Mathematical Modeling

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Mathematical Modeling

After recording tidal heights of Grindstone Island by the hour on 27 December 2003 (AST), the data below was recorded:

Time (AST)

00.00 01.00 02.00 03.00 04.00 05.00 06.00 07.00 08.00 09.00 10.00 11.00

Height (m)

7.5 10.2 11.8 12.0 10.9 8.9 6.3 3.6 1.6 0.9 1.8 4.0

Time (AST)

12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00

Height (m)

6.9 9.7 11.6 12.3 11.6 9.9 7.3 4.5 2.1 0.7 0.8 2.4

When you enter this data in a graphing software – time against height – you create the graph:

From the data above, it can be seen that the maximum height of the tide is 12.3m at 15.00, and the minimum height of the tide is 0.7m at 21.00. There is also an obvious logarithmic pattern to the tidal heights by time, where each period lasts 12 hours (21-9). There are also horizontal and vertical shifts in the data.

To find a function that applies to the data, we will apply the formulae below and discover the different variables (a, b, c, d) and collaborate it into the formulae, where a is the amplitude, b is affected by period, c the horizontal shift, and d the vertical shift.

y=a ∙sin ( b (x−c ) )+d

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However, it is important to remember that in real life, tides are affected by external factors unpredictable in the function formulae above, so the function can only predict (to a degree of accuracy) the tide heights at for that specific date, if without external forces.

Because we have knowledge on vectors, where the maximum height of the wave is 12.3 and the minimum height is 0.7, to discover the amplitude (a) we can apply the formulae below:

a=maximum−minimum2

a=12.3−0.72

a=11.62

a=5.8With the same knowledge on vectors, we can discover the vertical shift (d) of the function.

d=maximum+minimum2

d=12.3+0.72

d=132

d=6.5Because we know that one period in the graph is 12, we can also discover b in the function.

b= 2 πPeriod

b=2 π12

b=π6

We can also discover the horizontal shift (c) by using the interpolation law with our knowledge on the vertical shift (d), using the formulae below

c=x low+(d− y low

yhigh− y low

)∙(xhigh−x low)

To do this, we must discover where d lies on the y-axis margins, as the interpolation law uses the data from the y-axis to discover the x-axis through ratios.

low highTime (AST) 11.00 x at d 12.00Height (m) 4.0 d (6.5) 6.9

With this information, we can apply the interpolation formulae, therefore:

c=11.00+( 6.5−46.9−4 ) ∙ (12.00−11.00 )

c=11+( 2.52.9 ) ∙ (1 )

c=11+2529

c=34429

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With the knowledge of the variables a, b, c, and d, we therefore have the updated function:

y=5.8 ∙ sin( π6 (x−

34429 ))+6.5

When we apply this updated function to the graph, we get this result (function is black line):

Time Height (m) Manual HeightError (%) in manual

height00.00 7.5 6.9 7.7501.00 10.2 9.8 4.3602.00 11.8 11.7 0.6903.00 12.0 12.3 2.3704.00 10.9 11.3 3.6805.00 8.9 9.0 1.4606.00 6.3 6.1 3.4707.00 3.6 3.2 9.8608.00 1.6 1.3 19.9409.00 0.9 0.7 20.5410.00 1.8 1.7 5.5911.00 4.0 4.0 0.7512.00 6.9 6.9 0.2713.00 9.7 9.8 0.5714.00 11.6 11.7 1.0315.00 12.3 12.3 0.1216.00 11.6 11.3 2.58

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17.00 9.9 9.0 8.7918.00 7.3 6.1 16.6919.00 4.5 3.2 27.8920.00 2.1 1.3 39.0121.00 0.7 0.7 2.1622.00 0.8 1.7 112.4323.00 2.4 4.0 65.42

Average Error (%) 14.89

The function fits the data’s vertical shift, horizontal shift, and amplitude, however it is not as

appropriate to the period and horizontal shift. With 34429

,6.5 as the beginning of the function,

it is visible that as the function moves either direction, the error increases, meaning that the fault is that the value of b is too low.

Considering these aspects, I propose this improved function, where the value of b has been increased to make the period of the function more appropriate.

y=5.8 ∙ sin( π6.2 (x−

34429 ))+6.5

This is the graphed illustration of the first modification (m1).

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The previous function remains indicated with the black line, whilst the improve function is indicated with the red line. With his improved function, the percentage of error has decreased 5.2%, (from 14.89% to 9.69%) making the improved function more accurate than the manual.

Time Height (m) Modified 1 (m1) Error (%) in m100.00 7.5 8.1 7.4901.00 10.2 10.6 3.6902.00 11.8 12.1 2.2603.00 12.0 12.2 1.3104.00 10.9 10.8 0.6705.00 8.9 8.4 5.5106.00 6.3 5.5 12.5107.00 3.6 2.9 20.4908.00 1.6 1.1 29.5609.00 0.9 0.7 17.5410.00 1.8 1.8 0.2311.00 4.0 4.0 1.1612.00 6.9 6.9 0.0713.00 9.7 9.7 0.3914.00 11.6 11.6 0.2115.00 12.3 12.3 0.0116.00 11.6 11.5 0.7217.00 9.9 9.5 4.3118.00 7.3 6.7 8.4619.00 4.5 3.8 14.5320.00 2.1 1.7 20.1521.00 0.7 0.7 2.7922.00 0.8 1.2 51.8723.00 2.4 3.0 26.61


Considering the graph above, c, or the horizontal shift, needs to be adjusted so that it matches with the tidal heights against the time more. Through this visual analysis, I increased the value of c so that it would move slightly to the right, proposing the following function

y=5.8 ∙ sin( π6.2 (x−

35029 ))+6.5

Notice that 34429

has been modified into 35029

, adjusting the graph so that it moves to the right.

With this second modification (m2), the following graph is produced, with the green line as the line representing the function:

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Time Height (m) Modified 2 (m2) Error (%) in m200.00 7.5 7.5 0.4201.00 10.2 10.1 0.7702.00 11.8 11.9 0.5503.00 12.0 12.3 2.1704.00 10.9 11.2 2.8205.00 8.9 9.0 0.8106.00 6.3 6.1 2.9307.00 3.6 3.4 6.8108.00 1.6 1.4 13.4309.00 0.9 0.7 22.1410.00 1.8 1.5 18.1311.00 4.0 3.5 12.2612.00 6.9 6.3 8.7313.00 9.7 9.1 5.8214.00 11.6 11.3 2.4815.00 12.3 12.3 0.1716.00 11.6 11.8 1.6717.00 9.9 10.0 0.7818.00 7.3 7.3 0.1619.00 4.5 4.4 2.21

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20.00 2.1 2.0 2.8421.00 0.7 0.8 14.4422.00 0.8 1.0 24.2523.00 2.4 2.6 7.11


With this modification, the average error has decreased 3.28% (from 9.69% to 6.41%), and a total of 8.48% from the original function (from 14.89% to 6.41%). This means that the second modification (m2) of the function is the most accurate than the original and the first modification (m1).

Using a graphical software, the suggested sinusoidal regression is as follows:y=5.7132 ∙ sin (0.5064 x+0.1956 )+6.5867

On the graph below, the blue line represents the sinusoidal regression.

Time Height (m)Sinusoidal Regression

Error (%) in sinusoidal regression

00.00 7.5 7.7 2.6301.00 10.2 10.3 0.7402.00 11.8 11.9 1.0903.00 12.0 12.2 2.0104.00 10.9 11.1 2.14

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05.00 8.9 8.9 0.1706.00 6.3 6.1 3.8207.00 3.6 3.4 6.4908.00 1.6 1.5 7.4009.00 0.9 0.9 2.4210.00 1.8 1.7 5.1311.00 4.0 3.8 5.9512.00 6.9 6.5 5.4313.00 9.7 9.3 4.0914.00 11.6 11.4 1.7215.00 12.3 12.3 0.0916.00 11.6 11.7 1.2617.00 9.9 9.9 0.0818.00 7.3 7.2 0.8719.00 4.5 4.4 2.1820.00 2.1 2.1 0.7421.00 0.7 1.0 35.9522.00 0.8 1.2 50.2723.00 2.4 2.8 16.84

Average Error(%) 6.65

Below is a comparative table of m2 and the software derived sinusoidal regression.

Time Height (m)Modified 2

(m2) Error (%) in m2

Sinusoidal Regression

from software

Error (%) in sinusoidal regression

00.00 7.5 7.5 0.42 7.7 2.6301.00 10.2 10.1 0.77 10.3 0.7402.00 11.8 11.9 0.55 11.9 1.0903.00 12.0 12.3 2.17 12.2 2.0104.00 10.9 11.2 2.82 11.1 2.1405.00 8.9 9.0 0.81 8.9 0.1706.00 6.3 6.1 2.93 6.1 3.8207.00 3.6 3.4 6.81 3.4 6.4908.00 1.6 1.4 13.43 1.5 7.4009.00 0.9 0.7 22.14 0.9 2.4210.00 1.8 1.5 18.13 1.7 5.1311.00 4.0 3.5 12.26 3.8 5.9512.00 6.9 6.3 8.73 6.5 5.4313.00 9.7 9.1 5.82 9.3 4.0914.00 11.6 11.3 2.48 11.4 1.7215.00 12.3 12.3 0.17 12.3 0.0916.00 11.6 11.8 1.67 11.7 1.2617.00 9.9 10.0 0.78 9.9 0.0818.00 7.3 7.3 0.16 7.2 0.8719.00 4.5 4.4 2.21 4.4 2.1820.00 2.1 2.0 2.84 2.1 0.74

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21.00 0.7 0.8 14.44 1.0 35.9522.00 0.8 1.0 24.25 1.2 50.2723.00 2.4 2.6 7.11 2.8 16.84



The average error in the sinusoidal regression from the software is 6.65%, which is higher than the average error in the second modification (m2) of the manual function, which is 6.41%. My final function is therefore 0.24% more accurate than the software proposed sinusoidal function. Both the function and the sinusoidal regression share similar period lengths and are place similarly on the horizontal axis in the graph. The most apparent difference in the graph is how the function’s amplitude reaches both the crest and the trough of the data, appropriated with the vertical shift, as compared to the amplitude of the sinusoidal regression, where the troughs do not reach the minimum height of the tides. From this alone, it is appropriate to conclude that the function is more accurate than the sinusoidal regression. However, in the data table above, m2 is more accurate lesser times than the software suggested sinusoidal regression, in a ratio of 7:17 for accuracy. This creates a contradiction; after looking at the data, the contradiction is plausible because t hours 21.00 and 22.00, the sinusoidal regression has large errors of 35.95% and 50.27% respectively. This huge shift in data makes the average error for the sinusoidal regression larger than the error for the m2 function. In this, the effects of an amplitude that is too short is apparent, because the sinusoidal regression compromises overall accuracy for the accuracy of individual points, whilst the function focuses on the wholeness of the function and its application to the data instead of focusing solely on individual figures. So, although the sinusoidal regression is more take more accurate to the data than the m2 function, it is overall less accurate. Also observable is that the percentage errors of m2 increases where the graph concaves up, and most accurate when the graph concaves down. These figures prove that m2 is more accurate by average, and the sinusoidal regression is more accurate by value, because it compromises average accuracy for individual accuracy.

Before 17 December 2003 – the day the data was recorded – the tidal range between 16:01 and 22:36 is 6 meters, with the following as the representing data:

Tuesday

Time Height of tide (m)04:23 4.609:56 0.916:01 5.822:36 0.7

If, supposedly, there is strong win to the shore in Nova Scotia on 27 Dec. 2003 from 01.00-04.00. If the wind were to come immediately on the shore, there would be no prominent effect on the tides, except that the tide would bear more waves. This is because the nature of tides is that they are caused by the gravitational pull of the sun and the moon, and for waves to turn into tides, it would require time and distance for the tides to grow. If the wind were immediately on the shore, the short distance would not provide enough time for the tides to grow, because the tide and their waves would break against the shore before having the opportunity to grow. There would be insignificant difference in the tidal heights between the hours. However, if the wind came a distance from the source, the change would be more apparent, depending on the source of the wind and the intensity of the wind – these are called

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swells. Below is a table illustrating the relationship of wells and how they affect tides, depending on source and strength of the wind.

WellsStrength of the Wind (“Sea Waves, Swells and Other Effects.”)

Weak Strong

Sou

rce

Far

fro

m th

e co

ast

The weak strength of the wind would create smaller wells, with smaller amplitudes. The wells would not travel far. If the well manages to reach the coast, it would only affect the wavelength of the waves slightly and only for a brief period, before the tide pertains its natural tidal wave. Because the well is far from the coast, the well would not significantly change the amplitude or wavelength of the tide. Because of this, the figures of this recording if subject to weak waves far from the coast would not be significantly different from the present data.

The strong strength of the wind would create large wells, with large amplitudes and large frequencies. Although these waves will travel far, their position means that when they reach the tide at the coast, they will not be as powerful. Therefore, the wavelength of the wells would decrease. This means that the wavelength (or period) of the tide would be affected if the well’s wavelength were different. If the wind were extremely strong, then the amplitude of the wells would also be significantly large, and because travelling decreases the amplitude lethargically, there could be a change in data amplitude. The strong nature of the wind also means the affects could be long lasting.

Clo

se to

the

shor

e

The weak strength of the waves would create smaller wells, with small amplitudes. Because the wells are close to the shore, the effect of the weak waves would be more immediate. However, because the winds are weak, the changes would not be quite significant, and the wells would quickly dissolve, creating little to no change to the overall tidally. Hence, the data recorded would not be significantly different from the present.

The strong strength of the wind would create large swells, with large amplitudes and large frequencies. Because the wells are close to the shore, the effect of the wells would be significant. When wells are created in these circumstances, the wells would be strong and last long. Extremely strong winds create wells with larger amplitudes, so the amplitude of the tides would be immediately affected. Depending on the frequency of the wind, the frequency of the wells could increase or decrease, creating different effects. Winds in this condition will create significant impact in the data, changing amplitudes and periods, with the potential of lasting for several days because of their magnitude.

Hence, the shift in data if strong winds hit the shore of Nova Scotia on 27 December 2003 (01.00-04.00) would be dependent on the source of the wind and the strength of the wind. Weak winds from a distance would have little to no effect, and for a brief period of time, whilst strong winds close to the shore would have significant effects that lasts for a long time.If this were to happen – strong winds, close range – then neither the function nor the sinusoidal regression would be appropriate to estimate the height of the tidal waves.Besides strong winds at a close distance, other factors that would affect the accuracy of the function against the data would be the day the data was recorded. For example, below is a

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graph of the recorded data (red points), the m2 function (green line), and the recorded data of the day after, 28 December 2003 (blue lines). Because the 28th follows after the 27th, the numeric follows the 27th, so 00.00 on the 28th is 24.00 on the 28th, and so forth.

Time (AST)

24.00 25.00 26.00 27.00 28.00 29.00 30.00 31.00 32.00 33.00 34.00 35.00

Height (m)

5.0 7.9 10.2 11.6 11.6 10.5 8.5 6.0 3.5 1.7 1.2 2.2

Time (AST)

36.00 37.00 38.00 39.00 40.00 41.00 42.00 43.00 44.00 45.00 46.00 47.00

Height (m)

4.4 7.2 9.7 11.3 11.8 11.1 9.4 7.0 4.4 2.2 1.0 1.3

Time Height (m) m2 Error (%) of m200.00 7.5 7.5 0.4201.00 10.2 10.1 0.7702.00 11.8 11.9 0.5503.00 12.0 12.3 2.1704.00 10.9 11.2 2.8205.00 8.9 9.0 0.8106.00 6.3 6.1 2.93

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07.00 3.6 3.4 6.8108.00 1.6 1.4 13.4309.00 0.9 0.7 22.1410.00 1.8 1.5 18.1311.00 4.0 3.5 12.2612.00 6.9 6.3 8.7313.00 9.7 9.1 5.8214.00 11.6 11.3 2.4815.00 12.3 12.3 0.1716.00 11.6 11.8 1.6717.00 9.9 10.0 0.7818.00 7.3 7.3 0.1619.00 4.5 4.4 2.2120.00 2.1 2.0 2.8421.00 0.7 0.8 14.4422.00 0.8 1.0 24.2523.00 2.4 2.6 7.1124.00 5.0 5.1 2.6925.00 7.9 8.0 1.8026.00 10.2 10.6 3.5527.00 11.6 12.1 3.9728.00 11.6 12.2 4.8529.00 10.5 10.8 3.2530.00 8.5 8.4 0.8431.00 6.0 5.5 7.8132.00 3.5 2.9 17.7733.00 1.7 1.1 33.2534.00 1.2 0.7 38.3635.00 2.2 1.8 18.5336.00 4.4 4.0 8.4637.00 7.2 6.9 4.3838.00 9.7 9.6 0.5739.00 11.3 11.6 2.7940.00 11.8 12.3 4.2341.00 11.1 11.5 3.8442.00 9.4 9.5 0.9643.00 7.0 6.7 4.2544.00 4.4 3.9 12.1845.00 2.2 1.7 23.2746.00 1.0 0.7 27.8847.00 1.3 1.2 7.18

Average Error (%) 8.14It is visibly apparent that the function is not appropriate for the data on the 28th. The error has increased, from 6.41% when applied to the 27th data to 8.14% when applied to the 27th-28th data. In the percentage error, it is also visible that the largest errors are in the regions where the data concaves up and concaves down, where the errors are larger near the vertices. Hence,

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the amplitude (a) parameter of the function must be changed to be more appropriate to the data. To change this, we must reconsider finding the amplitude and make it appropriate to the new data, while still maintaining the accuracy of the previous data

a=maximum−minimum2

a=11.8−12

a=10.82

a=5. 4We discover that the amplitude for the 28th is 5.4. Keeping in mind that the function for the first day is 5.8, we should find the average between the two.

a=a27+a28

2a=5.6

Hence, the function becomes:

y=5.6 ∙ sin( π6.2 (x−

35029 ))+6.5

Below, the function is applied to the graph (magenta line):

Time Height (m) m3 Error (%) of m300.00 7.5 7.4 0.87

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01.00 10.2 10.0 1.9902.00 11.8 11.7 1.0103.00 12.0 12.1 0.5104.00 10.9 11.0 1.3305.00 8.9 8.9 0.1506.00 6.3 6.1 2.7207.00 3.6 3.5 3.7908.00 1.6 1.6 2.4009.00 0.9 0.9 0.0810.00 1.8 1.6 8.5011.00 4.0 3.6 9.6812.00 6.9 6.3 8.6313.00 9.7 9.0 6.7514.00 11.6 11.1 3.9115.00 12.3 12.1 1.7916.00 11.6 11.6 0.0917.00 9.9 9.9 0.4318.00 7.3 7.3 0.5419.00 4.5 4.5 0.6120.00 2.1 2.2 4.4821.00 0.7 1.0 42.5122.00 0.8 1.2 47.9823.00 2.4 2.7 12.7524.00 5.0 5.2 3.6425.00 7.9 8.0 1.1226.00 10.2 10.4 2.1727.00 11.6 11.9 2.3228.00 11.6 12.0 3.1629.00 10.5 10.7 1.8230.00 8.5 8.4 1.6231.00 6.0 5.6 7.2532.00 3.5 3.0 14.2033.00 1.7 1.3 22.3734.00 1.2 0.9 21.8135.00 2.2 2.0 11.1536.00 4.4 4.1 6.5237.00 7.2 6.9 4.5638.00 9.7 9.5 1.6839.00 11.3 11.4 1.2340.00 11.8 12.1 2.5441.00 11.1 11.4 2.2842.00 9.4 9.4 0.1443.00 7.0 6.7 4.3544.00 4.4 4.0 10.1145.00 2.2 1.9 15.7346.00 1.0 0.9 7.95

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47.00 1.3 1.4 6.86Average Error(%) 6.67

In the graph, it is visible that the modification has made the function more appropriate for the data between the 27th-28th. The average error has also decreased 1.47% (from 8.14% to 6.67%), meaning that the function has become more accurate.

Using a graphical software, the suggested sinusoidal regression for the 28th’s recorded data is:y=5.525 ∙ sin (0. 5052 x+0.2037 )+6.57 4 4

On the graph below, the blue line represents the sinusoidal regression.

Time Height (m) m3Error (%) of

m3Sinusoidal Regression

Error (%) of Sinusoidal Regression

00.00 7.5 7.4 0.87 7.7 2.5601.00 10.2 10.0 1.99 10.2 0.2802.00 11.8 11.7 1.01 11.8 0.4103.00 12.0 12.1 0.51 12.0 0.3204.00 10.9 11.0 1.33 11.0 0.5505.00 8.9 8.9 0.15 8.8 1.2806.00 6.3 6.1 2.72 6.1 3.8207.00 3.6 3.5 3.79 3.5 3.85

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08.00 1.6 1.6 2.40 1.6 2.5809.00 0.9 0.9 0.08 1.1 17.0510.00 1.8 1.6 8.50 1.8 2.5011.00 4.0 3.6 9.68 3.8 4.5512.00 6.9 6.3 8.63 6.5 6.0913.00 9.7 9.0 6.75 9.2 5.5114.00 11.6 11.1 3.91 11.2 3.4215.00 12.3 12.1 1.79 12.1 1.7516.00 11.6 11.6 0.09 11.6 0.0917.00 9.9 9.9 0.43 9.8 0.5918.00 7.3 7.3 0.54 7.3 0.3219.00 4.5 4.5 0.61 4.5 0.8220.00 2.1 2.2 4.48 2.3 9.8021.00 0.7 1.0 42.51 1.1 63.0522.00 0.8 1.2 47.98 1.3 66.7923.00 2.4 2.7 12.75 2.8 18.1924.00 5.0 5.2 3.64 5.3 5.4525.00 7.9 8.0 1.12 8.0 1.6926.00 10.2 10.4 2.17 10.4 2.2627.00 11.6 11.9 2.32 11.9 2.2728.00 11.6 12.0 3.16 12.0 3.2429.00 10.5 10.7 1.82 10.7 2.2730.00 8.5 8.4 1.62 8.5 0.4731.00 6.0 5.6 7.25 5.7 4.8232.00 3.5 3.0 14.20 3.2 9.2133.00 1.7 1.3 22.37 1.5 12.1834.00 1.2 0.9 21.81 1.1 10.1735.00 2.2 2.0 11.15 2.0 7.4536.00 4.4 4.1 6.52 4.1 6.1837.00 7.2 6.9 4.56 6.8 5.1238.00 9.7 9.5 1.68 9.5 2.3739.00 11.3 11.4 1.23 11.4 0.7640.00 11.8 12.1 2.54 12.1 2.5441.00 11.1 11.4 2.28 11.4 2.9942.00 9.4 9.4 0.14 9.6 1.6143.00 7.0 6.7 4.35 6.9 1.0444.00 4.4 4.0 10.11 4.2 4.2245.00 2.2 1.9 15.73 2.1 4.9446.00 1.0 0.9 7.95 1.1 8.8547.00 1.3 1.4 6.86 1.5 12.02



In the graph, the sinusoidal regression has a smaller period than m3, however it is visible that they have different amplitudes. In the data, it is proven that m3 is 0.21% more accurate than the graphic program derived sinusoidal regression. Unlike the m2 function against the data of

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the 27th, the m3 function and the sinusoidal regression have equal times of being more accurate than the other when applied to the 27th-28th data. Hence, the higher accuracy of m3 is because the figures are generally more accurate. Because of the 6.67% accuracy of m3, the function fits the data (as it is also visibly observable in the graph).

The reason such alterations had to be made from the original function (m2) into a function appropriate to both days (m3) are because of external factors that impact tides. Aside from strong winds that are capable of changing tidal heights and periods, the lunar cycle is also a significant external factor. Depending on the position of the moon, tidal heights change, through tractive forces created by the moon’s gravity heightens water surfaces, making for higher tides. Rather than revolve around the earth, the moon and the earth share a system that orbits the center mass of gravity. Hence, the earth rotates on its axis, and the moon seemingly revolves around the earth. This system allows the moon to change its position against he equator, and it regularly shifts 28½ degrees above and below it. Different coordinates will receive different amounts of exposure to the moon’s gravity forces daily, increasing or decreasing tractive forces and therefore changing tidal height patterns. The sun, which has its own gravitational force, also affects tidal heights. The moon-earth system revolves around the sun, which causes another variety in tidal heights. The tilting degree of the earth as it rotates around its axis is also something to consider (Garrison). For these reasons, it is difficult to determine an absolute function that can predict tidal heights in certain locations, as there the external factors of wind and gravitational forces of other orbitals.

ReflectionThe results are relevant to discovering a function that is applicable to predict the tidal heights of Grindstone Island on the 27th December 2003, as the recorded data is an isolated recollection. Because of its exclusive nature, it is possible to discover a function to predict this. Modifying the function to predict the tidal heights between the 27th-28th of December is also possible because the recorded data is also isolated. The method used to discover these functions (m2, m3) are effective to their purpose, because it considers the given information and makes slight modification according to visual observations. These methods are also accurate because they have percentage errors of 6.41% and 6.67%, both relatively low figures. Because of that, the method of discovering the functions is appropriate and relevant to predict the tidal heights in the isolated data recorded between 27th-28th December. Hence, the results achieve an acceptable degree of accuracy, as they have an overall percentage error lower than 10%, as well as achieve the goal of predicting tidal heights for those two days. However, in the future, besides overall average, the accuracy of each individual figure should also be more attended to receive even more accurate results, which will require further investigation on sinusoidal functions and the manipulation of. In real life, the sinusoidal functions are useful in predicting tidal heights. However, it must be remembered that the figures are only predictions, and can only provide accuracy if the external factors (like wind) are not involved or if those external factors are considered (like lunar cycles). When this is achieved, sinusoidal functions is a practical way of predicting tidal heights, therefore useful for preventing natural catastrophes like tsunamis or floods. For this, I will need to do more research on sinusoidal functions, to receive a higher degree of accuracy.

Works Cited

Garrison, Tom S. “Tides.” Essentials of Oceanography. Brooks Cole: 2012. Print.

Nadia Eliora 18

“Sea Waves, Swells and Other Effects.” The Weather Window. Malasail. Web. 25 October

2015. <http://weather.mailasail.com/Franks-Weather/How-Waves-And-Swell-Form>

Mathematical Modeling

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