so only local observation reveals them.

Did the machines “compute” like a computer? They were analogue computers: continuous shafts and wires instead of digits. No rounding error, but plenty of backlash and friction — operators learned each machine’s personality.

If you remember one thing

Remember the miniature above: a tide is a column of numbers being added, and for ninety years the adding was done by wire and pulley. Everything else — the ten cranks of 1872, the thirty-seven of 1912, Doodson’s sixty-wave analyses — is that idea scaled up. For the machines’ full life story, read the history pillar; to meet the waves by name, read Why 37?.