Calculate formula needed for complex sine wave that models major and minor tides

[SOmarN]

I am trying to find a summed sine wave for my assignment that fits the data points of tide readings from a specific coastal region (Avalon Beach).

The requirements are as follows:
- both tides must touch their averages
- Major (first): 1.82, 0.245
- Minor (second): 1.6175, 0.2725
- peaks should align with data points on the x-axis
- equation should be [math]asin(bx+c)+dsin(ex+f)+g[/math] where 2b ≈ e and d>a

The data points, average lines and other info is at https://www.desmos.com/calculator/ahlz9vqimg

I remember doing a bit of summed sine waves before, but not at this level, only adding 2 equations to get an answer, and not the other way around. I'm not sure what a summed sine wave "model" is.

Things I previously tried:
- Making 1 uniform sine wave that hits the x-axis of the points and the average of the averages of the y-axis of the points (was told to make complex graph)
- Adding 2 sine waves that were zeroed where the other wave was meant to be shown (increasing height centre of 1 increased the height centre of the total)
- Adding 2 sine waves that were "Noned" (i.e. no value/line) where the other value was meant to be shown (cut-off graphs, was told to sum sine waves instead)

I am stuck with the summing because the graph moves in an unusual way, where moving 'c' or 'f' separate from each other changes the height, and I am unsure of how to change the height centre since it's different for both tides.

 

[blamocur]

I don't know how to interpret the data in the online calculator. E.g., [imath]\left(7\frac{13}{15}+12, 1.67\right)[/imath] the same as [imath]\left(19\frac{13}{15},1.67\right)[/imath], or is there some special meaning in that unreduced expression?
Also, not sure what "peaks should align with data points on the x-axis" means: do you need something different from a simple minimum square approximation?

 

[SOmarN]

1. The [math]7\frac{13}{15}+12[/math] is for convenience purposes, since it's 7:52pm, which is 12 hours after 7:52am, and so on.
2. the peaks of the compound sine wave should have a similar x-coordinate to the data points, since they're the high and low tide points of the major and minor tides.

 

[blamocur]

1. The [math]7\frac{13}{15}+12[/math] is for convenience purposes, since it's 7:52pm, which is 12 hours after 7:52am, and so on.
2. the peaks of the compound sine wave should have a similar x-coordinate to the data points, since they're the high and low tide points of the major and minor tides.

Thanks. Several more questions/requests:

1. Is the second value the height of the tide ?
2. Is there any significance to the colors?
3. Could you convert the data from the calculator format to a text which can be copied and pasted? I'd rather not retype all your values by hand.
4. Do I understand correctly that the times are more important than heights (y-values?)?
5. What is the criterion for the best match? Minimum sum of squared errors? Minimal max. error?

Thanks again.

 

[SOmarN]

1. yes. in this case it doesn't matter, since the compound sine wave is mean to hit the average line
2. no
3. that's convenient, i did that as well
Table:

1\frac{11}{30}	0.2725
7\frac{52}{60}+12	1.6175
2\frac{2}{60}+24	0.2725
8\frac{41}{60}+36	1.6175
2\frac{43}{60}+48	0.2725
9\frac{32}{60}+60	1.6175
3\frac{26}{60}+72	0.2725
10\frac{26}{60}+84	1.6175
7\frac{39}{60}	1.82
1\frac{47}{60}+12	0.245
8\frac{21}{60}+24	1.82
2\frac{38}{60}+36	0.245
9\frac{5}{60}+48	1.82
3\frac{29}{60}+60	0.245
9\frac{51}{60}+72	1.82
4\frac{24}{60}+84	0.245

Array (2-dimensional):

data = [
[1+11/30, 0.2725],
[7+13/15+12, 1.6175],
[2+1/30+24, 0.2725],
[8+41/60+36, 1.6175],
[2+43/60+48, 0.2725],
[9+32/60+60, 1.6175],
[3+26/60+72, 0.2725],
[10+26/60+84, 1.6175],

[7.65, 1.82],
[107/60+12, 0.245],
[8.35+24, 1.82],
[79/30+36, 0.245],
[109/12+48, 1.82],
[209/60+60, 0.245],
[9.85+72, 1.82],
[4.4+84, 0.245],
]
#end of array

4.yes
5.i'm not sure what those mean. the teacher i asked said it should match the time 'on the dot' (figuratively).

No, thank you. (not 'no thank you' i.e. 'no thanks', but rather "i'm the one thanking you instead")

 

[blamocur]

1. yes. in this case it doesn't matter, since the compound sine wave is mean to hit the average line
2. no
3. that's convenient, i did that as well
Table:

1\frac{11}{30}	0.2725
7\frac{52}{60}+12	1.6175
2\frac{2}{60}+24	0.2725
8\frac{41}{60}+36	1.6175
2\frac{43}{60}+48	0.2725
9\frac{32}{60}+60	1.6175
3\frac{26}{60}+72	0.2725
10\frac{26}{60}+84	1.6175
7\frac{39}{60}	1.82
1\frac{47}{60}+12	0.245
8\frac{21}{60}+24	1.82
2\frac{38}{60}+36	0.245
9\frac{5}{60}+48	1.82
3\frac{29}{60}+60	0.245
9\frac{51}{60}+72	1.82
4\frac{24}{60}+84	0.245

Array (2-dimensional):

data = [
[1+11/30, 0.2725],
[7+13/15+12, 1.6175],
[2+1/30+24, 0.2725],
[8+41/60+36, 1.6175],
[2+43/60+48, 0.2725],
[9+32/60+60, 1.6175],
[3+26/60+72, 0.2725],
[10+26/60+84, 1.6175],

[7.65, 1.82],
[107/60+12, 0.245],
[8.35+24, 1.82],
[79/30+36, 0.245],
[109/12+48, 1.82],
[209/60+60, 0.245],
[9.85+72, 1.82],
[4.4+84, 0.245],
]
#end of array

4.yes
5.i'm not sure what those mean. the teacher i asked said it should match the time 'on the dot' (figuratively).

No, thank you. (not 'no thank you' i.e. 'no thanks', but rather "i'm the one thanking you instead")

I thought about your problem, and I cannot see a simple generic solution. Nor am I an expert in the field of tides' computing. I.e., take my suggestions with a grain of salt.
I am skeptical about the requirement that "peaks should align with data points on the x-axis" : sine waves have zero derivatives at the peaks, so empirical measurements of X locations are bound to be imprecise.
If I had to solve this numerical problem I'd treat it as a minimum square error approximation problem. You have a general function [imath]F(a,b,c,d,e,f,g,x) = a\sin(bx+c)+d \sin(ex+f)+g[/imath], and you want to minimize your error [imath]E(a,b,c,d,e,f,g) = \sum_k \left(F(a,b,c,d,e,f,g,x_k)-y_k\right)^2[/imath]. I would try to use some version of a gradient descent optimizer. To speed things up I would carefully select the initalial values of [imath]a,b,c,d,e,f,g[/imath] which look like an approximation. E.g., compute initial values of [imath]a[/imath] and [imath]b[/imath] from the spacing of the peaks, etc.
There might be more specialized methods in your field which I am not familiar with. But if you decide to use my approach feel free to ask questions. Either way good luck with you project!