tidal movements modelling

An Investigation into Mathematical Modelling

Developing a function describing Tidal Movements in

2014 in Aberdeen, Scotland

Matylda Skupnik, Mathematics SL7.06.2015

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Introduction

The tide is caused by the sun and the moon gravitational effect on the earth, which leads to the cyclical movements of the seas. Also, the Earth rotation plays a significant role in sea level changes. Though tidal effects act upon the entire Earth surface, they become perceptible to the eye only on water. Tidal end result is said to be influenced by distinct factors, which are tidal constituents. In other words, these are the combination of powers determining tidal changes. Primary constituents refer to Earth's rotation, moon and sun alignment along with Moon's elevation over Earth's equator.The usual course of tidal changes involves four basic stages.Firstly, in a couple of hours the level of seas climbs to achieve the alleged intertidal zone.Secondly, for a limited time only, the top level, specifically high tide is reached.Thirdly, ebb tide follows - the seas level decline, also during several hours.Finally, water enters the inferior level, thus low tide occurs. The tidal force, responsible for the tides, includes force transition, resulting in uneven distribution of gravity force influence. Therefore, the attraction on the oceans located nearest the moon would display the greatest effect. Since then, the farest oceans are less attracted.

This topic got my interest since I enjoy sailing and sometime ago while I was on a yacht docked in a port, I observed tidal changes on the water surface.I wanted to know how frequently tides come, what is the pattern of the moon influence and by what kind of a function this proccess can be described with.In order to answer numerous questions emerging to this topic, I consulted the British Oceanographic Data Center, for the annual tidal records in Aberdeen, Scotland, from which I could form graphs.

This natural phenomena is cyclic, so it surely can be described by a mathematical function, so that for each x one and only y can be found. I supposed this would be a trigonometric function, since tides are referred to as a ‘periodic phenomena’, which means a physical proccess exhibiting a periodic behaviour.Consider this, I knew I would need to develop a model in order to investigate the character of function needed. The computed mathematical models of high and low tides comprise predictions tables, published in order to inform all those interested about the foreseen changes. Adequately, data gathered from the British Oceanographic Data Center, concerning the tidal records in Aberdeen would be my raw data. Water altitude was recorded once per 15 minutes each day throughout the year, providing a total of 35040 recordings.

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Firstly, it was essential to investigate what type of a trigonometric function this would be - sine or cosine? In order to answer this question, I collected the records from the first three days and constructed a graph out of them.

Graph 1 - the changes in the level of water from 1.01.2014 00:00:00 to 3.01.2014 23:45:00

This curve resembles a sine function, therefore at this point it is highly probable that the function I aim to obtain would be described by a formula

y=asin(b(x−c))+d ,where a≠0 , b>0

At this point I would like to demonstrate the step-by-step method of finding out the function formula.What needs to be done in the first place, is determining the period of the function.A point was taken (A) and the time (k) needed for another point (B) to be found in the identical position in terms of water height (namely, the interval ) was measured.

3 :45 :00+k→16 : 45 :00k=13hoursperiod=13

now,

period=2 πb

13=¿2πb

b=2π13

b=0.4833219 .. .

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b≈0.483

Knowing what the b parameter was equal to, I can proceed to the amplitude of the sine function. The amplitude of a=sinx represents half the distance between the maximum and minimum values of the function.

a=max−min2

a=4.753−0.7132

a=2.02

The d parameter, known as the principal axis is midway between the maximum and minimum, so

d=max+min2

d= 4.753+0.7132

d=2.73

To obtain the full sine function formula, parameter c responsible for the horizontal translation of the graph is essential.I decided to calculate it from the yet incomplete formula.But, to do this,a height of water at a certain point must have been inserted to the formula, time acting as x and water level acting as y.

f (00 :00 :15)=4.409

for the sake of the calculations, the time units must be converted into decimal units, therefore

f (0.25)=4.409

now we substitute the already calculated parameters into the formula

4.409=2.202(sin 0.483 (0.25−c))+2.73(4.409−2.73) :2.202=sin 0.483(0.25−c)

1.679 :2.202: sin 0.483=0.25−c−c=0.762:sin 0.483−0.25=1.3917 ...

c ≈−1.39

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The general sine function calculated from data corresponding to the first 3 days of year 2014 is equal to :

y=2.202(sin 0.483(x+1.39))+2.73

From that calculations I came to a conclusion, that the period of this function is affected by two phenomenona, namely the high tide and the low tide, followed by another high tide.However, to be absolutely sure, whether the period of the function is constant, I undertook calculating it once again, this time for the first week of January 2014.


As it can be observed, two points (A,B) were considered as the beginning and the end of a period and the time (k) between them was measured.

A→3:15 :00B→16 :15 :00

3 :15 :00+k→16 :15 :00k=13hoursperiod=13

Proven this, one can be sure, that the period of this function is constant, thus this occurrence stands for a sinusoidal function.When the b parameter was defined, I entered the function formula to Geogebra. My aim was to examine the graph of the function ploted by the formula I calculated from the other graph, therefore ensure about the correctness of the formula.

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Graph 3 - function y=2(sin(0.483(x+1.39))+2.73 ploted in Geogebrapoints A and B added to demonstrate the period equal to 13,

for the sake of calculations sin 0.483 was approximated to 0.48

It can be seen, that the formula is adequate to the real-life situation, since the outlined graph corresponds to the high and low tides levels, besides, it must be acknowledged that it has its limitations, no negative value will ever be reached by this function, since even during an extremely low tide, water cannot go below its own surface level, that is 0 metres. The domain of this function would be equal to x ϵ Z❑+¿ ¿, since we must recognise time as x-value and time cannot be negative.

Now I would like to determine the frequency of high tides and low tides. Again, I am going to demonstrate my calculations, basing on a graph of the first week of January 2014.

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Graph 4 - the changes in the level of water from 1.01.2014 00:00:00 to 7.01.2014 23:45:00, with the respect to low and high tide stages

A→6 :15: 001.01 .2014B→13 :00 :001.01 .2014C→19 :30: 001.01 .2014D→2 :00 :002.01 .2014E→8: 00 :002.01.2014F→14 :45 : 002.01 .2014

k=13 :00 :00−6 :15: 00=6hours 45minutesk❑1=19 :30 :00−13 :00 :00=6hours30minutesk❑2=2:00 :00−19 :30 :00=6hours30minutes

k❑3=2:00 : 00−8 :00 :00=6hoursk❑4=8 :00 :00−14 : 45: 00=6hours 45minutes

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yet, the units must be changed from the sexagesimal system to decimal system, so that the the calculations can be carried out and the mean time needed for the

water to go from high tide to low tide (adequatedly, the other way round, too) can be rated

6.75+6.5+6.5+6.+6.756

=32.55

=6.5

Conclusively, the mean time neccessary to go from the highest to the lowest point equals 6 hours 30 minutes.From the previous calculations we already know the value of the period, therefore it can be stated that the period equals twice the interzone between a high and low tide.This relationship gives evidence dor the diurnality of tides in Aberdeen, by way of explanation, two high tides and two low tides follow each day (taking into consideration the margin of error, concerning ≈2 hours of the next day.

Since, we already accomplished the first two bulletpoints of my investigation aim, the last task to be resolved is the influence of the moon on the periodicity of the monthly water movements. At this point, I would like to demonstrate charts outlining the water peaks variance and point out the pattern caused by the impact of the moon.Let us consider this six-month period.


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Surprisingly, we observe a specific type of pattern throughout the months.This led me to a conclusion, that moon as a primary source influences the changes directly. When the bulges (the relatively highest peaks and lowest points) on the graph are taken into consideration, it can be seen, that they last approximately 7 days each month,at times this may be also interrupted because the month ends, and so does the graph. However, from astronomical knowledge it can be deduced,that these seven days periods are also an estimated time of a moon singular phase. Thus, comparing the sinusoidal function and the seeming movement of the moon on the sky, the highest tides in terms of a distinct moon phase can be stated.

For instance, using internet I determined the moon phases for each month presented on the charts above. Respectively, in UK there was Full Moon phase from 5th to 12th January, explaining why the tidal amplitude was the highest throughout the month,then in February the same phase lasted from 14th to 21st.How is it mathematically possible?

Picture 1 - the four moon phases seen, moving on an arc scheme

The moon moves on an arc scheme, therefore each phase referrs to as 14

of this

arc.Since we already know, that the sinusoidal function presented on graph will reach the greatest values in the full moon phase, it is worth consideration, why is it so.

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Picture 1 - the four moon phases, considering their influence on tides on the Northern Hemisphere

We observe a final settlement on the second and eighth chart. Hence, the first and the third quarters will result in the equal amplitude of the sinusoidal function concerning the tidal changes. Moreover, since the moon follows an arc scheme, these relationships, concerning ‘bulges’ on the function graph would follow endlessly, therefore suggesting the periodicity not only of the tides itsfelf, but also of the pattern affecting the high and low tides values.

To conclude, tidal variations in the depth of water can be described by a trigonometric function. However, there were some serious limitations throughout the experiment. Yet by use of simple mathematics a model could be estimated, this must be treated critically, since other relevant to the outcome of calculations factors were omited (solar influence, earth movements).Yet, developing a model of a function turned out to be incomparably more complicated than reading and the analysis of a ready model.I believe I have reached my goal by presenting a real-life situation by use of mathematics, especially since I had the possibility to proccess an authentic data.

Bibliography :

http://www.calendar-uk.co.uk/lunar-calendar/http://en.wikipedia.org/wiki/Lunar_phasehttp://noc.ac.uk/http://oceanservice.noaa.gov/education/kits/tides/lessons/tides_tutorial.pdf

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