Chapter 4: Systems Based on Numbers

Section 6: Mathematical Functions

Substitution systems [and sine sums]

Cos[a x] - Cos[b x] has two families of zeros: 2 π n/(a+b) and 2 π n/(b - a). Assuming b > a> 0, the number of zeros from the second family which appear between the nth and (n+1)th zero from the first family is

(Floor[(n+1) #]-Floor[n #])&[(b-a)/(a+b)]

and as discussed on page 903 this sequence can be obtained by applying a sequence of substitution rules. For Sin[a x]+Sin[b x] a more complicated sequence of substitution rules yields the analogous sequence in which -1/2 is inserted in each Floor.

From Stephen Wolfram: A New Kind of Science [citation]