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Modelling tides: What is the effect of a full moon?

Let’s have a look at the effect of the moon on the tides in Phuket.  The Phuket tide table above shows the height of the tide (meters) on given days in March, with the hours along the top.  So if we choose March 1st (full moon) we get the following graph:

Phuket tide at full moon:

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If I use the standard sine regression on Desmos I get the following:

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This doesn’t look a very useful graph – but the R squared value is very close to one – so what’s gone wrong?  Well, Desmos has done what we asked it to do – found a sine curve that goes through the points, it’s just that it’s chosen a b value of close to 120 – meaning that the curve has a very small period.  So to prevent Desmos doing this, we need to fix the period first.   If we are in radians the we use the formula period = 2pi / b.  Therefore looking at the original graph we can see that this period is around 12.  Therefore we have:

period = 2pi/b

12 = 2pi/b

b = 2pi/12 or pi/6.

Plotting this new graph gives something that looks a lot nicer:

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Phuket tide at new moon:

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Analysis:

Both graphs show a very close fit to the original data – though both under-value the tide at 2300.  We can see that the full moon has indeed had an effect on the amplitude of the sine curves – with the amplitude of 1.21m for the full moon and only 1.03m for the new moon.

Further study:

We could then see if this relationship holds throughout the year – is there a general formula to explain the moons effect on the amplitude?  We could also see how we have to modify the sine wave to capture the tidal height over an entire week or month.  Can we capture it with a single equation (perhaps a damped sine wave?) or is it only possible as a piecewise function?  We could also use some calculus to find the maximum and minimum points.

There is a very nice pdf which goes into more detail on the maths behind modeling tides here.  There we go – a nice simple investigation which can be expanded in a number of directions.

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

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I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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