What are tidal harmonic constituents?

A tide is the sum of many sinusoidal components, each driven by a specific astronomical motion. M2 is the principal lunar semidiurnal, S2 the principal solar semidiurnal, K1 and O1 the main diurnal terms. Their amplitudes and phases are published per station.

How accurate is a four-constituent estimate?

M2, S2, K1, and O1 capture the bulk of the tidal signal at most ports, so the shape and timing are usually close. Full predictions use dozens of constituents, so treat this as an approximation, not a replacement for an official tide table.

What speeds does the tool use for each constituent?

It uses the standard angular speeds in degrees per hour: M2 = 28.984, S2 = 30.000, K1 = 15.041, O1 = 13.943. These are fixed astronomical rates, so you only supply amplitude and phase.

What is the phase lag g?

The phase lag, in degrees, is how far each constituent is delayed relative to its astronomical forcing at a reference meridian. It is published alongside the amplitude in a station's harmonic constants and sets when each component peaks.

Why might my predicted high water be off by an hour?

This simplified model treats the start of the day as the common phase reference and omits nodal corrections and minor constituents. Errors of tens of minutes in timing are normal. Always defer to official predictions for navigation.

What is the Tidal Height Harmonic Estimator?

Predict tidal height over 24 hours by summing the four main harmonic constituents M2, S2, K1, and O1 from their amplitudes and phases. Built for mariners, coastal engineers, and divers when official tide tables are unavailable. It runs free in your browser on Gera Tools, with nothing uploaded.

Tidal Height Harmonic Estimator

Name: Tidal Height Harmonic Estimator
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

When official tide predictions are out of reach, you can still reconstruct a useful tide curve from a station’s published harmonic constants. This estimator sums the four dominant constituents — M2, S2, K1, and O1 — using their fixed astronomical speeds and your supplied amplitudes and phases.

How it works

Each constituent contributes a cosine wave, and the total height at hour t is the datum plus the sum of those waves:

h(t) = Z0 + Σ Hᵢ × cos( ωᵢ × t − gᵢ )

ω(M2) = 28.984°/h   ω(S2) = 30.000°/h
ω(K1) = 15.041°/h   ω(O1) = 13.943°/h

Hᵢ is the amplitude in metres, gᵢ the phase lag in degrees, and ωᵢ × t is the constituent’s astronomical argument advanced from the start of the day. Degrees are converted to radians internally before the cosine is taken.

What each constituent represents

The four constituents used here are not arbitrary: they are the four largest tidal components at most of the world’s coastal stations, and together they determine whether a place has a predominantly semidiurnal (two highs and two lows per day), diurnal (one high and one low), or mixed tidal regime.

M2 — the principal lunar semidiurnal constituent — is driven by the gravitational pull of the Moon on a 12.42-hour cycle. It is the largest constituent at the vast majority of Atlantic and Pacific coast ports. Its amplitude alone gives you a rough idea of the tidal range at a location: an M2 amplitude of 2 metres means a spring tidal range of around 4 metres when M2 and S2 reinforce at new and full moon.

S2 — the principal solar semidiurnal constituent — follows the same 12-hour rhythm as the solar day and interacts with M2 to produce the spring/neap cycle. When M2 and S2 are in phase (at new and full moon), they add together for spring tides. When they are opposed (at the quarters), they partially cancel for neap tides. The ratio of S2 amplitude to M2 amplitude controls how pronounced this variation is.

K1 and O1 — the two main diurnal constituents — are on roughly 24-hour cycles and are responsible for the once-daily inequality in semidiurnal tides and for the dominant signal at diurnal ports in places like the Gulf of Mexico, parts of the Pacific, and Southeast Asia. At a port where K1 and O1 are large relative to M2 and S2, the tide may alternate between large and small highs (or even produce only one high and one low per day) depending on the phase relationship.

Example and worked illustration

A semidiurnal port with the following harmonic constants:

Z0 = 0.0 m (mean sea level above chart datum)
M2: H = 1.8 m, g = 200°
S2: H = 0.6 m, g = 240°
K1: H = 0.15 m, g = 60°
O1: H = 0.10 m, g = 30°

This port would show two distinct high waters and two low waters per day, with a maximum range on spring tides of roughly 4.8 m when M2 and S2 add constructively. The small K1 and O1 amplitudes produce only a modest diurnal inequality — the two highs will be close in height but not identical.

Where to find harmonic constants

Published harmonic constants are available from national hydrographic offices and oceanographic data centres. NOAA publishes them for US stations; the UK Hydrographic Office maintains them for British and many international stations; the IOC/PSMSL global tide gauge network publishes data for many academic stations. Constants are typically listed by station name and include the amplitude in metres or centimetres and the phase lag in degrees referred to the Greenwich meridian.

Notes on accuracy and limitations

Because this model omits nodal factor corrections and the many minor constituents that a full tidal analysis includes, expect height errors of a few tens of centimetres and timing errors of tens of minutes. The nodal cycle — an 18.6-year variation in the Moon’s orbital plane — modulates the amplitudes of M2 and other constituents by a few percent, which this simplified model does not account for. Use it for planning intuition, never for navigation where an official prediction exists.