Beat frequency universal mathematical proof?by dunnoe Tags: beat, frequency, mathematical, proof, universal 

#1
Feb212, 05:28 AM

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It is easily derived from trigonometric identities that
cos(2*pi*f1*t)+cos(2*pi*f2*t)=2cos(2*pi*(f2+f1)*t)*cos(2*pi*(abs(f2f1)*t)) which proves that superposition of two cosine wave will generate a beat frequency, but what about about a universal proof that applies to any kind of periodic waves. Rather I am more curious about the insight behind why superposition of two frequency will generate a beat frequency. 



#2
Feb212, 08:30 AM

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You don't always get beats though  that's just when the two waves are close in wavelength. Far apart in wavelength you get amplitude modulation and inbetween you get a nasty looking ... thing. 



#3
Feb212, 09:17 AM

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#4
Feb212, 09:55 AM

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Beat frequency universal mathematical proof?There is a transformation between the sum and product of two sinusoids, one is called beats, the other is called amplitude modulation. You might want to review the discussion in this recent thread. http://physicsforums.com/showthread.php?t=567089 



#5
Feb212, 09:34 PM

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But you must realize that geometry is as basic as you can get. eg. "Geometry, coeternal with God and shining in the divine Mind, gave God the pattern... by which he laid out the world so that it might be best and most beautiful and finally most like the Creator."  Johannes Kepler [1] The effect you see is due to constructive and destructive interference  if the two waves have a different frequency then it follows that there must be a place where a trough of one must line up with a peak of the other and give you zero while at other places there must be two peaks or two troughs matching up and inbetween you must get the, well, "in between" case. What you are graphing is more: y=f(x).sin(kx) ... where f(x) happens to be another periodic with a different k. It doesn't have to be  all f(x) is doing is determining the amplitude of the sinusoid. Try f(x)=1/x for another famous example (you'll have to deal with what happens at x=0 though.). You can also examine the case where the two waves have exactly the same frequency but different phases. It's the trig ID that has you thrown off balance a bit? It is a shortcut to actually doing the addition directly  with it's own separate proof. One way of exploring underlying principles is to use the phasor representation for the waves  it's much more general.  [1] Quoted by Field J. V. in: Kepler's Geometrical Cosmology (1988), p. 123 quoted in Kepler's Geometrical Cosmology (1988), p. 123] 



#6
Feb212, 10:39 PM

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It is easiest to demonstrate what I'm talking about when you try it with soundwaves that are audible  using a multivibrator you can change the relative frequencies continuously to move from beats to AM ... in between the two there is a horrible noise. The waveform on an oscilloscope does not look like beating nor AM. That's what I was talking about. What I was trying to get across was that multiplying two periodic waves does not always produce beats. It was not my intention to be specific about every possible result. Hope that clears up any confusion. 



#7
Feb212, 10:44 PM

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Wow, i never though of it in term of fourier series. 



#8
Feb212, 11:06 PM

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Hopefully you now have some of the deeper understanding you were after. 



#9
Feb312, 03:59 AM

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However the relationship of the physical phenomena of beats and amplitude modulation is not quite the same. This is because there is an extra term in the maths for amplitude modulation that does not appear in the trigonometric transformation or beat frequency derivation discussed above. This results in there being two frequencies ( at any stage) only being involved in the sum or product but three in amplitude modulation. 


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