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...
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]
** 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]