When we have 2 sinusoidal waves with slightly different wavelengths and the same amplitude, the resulting wave has a changing amplitude (amplitude = intensity) which varies from 2*Amplitude to 0 Suppose we have two different sources of visible light (slightly different wavelength: different color) producing an electromagnetic wave in 3D. This will result from point to point a wave similar to the sum from the picture. Now let's say we have a 2d surface (let's say a paper) where this (superposition of waves) can be projected (and finally a human can perceive it ). What will appear in this surface?

enter image description here

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    $\begingroup$ Welcome to physics.se, Chris. You'll need to enlarge and clarify your question. For a start, we live in 3 space dimensions, not 2. Secondly, the term "color intensity" doesn't mean anything, and the term "envelope" is not clear in this context. Try with more words. $\endgroup$ – akrasia Aug 31 '14 at 13:37
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    $\begingroup$ You are correct, my knowledge is restricted and hence i can't describe it greatly! is this better? $\endgroup$ – Cris Tsan Sep 2 '14 at 12:55
  • $\begingroup$ I'm still not sure what you are asking - if you shine two light sources onto the same point on a canvas, you will see precisely that which you would see if you shone a source there with their sum as intensity pattern. What will appear on the surface is a dot of light, but somehow I have the impression that you think there should be another answer. $\endgroup$ – ACuriousMind Sep 2 '14 at 13:50
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    $\begingroup$ I think this question is now answerable. $\endgroup$ – akrasia Sep 2 '14 at 15:09
  • $\begingroup$ @ACuriousMind i expect one color (because we have one wavelength) and different intensity of this color from point to point (because the amplitude differs from point to point) due to this "envelope" (the blue wave in the picture). What do you mean a dot of light? I believe the exact association of light with a wave of specific wavelength would clear things up. $\endgroup$ – Cris Tsan Sep 2 '14 at 18:24

The full formula for the sum of two waves of different frequencies $f_1$ and $f_2$ is:

$$\cos(2\pi f_1t)+\cos(2\pi f_2t)=2\cos\left(2\pi\frac{f_1+f_2}{2}t\right)\cos\left(2\pi\frac{f_1-f_2}{2}t\right)$$

In other words, the combined wave can be thought of as oscillating with a frequency that is the average of the two frequencies that produced it, while its intensity oscillates with a frequency equal to half of the difference of the two frequencies that produced it.

In order to perceive this oscillation, it must be slow enough that your eye can process it. Your eye has an effective "frame rate" of around 60 images per second (this is why movies work - despite the fact that they're displaying a flickering image, they're switching frames faster than your eye's flicker fusion threshold, so it appears to be a continuous image). Any brightness oscillation that is faster than around 60 Hz will not be perceptible by the eye.

So, if you somehow had two sources that produced light at frequencies that were exactly, say, 4 Hz apart, and you set up this arrangement, then you would see a point on the paper brighten and darken 2 times per second. The phase of the brightness oscillation would depend on the path length that the light took to reach your eye, so different points on the paper would appear differently bright at any given instant. It would definitely be a pretty cool pattern, if you could get it to work.

The problem is that for any real light source, the frequencies that it produces are spread out. For example, a helium-neon laser doesn't just produce light at just one frequency; it has a bandwidth of around 1.5 GHz (which is still extremely small, considering its frequency is in the THz range), so it produces significant amounts of light with frequencies that are up to 1.5 GHz away from the center frequency. Two frequencies that are 60 Hz apart (the maximum possible difference perceptible to the human eye), but which can vary by up to 1.5 GHz, are completely indistinguishable. The 60-Hz difference is totally insignificant compared to the variation. In short, we need much better light sources in order for this to actually be practical.

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    $\begingroup$ Only a minor correction: 60Hz for the human eye is WAY too high. Traditional cinema is 24Hz and the screen flickers only in peripheral vision. An estimate from an old cinema theory book I remember is 10-15Hz for the central spot. $\endgroup$ – fraxinus May 20 at 19:45
  • $\begingroup$ @fraxinus I was intentionally being very generous, as there are several sources that say that the flicker fusion threshold in certain circumstances is a quite a bit higher than 10-15 Hz. $\endgroup$ – probably_someone May 20 at 19:47
  • $\begingroup$ Well, it is pretty much unrelated to the topic, the best visible lasers being 100's of MHz wide. We can probably arrange to see moving at visible speed interference patterns using a moving semi-transparent mirrors. $\endgroup$ – fraxinus May 21 at 6:02

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