Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Can a sum of sine waves predict the tides?

AA SLAA HLAI SLAI HL Solid Also in: Modelling

Research question

How accurately can tide heights at [a harbour] be predicted by one sinusoid with a period of about 12.42 hours, and how much does adding a second component with a period of about 12 hours improve the model?

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Why it makes a good exploration

Tides are caused by the Moon and Sun, whose periods differ slightly, producing spring and neap tides. Building a model from sine waves and testing it against six-minute measurements is modelling at its most satisfying.

The mathematics you'll need

Sinusoidal functions and parameters
Sum of sinusoids and beats
Fitting with technology; residuals
Predicting future tides and measuring error

Modelling the data, step by step

This idea usually needs a sinusoidal model. See it worked step by step, with a criterion tip at every step: Sinusoidal y = a sin(b(x − c)) + d · Choosing and comparing models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Download water levels from NOAA Tides & Currents for a station (or your national tide authority).

NOAA Tides & Currents — Water levels every 6 minutes at hundreds of stations, CSV download.

A possible outline

Explain the lunar (M2) and solar (S2) tides.
Fit one sinusoid to a week of data.
Add the second component and compare.
Predict the next week.
Reflect on weather surges and local geography.

Pitfalls that cost marks

Choosing the period by eye without justification.
Fitting too short a window.
Ignoring datum definitions.

Showing personal engagement

Choose a harbour you have visited.
Predict a spring tide and check.
Explain the beat period of about 14.8 days.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Sinusoidal functions and parameters; Sum of sinusoids and beats
AA HL	Good fit	Sinusoidal functions and parameters; Sum of sinusoids and beats
AI SL	Good fit	Sinusoidal functions and parameters; Sum of sinusoids and beats
AI HL	Good fit	Sinusoidal functions and parameters; Sum of sinusoids and beats

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: choosing the period by eye without justification; and fitting too short a window. Say how each one affects your answer.

Use of mathematics (E)

SL: sinusoidal functions and parameters; sum of sinusoids and beats; fitting with technology; residuals; predicting future tides and measuring error, each explained in your own words and carried out correctly.

How environment explorations reach the top bands in general: Environment & climate IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Add the diurnal components and compare with the official tide prediction.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL environment exploration adds.

See a complete IA, marked

When can we launch? Modelling the tide at our sailing club's slipway (AA SL) explores a question like this one, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Sinusoidal functions and parameters, sum of sinusoids and beats, fitting with technology; residuals and predicting future tides and measuring error.

Where can I get data for this IA?

Download water levels from NOAA Tides & Currents for a station (or your national tide authority). Checked sources and tools: NOAA Tides & Currents.

Turn this idea into your IA

The planner opens with “Can a sum of sine waves predict the tides?” filled in: the question, the maths and the data sources.

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