# Finding Extrema of Sum of Three Sines

by Rick66
Tags: extrema, sines, wave equations
 P: 6 Hello everybody, I'm new to this forum so thanks for having me. I'm trying to find the times when the extrema occur for a periodic wave f(t) equal to the sum of three sine waves. Given f(t) = sin(2∏at) + sin(2∏bt) +sin(2∏ct) where a, b and c are whole numbers in lowest form (i.e. the wave has a frequency of 1), the first derivative test for extrema gives f'(t) = 2∏acos(2∏at) + 2∏bcos(2∏bt) +2∏ccos(2∏ct) = 0. Solving this for t would produce the times at which the extremas in f(t) occur. Since the wave is periodic, we can restrict the domain to 0 ≤ t ≤ 1. I've tried solving this using Euler substitutes, trig ID's, inverse trig functions, manual calculations of real examples etc but just can't see my way to a solution. This problem is really holding me up so any help would be greatly appreciated. Thanks Rick66
 HW Helper Sci Advisor Thanks P: 9,060 It is an interference problem and won't have a simple analytical solution for any set of three frequencies. When the three frequencies (a,b, and c) are close together you get a beat-like pattern, and when they are far apart you can get a kind of modulation ... in between the pattern of peaks can be chaotic. Have you tried this for just two sine waves? You should experiment plotting the function for different frequencies to get a feel for it. You could also have a look at fourier transforms.
 P: 6 Hi Simon, Yes I've solved the problem for 2 sines. In fact the three pairs can be reduced to the form: sinX + sinY + sinZ = sin((X+Y)/2)cos((X-Y)/2) + sin((X+Z)/2)cos((X-Z)/2) +sin((Z+Y)/2)cos((Z-Y)/2) However, adding the extrema of the individual terms here doesn't produce the extrema of the entire wave. The FT just reproduces the original 3 frequencies so that gives me nothing to work with either. But thanks anyway.
HW Helper