Beat frequency- universal mathematical proof?

dunnoe

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(f2-f1)*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.

Simon Bridge

That was it.
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 in-between you get a nasty looking ... thing.
srsly? do you also wonder how two lengths at an angle can make an area? It's geometry.

dunnoe

At first, I was skeptical that two periodic waves which are close in wavelength regardless of their shape will generate a beat frequency given only proof of the cosine-cosine case. Later, I toyed with matlab and plotted sine-sine, cosine-cosine & sine-squarewave graph which showed beat frequency as well. Surely there's a reason beyond geometry it's applicable for waves of all shapes . I don't think being able to derive a formula is the same as understanding the formula.

Studiot

Are you sure about this?

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

Simon Bridge

The reason you need only do the proof for the sinusoidal case is that every other periodic function can be expanded as a linear sum of sinusoids. Which is a separate proof.Can you provide an example in math where there is a reason "beyond geometry" for something?

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 in-between 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.

Well that's correct.

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]

Simon Bridge

Quite sure :) worked with radio for ages. I know - these are two classes of the same math - and there is a continuous transform from one to the other.

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 wave-form 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.

dunnoe

Thank you for the very satisfying response.

Wow, i never though of it in term of fourier series.
I admit that I have a poor understanding on the definition of geometry back then.

Yup, definitely got confused by the trigonometry identity.

Simon Bridge

No worries - it helps to make links between different parts of your knowledge.

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

Studiot

The maths of the sum or product of sinusoids is as you say, classes or opposite sides of the same equation.

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.