Modeling Tidal Patterns: Finding an Equation

[Ry122]

I need to determine an equation that models tidal patterns.
High tide is at 12:00am
Low Tide 6:10pm
High tide returns at 12:24pm
High tide is 5.4m and low tide is .1m

Therefore the amplitude is (5.4-0.1)/2 = 2.65
The mean value is 2.65+0.1=2.75
Therefore
y=2.65cos(Bx+c)+2.75

Im yet to determine B (Period) and c (Phase Shift).
Also, I am not sure whether to use cos or sin.

 

[WolfOfTheSteps]

Sounds like a fun problem.

As for your last question, use whichever one you want. You will just end up with different phase shifts... So either cos or sin will be fine. Also, it looks like your period will be 12 hours and 24 minutes. (convert this to one unit, like X minutes, or x.xx hours. Minutes is probably better) And remember that B is the angular frequency. So divide 2\pi by your period, T, to get the correct value for B.

Hint: cos is probably better if you start with time = 0 at 12:00am. Try to figure out why.

Also, I just noticed something... Low tide should be 6:10am, shouldn't it? And even if that is the case, your wave is a little skewed. If low tide was at 6:12am it would be a perfect cos or sin wave.

 

[Ry122]

Hint: cos is probably better if you start with time = 0 at 12:00am. Try to figure out why.

Is it because the wave starts at 5.4m and because cos0=1 and sin0=0, cos is closer to 5.4 therefore u will need less phase shift?

 

[WolfOfTheSteps]

Yes. In fact, you will need NO phase shift. :)

 

[Ry122]

But there would still have to be some phase shift wouldn't there?
2.75 + 1 = 3.75, not 5.4

 

[WolfOfTheSteps]

But you are forgetting your amplitude of 2.65!

2.75 + 2.65*cos(0) = 2.75 + 2.65*1 = 5.4

 

[AlephZero]

If this is a question about "real" tides, you can't just use three times to get a useful prediction.

You can see from your numbers that the two time intervals between high and low are 6h 10m and 6h 14m. For a single sin or cos function, they would be exactly the same. Also the height of high and low tides varies with each tide.

The two biggest effects are the sun and the moon. If the Earth's orbit round the sun was circular (it isn't) the tides from the sun have a period of exactly 12 hours. If the moon's orbit round the Earth was circular (it isn't) the lunar tides would have a period of 12 hours 25 minutes.

The actual tide is the sum of two sin (or cos) waves with these different frequencies, and also different amplitudes and different phases.

The next biggest effect is that the Earth's axis is tilted relative to the plane of rotation of the sun and moon. In the northern hemisphere in March, that causes high tides in the evening to be higher than those in the morning, and the opposite in the southern hemishpere). In September, this is reversed (Please don't ask why...).

The elliptical orbits also make a difference. For example varying speed of the Earth's orbiting round the sun changes the times of solar tides by about 20 minutes compared with a circular orbit, during the year. Lunar tides vary in a similar way during each lunar month.

Professional tide prediction uses many more than just these effects. The "standard model" contains 69 different frequency components, plus 77 more for shallow water effects near the coastline!

For example see three papers by Foreman, 1996 - links at the bottom of the this page: http://www.pac.dfo-mpo.gc.ca/SCI/osap/publ/online_e.htm

Here's a set of real data (for Kings Lynn, UK, March 2007) that shows just how "untidy" the numbers are. Note the time between successive high tides varies from about 12h 20m to 13h 10m, and the highest "evening" tides are 0.5m higher than the "morning" ones.

Heights in meters

05:53 2.0 10:29 5.2
18:17 2.0 22:55 5.0
06:45 2.1 11:22 4.8
19:13 2.0 00:03 4.6
07:46 2.2 12:50 4.5
20:18 2.0 01:59 4.5
08:54 2.2 14:34 4.6
21:36 1.9 03:17 4.8
10:25 2.1 15:37 5.1
23:12 1.7 04:10 5.3
11:50 1.9 16:25 5.7
00:20 1.3 04:56 5.9
12:47 1.6 17:09 6.4
01:13 0.9 05:37 6.4
13:35 1.5 17:50 6.9
01:59 0.7 06:16 6.7
14:18 1.4 18:30 7.4
02:42 0.7 06:56 7.0
14:59 1.4 19:11 7.6
03:23 0.8 07:36 7.1
15:38 1.5 19:53 7.6
04:03 1.0 08:18 7.0
16:15 1.7 20:36 7.4
04:40 1.3 09:00 6.7
16:49 1.8 21:21 6.9
05:17 1.6 09:46 6.3
17:24 2.0 22:10 6.3
05:55 2.0 10:38 5.7
18:10 2.1 23:10 5.6
06:40 2.2 11:44 5.2