Tide Mechanics

1A tide is a sum of waves

Each constituent is a pure cosine with its own period and size. Add them and you get the tide. Turn them on and off below. The amplitudes start at Boston-like values. These six are the largest at a port like Boston, out of a much bigger set: section 2 lists all 37 that NOAA publishes.

Three days. Each lane is one constituent on a shared scale; the bottom lane is their sum. Raise K1 and O1 to about 0.5 m and successive high waters become unequal: that is diurnal inequality.

Thirty days of the same sum. M2 (12.42 h) and S2 (12.00 h) drift in and out of step every 14.77 days. In step they add: spring tides. Out of step they cancel: neap tides. The dashed line is the M2+S2 envelope. N2 adds a monthly swing on top.

2What a constituent is

A constituent is one astronomical rhythm. Its speed in degrees per hour comes from the motions of the Moon and Sun, so it is the same at every port. The Doodson number encodes that speed as a sum of six basic astronomical frequencies.

The tidal spectrum at a Boston-like port, all 37 constituents NOAA publishes. Constituents cluster into species: diurnal near 15°/h (once a day), semidiurnal near 30°/h (twice a day), and shallow-water overtides at 3, 4, 6 and 8 times a day. The ticks under the axis mark every constituent; the number over each band counts them. The lower panels zoom into the two crowded clusters. Most lines are only millimetres tall: here M2, S2, N2, K2, K1, O1, P1 and Q1 carry most of the tide's variance. The nine constituents of the table below keep their illustrative amplitudes; the other 28 use NOAA's published amplitudes for Boston (station 8443970).

How many constituents?

There is no fixed number. Each source publishes the set its own analysis resolved:

• NOAA CO-OPS: the same 37 at every station, even where some round to zero (Boston's Mm, MSf and Mf are 0.000 m).
• JMA: 60 at the 70 stations JMA runs; only M2, S2, K1 and O1 at the other stations of its tide tables.
• Kartverket and LINZ: a list per station. Kartverket gives Bergen 55, Stavanger 52 and Trondheim 57; seven LINZ ports we checked have 16 to 30.
• OTC's own fit of GESLA records tries up to 45: NOAA's 37 plus 8 more. Record length sets how many fit (section 6).
• The prediction engine's constituent table names 176, so it can evaluate whatever a source publishes.

In practice a few carry most of the tide: M2, S2, N2, K2, K1, O1, P1 and Q1. The shallow-water overtides (M4, MS4, M6 and the rest) are small at the open coast but grow in estuaries and shallow harbours, where they make the tide rise and fall at different rates.

3The prediction formula

Every tide predictor evaluates this sum. Σ runs over every constituent a station publishes: 37 for a NOAA station, up to 60 for JMA. The six in the figures carry most of the tide. Select a term to see it on the wave.

• Z₀ Mean water level above the chart datum. One number per station. OTC
• H Amplitude: half the crest-to-trough height of this constituent. OTC
• g Phase lag: how many degrees the local crest trails the Moon or Sun's pull. OTC
• ω Speed in °/h. Identical worldwide. astronomy
• V₀ Where the astronomical driver sits at t = 0. Depends only on the date. SDK
• f and u Nodal corrections for amplitude and phase over the 18.6-year lunar node cycle. SDK
• t Hours since the time origin, in UTC. Σ adds every constituent. input

One M2 wave. The dashed curve is the equilibrium tide, the astronomical driver set by V₀ + u. The real crest arrives g degrees later; for M2 every 28.98° of lag is one hour.

The nodal factor f across one lunar node cycle, centred on today. M2 moves about ±3.7%. K1 moves about ±11.5%, and the two run in opposite directions. The SDK computes f and u from the date, so OTC never stores them.

4Phase lag g and the time-base trap

g is measured against a clock. Move the clock by Δt and every phase moves by its own speed times Δt. One hour is 28.98° for M2 and 15.04° for K1. A wrong time zone therefore leaves a fingerprint: phase errors that scale with each constituent's speed.

Clock offset. Ink needle: true Greenwich g. Magenta needle: what a fit reports if the record's clock runs Δt ahead of UTC. The shift is ω·Δt, mod 360. At 12 h, S2 lands exactly where it started while M2 is off by 12°, so S2 alone cannot catch this error.

Local meridian. Some agencies publish κ, the lag relative to the station's own meridian. Converting to Greenwich uses g = κ − p·λ, where p is the species: 0 long-period, 1 diurnal, 2 semidiurnal (λ east-positive). Here M2 and S2 shift by the same 2λ. A clock error shifts them by different amounts, and that difference tells the two traps apart.

5Harmonic analysis: getting H and g from a record

Analysis runs the formula backwards. Given hourly water levels, a least-squares fit finds the Z₀, H and g that best explain them. Below is a synthetic record built from known constants plus noise; the fit tries to recover them. Real analyses fit dozens of constituents, not the handful in this toy: OTC's fit tries up to 45.

Grey dots: observed hourly values. Magenta line: the fitted sum. The lower strip shows the residual, observed minus fit. The fit includes only constituents the record length can separate (section 6). Below about 15 days, S2 is left out and its energy leaks into M2.

This toy omits nodal corrections, so its true values stand for f·H and g − (V₀ + u). A real fit applies f and u for each observation time.

6Record length decides what you can separate

Two constituents with close speeds look like one wave until they drift a full cycle apart. The Rayleigh criterion sets the minimum record: T ≥ 360° / |ω₁ − ω₂|. The figures below compute T from the speeds.

Record length is what limits how many constituents a fit can hold. Applied to OTC's 45 candidates, the rule keeps 29 for a 30-day record, 43 for one year and all 45 for 19 years. A noise test can then drop more.

7What it applies to

8From source to SDK

The OTC pipeline, top to bottom. Every record survives to the end; the audit adds labels and never deletes.

1. in

   Sources

   Water-level records and official constants from 16 sources, GESLA among them.

   GESLANOAAKartverketLINZ+ 12 more
2. Δt

   Per-record time-base audit

   Checks each record's clock and meridian convention against section 4's fingerprints, then annotates it. No record is dropped.

   verifiedcorrectedunverifieddisputed
3. fit

   Harmonic fit

   Least squares over the hours present. Record length decides how many constituents enter (section 6).

4. ✓

   Canary check

   Compares fitted H and g with official constants at overlapping stations to catch systematic errors.

5. db

   Combined dataset

   Z₀, and H and g per constituent, per station, with the audit labels attached.

6. h(t)

   SDKs

   Python, Ruby, TypeScript and C read the dataset, compute V₀, f and u for the requested time, and sum the formula.

   PythonRubyTypeScriptC