Summing sinusoids of different frequency

AI Thread Summary
Combining sinusoids of different frequencies into a single sinusoid is not possible, as they can yield non-sinusoidal shapes when summed. The discussion highlights that while any repeating function can be represented as a sum of sinusoids through Fourier analysis, there is no straightforward transformation to condense two different frequency sinusoids into one. The sum of sinusoids with different frequencies can lead to complex waveforms, and the resulting period is determined by the least common multiple (LCM) of their individual periods. Additionally, if the frequency ratio is irrational, the sum will not be periodic. Ultimately, the consensus is that two distinct sine wave frequencies cannot be combined into a single pure sine wave frequency.
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I'm curious if there is a way to combine sinusoids of multiple frequency into a single sinusoid?

For example, I'm looking for a way to combine:

Acos(2παt + φ) + Bcos(2πβt + χ) where A, α, φ, B, β, χ are scalar constants and α & β represent the frequency.

I'd even be interested in a less complex solution to:
cos(2παt) + cos(2πβt)
if this is much simpler than the general form above.

I found an incomplete explanation on here that does a transform and then recombines using complex numbers ending up with a cosine with an arctangent inside.
https://www.physicsforums.com/showthread.php?t=372263

However, explanations of the process are probably better than just a discrete example alone.
 
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cos(a+ b)= cos(a)cos(b)- sin(a)sin(b)

cos(2\pi\alpha t+\phi)= cos(2\pi\alpha t)cos(\phi)- sin(2\pi\alpha t)sin(\phi) and
cos(2\pi\beta t+\chi)= cos(2\pi\beta t)cos(\chi)- sin(2\pi\beta t)sin(\chi)
 
HallsofIvy said:
cos(a+ b)= cos(a)cos(b)- sin(a)sin(b)

cos(2\pi\alpha t+\phi)= cos(2\pi\alpha t)cos(\phi)- sin(2\pi\alpha t)sin(\phi) and
cos(2\pi\beta t+\chi)= cos(2\pi\beta t)cos(\chi)- sin(2\pi\beta t)sin(\chi)

I don't think trig identities are going to be much help.

Acos(2παt + φ) + Bcos(2πβt + χ) = A(cos(2παt)cos(φ) - sin(2παt)sin(φ)) + B(cos(2πβt)cos(χ) - sin(2πβt)sin(χ))

Which has turned a sum of 2 sinusoids into 4 products and 3 summing operations on 8 sinusoids. This is kind of the opposite direction of where I was looking to go. I'm trying to figure out if there's a transformation I can use to get this all inside of a single sinusoid function, even if there is a trig function inside the sinusoid.
 
Summing sines of different frequencies can yield very NON sinusoidal shapes. Why do you expect to use a single sine function to describe non sine shaped functions?

i.e. The sum of all odd harmonics yields a square wave.
 
Integral said:
Summing sines of different frequencies can yield very NON sinusoidal shapes. Why do you expect to use a single sine function to describe non sine shaped functions?

i.e. The sum of all odd harmonics yields a square wave.

I don't expect to, I asked if it was possible.

I know it is possible to write any repeating function as a sum of sinusoids (Fourier?) but I was just curious if there is a process to combine a sum of 2 into 1. I'm mainly interested for representing purposes, I just wanted to see if there was a way to represent the sum mathematically without both sinusoids needing to be shown. Maybe it could be represented as a series?
 
Functions that are called linearly independent cannot be summed into a single one.

Sines of different frequencies are no different in this regard than trying to sum together, say, the functions Bz and Bz^2 into a SINGLE power in z.
 
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What about the period of the sum of sinusoids of different periods? What would that be?
 
LuizHP said:
What about the period of the sum of sinusoids of different periods? What would that be?
It would be a repeating waveform, of period equal to the LCM of their periods. Try some sketches and discover for yourself.
 
NascentOxygen said:
It would be a repeating waveform, of period equal to the LCM of their periods. Try some sketches and discover for yourself.
To give an example, if the ratio of the periods is not a rational number, the sum will not be periodic.
##f(x)=\sin(x) + sin(\pi x)## is not periodic.
 
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The short answer is no. Two different pure sine wave frequencies can not be summed to another pure sine wave frequency.
 
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