What are Harmonic Constituents?

Tides are created by the gravitational forces of the Moon and Sun, acting upon the waters of the Earth. Those gravitational forces change as the relative positions of the Earth, Sun, and Moon change. We can visibly see these position changes in the rise and set of the Sun and Moon, the changing phases of the Moon, and the changing seasons of the years. Each of these changes is cyclical, repeating over time; and each change also has a measurable effect on the tides we experience on the ocean’s coast.

There are hundreds of periodic motions of the Earth, Sun, and Moon that are identified by astronomy. Each of these motions or “constituents” in a set of harmonic constants is a mathematical value describing the effect that cyclical motion of the Earth, Sun, Moon system has on the tides. There are 37 which normally have the greatest effect on tides and are used as the tidal harmonic constituents to predict tidal conditions for a location.

A few examples:
  • M2 – The largest lunar constituent – is related to the direct gravitational effect of the Moon on the tides. The Earth rotates on its axis every 24-hours, but the Moon is orbiting in the same direction as the Earth’s rotation. It takes a location on the Earth an additional 50 minutes to “catch up” to the Moon. This results in a tidal signal (M2) which has 2 peaks every 24-hours and 50 minutes.
  • S2 – The largest solar constituent – is related to the direct gravitational effect of the Sun on the tides. The Earth rotates on its axis every 24-hours. This results in a tidal signal (S2) which has 2 peaks every 24-hours.
  • SA – Solar Annual constituent – is related to the changing positions of the Earth and Sun on an annual basis, every 365.25 days.
Each constituent can be represented as a Cosine Curve, with the values providing the amplitude and phase difference for one of the 37 periodic motions. Tide predictions are a calculation, summing the effects of the 37 Cosine Curves for a set of harmonic constants; resulting in the complex curve of the tides.

How are Harmonic Constants generated?

Harmonic constants can only be calculated through the analysis of tidal data collected at a location. A minimum of 30 days of data is used in order to observe the majority of the lunar and solar cycles. A year of data is necessary to directly observe all of the 37 tidal harmonic constituents. Stations which have longer series of data will typically use harmonic constants based on multiple years.

This page will not try to describe the mathematical process for this analysis. Instead, we recommend that you review the information from the following publications, which go into that process in much more detail than can be provided here.

(PDF) NOAA Special Publication NOS CO-OPS 3 - Tidal Analysis and Predictions
(PDF) Special Publication No. 98: Manual of Harmonic Analysis and Prediction of Tides

How are Harmonic Constants used to calculate tide predictions?

Each harmonic constituent provides the mathematical values which describe a specific cosine curve. Tide predictions are a calculation, adding the effects of each of the 37 cosine curves, to produce the complex curve that is the predicted tides.

The formula for that calculation is: h = Ho + Sum{ƒH cos[at + (Vo+u) - K]}

The terms of the equation are defined as:
h = height of tide at any time t.
Ho = mean height of water level above datum used for prediction.
H = mean amplitude of any constituent A.
ƒ = factor for reducing mean amplitude H to year of prediction.
a = speed of constituent A.
t = time reckoned from some initial epoch such as beginning of year of predictions. (Vo+u) = value of equilibrium argument of constituent A when t = 0.
K = epoch of constituent A.

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