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I have two waves interfering with different amplitudes, frequencies and phases.

$$ x(t) = X_1 \sin(\omega_1 t+\varphi_1) + X_2 \sin(\omega_2 t+\varphi_2) $$

I am able to convert this wave into the form

$$ x(t) = A \cos(\delta t) \cos(\sigma t) + B \cos(\delta t) \sin(\sigma t) + C \sin(\delta t)\cos(\sigma t) + D \sin(\delta t)\sin(\sigma t) $$ with parameters

$$ \begin{aligned} \delta & = \frac{\omega_1-\omega_2}{2} &\sigma & = \frac{\omega_1+\omega_2}{2} \\ A & = X_1 \sin(\varphi_1)+X_2 \sin(\varphi_2) & B & =X_1 \cos(\varphi_1)+X_2 \cos(\varphi_2) \\ C & = X_1 \cos(\varphi_1)-X_2 \cos(\varphi_2) & D & = X_2 \sin(\varphi_2)-X_1 \sin(\varphi_1) \end{aligned} $$

What I am looking for is an expression of the envelope shape (see below) in terms of A,B,C,D,σ,δ.

Fun

Edit 1

The example above is from

$$ x(t)=0.8 \sin(20 t+(-1)) + 0.2 \sin(21 t+0) $$

It seems like an approximation by I have a Wolfram Plot which indicates that the envelope is $$e(t) = X_1+X_2 \cos((\omega_1-\omega_2) t+(\varphi_1-\varphi_2))$$

I am curious if there is an analytical way to prove that this is good approximation.

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2 Answers 2

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I have confirmed the envelope function to be

$$ e(t) = X_1+X_2 \cos((\omega_1-\omega_2) t+(\varphi_1-\varphi_2)) $$

based on two waves interfering. I tried it on a bunch of wave combinations and it seems to work reasonably well.

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I'm pretty sure the envelope function is simply $$ envelope(t) = X_1 \sin(\omega_1 t+\varphi_1) + X_2$$

I have added a link to a plot because you seem to be struggling to plot the graph http://www.wolframalpha.com/input/?i=plot+x%3D4sin%284t%29%2Bsin%28t%2F6%29%2C+x+%3D+sin%28t%2F6%29%2B4+

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  • $\begingroup$ Nope, that is not it. The envelope function shouldn't cross zero. It should vary between $X_1+X_2$ and $|X_1-X_2|$ $\endgroup$ Commented Jun 11, 2015 at 19:47
  • $\begingroup$ $ envelope(t) =X_1+ X_2 \sin(\omega_2 t+\varphi_2) $ is the function you want then. Which satisfies the equations you wrote. You didn't say which sine wave was bigger $\endgroup$ Commented Jun 11, 2015 at 19:55
  • $\begingroup$ Getting closer, but the envelope angular frequency is $\omega_1-\omega_2$ or half of that. $\endgroup$ Commented Jun 11, 2015 at 20:01
  • $\begingroup$ Based on what you stated a possible candinate is $$X_1+X_2 \cos((\omega_1-\omega_2) t+\varphi_1-\varphi_2) $$ $\endgroup$ Commented Jun 11, 2015 at 20:03
  • $\begingroup$ What I have stated is correct. Plot it and you will see it's exactly what you want. $\endgroup$ Commented Jun 11, 2015 at 20:04

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