Daylight hours are dependent on sunrise and sunset times for each day which are dependent on seasonal change. The number of daylight hours can be represented by a periodic function. This periodic function can help Alaskan Council predict daylight hours for tourist travelling to watch their Bore Tide. A Bore Tide is a tidal phenomenon in which the leading edge of the incoming tide forms a wave (or waves) of water that travels up a river or narrow bay against the direction of the river or bay's current. Anchorage, Alaska is known for its famous Bore Tides. These bore tides occur at least once a day during high and low tide. Yet, health warnings are applied to viewing bore tides during low tide. Tourists have died by getting stuck in glacial silts by drowning due to incoming bore tides. Glacial silts are created when glaciers scrape bedrock into fine, round particles of rock or clay which are then carried away by tides or rivers. This substance becomes incredibly sticky therefore trapping tourists whom walk in them. The significance of tides is that they can be represented by a sinusoidal periodic function. The function will then help predict when low time is coming and help council roster park ranger to those times to prevent tourist wondering onto glacial silts therefore avoiding fatal accident. Picture + Diagram
Sunrise and sunset hours for locations around the world are all dependable on various variables such as geographical location of the location and seasonal change. With these factors in mind the daylight hours can be modelled using a sinusoidal periodic function. Question 1a:
The information used to generate the model for Anchorage, Alaska was gathered from www.timeanddate.com. (Refer to Appendix 1 for a sample of raw data). Information from the year 2011 was gathered and the day light hours of each day was converted into minutes and the dates were converted into the number of day within the year it is. (Refer to Appendix 2 for Tabulated data) Question 1b:
When the daylight minutes were obtained it was tabulated on excel and then graphed on Graphmatica to give an idea of how the daylight hours looked graphed. (Refer to Appendix 3 for a graph of daylight hours.) The graph displayed sine curve patterns. Question 1c:
Due to the fact the graph on graphmatic revealed makings of a sin curve it was realised that that data of daylight hours in Anchorage can be modelled using a sinusoidal periodic function using the base; L=Asin[B(t-C)]+D. Where;
L = the length (daylight hours) of the day
t = the day of the year since December 30th 2010 (eg 23rd of Jan 2011 is day 23 but 23rd of Jan 2012 is day 388 (365+23) A = the measurement of the amplitude of the graph. This is how much the graph is vertically stretched in comparison to a normal sin curve. To find the amplitude of the graph the highest number (longest day of the year) minus the lowest number (shortest day of the year) was divided by two. This finds the mean of the daylight hours which is the amplitude. So, (1161-327)/2=417
Therefore the Amplitude is 471
B = The period of time that it takes to complete one cycle (a year). The period of time to finish one cycle is found by dividing 365 (if working in degrees) by 365 (number of days in a year). If working in radians 2π/365 was used. Radian form was used for simplicity throughout the investigation. So, B = 2π/365
D = is how much the sin curve is shifted up or down.
To find D the same procedure used to find A was used but instead of subtracting the longest day from the lowest day these two figures were added then divided by two, therefore finding the vertical shift. So, (1161+327)/2=744
Therefore the Vertical shift is 744
C = is the horizontal shift of the graph. This is how much the daylight hour graph has shifted horizontally in comparison to a regular sin graph. To find C was found by locating the D value on appendix 3s graph. The x intercept of the D value will be equal to...