Friday, November 21, 2014

wave interpolation (with natural number harmonics of different amplitudes)

I noticed that it's possible to find solutiond for waves that are 0 at the endpoints of the interval and that pass through a certain number of points.
Like for instance, say you want to know the waves that pass through these points:
[[.5, 1], [1, 0], [1.5, -1]]
Since that is three points, I start by considering waves of the form:
Maple can solve this by entering...
You can see that there are many waves satisfying these conditions, b and c can have any possible value and a just has to be 1+ whatever c is chosen to be... for example a=5.5, b=3, c=4.5
The thing I noticed is that there are two degrees of freedom in the solutions for this wave because the t values interpolated through .5, 1, 1.5 are all multiples of the first value of .5. We can remove a degree of freedom in the solutions by interpolating through .5, 1.2, 1.5, here see that 1.2 is not a multiple of .5 but 1.5 still is...
The solutions only have one degree of freedom now so that c can be chosen as anything. Interestingly the solutions involve the golden ratio! 1.618033... is usually known as phi or the golden ratio. I'm not sure exactly why that is.

So the next step of course is to look to see that a case where the t values are all not multiples of each other yields only one solution...Let's try [.7, 1.2, 1.6]
There is only one solution for those heights through those t values.

** Finding unique solutions for an unknown wave though a number of points**
The trick for this is as you can see above to sample your unknown wave so that none of the t values are multiples of each other... There might be many ways to do this. It's kind of an interesting problem in itself.
One way is to use prime numbers...if you have n points you would use fractions of the (n)th prime like for example if you have six points you could use for t values:
[1/13, 2/13, 3/13, 5/13, 7/13, 11/13]
For example, if our 6 heights at those t values are [1, 1.5, .5, 0, -.65, -.8]

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