Wave Convergents Final Form

Hello all,

A few weeks ago Steve Martin pointed out that my previous method was not always giving the correct results. I've now corrected this. I won't give any proofs now but just the method. I've attached a file which shows that the peaks of the wave do indeed occur at the GCD's between the (wave) convergents.

Given the sum of two sines of the form

f(t) = sin[2pi at] + sin[2pi bt],

where a/b is either coprime or irrational, then its extrema occur at the times t = (2k +/-1)/2(a + b) where

(2k +/-1) = (Round to even[j*2(a + b)/(a - b)]) +/- 1, j = 0,1,2,3,...

The wave convergents are given by

Round[2a/(a - b)]/Round[2b/(a - b) = p/q.

The ~ GCD = (a + b)/(p + q) or

2(a + b)/(Round[2a/(a - b)] + Round[2b/(a - b)]).

You can see some examples on the file I've posted called "Round Odd eg's". It shows the intervals a/b = 81/64, 19/16, 32/27 and 51/32 respectively. There you'll see that the black dots occur at all the wave extrema and that the time difference between them correspond to the GCD fundamentals between the wave convergents.

Cheers

Rick