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What goes UP must come DOWN The mathematics of

tide prediction

Rex Boggs, Glenmore State High School Kiddy

Bolger, The Rockhampton Grammar School

Tides are defined as The periodic rise and fall

of water resulting from gravitational

interactions between the Sun, Moon, and Earth.

A model for tide height often used in the

classroom is a sine function of the form

But actual tides are much more complex than a

simple sine function.

The modern method used to generate an accurate

mathematical model for tides is called harmonic

analysis. The model generated by harmonic

analysis is a sum of cosine terms, plus a

constant.

The tide function for a given port typically

has upwards of twenty cosine terms. Each cosine

term is called a harmonic constituent. The

constant term Z0 is called the chart datum and is

the base from which tides are measured.

Until 1965, tide heights were calculated by

machines similar to this one designed by Sir

William Thomson in 1873.

The photo shows one such machine for calculating

tides. In the mid-1960s, these machines, were

replaced by computers.

Consider the tides at Mooloolaba on January 1,

2002. Here are the data from the Official Tide

Tables and Boating Safety Guide, 2002

We wish to develop a mathematical model that

accurately predicts these tide heights.

Our first tidal model will use only one harmonic

constituent, M2, which is the gravitational

influence of the Moon. This is the constituent

with the greatest influence on the tides.

From the Australian National Tide Tables for

2002, we obtain the following information for

Mooloolaba on 1 Jan 2002

Chart datum for Mooloolaba is 0.94 metres.

Substituting into the tide function gives us our

first model for the tides at Mooloolaba on 1

January 2002

Note that the formulas for tide prediction are

in degrees (and not radians)

Here is the graph of our model over a twenty-four

hour period.

This model predicts a low tide of 0.336 metres at

319 a.m., while the tide tables show a low tide

of 0.06 m at 243 a.m. Our model needs a bit of

refinement!

The next most important constituent is S2, the

gravitational influence of the Sun. At new moon

and full moon, the Sun and the Moon combine to

give larger tides.

In the first quarter and third quarter, the Sun

and the Moon are pulling at right angles to each

other, resulting in smaller tides.

This diagram nicely shows the effect on the tide

height of the different phases of the moon over a

thirty-day period.

From the Australian National Tide Tables for

2002, we obtain the following information for

Mooloolaba on 1 Jan 2002

This results in a more accurate model for the

tide height.

Here is the 24 hour graph

This model predicts a low tide of 0.219 metres

(compared to 0.06) at 309 a.m. (compared to 248

a.m.).

This is a better model, but still not

sufficiently accurate. We havent yet

accounted for the fact that successive high tides

usually are not equal.

In Mooloolaba on 1 January 2002, the high tides

differ by 0.72 metre, and the low tides differ by

0.23 metres. Such differences are a common

occurrence at mid-latitudes.

The high tides differ by 0.72 m

This is largely caused by the declination of the

moon (i.e. the angle of the moon above or below

the equator).

A person at A would experience a relatively high

tide. Slightly more than twelve hours later, the

person would be at B (due to the rotation of the

Earth) and the tide would be smaller.

This is called a diurnal effect, since its period

is approximately one day. There are two harmonic

constituents needed to account for this, called

K1 and O1.

Including these constituents gives us the

following model for tide height

Here is the graph of this function

This model predicts a low tide of 0.05 metres

(compared to 0.06) at 241 a.m. (compared to 248

a.m.).

The model also predicts a high tide of 2.02

metres (compared to 2.03) at 930 a.m. (compared

to 929 a.m.).

The tide model, using only four harmonic

constituents, has an error of less than 1 for

both time and height !

For each port in Australia, the tide models

constructed at the National Tidal Facility at

Flinders University typically include over twenty

constituents. It is interesting to learn that

tide models based solely on the four constituents

M2, S2, K1 and O1 will have a maximum error in

both height and time of no more than 5 (and

often much less than that).

The presenters wish to thank Mike Davis from

the National Tidal Facility at Flinders

University and Captain Arthur Diack from the

Maritime Division of Queensland Transport for

their technical assistance in the development of

this presentation.

Appendix A A more general model for tide

prediction

The tide function given earlier was a

simplified version of the more general model

so each constituent has the form

The amplitude of each constituent is actually the

product of two factors, H and F. H is the base

amplitude. Determining this value requires

observations to be taken over many years.

The amplitude of each constituent is actually the

product of two factors, H and F. F is a slowly

varying function, having a period of about

nineteen years. This function is based on the

precession of the plane of the Moons orbit,

which has a period of about 18.6 years.,

The phase angle has two components, g and A,

which are added together. g depends only on the

site at which the observations are made. A is

the phase of the constituent at Greenwich when t

0 (i.e. at midnight).

- In summary
- The values of Z0, H and g depend only upon the

port. Values for each Standard Port are given

in Table II of the National Tide Tables. - The values of A and F for the four main

constituents depend only upon time. Values for

each day of the year are given in Table III of

the National Tide Tables.

As an example, here are the calculations for the

M2 constituent at Mooloolaba on 1 January 2002.

From the Australian National Tide Tables H for

Mooloolaba is 0.539 F on 1 January 2002 was

1.12 Multiplying these factors gives 0.60368,

which was the amplitude given previously.

From the Australian National Tide Tables g for

Mooloolaba is 234.9 A on 1 January 2002 was

041 Adding these factors gives 275.9, which was

the phase angle given previously for the M2

constituent.

Appendix B Further information about tides

Now a bit more about tides. This diagram show

why successive tides at the equator tend to be a

similar height.

This diagram show why there tends to be only one

high tide per day at higher latitudes. At B, the

tide is negligible.

Not all tides are semi-diurnal. Here are 30-day

tide charts from other parts of the world.

These diagrams show why typically there are two

high tides per day. One is caused by the

gravitational attraction of the moon, whilst the

other is due to centrifugal force caused by the

rotation of the Earth/Moon combination about

their centre of gravity.