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math modelling

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Introduction

Aim: in this task I will develop a model function for the relationship between the time of the day and the height of the tide.

Graph2.1 representing time of day against the height of the tide in the Bay of Fundy in Novia Scotia in Canada.

image00.png

Tides are the vertical rise and fall of the oceans of the world. Tidal currents are the horizontal movement of the water caused by these tides. Tides can be described as the alternating rise and fall of sea level with respect to land, as influenced by the gravitational attraction of the moon and sun.

Other factors influence tides; coastline configuration, local water depth, seafloor topography, winds, and weather alter the arrival times of tides, their range, and the interval between high and low water.

Predicted tidal heights are those expected under average weather conditions. When weather conditions differ from what is considered average, corresponding differences between predicted levels and those actually observed would occur.

Generally, prolonged onshore winds (wind towards the land) or a low barometric pressure can produce higher sea levels than predicted. Whilst offshore winds (wind away from the land) and high barometric pressure can result in lower sea levels than predicted

The world’s oceans are in constant flux. Winds and currents move the surface water causing waves.

...read more.

Middle

As it can be seen a sin wave graph has been formed.

The general rule for a sin graph is y   = A sin (B*t) + C

Where A is the amplitude and equals = (maximum-minimum)/2

                                        = (12.3-0.7)/2

                                        = 5.8 = A

And B is the period and equals      13 = 2∏/B

                                         B= 2∏/13

To find C we may take the maximum point so:

12.3 = 5.8 sin (15 * 2∏/13) + C

12.3 = 5.8 (1) +C

C     = 6.5

So the function for the graph is: 5.8 sin( t * 2image02.png/13) + 6.5 = y and lets call it function 1

Now to see how much the new function fits on the original graph we will place the 2 functions together:

Graph 2.1 representing function 1 with the graph of 1.1

image03.png

The blue curve is of function 1 while the scatter plot is of the original values of the tide.

As it can be seen the graph of function 1 does not fit much on the original values of the tide height.

So maybe if the graph form changed to this form y   = A sin (B*t + D) + C it may fit more on the original graph:

Now D has a rule so we can find it

D = (maximum point + minimum point) / 2

D = (12.3+0.7)/2

D = 6.5

So the new rule is 5.8 sin (t * 2∏/13 +6.5) + 6.5 and lets call it function 2

Graph 3.1 representing function 2 with the graph 1.1

image04.png

Now this graph will be produced and as it can be seen, this function (function 2) fits more on the original graph than function 1.

...read more.

Conclusion

So if we consider the function in the following shape:

y   = A sin (B*t + D) + C

So     A=5.34

        B=0.51

        D=-0.26

        C=6.58

So the general function for December 28 2003 is:

5.34 sin (0.51t – 0.26) + 6.58

Now it is clear that at different days in the year we have different general functions showing the relation between the time of the day and the height of the tide and this is due to many reasons such as:

  • prolonged onshore winds (wind towards the land), increase the sea level
  • high barometric pressure can result in lower sea levels than predicted
  • The position of the moon compared to that of the earth, because the moons attraction to the water this pull causes the water to bulge toward the moon. And at different times the moons position is different causing its gravitational pull to be altered to a certain body of water
  • The position of the sun compared to that of the moon since the sun also has a gravitational on earth, so if the moons position was between earth and sun the highest gravitational force will be exerted and so the highest tide
  • The friction between water and the waters body ground, the higher the friction the lower the tide height
  • The temperature so at higher temperature water expands, increasing the volume of water making tides higher

...read more.

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