2. The lecture notes on differentiation of trigonometric functions show how trigonometric functions can be used to generate tide tables. Tides are governed primarily by the forces of gravity from the moon and the sun. However, it is much more complicated than simply just adding these two forces. Many of the tide tables are generated using the sum of 12 or more trigonometric functions based on varying forces due to the elliptical orbits of the Earth and moon and the revolution of the Earth. In this problem, we will use a reduced set of functions to approximate the tides for San Diego in October 2000. There are four dominant forces affecting the tides. The diurnal components (once per day) are denoted K1, the lunisolar force, and O1, the main lunar force. The semidiurnal components (twice each day) are given by M2 the main lunar force, and S2, the main solar force. The table below gives the period, Pi, in hours for each of these components. This lab uses Excel’s Solver to find the amplitude, Ai, in feet, and phase, fi, in radians needed to predict the tides for October 2000. Because of the semidiurnal forces, there are two high tides and two low tides each day.
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The mathematical model for the height, h(t), of the tides is given by the formula:
a. The first task in this lab is to complete the table above by finding the least squares best fit of the model to data from actual tide tables for San Diego for October 2000. As we have done in previous lab exercises, we use Excel's solver to find this best fit of h(t) to the data by varying the coefficients, Ai and fi, in the formula above and minimizing the square of the error between the model and the data for high and low tides. To save you considerable time a spreadsheet of the data and some entries for the model are provided to help you with the initial part of the problem. The Help sheet will provide additional guidance to aid you in completing the entries and completing the minimization process. In your lab write up, give the values of the coefficients Ai and fi, including A0, that are determined by Excel's solver routine. Also, write the value of the sum of the squares of the error calculated on the spreadsheet.
b. Produce two graphs showing the second and fourth weeks of October of tides (Oct. 7-13 and Oct. 21-27), including both the model and the data points for the actual high and low tides. Do NOT forget that the time used in the spreadsheet is in hours, so you will need to convert the hours back to days by when you produce your graphs with corresponding dates on the horizontal axis. Determine the date for October 2000 of the highest high tide and lowest low tide and find the specific height and time of these events. Compare your results to the actual tide tables.
c. Make a graph for October 18, 2000, showing the model and the data points. Determine the heights and times of the high high tide, low low tide, low high tide, and high low tide. Compare your theoretical results to the actual values. Write which values give you the largest and the smallest errors in both the timing and the magnitude of the tides. (Use absolute values and not percent errors.)
d. Which of the coefficients of force is the largest? Thus, what force is most significant in determining the height of the tides? Use the lunar dates and times below to see how your graphs of the tides compare with the phases of the moon (new, first quarter, full, and last quarter). Discuss what you observe (in height and times) about the tides on those dates.
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FIRST QUARTER |
FULL MOON |
LAST QUARTER |
NEW MOON |
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http://aa.usno.navy.mil/AA/data/docs/MoonPhase.html#y2000 http://www.saltwatertides.com/pickpred.html