Finding Extrema of Sum of Three Sines

[Rick66]

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

 

[Simon Bridge]

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.

 

[Rick66]

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.

 

[Simon Bridge]

OK. Do you also understand what I said in the first two sentences?

 

[Rick66]

That it doesn't have a simple analytical solution? Sorry Simon, I did overlook this statement. I didn't know this when I started but I certainly do now. If the frequencies and/or sum frequencies bear a rational relation, for eg, then some solutions are analytic. But these are not necessarily the ones corresponding to the extrema.

The problem comes from acoustics and the only real condition is that the 3 frequencies are unequally spaced. Now, it did occur to me that this freedom might allow me to choose a form that *can* be solved by analytic means. For eg, solving f '(t) = cos[2pi at] +cos[2pi bt] +cos[2pi ct] = 0 and "choosing" the integral as the original f (t). I'm just wondering. Are you aware of any standard way to simplify the assumptions so as to make the solution analytic? Perhaps finding the closest rational approximations, introducing phase or amplitude?

Much appreciated

Rick

 

[Simon Bridge]

I don't know of any way to work out the frequency ratios systematically no.
But you will underestand it better by starting with two sine waves and graphing different situations - if the frequencies are close together, then you get beats, and the local maxima periodically spaced. If the frequencies are far apart, then you get the amplitude modulation situation, and the the local maxima are also periodically spaced. There is an in-between case that I have not looked at in detail.

The main wrinkle I can think of is that sometimes destructive interference will remove some of the local maxima - producing a gap in the periodicity. Figuring where this would occur would be a headache. If I were you I'd go back to the context of the problem and it's metadata for a guide for how to proceed.

 

[Rick66]

Hi Simon,
sorry for the late reply. Yes I tend to agree. If the sines are equally spaced then the problem is easily dealt with using the usual methods of carrier and modulation frequencies. Now a promising lead was that it might be possible to solve the problem by trying to find a difference freq (modulation) that divides into all three. For eg, the frequencies might represent the 5th, 7th and 12th harmonics to some fundamental. From here it should be possible to use this fundamental, or the difference freq, to find where the extrema occur. However, it ends up that in terms of the problem that I was working on - it has to do with speech patterns - this line of inquiry was no longer necessary. Still, I might come back to it for its own sake when I've got the time.
Anyway, thanks for your thoughts.
Rick66